LIBRARY UNIVERSITY OF CALIFORNIA. C/J.vs Ctontom $rm A TEXT BOOK OF ALGEBRA ALDIS HENRY FROWDE OXFORD UNIVERSITY PRESS WAREHOUSE AMEN CORNER, E.C, ss TEXT BOOK OF ALGEBRA BY W. STEADMAN ALDIS, M.A. TTNIVEBSITY COLLEGE, ATTCZLA1TD, NEW ZEALAND AT THE CLABENDON PKESS M DCCC LXXXVII [ All rigUs reserved ] PREFACE. THIS book is the outcome of lectures delivered during some years to students in the College of Physical Science at Newcastle-upon-Tyne. The Author has endeavoured to place the subject on a foundation of strict reasoning 1 , and a comparatively large amount of space is therefore devoted to the discussion- of first principles. It is hoped that on this ground the book may be of service to persons who have not the opportunity enjoyed by students in the old Universities of access to large libraries, or intercourse with other mathematical scholars. The Author desires to thank his brother, Mr. > T. S. Aldis, M.A., of Trinity College, Cambridge, for revising the proof-sheets, a task which the remoteness of the Author's present residence rendered it impossible for himself to fulfil. UNIVERSITY COLLEGE, AUCKLAND, NEW ZEALAND. 45423 CONTENTS. The numbers indicate the Articles. CHAPTER I. PACK ARITHMETICAL NOTIONS 1 Addition; Subtraction; multiplication; commutative, associa- tive and distributive laws, 21-26. Fractions, 29-40. Division, 41-46. Division of fractions, 47, 48. Zero, 51. CHAPTER II. ALGEBRAICAL LAWS 33 Summary of formulae, 53. Negative expressions, 54, 55. Rule of signs, 56-58. Powers, indices, index law, fractional and nega- tive indices, 62-70. Extensions of formulae, 72-80. CHAPTER III. ADDITION AND SUBTRACTION 62 Definitions, 81-88. Addition, 89-94. Subtraction, 95-97. Brackets, 98-101. CHAPTER IV. MULTIPLICATION . 61 Single terms, 103. Rule of signs, 104, 105. General rule of multiplication, 106-110. Special examples, 114-137. viii CONTENTS. r -^ CHAPTER V. PAGE DIVISION 81 Preliminary, 138-148. Examples; definition of quotient and remainder, 149-158. General theorem for division by a binomial, 163-172. CHAPTER VI. HIGHEST COMMON DIVISOR AND LOWEST COMMON MULTIPLE . . 100 Definition of Highest Common Factor, 175. Monomials, 177. General process, 180-185. Examples, 186-189. Definition of Lowest Common Multiple, 191. Monomials, 192, 193. General method, 195, 196. Examples, 197, 198. CHAPTER VII. FRACTIONAL FORMS 116 General definition ; repetition of laws, 199, 200. Reduction of fractions, 203. Addition and subtraction, 205. Multiplication and division, 208-210. Compound fractions, 211. CHAPTER VIII. INDICES 128 Four laws, 214. Incommensurable quantities, 215-225. Inter- pretation of fractional and negative indices and proof of the four laws with this interpretation, 227-248. CHAPTER IX. SURDS 146 Definition, 249-251. Simplification of surds, 252-256. Binomial surd, 257. Rationalising factor, 258. Product of binomial surds, 261-263. Square root of binomial surd, 264-266. Impossible or operational quantities, 270. Interpretation, 272-276. Cube roots of unity, 277-283. CONTENTS. IX SECTION II. EQUATIONS. CHAPTER X. PA(JE SIMPLE EQUATIONS WITH ONE UNKNOWN 166 General principles, 284-287. Examples, 288-296. General type, 297, 298. Problems, 299-304. CHAPTER XI. SIMPLE EQUATIONS WITH MOEE THAN ONE UNKNOWN . . . 183 Example, 306, 307. General type of equations with two un- knowns, 308. Elimination, 309. Three methods, 310-316. Equations with three unknowns, 321-323. Problems, 324-330. CHAPTER XII. DETERMINANTS 202 General formulae for solution of two equations with two unknowns, 334-336. Definition of determinant, constituent, row, column, 337-351. Evaluation of determinant of third order, 344. Inter- changes, 345, 346. Minor determinants, 348. Properties of determinants, 349-354. Solution of equations with three un- knowns, 355-358. Inconsistency or independence of three equations, 361-364. CHAPTER XIII. SQUAEE ROOT 224 An inverse process, 366. General method, 367-369. Examples, 370-376. Approximate square root, 377, 378. Square root of a number, 379-385. Examples 386-388. Approximation by division, 389-392. Square root of fraction, 394-400. CHAPTER XIV. QUADEATIC EQUATIONS 244 Definition, 401. Type form, 402. Preliminary theorem, 403. General method, 406-409. Discriminant, 410-413. Sum and product of roots, 414. Resolution of quadratic expression into factors, 415. Another ii , r estigation of method of r solution, 416-421. Equations with given roots, 424. Symmetrical functions of the roots, 426, 427. Equations like quadratics, 428-434. CONTENTS. CHAPTER XV. PAQB CUBE ROOT AND CUBIC EQUATIONS 268 General method for cube root, 435-439. Examples, 440-442. Cube root of a number, 444-449. Approximate cube roots, 450-451. Cubic equation, 452-459. CHAPTER XVI. SIMULTANEOUS EQUATIONS ABOVE THE FIEST DEGREE . . . 283 Equations in which one unknown occurs only in the first power, 462. Homogeneous equations, 464. Symmetrical equations, 466-469. Two equations of second degree, 471, 472. Elimina- tion, 473. Problems, 474. SECTION III. SERIES. CHAPTER XVII. PEEMUTATIONS AND COMBINATIONS 297 Definition of series, factorial, 475-477. Permutations, 480-484. Combinations, 485-498. Arrangement of n things some of which are alike, 499. CHAPTER XVIII. AEITHMETICAL AND HAEMONICAL PROGRESSIONS .... 316 Definition of Arithmetical Progression, 503. Formulae, 505, 506. Problems, 507-509. Arithmetic means, 510, 511. Reci- procal, 515. Harmonical Progression, 516. Means, 517, 518. CHAPTF GEOMETRICAL PROGRESSION . 327 Definition, common ratio, = ilse, 521, 522. Means, 523-527. Sum to infinity, f epeating decimals, 532 533. Interest, 534-538. Ai 1-541. CONTENTS. xi CHAPTER XX. PAGE BINOMIAL AND POLYNOMIAL THEOBEMS 345 Product of unlike binomial factors, 543. Binomial theorem, 544, 545. General term, 547. Examples, 548, 549. Polynomial theorem, 550-553. Mathematical induction, 554. Another proof of Binomial Theorem, 555-557. Greatest term, 558. CHAPTER XXI. GENEBALTSATION OP THE BINOMIAL EXPANSION . . . .360 Explanation of the equivalence of the power of a binomial and the infinite series, 560-572. Convergency of series, 573-576. General term, 577-579. Approximate roots, 581. Extension of polynomial theorem, 582-584. CHAPTER XXII. INDETEBMINATE COEFFICIENTS AND RECURRING SERIES . . .377 General principle of undetermined coefficients, 585-587. Recur- ring series, scale of relation, 588-590. Resolution into partial fractions, 591. General term, 593, 594. Sum to n terms and to infinity, 595. Homogeneous products, 597, 598. CHAPTER XXIII. SUMMATION OF SOME SPECIAL SERIES 389 Series in which each term is the product of factors in arithmetical or harmonical progression, 599-604. Sums of powers of natural numbers, 605-609. General term, any rational integral function of n, 6 1 o. Method of differences, 6 1 1 . CHAPTER XXIV. LOGARITHMS AND EXPONENTIAL SERIES 404 Limit defined, 612. Examples, 613. Value of , 614-616. Ex- ponential theorem, 617, 618. Logarithm defined, 619. Series for logarithm, 622, 623. Properties of logarithms, 624-626. Base, characteristic, mantissa, 628-632. Different bases, 633-635. Modulus, 636. Other series, 637, 638. Calculations, 639-643. *ii CONTENTS. SECTION IV. ARITHMETICAL APPLICATIONS. CHAPTER XXV. PAGE RATIO, PROPORTION AND VARIATION 424 Definition of ratio, 644-646. Proportion, 647. Algebraical ex- pression of, 648-650. Derived results, 651-655. Rule of Three, 656. Compound ratio, 657. Duplicate ratio, 658. Sum, of ante- cedents has same ratio to sum of consequents, 659. Variation, definition of, 660-662. Inverse, 663. Joint, 664, 665. Double rule of three, 666. CHAPTER XXVI. CONTINUED FRACTIONS 440 Reduction of fraction to form of continued fraction, 667-669. Convergents, 670-677. Quadratic surd, 678-680. Repeating continued fraction, 681, 682. CHAPTER XXVII. INDETERMINATE EQUATIONS 454 Statement of problem, 683-685. Methods of solution, 686-689. Use of continued fractions, 690-692. Number of solutions, 693, 694. Three unknowns, 695. Special equation of two dimensions, 697. CHAPTER XXVIII. INEQUALITIES 466 ^ Definitions, 698, 699. Arithmetic and geometric mean, 700, 705. Maxima and minima values, 701-704, 706. Other theorems, 707-709. CHAPTER XXIX. THEORY OF NUMBERS 474 Modulus and residue, 710. Scale, radix, digits, 711-713. Ex- amples of change of scale, 714-717. Numbers divisible by certain factors, 718-722. Natural numbers arranged in sets ; prime numbers, 723-730. Divisors of a given number 731, 732. Numbers less than given number and prime to it, 733-736. Repeating decimals, 738-740. Extension to any radix, 741-745. Fermat's theorem, 746, 747. Product of any n consecutive integers divisible by \n, 748. Second proof j; Fermat, 749, 750. CONTENTS. Xlll CHAPTER XXX. PAGX PROBABILITIES . 497 Measure of chance 751-756. Compound chances, 757-762. Inverse chances, 763, 764. Value of expectation, 765, 766. Life assurance, 767, 768. Present value of assurance, 769, 770. Of life annuity, 771. Commutation Tables, 772, 773. Annual premium, 774. ANSWERS TO EXAMPLES . .513 MISCELLANEOUS EXAMPLES 553 ANSWERS TO MISCELLANEOUS EXAMPLES 575 ERRATA. Page 20, line u, for multiple read multiplier. 28, 2, for Art. 45 read Art. 44. >, 34> ipj/orlawsrazcHaw. 45, 10, omit of. 5 J > 8, /or >/ - read */l - c. 52, 2 from bottom, for the value reatf these values 77 9,for y + x ready + z. 81, 9 from bottom, insert the before highest power. 91, 4th line from bottom, for 161 read 160. 106, line 24,/or a?-Zx-12 read # 3 -3a? + 2. CHAPTER I. ARITHMETICAL NOTIONS. 1. THE laws which regulate the relations of different numbers to each other constitute the science of Arithmetic. The application of these laws to practical purposes con- stitutes the corresponding art, an art which in too many cases conceals from its possessors the science which under- lies it. The laws of Algebra are in most cases identical, in form at least, with those of Arithmetic, and for this reason it is desirable to obtain clear ideas of the latter before proceeding to the former. 2. A single thing of any kind is called a unit or one, and is denoted by the symbol 1 . If another thing of the same kind be placed with the first, we say that there are two of them, and denote the word two by the arbitrary sign 2. To indicate the juxtaposition of two things we use the s^gn +, and can thus also denote the two things when placed together by the combination of symbols 1 + 1. The fact that this symbolical representation is equivalent to the former is expressed by the use of another sign =. Thus we have what is called an Arithmetical equation, 1 + 1 = 2, the sign = meaning that the numbers represented by the symbols on either side of it are equal. B 2 Arithmetical Notions. [3. 3. If another thing- of the same kind be placed with the former two it is said that there are three of them, and their number is denoted by the arbitrary symbol 3. It is equally denoted by either combination of previously known signs 2+1 or 1 + 1 + 1. We have thus 2 + 1 =3 or 1 + 1 + 1=3. '4. By a similar process the ideas represented by the symbols 4, 5, 6, 7, 8, 9 are obtained, each of these in succession being- denned as equal to its predecessor with one more put to it. Thus 4=3 + 1, 5 = 4 + 1, 6 = 5+1, and so on. At this point the introduction of new arbitrary symbols to represent new numbers ceases, and the remaining numbers are represented by placing the symbols already named in different positions. The result of adding one to nine, which may be denoted by 9+ 1, is, it is true, called by a new and arbitrary name ten, but it is denoted by writing- the sign for one in a different position, so that it shall indicate not one unit but one ten. To effect this alteration of position another symbol 0, signifying nought or nothing- and spoken of as nought or zero, is used, and ten is denoted by the combination of 1 and 0, thus, 1 ; the combination indicating that the number consists of one ten and no units. 5. All succeeding numbers are similarly represented by writing down the number of tens they contain in one place and the number of units over to the right. Each number is still defined as being equal to its predecessor with a unit added to it. When a number which contains ten times ten is reached, a new name, a hundred, is given to it, and a new place, the third from the right, is reserved for it. Ten hundred is called a thousand, and the sign which represents the number of thousands in a given number is 8.] Arithmetical Notions. 3 put in the fourth place from the right. This process can be continued indefinitely. Thus in the symbol 2456789 the 9 means nine units, the 8 means eight tens, the 7 means seven hundreds, the 6 six thousands, the 5 five ten- thousands, the 4 four hundred-thousands and the 2 two thousand-thousands, to which latter accumulation of numbers the name of a million is given. 6. The symbol 2456789 thus carries on its face, so to speak, the manner, or rather a manner, in which the number which it represents is composed. It does not however tell all the different ways in which the number might be made up. 7. If three things of a certain kind be lying on a table and four more of the same kind be placed with them, the result is denoted by the combination of signs 3 + 4, the sign + merely signifying that the number represented by the symbol which follows it is to be put to, or added to, that represented by the symbol which precedes it. If the original three things be removed, there will evidently remain four, and the result of replacing the three will be properly represented by 4 + 3. Hence 3 + 4 =4 + 3. This proof does not in any way depend on the particular numbers 4 and 3, but it will similarly follow that if a and I be taken as symbols to represent any two integers whatever, a + b = b + a. 8. It is also tolerably evident that if there be three groups of things, one containing three, the second four, and the third five, and the three groups be all collected into one, the total number of articles in the combined group will be the same whichever of the three groups we bring to the other and in whatever order we move the two that are moved. B 'a 4 Arithmetical Notions. [9. The number of things in the final group will be represented by one or other of the sets of symbols, 3 + 4 + 5, 3 + 5 + 4, 4 + 3 + 5, 4 + 5 + 3, 5 + 3 + 4, 5 + 4 + 3, and we see that any one of these sets is equal to any other. The same thing- would be true if any number of different numbers be combined by addition ; the order in which the additions are effected is a matter of perfect indifference. The result is known as the Commutative law in addition. The student can easily convince himself that the proof of the truth of the principle does not in any way depend on the particular numbers 3, 4, 5, that have been employed, but that it is equally true for all numbers. Thus if a, b, c represent any three integers, the value of a + b + c is the same as that of either # + c+0, a + c + b, b + a + c, c + a + b or c + b + a. 9. When two groups of similar articles, the number in each group being known, are placed together, the process of discovering the number of articles in the combined group is called addition, and is denoted by writing the symbol + between the numbers which represent the groups. The resulting number is called the sum of the two original numbers, and can always be discovered by a process of reasoning depending on the definitions of the original symbols ; thus if the number in one group be 3 and the number in the other 4, the number in the united group is 3 + 4, or 4 + 3. But 3 = 2+1 = 1 + 1 + 1 Therefore 4 + 3 = 4+1 + 1 + 1 = 5+1 + 1 = 6 + 1 = 7 ii.] Arithmetical Notions. 5 The collection of symbols 4 + 3 can thus be replaced by the single symbol 7. 10. If from a group of articles containing a given number we take a part away, the number of the latter being also known, the process of ascertaining the number left is called subtraction. The symbol for the operation of taking away one number from a greater is . Thus if there be originally 5 articles, and 2 be taken away, the operation is denoted by the combination of symbols 5 2. The process of subtraction is the reverse of that of addition : the result of a subtraction has to be obtained by remember- ing known results of addition. Thus when we have ascertained from addition that 3 + 2 = 5, it follows that 3 is the number left when we take 2 from 5. Hence we may write 5-2 = 3. The equation 5 = 3 + 2 shows also that if 3 be taken from 5, 2 will be left. Thus also 5 3 = 2. If the results of addition of all possible pairs of numbers were tabulated, the table would also give the remainder after subtracting any number from any larger number. By the help of memory and certain rules depending on the ordinary system of decimal notation the necessity for the actual formation of such a table is obviated. 11. A collection of symbols such as 4 + 3 or 5 2 will in future be called an expression. The expression 4 + 3 may be always replaced by the number 7, and the expression 5 2 by the number 3 ; but it is sometimes desirable to retain the expression as a record of the manner in which the number was obtained. When it is required to perform any operation on the number which is represented by the expression, the latter quantity is frequently enclosed in a 6 Arithmetical Notions. [12. bracket thus (4+3) or {4 + 3} or [4 + 3]. Thus the compound expression 6 + (4 + 3) implies that the number represented by 4 + 3 is to be added to 6 ; the compound expression 8-(5-2) means that the number represented by (5 2) is to be subtracted from 8. The parts of an expression which are connected by the sign + or are called the terms of the expression. 12. The expression (4 + 3) denotes a group formed by putting three things along with four things of the same kind. The expression 6 + (4 + 3) denotes a group formed by putting the group (4 + 3) along with a group of 6 things. It is clear that the final result will be the same if the four things be first carried to the six and the three remaining things fetched afterwards. The result of this process would be denoted by the expression 6 + 4 + 3. Hence 6 + (4 + 3) = 6 + 4 + 3 Similarly it follows that 6 + (4 + 3 + 5) = 6 + 4 + 3 + 5, for the one expression denotes the result of taking three groups of 4, 3 and 5 things respectively and putting them to a group of six all at once, while the other represents the result of taking them in succession ; which operations must lead to the same final result. Hence if an expression all whose terms are preceded by a + sign be added to a number, the bracket may be removed and the terms of the expression added to the number in succession. 13. The expression 9-(4 + 3) means that from a heap of nine things the number of things denoted by the expression (4 + 3) is to be taken. 15.] Arithmetical Notions. 7 Clearly the result will be obtained equally well by first taking four of the things, and then going back and fetching three more. The result of taking the four things away is denoted by the expression 9 4 ; the result of taking the three other things by 9 4 3. Hence 9 _( 4 + s) = 9-4-3. Similarly 9__(4 + 3 + 2) = 9-4 3-2, the former denoting that (4 + 3 + 2) things are taken away at once, and the latter that 4 are first removed, then 3, and then 2. Hence if an expression all whose terms are connected by the sign + be subtracted from a number, the bracket may be removed provided the sign be written before each term. 14. Again, consider the expression which means that (5 2), that is, the remainder after taking 2 from 5, is to be added to 8. Let the numbers represent pounds sterling : the process required to be effected is to pay (5 2) to a person who already has 8. If a payment of 5 be made, the payee has evidently received too much by 2, which he must give as change to the payer. The net result is that the number of pounds the payee has is 8 + 5 2. Hence 8 + (5 2) = 8 + 5 2. 15. Suppose that a person A employs a messenger B to collect debts of 3 and 4. due to A and on the same errand to pay a bill of 5 due by A. B receives (3 + 4) pounds and pays 5 pounds. He thus brings back to A a number of pounds denoted by (3 + 4 5) and if A originally had ^8, he now has 8 + (3 + 4 5) pounds. - If A had gone on the errands himself, after receiving the first payment he would have 8 + 3 pounds ; after 8 Arithmetical Notions. [16. receiving 1 the second, he would have 8 + 3 + 4 pounds. He then pays 5 pounds, and consequently has 8 + 3 + 4 5 pounds left. Clearly however his final financial state is the same as when he employed the messenger. Hence 8 + (3 + 4 5) = 8+ 3 + 4-5. By this and similar instances it can be seen that when an expression is to be added to a number the bracket may be removed and all the terms written down with their original signs. 16. It may be noticed here that when A went to collect his debts himself it would make no difference if he paid the sum of a^Ps before he received either or both of the sums of money due to him. Thus all the expressions 8 + 3 + 4-5, 8-5+4+3 \ 8 + 3-5 + 4, 8 + 4-5 + 3 I (l) 5) 8-5 + 3 + 4, 8 + 4 + 3 5 since they merely denote the result of the same operations conducted in different orders, are identical. 17. In the expressions in (l) we see that a term which has a + sign before it indicates a payment made to A, while a term with a sign before it indicates a payment by A to another person. Here we have a first hint of a wider interpretation than that with which we started of the signs + and , a suggestion that besides indicating the two opposite processes of addition and subtraction, they may indicate two opposite qualities in the quantities before which they stand. 18. Suppose now that A keeps his capital in a bank, and that he pays 5 by a cheque on his banker, while the debtors pay 3 and 4 respectively into the bank to A's credit. Provided that A'B original balance exceed 5, the order in which the payments are made to and from the banker will make no difference to the final state of 20 .] Arithmetical Notions. 9 affairs. If then we regard the sign to denote that the number before which it stands is to be subtracted from some number which need not be stated, while + denotes a similar process of addition, the operations denoted by + 3-f4-5, +4-5 + 3, -5 + 3 + 4 + 4 + 3-5, +3-5 + 4, -5 + 4 + 3 all give the same result, namely in this case an increase of 2. 19. With this understanding as to the meaning of the signs + and when they stand without any preceding terms, the theorem of Articles 7 and 8 may be extended, and it may be asserted that the value of any expression such as those which we have been considering is inde- pendent of the order of the terms. In writing the ex- pression the sign + is usually omitted if it occur before the first term ; the sign , on the other hand, must be carefully inserted. A term with a sign + before it is called a positive term, one with the sign preceding is called a negative term. 20. The expression 8 (5 2) denotes that the expression (5 2) is to be subtracted from 8. If 5 be subtracted from 8, the result is denoted by 8 5. Evidently however this result is too small, because more has been taken from 8 than ought to have been ; to obtain the correct result we must add that which has been subtracted in excess, that is, 2. Hence the final result is .8 5 + 2, or 8-(5 2) = 8 5 + 2. Suppose again that a person A employs a second person to pay his debts and to receive the money due to him. A owes 3 to >4' 11" VT' and thus it follows that the distributive law (Arts. 25, 26) holds when any or all of the numbers involved are fractions. 41. A number expressed in the form of a fraction may be really an integer. Thus the fraction ^ means that 42 parts are to be taken such that 7 of them make a unit. Evidently these 42 parts can be arranged in 6 sets each containing 7 parts. Each set of 7 parts makes a unit. Hence the whole 42 parts make 6 units. Hence But 6 was shown in Art. 28 to be the quotient when 42 is divided by 7, and the equation 42 -r- 7 = 6 was proved from the definition of division and quotient. Hence 42 = 42-:- 7. 42. Similarly the fraction 45_42 + 3 3 Y 1 7 " t ~7* In this way any fraction whose numerator exceeds its de- nominator can be reduced to the sum of an integer and a fraction whose numerator is less than its denominator. For we can parcel off from the whole number of parts taken, a number of parts represented by the denominator, which will make up a whole unit, and repeat this process until either none are left or a number of parts less than the denominator. A fraction whose numerator is less than its denominator is called a proper fraction. It evidently represents a quan- tity less than the unit. A fraction whose numerator is 45-] Arithmetical Notions. 27 greater than its denominator is called an improper fraction, and evidently represents a quantity larger than a unit. 43. We are now able to return to the subject of division treated of in Art. 28. The word division is used in most treatises on Arithmetic in two entirely different senses which should be carefully distinguished as logical processes, although they lead to results identical in appearance and numerical representation. In the sense already explained it is said that 42-r-G 7 when it has been ascertained that 42 units of any kind can be assorted into 7 sets of 6 units. The quotient 1 is strictly a number of times. There is an entirely different sense in which the same numerical result 7 can be obtained. Suppose a line 42 inches in length, and let it be divided into 6 equal parts : the length of each of these parts is easily seen to be 7 inches, since 6.(7) = 42. Here a process of division performed on 42 with the help of the number 6 gives the same number 7 as before, but the 7 now means a length of 7 inches and not a number of times. 44. In this second sense we can represent the process of division of any number, as 45, by any number, as 7, as equi- valent to the operation of taking \ of 45, so that we may say that 45 divided by 7 is \ of 45, or = ^.(45) (Art. 33). 1 AK q But i . (45) = ~ = 6 + - , (Art. 42). That is to say, that if a line of 45 inches long is divided into 7 equal parts, each part will be six inches and f of an inch in length. 45. Referring back now to the former meaning and taking 45 as the number to reason upon, let i-he 45 units be continually parcelled off 1 into sets of 7 units. When 6 of these sets have been taken, there will be 3 units left and 28 Arithmetical Notions. [46. it may be said that 45 contains 7, 6 times with 3 remain- ing. The number 6 occurs here as in Art. 45, but here as strictly a quotient, a * how many times.' The number 3 occurs here as a remainder after the complete sets have been taken away. It is desirable however for many purposes to make the two results agree formally in all cases, as it has been seen they do when the quotient is an integer. This can be done by a somewhat similar enlargement of the word * times ' to that which was made of the word ' multiplica- tion/ By the definition of a fraction, 3 units is f of 7 units, since if a length of 7 units taken as a whole- be divided into 7 equal parts, each of them is a unit, and if 3 of these parts be taken the result is 3 units. It may be agreed to say that 3 contains 7, f of a time, and then it will follow that 45 contains 7, (6 + f) times, and that therefore we may write in this sense also, 46. The important result is however that the result of division, when multiplied into or by the divisor, always gives numerically the dividend, and as far as numerical results are concerned it is perfectly true that in all cases 47. It may be required to determine how many times, with the above extension of the word ' times,' one fraction of a unit contains another, or in other words to divide one fraction by another. Suppose it be required to divide T 6 T by -?. The fractions when reduced to a common denominator become re- (6x7 Actively L that is to say, the first contains (6 x 7) or 42 parts of a size such that (11 x 7) of them make up a unit while the other 4 8.] Arithmetical Notions. 29 contains (5x11) or 55 parts of the same size. Hence the first fraction is |f of the second or may be said to contain the second if times. Hence we may write 6 5_42__ (6x7) 6 7 11 " 7 ~ 55 ~ (TnTs) ~ TT x 5 (A so that the number of times one fraction contains a second is obtained by inverting the second fraction and using it as a multiplier. Here too it is easily seen that the result of division when multiplied by the divisor gives back the dividend. For (11x5) 7~(llx5)x7 . since the value of a fraction is not altered by multiplying or dividing both numerator and denominator by any integer, and (7 x 5) = (5 x 7) (Art. 22). 48. We can now give a meaning to an expression such jj_ as -^-, that is, an expression assuming the form of a frac- tion, in which the numerator and denominator have been replaced by fractions. Such an expression is often called a compound fraction, and as no interpretation has yet been given to it, it is permissible to assume it to have any meaning consistent with former results. It has been seen that whichever of the two possible meanings be given to the word 'division' 45-7-7 = -y 1 . By analogy it may be assumed that A 6 5 T 11 7' This meaning will obviously be consistent with previous definitions in any case in which the fractions, which form 30 Arithmetical Notions. [49- what by analogy with the definition of a fraction are called the numerator and denominator of the compound fraction, become really integers. 49. From this definition of the meaning- of a compound fraction it will follow that the value of such a fraction is not altered by multiplying its numerator and denominator by any the same number whether integral or fractional. Thus if we multiply the numerator and denominator of the fraction in the last article by the fraction f , we have 3 _6_ (3x6) 4 X IT _ (Txll) _ (3x 3 5 (3x5) "" (4 x 11) ' (4 X 7) 4 X 7 (4x7) (Art 48) V (3X6) X (4X7 = (4X11)X(3X5) (A (3X4)X(6X7) . -(4X3) X (11X5/ A (Art. 31) \ ' /I 1 v since (4 x 3) = (3 x 4) - JL I- A ^5 ~TT x 5 "ir^r _T 6 T 50. The addition of compound fractions is most easily effected by reducing each of them to a simple fraction as in Art. 47, and then using the method of Art. 39. The particular case in which the fractional denominators of the two compound fractions are the same demands a little consideration. 5i.] Arithmetical Notions. 31 Let it be required to add the two fractions 4- and IT -. TT We have -f- + -f- = ! x" +4x^ TT = (i+D by Art. 48. TT 51. In Art. 4 reference has been made to a symbol used in the ordinary system of decimal notation to fill up a place corresponding to which there is no actual number. This symbol is one of great importance and one with which the student should early familiarise himself. In its strict use it signifies that absolutely none of the quantities con- sidered are taken. Hence each of the expressions (2x0), (0 x 2), which signify respectively that the number 2 is taken no times and that no part of two units is taken, give the same result, namely nought or zero. Similarly the expression ^ . (0) which directs that one seventh part of nothing is to be taken, must equal nought, and the expression O-r-7, whether it is to be taken to mean the number of times that contains 7 or the value of one seventh part of nothing, has also a null or zero value. Hence if in any product one of the factors is zero, the product must also be zero. It obviously follows that if the product of any number of arithmetical factors is zero, one of the factors multiplied together must be zero. 32 Arithmetical Notions. EXAMPLES. 1. Simplify (J_I) X (* + i)+^ + i^|. 2. Divide 22557 days 19 hrs. 30 min. 48 sec. by 3432, and also by 57 min. 12 sec. Explain clearly in what respects the two processes differ and what is the difference in the nature of the results. 3. Add together f , T 4 T , f ; also 3 T , -ft-, T 6 T , and multiply their difference by 1J if 4. Find the simplest form in which 6s. 9d. can be expressed as a fraction of .1. Express also 6s. 9d. as a fraction of 11. Os. 3d. Show in each case the size of the parts into which the whole has been divided. 5. Illustrate by taking the example of a pound sterling as the unit, that f of f = f of f = -fa . 6. If a = 10, b = 2, c = 3, d = 4, illustrate the truth of the equations a (b + c d) = a bc + d, a . (6 + c cZ) = a . 6 + a. c a .d. 7. If a = 1, 6 = 2, c = 3, d = 4, find the values of d CHAPTER II. ALGEBKAICAL NOTATION. 52. IN the last chapter a proof has been given of the principal laws which govern the combinations of different numbers with one another. In proving each, definite numbers have been used, just as in the figures used in geo- metrical propositions one particular triangle is made to do duty for all possible triangles. The essence of the proof in the arithmetical proposition, as in the geometrical, is indepen- dent of the particular instrument employed, and thus all the propositions of the last chapter are true for all numbers which satisfy the conditions supposed to exist in each case. It is possible by means of a new notation to enunciate the propositions in a more general form. 53. Let the letters a, b, c, d, p, &c. be employed to denote numbers whether integral or fractional, then, retaining the meanings already given to the signs +, ,x,-f-, .,=, which indicate operations, or are equivalent to abbreviations for words, the following laws have been proved to hold : (1) a + b = b + a, the commutative law in addition ; (Arts. 7, 39) (2) a + b + c = b + c + a=c + b + a = a + c+b = b + a + c = c + a + b', (Art. 8) (3) a + b c = c + a + b = ; (Art. 18) (4) a + (l + c) = a + b + c; ' (Art. 12) (5) a + (b-c) = a + bc; (Art. 14) (6) a (b + c) = a-b-c-, (Art. 13) 34 Algebraical Notation. [53. (7) a (b c) = a 1+ (Art. 20) (8) bxa = axb-, (Art. 22) (9) a.b = b.a, (Arts. 32, 35, 36) which last two equations are different ways of stating- the commutative law in multiplication. In the last form the dot between the letters is usually omitted and ab written for the product of b and a. Thus a third way of stating- the commutative law is ab = ba. (l 0) a . (be) = (ab) . c ; the associative law. (Arts. 24, 38) From this it follows that either of these products may be written abc and the factors taken in any order. Thus, similarly, the product of any number of factors can be shown to be a definite quantity quite independent of the order in which the factors are taken. (11) a(b + c) = ab + ac; (Arts. 25, 40) (12) a(bc) = abac; (Arts. 26, 40) These two equations constitute the distributive laws in multiplication. (13) =|f; (Arts. 31, 49) < 14 ) l + l=^ ; (Arts. 39,50) ( 15 ) 1-1 = ^' (Arts. 39, 50) a c ac c a . . ( 1G ) rJ-5-J-i- (Art37) As a particular case of this 3=?=i--' (Arts.34,35) (17) j=a+b; (Arts. 41, 42, 45, 48) 54-] Algebraical Notation. 35 (18) 6.=a&-*-6=a; (Arts. 28, 46, 47) X(20) 0x0=0x0 = 0; (Art. 50) (21) 0-s- = 0. (Art. 50) To these algebraical relations may be added the four following axioms or general principles of operation. (1) If equals be added to equals the wholes are equal. (2) If equals be subtracted from equals, the remain- ders are equal. (3) If equals be multiplied by equals the products are equal. (4) If equals be divided by equals the quotients are equal. It may be noticed that the single letters a, b, c, &c. in the above algebraical formulae may stand for the sum or difference of any number of numbers, may in fact represent any expression (Art. 11) formed by combining the symbols for any numbers. A relation such as 7 + T = = , or any of those in b b b this article, is called an algebraical formula. The signs <-*', > , < , which have not yet been introduced will be occasionally used in the following senses : (a) a <*+/ b means the numerical difference between a and b obtained by subtracting the smaller from the larger ; (/3) a > b means that a is greater than b ; (y) a < b means that a is less than b. 54. In the equations of the last article no limitation is imposed on the values of the letters #, &, c, except in those formula) which involve subtraction. It will easily be seen that the proofs given of these will only hold good on the D 2, 36 Algebraical Notation. [55. supposition that the subtrahend, or number subtracted, is numerically less than the number from which it is sub- tracted. It will be desirable to remove this restriction before proceeding to deduce any consequences from these laws. This can be done by somewhat enlarging 1 and making more precise the definition of a negative quantity hinted at in Articles 17, 18, 19. Let it be agreed that the sign written before a term shall signify some quality of an opposite kind to that which is indicated by the term when standing alone or with the sign + before it. For instance, if the term alone or with a sign + signify an asset, a term with a sign will represent a debt. Thus let the sum of a person's assets be represented by a and the sum of his debts by b. If a be greater than b, a b is the net amount of the balance in the person's favour. If, on the other hand, b be greater than a the balance of debt is evidently b a, and with the understanding that a debt is to be indicated by placing the sign before its numerical representative, the state of the man's finances is indicated by saying that he has a balance -(*-4 Now if c be any quantity numerically larger than b a, c-(b a ) = c-b + a (Art. 20) = c + a b (Art. 18) = c + (aV). (Art. 14) Hence as far as regards the effect of placing them along with any other number e produced to C' and BC' be made equal to BC, OC' is evidently represented by b c, and if the parallelo- gram AQC'F' be completed it is represented by a.(b c). But AOC'F'= D OAIIB- D BIIF'C', .-. a.(bc) = abac. 80. These illustrations have been given now, not with the intention of following up the interpretation thus suggested, but that the student may see more clearly that the equations of Art. 53 admit of still greater generality of meaning than has been given even by the introduction of the negative quantity, and that Algebra is something very much wider in its scope than a mere substitution of letters for numbers to aid in the solution of general arith- metical problems. EXAMPLES. 1. If a = 4, b = 3, c = 2, d = 1 ; find the values of (1) a + b- (3) 2. With the same values of the letters, show, by finding the arithmetical value of each of the quantities involved, that (1) be 3. Find the values of (a + 6 + c)*; a, b, c, d having the same values as before. 8o -] Algebraical Notation. 51 4. Illustrate the formulae (1) (ab) n =a n b n ; (2) a m xa* = a m + , in the cases when a = 2, b = 3, m = 4, n = 5, by finding the numerical value of each side in each equation. 5. If a = 7, b = 5, c = 4, find the values of (1) V (3) * 6. If x = 3, y = 2, find the values of (1) (* 2 - (2) * 4 - (3) (4) E 2 CHAPTER III. ELEMENTARY OPERATIONS. ADDITION AND SUBTRACTION. 81. Any combination of symbols, such as a, b, c, when connected by the signs of operation x , . , H-, + , , &c., is called an algebraical expression. Those parts of an expression which are connected by the signs + and are called the terms of the expression (Art. 11). 82. An expression, such as which does not assume a fractional form, is called an integral expression. If none of the terms involve factors raised to fractional indices the expression is called a rational expression. If the expression assume a fractional form , as -- 7-, it is called a, fractional expression. 83. It may of course happen that an integral algebraical expression may, for some values of the letters, have a fractional value, while the fractional expression may have an integral value. Thus if a have the value , and b the value , the integral expression a 2b has the value f 2x1; that is, f 1 or J. On the other hand, the fractional expression j has for the value of a and b the value ~ , which is 3. The words ' fractional ' and ' integral/ 86.] Addition and Subtraction. 53 as applied to algebraical expressions, thus apply to the form of the expression and not to its value for particular values of the letters involved. 84. A single term may consist of one factor, as a, or may be the product of two or more factors, as a 2 , abc, a 2 c. If one of the factors is a definite number, as the factor 3 in the term 3 abc, that factor is usually written first, and is called the coefficient or sometimes the numerical coefficient of the term. 85. Terms which involve the same combination of literal factors, as 3abc, 7abc, are called like or similar terms. Terms which involve different combinations of literal factors are called unlike or dissimilar terms ; as 5 3 , 7 a?b. 86. A term involving a single literal factor, as 5 a, 7 b, c, is called a term of one degree, or of one dimension, or a linear term. The last two designations arise from the fact that single letters, as a, b, c, are often employed to denote the lengths of lines. A term involving the product of two literal factors, as a 2 , ab, be, is called a term of two dimensions, or of the second degree. In Articles 23 and 77 it is seen that a product, such as ab, may represent an area, having two dimensions, length and breadth. A term involving the product of three literal factors is similarly called a term of three dimensions or of the third degree, as a 3 , a?b, abc. In Articles 24 and 78 it is shown that a volume having three dimensions length, breadth and thickness may be represented by the product of three factors each of which represents a line. Most people are unable to conceive of actual physical quantities possessing more than three dimensions, but the phrase is a convenient one and in analogy with the language used up to this point; a term containing four 54 Elementary Operations. [87. literal factors is called a term of four dimensions or of the fourth degree, and so on. Thus the terms 7 3 6 2 and ll# 2 3 c 4 are of five and nine dimensions respectively, for 3 b 2 is merely an abbreviation for aaabb, and a 2 b z c* is an abbreviation for aabbbcccc, and these terms contain five and nine literal factors respectively. 87. An expression consisting 1 of one term is called a monomial: one consisting- of two terms, as a + b, or 2 a 3b, a binomial; and the words trinomial, quadrinomial, polynomial are applied to expressions containing three, four, or many terms respectively. 88. An expression of which all the terms are of the same dimensions is called a homogeneous expression. Thus a + b + c is a homogeneous expression of one dimension or the first degree or a linear expression ; 3 a 2 + 2 ab - 5 b 2 is a homogeneous expression of two dimensions or of the second degree. A homogeneous expression will be spoken of as an expression of four, five, &c., as the case may be, dimensions. 89. Before proceeding to explain and illustrate the processes of addition and subtraction as applied to Alge- braical expressions, it will be well to reproduce those formulae of Art. 53 which will be chiefly employed. They are, retaining the numbering of that Article, (4) a + (b + c) = a + b + c, (5) a + (b c) = a + b c, (6) a (b + c) = abc, (7) a (b c) = ab + c, (9) ab = la, (11) a(b + c) = ab + ac, (12) a(b c) = ab ac. 90. The addition of two algebraical expressions, if none of the terms are like, can only be represented by connecting the second expression with the first by means of the 93-] Addition and Subtraction. 55 sign -f . Thus if it be required to add a + b c top + q + r the process and the result can only be represented in the form or, by (4) and (5), p + q + r + a + b c. 91. If however some of the terms in the one expression are like to some in the other, a considerable simplification may be effected. Thus let it be required to add 3# 2 + 4o?+5 to 2# 2 -3tf 2, the result is expressed as 5-2 -}-(4-3),2? + (5-2) by (11) and (12) Two like terms can be replaced by a single term like to them, its coefficient being the sum or difference of the coefficients of the original terms, according as those terms have the same or opposite signs. 92. The above process is more conveniently represented by writing down the expressions to be added in two horizontal lines, like terms being placed in the same vertical line, thus 2 a; 2 3 a? 2 The sum is then obtained by adding or subtracting the coefficients of like terms. 93. It is convenient for many purposes to alter somewhat the phraseology hitherto employed in regard to subtraction. The expression a b has been said to denote the process of subtracting b from a: it will in future be 'more usually spoken of as the result of adding b to a, and will be called the algebraical sum of # and b. r T ' 56 Elementary Operations. [94. Thus a + l c d is the algebraical sum of a, l, c and + c+ d+... a? _|_ ab + ac + ad + . . . ab + l 2 +bc + ... 2 (a 2 ) + 2 2(00.) Then the product is obtained (Art. 106) by multiplying each term in the upper row by every term in the lower row and adding all the results. The two terms multiplied together must either be like or unlike (Art. 85). The product of two like terms gives aa, lib, cc, ... or a 2 , b 2 , c 2 ,... The sum of all these may be denoted by the convenient symbol 2 (a 2 ), where 2 may be 70 Elementary Operations. [120. taken as an abbreviation for the words ' the sum of all such quantities as.' The product of two unlike terms gives a term of which ab is the type. This particular term will occur twice, namely as the product of the a in the lower row into the b of the upper row, and the b in the lower row into the a of the upper row. Hence the sum of the products of unlike terms may be written as 2 2 (ab}, where 2 has the same meaning as before. Hence on the whole 120. As an example, suppose it be required to find the value of 2 (a 2 ) = I 2 + (#) 2 + (a 2 ) 2 + (# 3 ) 2 + (a? 4 ) 2 = 1 +% 2 + x* + x 6 + x 8 by (24) of Art. 72. The value of 2 2 (ab) is most easily obtained with accuracy by taking first the product of the first term into all that succeed it, then the product of the second term into all that follow it, and so on. Thus = 2x + 2x* and since 2 (a 2 ) =1 + x* + ar 4 + 6 + x * it follows that 121. A similar formula holds for the cube (Art. 62) of any polynomial. Let the three factors to be multiplied together be written in separate horizontal lines one below the other : i2i,] Multiplication. 71 Then, by an extension of the rules of Articles 104, 106, the final product must be obtained by multiplying- each term in the third row by every product of one term in the second and one term in the first row. Three cases may arise : (1) The three terms multiplied together may be all alike. This will give aaa, bbb, ccc, ... or a 3 , 3 , c 3 , . . . ; and the sum of all these terms may be denoted, as in Art. 119, by 2 (,*'). (2) Two of the terms taken may be alike and the third different. This will give a term of the type (fib. This particular term may arise in three ways, namely by taking- the unlike factor b out of the first row and the two like ones out of the second and third, or the factor b out of the second row and the two factors a out of the first and third ; or finally the two like factors out of the first and second, and the factor b out of the third row. Thus the sum of all this class of terms will be denoted by 3 2 (a 2 b). (3) All three terms taken may be unlike. The type of such a term is abc. This particular term abc occurs in six ways. For if a be taken out of the first row, b may be taken out of the second and c out of the third, or b out of the third and c out of the second. If a be taken out of the second row, there are similarly two ways of taking b and c, and two more if a be taken out of the third. Hence the sum of all this class of terms is 6 2 (abc). Thus on the whole 122. As particular examples (i) arranging the terms in descending powers of a. 72 Elementary Operations. [123. In this example no terms exist of the type ale, since the expression only contains two different terms. (2) (a + l + cy = a* + P + c* 3 a*c (3) +7o? 3 + 30* + 3 4 -f 3a; 4 -f 3x 5 123. In applying the formulae of Art. 119 and 121, the rule of signs in multiplication has to be carefully remembered. Thus in finding the square of (a + b c) by Art. 119, the square of c being the product of two negative factors will be positive (Art. 58) while the product of +a and c will be negative (Art. 56, 57). Thus (a + bcf = a 2 + b* + c 2 + 2al 2ac 2l)c. 124. The cube of (a + b c) can be deduced from the for- mula of Art. 121 with the same precautions. Thus -f 60( c) Now (-c) 2 =(--c)x(-c) = + c 2 (Art. 58) a 2 (-ab 2 + 4ac 2 . The proof again lies in the fact that this last expression multiplied by 3abc gives back the dividend. G 2 84 Integral Forms. [146. This process is conveniently conducted by writing- the divisor to the left of the dividend with a line to separate them, and a line being drawn below the dividend, writing the successive terms of the result below the terms of the dividend from which they arise : as illustrated below in the example already worked. Babe) 6a*b*c3a 2 b*c+l2a 2 bc* 2a 2 d ab 2 146. In the more difficult problem of the division of one polynomial by another, the same principles must be used. The result of division must be an expression such that when multiplied by the divisor it will give the dividend. If the two expressions do not involve the same dominant letter, the division can not be effected as an algebraical operation. It may of course be effected as an arithmetical process, provided special values be given to the letters, but the result can only be exhibited in algebraical symbols by a fractional form of which the dividend is the numerator and the divisor the denominator. See (17) of Art. 53. It is farther obvious that the division cannot be algebraically effected unless the dominant letter occur in the dividend to at least as high a degree as in the divisor, and that if it occur to the same degree in both, the result of division must be a number, or an expression not con- taining the dominant letter. 147. A farther difficulty requiring removal is found in the fact that, as has been seen in Art. 136, the term in- volving any particular power of the dominant letter in the product of two factors is obtained by adding together the products of several pairs of terms in the factors. The one exception is the case of the term involving the highest power of the dominant letter, which, we have seen, is formed by the product of the highest terms in the 150.] Elementary Operations. Division. 85 two factors (Art. 110). A similar statement applies to the product of the two lowest terms. By the phrases ' lowest ' and ' highest ' terms will in future be meant the terms involving" the 'lowest* and ' highest ' powers respectively of the dominant letter. 148. It becomes thus of fundamental importance to the process of division to arrange the dividend and divisor both in descending, or both in ascending, powers of their common dominant letter. This arrangement was found to be convenient in multiplication. It is practically indispen- sable in division. 149. Let the dividend be 2# 2 + 7# + 3 and the divisor 2#+l. The highest term in the other factor required must (Art. 147) be such as, when multiplied by 2#, the highest term in the divisor, will give 2# 2 , the highest term in the dividend ; that is, it must be x. To ascertain whether this is the whole of the required factor, the divisor is multiplied by x : the product is 2 # 2 + x which is not equal to the whole of the dividend. Hence something has to be added to the term x in order to give the full result of division. This 'something' must be such as, when multiplied by the divisor, will give the rest of the dividend. Hence the next step is to subtract 2 x 2 + x from the dividend. This leaves a remainder 6#+ 3. The next term in the required result must therefore be such as when multiplied by 2 a? shall give 6#, that is, it must be 3. Multiplying the whole of the divisor by 3, we get 6 a? +3, which is equal to the rest of the dividend. Hence it has been ascertained that x + 3 when multiplied by 2#+ 1 gives 2x 2 + 7#+ 3, whence we may write (2 a? 2 + 7x + 3) 150. The actual working of the above process is usually written more concisely as follows. 86 Integral Forms. [151. 6a?+3 The dividend and divisor are arranged in descending powers of their common dominant letter and the divisor written to the left of the dividend with a line separating them. A similar line is drawn to the right. The first term in the required result must, as before explained, be x. This is written to the right of the right- hand curved line. The divisor is multiplied by this term and the product written down under the dividend, the same powers of the dominant letter being for convenience written in a vertical line. This product is subtracted from the dividend. The remainder now becomes the dividend, for the problem now to be solved is to find what has to be used to multiply the divisor in order to produce this remainder. 151. In the last article the student will see that the product of a? into 2#+l has been first subtracted from the dividend and afterwards that of 3 into 2 #4-1. On the whole the product of x + 3 into 2 x + 1 has been subtracted and there is no remainder. The process has therefore not only ascertained what the factor must be which, when multiplied by 2x+ 1, shall give the dividend, supposing there to be such a factor, but has also shown that x+ 3 satisfies completely the required condition. In this case x-\- 3 is called by analogy with numerical results (Art. 28) the quotient of 2x z +7x+ 3 divided by 152. Suppose that it had been required to divide 2 a? + 7x+ 5 by 2z+l. The process up to the last stage is identical with the previous one. I53-] Elementary Operations. Division. 87 2 It is still obvious that if any factor can be found which, when multiplied by 2 x + 1 , shall give the dividend, the terms of that factor must be in succession x and 3. On subtracting the products of these terms into the divisor in succession from the dividend it is found that there is a number 2 still left over. It is obvious that no expression of integral form when multiplied by 2x+l can give 2 the number 2. A fractional form would do so (Art. 53 (18)), and the result of division might therefore be 2 written as x + 3 + It is usual, however, to speak of 2 as the remainder after the division has been effected, while the part of the result x+ 3, which assumes an integral form, is called the quotient. 153. As another instance let it be required to divide 3# 4 -# 3 + 2# 2 + #+l by # 2 -#+l. Arranging the divisor and dividend as in Art. 150, the process is represented below x* - sc+l The highest term in the quotient, as it may now be called, 88 Integral Forms. [154- must be such as when multiplied by x z shall give 3# 4 , that is, it must be 3# 2 . The product of the whole divisor by this term is written below the dividend and subtracted from it in accordance with the rules of Arts. 95, 96. The remainder is written below, and it now becomes obvious that the highest term in the rest of the quotient must be such as when multiplied by x 2 shall give 2# 3 , that is, it must be 2x. The whole of the divisor is multiplied by this term, written below the first remainder and subtracted from the latter. There is still left x 2 x+ 1 of the dividend, and it is obvious that the only remaining term required is 1 . The divisor being multiplied by 1 and the product written down below and subtracted from the second remainder there is nothing- left. 154. As in the previous example it may be noticed that the products of 3x 2 , 2#, and 1 into the divisor have been in succession subtracted from the dividend: that is, on the whole the product of (3x 2 +2x+l) into the divisor has been subtracted. There being no remainder the process has proved that (3# 2 + 2#+l). (x 2 x+l) = 3# 4 -# 3 or that (3x*-z* + 2x 2 +x+l)-:-(x 2 x+l) = 155. As another example let it be required to divide 3# 4 -# 3 +2# 2 -f 5#-2 by 3x 2 +2x+l. 3x 2 +2x+I )3# 4 x* + 2x 2 +5x 2( x 2 x+l 3x x x The first term in the quotient is evidently x 2 . The 1 5 7.] Elementary Operations. Division. 89 remainder after the product of x z into the whole divisor has been subtracted is 3x 3 + x 2 + 5x 2. The last term of these has not been written down in the lower line as experience has now taught us that it will not be affected by the next stage of the working. The next term in the quotient must be such a term as when multiplied by 3# 2 shall give 3 a? 3 . It must there- fore be x (Arts. 56, 57), and in multiplying the divisor by this term the rule of signs must be attended to. The third term in the quotient must evidently be 1 . When the product of the divisor by this term is finally subtracted there remains over 4 3. Evidently no integral form when multiplied by the dividend can produce 4x 3, and either the result of division must, as before, be written in 4/g _ 3 the form # 2 a?+ 1 H -- ; - > or it may be said that 3a^ + 2#+l the quotient is x 2 x+l and that there is a remainder 4# 3. 156. The process of the last Article has shown that if x 2 x+l be multiplied into 3# 2 + 2#+l and the product subtracted from the dividend, there will be a remainder 4# 3. It follows that if 4# 3 be added to the product of a? 2 m * m l^e the indices of Art. 62, but merely the position which each occupies in the series. Thus^? 7 means the seventh coefficient not reckoning the first, and is the coefficient of # n ~ 7 . The number which denotes the co- efficient increases by unity from term to term, that which denotes the power of #, or in other words the index of x (Art. 62) decreases by unity from term to term. The sum of these two numbers is therefore always the same and equal to n the degree of the expression (Art. 139). It is thus clear that the coefficient of x 2 will be p n -% , and so on. The dots which separate the two + signs denote the terms omitted between those which are written down, the number of which omitted terms, as it depends on n, must be left undetermined if the proposition is to be perfectly general. 167.] Elementary Operations. Division. 93 165. The expression (a) is the most general form of a rational integral expression of the n th degree in x. It will be sometimes conveniently denoted by the symbol /(#), where f is an abbreviation for the words ' a function of,' the term ' function ' being used in mathematics to express any quantity which depends on another in a definite manner. 166. Let, then, the expression (a) be divided by x a with remainder R and quotient Q. By Art. 158, R is of lower degree than x a in x ; that is, E cannot contain x at all, and is in fact a definite function of a and the coefficients of the expression (a). This will be obvious to those who have followed the examples in division previously given. Then, by Art. 157, This identity exists quite independently of the particular value of x, and will be true for all values. Let the value a be given to x. R which is independent of x remains unchanged by this substitution, Q will assume some special value, and x a will assume the value zero. Consequently (Art. 51) whatever may be the particular value of Q, the product Q (x a) certainly is zero. Hence it follows that that is, the remainder after dividing /(#) by x a is the value which f(x) assumes when for x the value a is substituted. 167. Hence if/(#) be zero when a is put for #, the remainder after dividing /(a?) by x a is zero, or in other words f(%) is divisible by x a without remainder, that is x-a is a factor of /(a?) (Arts. 138, 151, 154). Conversely if x a be a factor of /(#),/(#) must vanish when the value a is put for x ; for in this case f(x) is divisible by x a without remainder, and consequently R, or f(a), as we may write it, is zero. 94 Integral Forms. [168. This second result follows also from Art. 51 since the factor x a has the value zero when a is put for a?, and therefore the product must also be zero. 168. The expressions x n a n and x n + a n are of frequent occurrence. Suppose n to be a positive integer, whatever the value of n may be, x n a n vanishes when a is put for x. Hence for all integral values of n, x n a n has a factor xa. If n be even and a be substituted for #, x n a n becomes ( ) n n , or by Art. 127, since n is even, a n a n -, that is, zero. Hence, when n is even x n a n has also a factor x ( a) or x + a. In the expression x n + a n let a be put for x. The value becomes (a) n + a n , which is 2a n if n be even and if n be odd (Art. 127). Hence, if n be odd x n + a n has a factor x + a. The results of Articles 116 and 133 are particular cases of the theorems of this Article. 169. As another instance, take the frequently occurring expression aH-atf + ac 2 a 2 c + tfcbc 2 , (a) which may be written in either of the equivalent forms or bc(b c) + ca(c a) + at(a I), (y) as the student can easily verify by multiplying out. Regarding a as the dominant letter and putting for a wherever it occurs in (a) the value b, the expression becomes c or zero. Hence a I is a factor of (a). Similarly by putting for a the value c, the expression becomes b c 2 -l 2 c + c*-c* + tj 2 c be 2 or zero. Hence a c is a factor of the expression (a). 170.] Elementary Operations. Division. 95 Regarding now b as the dominant letter and putting for I the value c the expression (a) becomes a 2 c ac 2 + ac 2 a 2 c + c 3 c 3 or zero. Hence I c is a factor of (a). Now since each of the three factors (a b), (a c), (bc) is (Art. 88) a homogeneous expression of one dimension, their product (Art. 112) is a homogeneous expression of three dimensions, that is of the same dimensions as (a). Hence a can have no other factors involving the letters or ^o =Po>
  • y + 2Qx*y'* by 4a 3 - 2a. -24. by 2t/V by x^yt &) by a 2 a 6) + 2 by a by (a (a; + y + z) (xy + yz + zx) xyz by a). Elementary Operations. Division. 99 25. Divide 1 + y 3 + 3 3 3yz by 26. Find the quotient and remainder when is divided by x 1. 27. Find the quotient and remainder when x s 5X 6 + 4# 4 3 x 2 + 2 is divided (1) by a 1, (2) by ce 2 1. ' 28. Find the quotient and remainder when is divided by as 1. Deduce the remainder when 15030794 is divided by 9. 29. By the method of Art. 170 find the quotient when a 6 5x 5 a 4 + 26a3 2 2cc+31 is divided by cc 2 7z+10. 30. Divide a? 3 + 3^ 2 +3a 2 a? + a 8 -6 3 by a + a-6. 31. Resolve (x^ + ^-z^-u^-^xy-^zuf into four factors. 32. Resolve 4 (im + ^) 2 -(x 2 -2/ 2 -^ 2 + w 2 ) 2 into four factors. 33. If 2s = a + b + c, prove that s(s b)(s c) + s(s c)(s a) + s(s a)(sb) (sa)(s b)(sc) abc. 34. Prove that H 2, CHAPTER VI. ELEMENTARY OPERATIONS. INTEGRAL FORMS. HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE. 174. IF it were always possible to discover all the factors of any expression, it would be merely a matter of com- parison to see what factors were common to any two expressions. Thus we know that x*-a?= (as a) (x + a) (Art. 116), and we also know that a? 3 a 3 = (x a) (oP + ax + a 2 ) (Art. 133). It is obvious that the only factor common to the two expressions a? 2 a 2 and x 3 a? is x a. 175. If there be more than one factor common to two given expressions, it follows from the associative law in multiplication that the product of all the common factors will also be a common factor. This product will be of higher dimensions (Arts. 86, 111) or of higher degree (Art. 139) than any of the common factors which produce it. Hence it has the highest degree of all the expressions which are common factors of the two given ones. It is therefore called the highest common factor. 176. Strictly speaking, if the two given expressions have a common numerical factor, the introduction or omission of this factor will not affect the degree or dimensions of the product of the common factors. It is usual however to include such numerical factors as part of the highest common factor, and the usage has its advantages for some practical purposes, although not without some counter- balancing disadvantage in producing a confusion in the 178.] Elementary Operations. Integral Forms. 101 student's mind between the highest common factor of two algebraical expressions, a term which refers to algebraical form, and the greatest common measure of two numbers, a term which refers to numerical magnitude. 177. If the two expressions be monomials (Art. 87) the factors of each are obvious, and the highest common factor of the two is the product of the highest powers of all common letters which are contained in each. Thus the highest common factor of the two monomials a 3 b*c and ab 2 c 3 is ab 2 c. For a is the highest power of a which is a factor of both, b 2 the highest power of #, and c that of c. If the two monomials have numerical coefficients with a common numerical factor or factors, it is usual to multiply the highest common factor of the literal part by the largest numerical common factor of the coefficients, and to call the result the highest common factor or, more incorrectly, the greatest common measure of the two monomials. Thus the highest common factor of the two monomials 150 3 2 tf and I2ab 2 c 3 will be said to be 3ab 2 c. As far as regards its dimensions (Art. 86) this factor is no higher than ab 2 c. It is of course numerically larger. 178. The numerical magnitude of an expression may or may not increase as the dimensions are increased. Thus if a have the value 2, the expressions a 2 , a 3 , 4 ,. . . of successively higher dimensions have the successively larger numerical magnitudes 4, 8, 16 ... If, on the other hand, a have the value |, these same expressions have the successively smaller values , J, T V.--- The student must therefore remember that in investigating the form of the highest common factor we are not necessarily discovering the magnitude of the largest numerical factor for any particular values of the letters involved. 102 Highest Common Factor. [179. 179. The process for finding the highest common factor of two polynomial algebraical expressions is identical in form with that for finding the greatest common measure of two numbers in Arithmetic. The proof of the validity of the algebraical process also obviously includes that of the arithmetical one, for if all the algebraical factors represent positive integers the two problems are identical in substance as well as in form. The scope of the algebraical theorem is however much wider than that of the arithmetical one. ISO. The process may be briefly described thus. Let a and b represent the two polynomials, and let a be that one of the two which is not of higher degree in the common dominant letter than the other. If there be no common letter, obviously there can be no common algebraical factor. Take a as divisor and b as dividend and carry on the process of division until the remainder is of lower degree than a (Art. 158). Let the quotient be denoted by p and the remainder by c. / a) b (p pa *)*($ L d) c(r rd Reference to the examples of division in Arts. 152-156 will show that, whether the quotient p consist of one term or many, the divisor a has been multiplied in succession by the different terms of jo, and on the whole pa has been subtracted. Hence (Art. 157) 6=j>a + c; (1) or subtractii% pa from these equals, b pa = c. (2) 1 8 r.] Highest Common Factor. 103 The remainder c is now to be taken as divisor and a as dividend. Let q be the quotient and d the remainder, which must be (Art. 158) of lower degree than c. As before, we get the two relations a = qc + d, . (3) a qc = d. (4) This process can be carried on until one of the remainders divides exactly the preceding one. Whatever the degree of a or I may be, since the degree of each remainder is less, by unity at least, than that of the previous one, the process must come to an end either -by one of the remainders dividing exactly into the previous one, or by the last remainder being a number, independent of the dominant letter. In the latter case this remainder will algebraically divide exactly into the previous one if fractional numerical coefficients be admitted, and therefore in any case the process will terminate by one remainder dividing exactly, from an algebraical point of view, into the previous one. Suppose that d is the last remainder and that it divides exactly into c with quotient r. It obviously follows that c = rd. (5) 181. It can now be proved that d is the highest common factor of a and b. For the proof two consequences of the general laws of multiplication are necessary. (1) Every factor of any expression is also a factor of any multiple of that expression. For if a be a factor of b, b must equal qa, and therefore by Axiom (3) of Art. 53, pb, any multiple of b, must equal pqa. Hence a is a factor of pb. (2) Every common factor of two expressions is also a factor of their sum or difference. For if a be a factor of b and also of c, b must equal pa 104 Highest Common Factor. [182. and c must equal qa. Hence, by Axioms (l) and (2) of Art. 53, b + c must equal pa + qa which by (11) of Art. 53 is (p + q)a, and I c must equal pa qa which by (12) of the same Art. is (pq)a. Hence b + c and b c are both multiples of a. 182. By (5) of Art. 180, d is a factor of c. Hence, by (1) of Art. 181, d is a factor of qc, and therefore by (2) of Art 181, d is a factor of qc + d or a by (3) of Art. 180. Hence d is a factor of pa and therefore of pa + c or of b, by (1) of Art. 180. Hence d is a common factor of a and b. Again, every common factor of a and b is a factor of bpa or c, by (2) of Art. 181; and therefore of qc, and therefore of a qc or d. But clearly no expression of higher degree than d can be a factor of d. Hence d is itself the highest common factor required. 183. It has been noticed in Art. 158 that in carrying on the division to the point required in the different stages in the foregoing proof, it may happen that the coefficients in the quotient and remainders are fractional. This will make no difference to the degree of the resulting expression, but will produce it in a form not perhaps the most convenient. In order to avoid such fractional coefficients, certain precautions are adopted. In the first place, it is desirable before beginning the operation to discover and remove any numerical factors of either of the given expressions. If there be any common numerical factor, this can be restored at the end to the Highest Common Factor, as in Art. 176. Secondly, if at the end of any stage the remainder is found to have a numerical factor common to all its terms, this factor may be removed. For clearly, if the first precaution has been taken, this will not be in any sense part of the Highest Common Factor. 184.] Highest Common Factor. 105 Thirdly, if at any stage the division cannot be carried on without introducing- fractional coefficients and the second plan is not available, the dividend may be multiplied by such an integer as to make the coefficient of its first term exactly divisible by that of the first term of its divisor. 184. The student who has followed and understood the proof in Arts. 180-182 will see that neither of the plans suggested in the last Article will affect the algebraical form of the resulting Highest Common Factor. Suppose, for instance, that c, in Art. 180, contains a numerical factor #, so that c = xc'. Let c' be taken as divisor, and suppose that in order to avoid fractional coefficients in the next division it is neces- sary to multiply a by some number y, y not being a factor of c. The process will be as indicated in the accompanying form a)b (p pa c xc c')ya (q d)c'(r rd and the relations of Art. 180 will be replaced by # pa = c = sec', or ~b = pa + xc ya qc = d, or ya = qc' + d c' = rd. From these, as in Art. 180, it will follow that d is a measure first of c' and then of qc +d or ya. Now as it has been agreed to take away all numerical factors at each 106 Highest Common Factor. [185. stage of the process, in accordance with the second precaution of Art. 183, d cannot have any factor common to y, hence d must be a factor of a. Therefore d being a factor of a and c', is a factor of pa and a&c', and therefore ofpa + xc' or b. Again, all common factors of a and l> are factors of pa and d, and therefore of I pa or xc'. But a and I have no common factor which is a factor of x by supposition. Hence every common factor of a and b must be a factor of c', and therefore of qc', and therefore of ya qc' or d. Whence, as before, d is the highest common factor. 185. The proof in the last Article also shows that it is allowable at any stage of the process to remove any literal factor, provided that it is not a factor common to the two original expressions. The only condition required to be satisfied by the factor x removed from c in the last Article was that it was not a common factor of a and b. 186. As an example, let it be required to find the H. C. F. or Highest Common Factor of x* 3% + 2 and # 4 1. Taking the former as first divisor and arranging the division in the usual way, a quotient x and a remainder 3# 2 2 os 1 are obtained. The latter must be taken as divisor, and, in order to avoid a fractional coefficient in the quotient, x 3 3#~1>^is multiplied by 3 before it is taken as dividend. x 187.] Highest Common Factor. 107 3#2_ 2 #_i (3 101 10^10 x-l X*- X At the end of the first step of the second division, the remainder is 2# 2 8 a? + 6, and as a factor 2 goes into all the terms of this expression it is better to remove it before proceeding. Ordinarily it is better to carry on each division until the remainder is of lower degree than the divisor. In this case however it will avoid fractional coefficients or the introduction of a new multiplier to take this simplified remainder at once as divisor. We then obtain a quotient 3 and a remainder 10# 10. The factor 1 0, which is not a factor of either of the original expres- sions, may be removed, and xl taken as the next divisor, when it is found to go exactly. The H. C. F. required is therefore x 1. 187. It may be worth while to go through the proof in the particular example last worked, that as I is the H. C. F. The division shows that xl is a factor of a? 2 4 a? +3. It is therefore a factor of 3(# 2 -4#+3) or 3x?-12x+9, and being a factor of 10 a? 10 it is a factor of their sum or 3# 2 -2a?-l. Hence x 1 is a factor of 2# 2 8 x + 6 and of 3 x 3 2x 2 x, and therefore of their sum or 3 a? 3 9 #+6, that is of 108 Highest Common Factor. [188. 3 (tf 3 3a?+ 2). But x 1 is evidently not a factor of 3. Hence it must be a factor of a? 3 3 x + 2, and therefore of x(x z 3# + 2) or # 4 -3# 2 + 2#; and therefore of the sum of this last expression and 3# 2 2x I or of # 4 1. Hence a? 1 is a common factor of the two given expressions. Again, every common factor of the two given expressions divides # 4 1 and x(x* 3x+2) or a? 4 3# 2 + 2#, and therefore divides the difference of these, or 3# 2 2 a? 1. Hence it is easily seen from the second division that every such common factor divides 2 a? 2 8 x + 6 or 2 (a? 2 4 x + 3). But no common factor of them divides 2. Hence every common factor must be a factor of x 2 4#+ 3. From the third division it similarly follows that every such common factor is a factor of 10 a? 10 and therefore of a? 1, since none of them can be a factor of 10. Whence x 1 is the Highest Common Factor. 188. If there be more than one letter involved in the two expressions it may be desirable or even necessary to arrange them, as in the multiplications of Arts. 129-135 or the division of Art. 162.. Let it for instance be required to find the H. C. F. of and Taking x as the dominant letter, the expressions will be written as in the form annexed : 190.] Highest Common Factor. 109 At the end of the first stage a remainder is arrived at of the first degree in x, every term of which contains a numerical factor 3. This may be removed. A more formidable difficulty to proceeding is however to be found in the still remaining coefficient of #, namely y 2 +yz + z 2 . This is evidently not part of the H. C. F., since it could not divide an expression arranged in powers of x unless it were a factor of the coefficient of each power, which it clearly is not in the case of the two given expressions. Fortunately the term independent of x in the remainder, namely ^ 3 + 2y z z+2yz 2 + z 3 , is tolerably easily seen to be the product of y + z into y 2 +yz + z 2 . Hence the latter factor goes into both terms of the re- mainder, and not being part of the H. C. F. required may be removed (Art. 185). There is left x + (y + z), which must be taken as next divisor, and which is easily found to be the H. C. F. required. 189. In any similar case the student must examine whether a coefficient, such as y^+yz + z*, occurring as that of the first term in any expression which has to be taken as divisor cannot be removed in virtue of its being a factor of all the other coefficients. If it cannot, further pursuit of the H. C. F. is probably hopeless unless the factor in question goes exactly into the coefficient of the highest term in the corresponding dividend. 190. The discovery of the highest common factor of two expressions, when any such exists, leads to the resolution, in part at least, of each of them into its component factors. When this resolution can be completely effected it gives the complete solution of a problem almost of equal import- ance with that of finding the Highest Common Factor, namely, that of discovering the Lowest Common Multiple of two expressions. Any expression which contains another as a factor is called a multiple of the latter (Art. 22). 110 Lowest Conunou Multiple. [191. 191. By the Lowest Common Multiple of two expres- sions is meant the expression of lowest degree or dimen- sions of which they are both factors. An infinite number of Common Multiples can be found, but the Lowest Common Multiple, for which words the letters L. C. M. are usually written as an abbreviation, is a perfectly definite quantity, of which all the other Common Multiples are multiples. 192. In the case of two or more monomials (Art. 87) the L. C. M. is discoverable by inspection. It will be ob- tained by taking the product of the highest powers of each of the letters involved that occur in any of the monomials. Thus the L. C. M. of the monomials is evidently a 3 L 4 c 3 . For no lower powers than # 3 , 4 or _2) (0-3) (a- 4) (0-5), which form is for many purposes more convenient than that of the last Article. EXAMPLES. 1. Find the Highest Common Factors of (1) 15a 3 6 2 c, 25a 2 6c 2 , (2) 2. Find the Highest Common Factor of 18a 3 6c and 24a 2 3. Find the Highest Common Factor of 2 -4a;+3 and 2 - 4. Find the Highest Common Factor of x 3 3x^+3x-l and 3 Find the Highest Common Factor of the pair of expressions in each of the twelve following examples. 5. a 3 7&+G, as 3 4cc 2 + cc+6. 6. a; 3 _a; 2 -8a+12, a 3 + 4cc 2 -3cc-18. I 114 Elementary Operations, etc. 7. 9 8. llo: 4 -9aar'-aV-a 4 13a?* lOaa? 3 9. a? 10. x 4 11. a; 4 +4^+16 and 12. a? + 3pa?-(l+3 13. x m y m , x n y n ; (1) when ra and n are both odd numbers ; (2) when m is odd and n is even ; (3) when m and n are both even. 14. a 4 + (a + 6) a 3 + 2a6a 2 + a& (6 + c) a + ab\ and 15. 6a 6 +15a 4 6-4aV-10a 2 6c 2 , and 16. 3 (6-c)+6 3 (c-a)+c 3 (a-6) 5 and bc(b s - Find the Lowest Common Multiple of the expressions in each of the nine following examples : 17. acc-x 2 , a z -x\ 18. o? 2 - 19. 3 - 20. 21. a? 3 y 3 22. or 2 + 3 23. a; 2 ( 24. 9cc 4 - 25. a 2 - 9a-10, a; 2 - 7cc- 30, 26. Find the Highest Common Factor and Lowest Common Multiple of 6ar J -ar 2 -2z-15 and 2oj 3 - 27. Also of the expressions - 4 and 9a5 4 -3a- 28. Also of the three expressions a? 3 6a5 2 +Haj 6, cc 3 -8o; 2 Elementary Operations, etc. 115 29. Find the Highest Common Factor of 7x' 2 -3x-l5 and 30. Find the Highest Common Factor and the Lowest Common Multiple of a? 4 + a?x 2 + a 4 and x 4 2ax 3 + a*x 2 a 4 . 31. Find the Lowest Common Multiple of x 2 3ax+2a 2 , 3x 2 19acc+28a 2 , and x^ 32. Find the Lowest Common Multiple of ar 2 -6a;-8 } or 5 -2^ -a +2, and x* I 2 CHAPTER VII. FRACTIONAL FORMS. 199. The meaning of a fractional form, as -, > in any case when a and b are positive integers, has been explained in Art. 30. It has been shown in Arts. 43-46 that the original definition leads to another, namely that j- is the result of dividing a by b, so that -r may be taken to be a quantity which when multiplied by b gives a. In Art. 48 this second meaning has been taken as the definition of the form 7- when a and b are not integral numbers but fractions. The algebraical form j- can thus be defined as a quantity which when multiplied by or into b gives a. The quantity a is still called the numerator, and the quantity b the denominator, although the original reason for those names no longer applies. 200. With this meaning, the following are the principal laws which govern operations on Fractional forms. The numbering of the equations of Art. 53 is retained. <"> 203-] Fractional Forms. 117 a c ac c , , ( } and as a particular case, c ac c = a+b. (17) b.^ = j = ab+b = a. (18) a c _a d_ad . . * ' d~ b'c~ be' { } 201. These laws have been proved to be true when #, #, c, d, ^ are integral or fractional numbers. If they hold good for other meanings of a, b, c, d, p, all their conse- quences will also hold good for those other meanings. 202. The most important problems to be considered in dealing with fractional forms are, first, the reduction of any single one to its simplest equivalent form ; and secondly, the reduction of an expression involving several of them to one single equivalent form. The first of these problems is usually known as that of reducing a fraction to its lowest terms ; the other involves the processes commonly called addition, subtraction, multi- plication, and division of fractions. 203. In virtue of (13) it follows that the value of a fractional form is not altered by introducing any common factory in numerator and denominator. It equally follows that any common factor already existing may be removed without altering the value of the fraction. Hence if the highest common factor of the numerator and denominator of a fraction be found and the numerator and denominator be each divided by it, there will result an equivalent fraction the degree of both expressions of which is lower than in the original one. The resulting fraction 118 Fractional Forms. [204. is then said to be in lower terms than the original one, and as the remaining' numerator and denominator have now no common factor, no farther reduction can be made, and the fraction is in its lowest terms. 204. As an example take the fraction a; 3 6# 2 +ll# 6 a? 3 12# 2 + 47# 60 In Art. 197 the highest common factor of the numer- ator and denominator has been ascertained to be which = af x a The divisor - + x a x + a (x + a) 2 (x~a) 2 , ~ (x-a)(x + a) + (x-a)(x + a) > (x a r a 2 a 2 2 ax , f x + a x a^ Hence { -, -- ^ -. -- ^ + - - \ -^ \ - - + - { (x af (x + af % 2 a 21 l x a x + a* (x a)* (x + a) 2 (x a) (x + a) (a?-a)(a? + a) w / iq x 2 2 ) And removing, in virtue of (13), the common factors 2, a 2 ,xa and x + a from the numerator and denominator, this finally reduces to -. r-; . or -5 (x a) (x + a) x 2 a 2 One very commonly occurring case is that of the multi- plication of a fractional form by an integral form. This is included in the general case of (16), since an integral form may always be treated as a fractional form whose denominator is unity, for a = a-^-l = -, by (17), (or see Art. 34). , / OF THE > 122 Fractional Forms. [an. b ab Hence #- = c c Thus #-2 #-2 If the multiplier be a multiple of the denominator the result is an integral form. / 2 2\ Thus - x (x 2 a 2 ) = x a os a _(*+)()(+). Art . (116)! X OL = (x 2 + a 2 )(x+a) by (18) of Art. 200. This last case will be found to be of great importance in the solution of equations (Chapter X). 211. The reduction of a compound fractional form to a simple one is effected by previous methods. Thus, for instance, the compound form x a?+l (Arts. 205, 206), x x+\ , - x(x\) (*+!) , ( Art - 208 )' As a somewhat more complicated instance take the fraction 1 - 2i2.] Fractional Forms. 123 This = ^-- = - ~- ' b ^ ( 19 )> (Arts. 205, 206) by (19), a? 2 +l 212. An expression in a fractional form can be reduced to a partly integral and partly fractional form in any case when the degree of the numerator is as high as that of the denominator. Thus the form in Art. 204, -* - , can be written or 9# + 20 (a?2-9#+20 The numerator can always be divided by the denominator until the degree of the remainder is lower than that of the denominator (Art. 158). If the fractional form be denoted N by -=r, and Q be the quotient and R the remainder when N is divided by D, it follows from Art. 157 that N= 124 Fractional Forms. The converse process, of reducing a mixed form to a purely fractional one, can be conducted by exactly reversing this operation. Thus #-l+_L- x2 #-2 EXAMPLES. 1. Reduce to its lowest terms x z 4cc+3 Reduce to their lowest terms each of the fractions in the twelve following examples : 2 a "" a 3 qy-- ' ' ' 7. ~ ~~ ' . 3 9. -T 10 12. ^ a 2 6 2 (a-6) + 6V(6-c) + cV(c-) 13 ' a +(1 fl t!y Fractional Forms. 125 Eeduce to their simplest forms the expressions in the following examples: a + b a b 4ab 14. ; H 7 a b a + b a 2 < .^'-^-j 16- 2(a;-l) + ! < ; -2 + 2(a ; -3)' 17. 3 2 2 ;-l x-2 x-3 18. A + 6 5 ;-l !B-2 a-3 19. X+y y x + y y*x*y 1 a 2 C + 20. a;_2a ' X *-S at-aW + b* a + b a-b 1 a 6_56 -- a 3_ 6 3 ^p a2 _ 6 2- T T ^7 03 / / 24. SnnpHfy / . ... 25 . Slmphfy 26. Simplify |I + I + ^I' (a y (a-y) 2 ) 27. Simplify te_^U{^L + JL.I (x-y a^ + 2/ 2 i (ic- + y ic-y) 126 Fractional Forms. 28. Reduce to its simplest form - -\ xa 29. Reduce to its simplest form (x-b)(x-c) (x-c)(x-a) (x-a)(x-b) (a-b)(a-c) (b-c)(b-ay(c-a)(c-b)' /a + b ab\ /a + b a b\ 30. Sunphfy (_ + _).H (___). 31. Simplify gg + g + i - *(*+*)(+*)(*+c) a; c (a a) (x b) (x c) !+!_! l+I.-l 1+I..I ca a6 6c ab be ca be ca ab 33, Simplify - - --- - 1B+ 34. Simplify 35. Reduce to its simplest form -Ar + - a; a; x x 77 r~ Xs CC 36 - 38 . Simplify Fractional Forms. 127 t 3 b s c s -c) (6-c) (J-) ( e -o) (c-6) 12 3+11 y+12 and find its value when x = 1 and y = 2. 39. Reduce to its simplest form ic a a; 6 a? c (a? )(#&)(# c) *> tJU + r+ 3 ' x+b x+c 40. Reduce to its simplest form 212. -- 2 a CHAPTER VIII. FORMS INVOLVING FRACTIONAL OR NEGATIVE INDICES, OTHERWISE IRRATIONAL FORMS. 213. In Art. 66 it has been shown that the only mean- L ing that can be given to a symbol such as a q , consistent with the general index law (22), is that it must be such a quantity as when raised to the q ih power shall give a p as the result. 214. It remains to show that with this meaning given to the symbol the laws (22) Art. 63, (23) Art. 68, (24) Art. 72, and (25) Art. 73 will hold good for all fractional and negative values of m and n. These laws are (22) (23) (a m } n =a mn = (a n ) m , (24) (ab) n =a n l n . (25) They have been shown to hold when m and n are any positive integers, the only limitation to their values being that in (23) m is supposed to be numerically larger than n. 215. Before proceeding to the proof it is important to consider what meaning can be attached to a symbol such as afr in a case where it directs the performance of an opera- tion on a which cannot be exactly numerically performed. For instance, if a represent 2, the symbol 2* denotes a quantity such that the square of it that is, the product of itself into itself shall be 2. Now no such number exists 2 1 6.] Fractional or Negative Indices. 129 in the form either of an integer or of a fraction whose numerator and denominator are integers. The proof of this fact is contained in the next three articles. 216. It has been shown, in Art. 23, that if the sides of a rectangle AB and AC contain a and b units of length respectively, a and I being integers, the area of the rectangle contains ab unit squares. The same proposition holds good if AB contain - unit lengths and AC T P contain -, where p, q, r, s are positive integers. For - s q is the same thing as and - is the same thing as qs s qs Hence if a length of which (qs) make a unit be taken for a new unit, AB contains ps of these and AC contains qr. Hence the area ABDC contains ps . qr squares, of which the side is the new small unit length. But since the old unit length contains qs of the new one, the square whose side is the old unit contains (qs . qs) or (q 2 s 2 ) of the squares whose side is the new. Hence the area ABDC contains pqrs squares of such a size that the unit square contains q 2 s 2 of them, that is ABDC is represented by the fraction --% of a unit square (Art. 30). But -4~^> = > removing the common factors q 2 s 2 qs qs from numerator and denominator, or the area ABDC contains , that is - - unit squares. qs q s Hence if a and b represent the two sides of a rectangle, ab represents the area, whether a and b be integers or fractions. K 130 Forms involving A similar extension of Art. 24 can be made by the student. 217. Let now ABCD be a square the length of whose side is represented by a. Let AC be joined. Then, by Euclid I. 47, the square on AC is equal to the sum of the squares on AB and BC\ that is, the Square on AC is double the square on AB. Hence if x represent the length AC, in virtue of the last article we must have x l = 2 a 2 . Hence, in virtue of Axiom (4) of Art. 53, dividing these equal quantities by a 2 . or, in virtue of the kw ( 1 6) of Art. 53, Whence by the definition of a fractional index - = 2^. J a 218. By the last article it follows that 2^ can only be expressed in the form of a fractional number provided AB and AC can be both exactly expressed as integral multiples of the same length. It can be shown that this is not possible. Join AD and from AD cut off a length AE equal to AB. Draw EF perpendicular to AD to meet BD in F, and join BE. Then, since AB is equal to AE, the angle ABE is equal to the angle AEB. But the whole angle ABF is equal to the whole angle AEF, both being right angles. Hence the 219.] Fractional or Negative Indices. 131 remainder, the angle FEB, is equal to the angle FBE. Therefore BF is equal to EF. Again, the angle DEF is a right angle, and EDF is half a right angle, therefore DFE, the remaining angle of the triangle DFE, is half a right angle, and equal to FDE. Hence DE is equal to FE and therefore to FB. Thus if there be any line such that it exactly measures both AB and AD, DE their difference must also contain this line an integral number of times, and therefore also DF, which is the difference of DB or AB and BF or DE. Thus if AB and AD have any common measure, that is any length in terms of which they can both be measured by integers, that length must also be a common measure of DE and DF. But DE and DF are the side and diagonal of another square much smaller than ABCD, and by repeating the same reasoning, any common measure of DE and DF must also be a common measure of the side and diagonal of a square smaller again. It is evident that this process may be repeated until both the side and diagonal of the final square are less than any finite length whatever. Consequently no finite length whatever can be found such that both AB and AD can be expressed exactly as integral multiples of it. Hence such a symbol as 2* cannot be replaced by any equivalent fraction whose numerator and denominator are integers. f The same statement applies to any such form as a* } where a is a number, integral or fractional, which is not exactly the q th power of some other integer or fraction. 219. Quantities such as those represented by 2* are called incommensurable or irrational with regard to quan- tities represented by numbers such as were employed in Chapter I. The word incommensurable denotes that the K 3 132 Forms involving [ 220. magnitudes represented by the two kinds of numbers respectively have no common measure, the word irrational that their ratio to one another cannot be expressed by a single simple numerical relation. To the latter subject we shall return presently (Arts 644, 645). 220. It has been shown, by taking a specific instance in Article 218, that incommensurable lengths do exist, and that, if such lengths are to be represented by symbols of a numerical kind, the symbols representing one class cannot be expressed exactly in terms of those representing the other. Such a symbol as 2^ or \/2 (Art. 67) does how- ever represent a clear idea, when it has been connected with the geometrical figure of Art. 218, and any other symbols involving the index \ can similarly have their representative lengths geometrically constructed. There is no simple equivalent construction applicable to such a symbol as 2, but nevertheless the student will probably be able to convince himself that the symbol does, or may be taken to, represent a length intermediate between two lengths which can be represented by numbers commensur- able with unity. 221. That lengths must exist which can only be represented by incommensurable numbers seems tolerably evident from reasoning of the following kind. Let AB represent a straight line, and suppose a point to travel along it from A to B. The distance of this point from A~ P Q ~~B A changes in an absolutely continuous manner. Between any two points P and Q in AB, however near to one another they may be taken, there is an infinite number of interme- diate positions of the moving point. Hence if PQ be ^ th of AB, and AP contain r parts equal to PQ, and con- 223.] Fractional or Negative Indices. 133 sequently AQ, contains (r+1) such parts, between the y 7* 4- 1 two fractions - and of AB there must be an infinite n n number of numbers required to express all the lengths intermediate between AP and AQ. This is true however large n may be. Hence it seems reasonably to follow that some of these lengths cannot be expressed exactly as any fraction of AB, where numerator and denominator are integers, that is, there must be lengths expressible only by numbers incommensurable with unity. 222. It must be understood that for practical purposes all lengths can be sufficiently nearly expressed by numbers commensurable with unity. If P Q be taken so small that the human senses are incapable of taking account of it, all the lengths between ^Pand AQ will be, as far as human consciousness is concerned, represented with sufficient accuracy by the same number. That fact is however only a consequence of the imperfection of human faculties and would cease to hold good as a fact in the case of beings whose eyes had the power of an infinitely magnifying microscope. For the purposes of strict mathematical reasoning, as distinguished from practical calculation, it is necessary to take account of quantities represented by numbers incommensurable with unity. 223. Although it is not possible to find any fraction whose numerator and denominator are integers, which shall be exactly equal to any such number incommensurable with p unity as can be denoted by a symbol S where p and q are positive integers, it is possible to find two fractions with any arbitrary denominator and numerators 'differing by unity, one of which shall be greater and the other less than any such number. For let n be any integer whatever, and let a series 134 Forms involving [214. 123 of fractions - > - - ... be formed. Let each of these n n n fractions be raised to the power q. The resulting powers will form a series of magnitudes each successively greater than the previous one, and by taking the numerator sufficiently large the value of the power can be made to exceed any number desired. Thus there must be powers of two consecutive fractions, one of which is less than a p , while the other is greater than a p : for the supposition that a q is incommensurable with unity shows that none of these powers can be equal to a p . Hence there must be an integer r such that (-) is less than a? and that ( ) is greater than a p . Hence the number whose th power p is equal to a p , that is, the number denoted by 0, must lie r r+l between - and - n n As n may be taken of any magnitude whatever, it is possible to find an ordinary fraction which represents the value of an incommensurable number within any required degree of accuracy. 224. It will, for instance, be capable of proof that, taking n to have in succession the values 10, 100, 1000, &c., the quantity 2* is greater than \^ and less than |, or that it lies between \fa and \^l , or between \^ and iJ, and BO on (Art. 397). 225. There are incommensurable numbers of a different P kind from those represented by such symbols as a 1 . The ratio of the circumference of a circle to its diameter and the base of the hyperbolic system of logarithms are instances in point. About all such numbers however two 228.] Fractional or Negative Indices. 135 things may be assumed : first, that there is always some length which they may be taken to represent, any given length being unity ; and secondly, that, if the unit length be divided into any number n of equal parts, an integral number r can be found such that the length represented by the incommensurable number will contain more than r and less than r + 1 of these parts ; that is, that the incom- m mensurable number will be greater than the fraction - and less than the fraction of unity. 226. The phrase ' incommensurable number ' is con- veniently used as an abbreviation for the more strictly accurate phrase c number incommensurable with unity.' From the fact that incommensurable as well as commen- surable numbers may be taken to represent lengths on any given scale, all numbers of both kinds are sometimes called ' scalar quantities/ or, more shortly, ' scalars.' The quantities i and o>, which we shall meet with later on (Arts 271,277) and which cannot represent lengths on a scale, are called ' non scalar ' or ' operational quantities? The adjectives impossible, imaginary, and unreal are often used to denote these latter quantities. 227. Let now a and I one or both of them represent incommensurable numbers. It is necessary to interpret the symbol a . b. One method of interpretation is analogous to that used in the discussion of the multiplication of fractions (Arts. 33-36). In this method a.b is considered to denote the result of an operation on the line b exactly similar to that which, when performed on the unit line, produced the line a. 228. As an instance, let AB represent the unit line and let AC be the diagonal of the square described on AB ; then AC is represented by the number 2*. 136 Forms involving [229. If then on AC a square be described and AD be its diagonal, AD will be represented by 2* . (2*). But a perfectly easy geome- trical proof shows that AD is equal to twice AB, or AD is represented by the number 2. Hence 2* . (2*) = 2, as ought of course to be the case (Art. 64). 229. To take an instance in which the two factors are not equal. Let AB represent the unit line and let -it be produced to F, making BF equal to AB. On AF describe the equilateral triangle AFG, and draw AE perpendicular on FG. Then AF contains 2 unit lengths, or AF is represented by the number 2, FE by the number 1 ; and if AE be represented by the symbol #, it follows from Euclid I. 47 that 2 2 = I 2 + # 2 , or x 2 = 3 ; consequently AE must be represented by the symbol 3* (Art. 64). If BG be joined and in BG, BR be taken equal to AB, and AH be joined, AH is represented by the symbol 2*. Let EG\)Q produced to K and EK made equal to EA and AK joined. Then AK is derived from AE exactly as AH is derived from AB. Hence AK will be denoted by the symbol 2*. (3*). Again, join HF, IIK and BE. By the construction it is easy to see that All is equal to II F, and also that the angle AHFis a right angle, while the angle ARE or AKFis half a right angle. Hence // must be the centre of the circle which 231.] Fractional or Negative Indices. 137 passes round AKF. Therefore KH is equal to AH, and the angle HAK to the angle HKA. But the angle KAE is equal to the angle HAB, each being half a right angle, and therefore the angle KAH is equal to the angle EAB. Therefore the triangle KAH is equiangular to the triangle EAB. Hence AK is derived from AH by an exactly similar construction as that by which AE is derived from AB, that is, AK is properly represented by the symbol 3* . (2*). Hence 2*. (3*)= 3*. (2*). Hence the commutative law holds with regard to these two numbers. 230. The student can easily see that in any case if the triangle AEK is similar to the triangle ABH, it will easily follow that the triangle^//^Tis also similar to the triangle ABE. Hence if 0, b be any symbols representing the lengths of the lines AE, and API when AB is the unit, AK will equally be represented by a . I or b . a. Thus the commutative law holds for the multiplication of any two numbers incommensurable with unity, assuming that all such numbers can be represented by lines (Art. 225). It should be noticed here that the symbols , b only refer to the lengths of the lines, not, as in Arts. 7480, to their directions. The student can probably convince himself in a similar way that the distributive and associative laws, and there- fore the index law (Art. 63), hold for incommensurable as well as commensurable quantities. 231. Another method of arriving at the same conviction may be indicated. The product a . b may be interpreted in all cases to denote the area of a rectangle whose sides are represented by the symbols a and b. This has been shown to be true when a and b are integers (Art. 23) and when a and b are 138 Forms involving [232. fractions (Art. 216). It is reasonable to assume it when a and b are incommensurable quantities. The results of Arts. 223-225 could be used to show that any other supposition as to the area than this would lead to an absurdity. With this assumption the commutative and destributive laws obviously hold, and if the farther closely connected assumption be made that the volume of a right solid is represented in all cases by the product of the three num- bers which represent its edges, the associative law will hold also (Art. 24). It follows that the index law holds for incommensurable numbers when the indices are positive integers. 232. It can now be assumed that quantities such as a q , even if really incommensurable with unity, may be treated by the same laws as commensurable numbers. P It has been shown (Art. 66) that a q must be a quantity such that when raised to the q ih power it shall give a p . It follows at once that =(), (a) since either of these quantities when raised to the q ih power gives a p . It may be noticed once for all that in dealing with fractional indices there has been given as yet no indication as to whether such a symbol as may or may not have more than one value (Art. 64). This point will be considered later on. At present such an equation as the last given can at the most be taken as an assertion of the fact that one of the values of a q is the same as one of the values of (a*}*. 233. Again, let x = a. 2 35-] Fractional or Negative Indices. 139 Hence, raising both sides to the power #, which may be done in virtue of axiom (3) of Art. 53, it follows that x q = a. Again, raising both sides of this last equation to the powerj??, (?)*=*, or (x p ) q a p , by (24), Art. 214. Whence x p is a quantity which when raised to the power q gives a p , that is, p x p = a q ; or, replacing for x its value, 234. Again, let y a<* . Therefore, by definition, y 3 = p ; and raising both these to the power r, where r is any integer, qr _ a p r ty (16) of (Art. 200) a , a whence the law (22) holds if both indices be negative. 242. By the use of negative indices the value of law (23) is very much diminished. To divide by a n becomes in fact the same operation as to multiply by a~ n , and obviously with this interpretation (23) becomes a mere special case of (22). At the same time it is made more general by the removal of the restriction as to the relative magnitude of m and n. 243. It easily follows that 247.] Fractional or Negative Indices. 143 (*)-" = (^ 1 Hence law (24) holds good when either or both of the quantities m or n are negative. 244. Again, (ab}~ n = -, = - = = a~ n b~* v ' (ab) n a n b n a n t> n Hence law (25) holds when n is a negative quantity. 245. It may be noticed as an extension of (25) that = a n (b- 1 )", by (25) = a n b~, Art. 243 (/3) and that this result holds for all commensurable values of n, positive or negative, integral or fractional. 246. The value of a, namely unity, has been deduced in Art. 70. It is obvious that this value will enable the symbol a to satisfy the laws (22), (25), in any case when it enters into combination with other symbols. 247. It will be desirable to some extent to enlarge the idea of lower and higher powers which has been involved in some of the processes of division of integral expressions. Any power of a letter is higher than a second power of 1 44 Examples. [248. the same letter when the index of the second power sub- tracted from the index of the first leaves a positive remainder. Thus a 4 is a higher power than a!, because f when subtracted from 4 leaves a positive remainder f . On the other hand, a~ 5 is a lower power than a~ 3 because 3 when subtracted from 5 leaves a remainder 5 ( 3), or 5 + 3, or 2 (Arts. 54-55). 248. Thus the quantities 1 1 1 5 ' a a 2 a' ...a\a\a,\, -l, _ , -1- , ... each of which is obtained by dividing the previous one by a, are a series in descending powers of , for they may be written with the notation of negative indices, ...a*,a*,a l ,a,a-\a-*,a-*,... and the successive indices 3, 2, 1, 0, -1, -2, -3 are obviously each obtained by subtracting unity from the preceding one. EXAMPLES. 1. Find the product of a^, a%, a?, a*, a^z, and a%. 2. Multiply together XT, x~%, X&, x$, x~%, x~&. 3. Reduce to its simplest form ( ) . ( V. (- ^ . ^ c ' ^ a ' ^b ' 4. Divide ci* 6^ c* by d* b^ c^. 5. Multiply 2cc+4# x~ l by 3x~% 6x~% + x. 6. Multiply x~ 3 or 2 + 4 of 1 8 by x~ 3 + x~ z + 4x~ l + 8. 7. Multiply a^ + cfi b& + <&* b^ + $ by a^ fa. 8. Multiply a* a*b* + b% by at + a*b* + $, and the product by a a Examples. 145 9. Divide x y by x^y^. 10. Divide 64o3 1 + 27?/~ 2 by 4 03~ + 3 2/~*. 11. Divide a^ + 03^+1 by xs + x%+ 1, and divide the quotient by x% x*+l. 12. Divide x + 1x^y% + y z by 03' 13. Divide 2 yz -\-2zx -\-2xy x* y* 2 2 by 03 14. Find the Highest Common Divisor of and 031 15. Find the L. C. M. of 16. Simplify 17. Reduce to its simplest form 18. Simplify i 19. If a? = m, a v = n, and (a 2 )* = m v n x , prove that 037/2 = !. 20. Reduce to its simplest form a + (a?-V)l a-(a*-b*}* ^ r + CHAPTER IX. SURDS AND IMPOSSIBLE QUANTITIES. 249. In the last chapter it has been shown that such quantities as 3*, 2^, cannot be expressed exactly in the form of any fraction whose numerator and denominator are integers and are therefore incommensurable with unity. Incommensurable quantities of this class are called surd quantities, or more simply surds. 250. The other notation (Art. 67) for 3*, 2*, &c., namely A/3, J{/2, is more frequently used, and will be adopted in the discussions which immediately follow. The student should remember that the two symbols 3* and \/3 are strictly equivalent, and that the use of one rather than the other is merely a matter of convenience. When dif- ferent roots (Arts. 64, 67) of the quantities involved are employed, the index notation is the more useful ; but when there is only one kind of root, as the square root, the symbol V is sometimes easier to work with. 251. A surd quantity involving only the index J, or the square root, is called a quadratic surd. 252. In the case of quadratic surds, equations (25) and (26) of the last chapter become (ab}% = y Axiom 2 (Art. 53). Now c, a?x, b 2 y are commensurable with unity by supposition; consequently c a 2 xb 2 y is so- also. Hence 2abViFi/is commensurable with unity. But by supposition x and y are different numbers and neither of them contains a factor which is an exact square. Let p be the G. C. M. 152 Surds and Impossible Quantities. [265. of x and y, so that x = pu, y = pv> where u and v have no factors in common. Then xy = _p 2 uv, and therefore Vxy = p Vuv. But all the factors in u are different from any in v, and no factor in either u or v is an exact square. Hence uv cannot be an exact square, and therefore Vuv, and con- sequently Vxy, is incommensurable with unity. Hence the supposition was untrue ; and in exactly the same way it can be shown that Vc cannot be equal to a + b Vy. 265. It follows that if such an equation as the following hold between x, y, a and b, x + Vy = a + VI), where y and b are not exact squares, then x must equal a and y must equal b. For the equation can be written, subtracting a from both sides (Axiom (2) Art. 53), (x a) + Vy = Vb, which makes Vb equal to a rational quantity together with a surd, unless the rational quantity vanish. Hence x a must be zero, or x must equal a, and therefore^ must equal b. It further follows that if x + Vy = a + Vb, then x Vy = a Vb. 266. Let now a + Vb be the given binomial surd, and, if there be a binomial surd whose square is equal to a + /b, let it be Vx + Vy. Then ( Vx+ Vy} 2 = a + Vb, or x +y + 2 Vxy = a+ Vb. Art. 263 Hence x+y = a 2Vxy (a), Art. 265 xy = Vb 3 267.] Surds and Impossible Quantities. 153 From these equations by squaring each side of each, a legitimate process in virtue of Axiom (3), (Art. 53), Hence in virtue of Axiom (2), (Art. 53), subtracting the lower equals respectively from the upper ones, or, by Art. 1 1 $ (*-,) = o-8. 08) Hence if a 2 b be not an exact square, there can be no binomial surd whose square is a -f Vb, and further investi- gation is needless. If however a 2 b be the square of some number k, it follows that (/3) is satisfied if we take x-y = k. And since x+y = a, by adding these equals (Axiom 1, Art. 53) we obtain 2x = a + k, therefore x = \ (a + k) ; and by subtracting the upper equals from the lower ones, or y=\(a k). Hence the values of x and y are determined. 267. Let the given binomial surd be 83 + 1 2 A/35. The process of the last article gives us (\/i+N/y) 2 = 83 + 12V/35 ; whence x+y = 83, or, squaring both of these, = 6889, = 5040, and subtracting, &-2xy+f = 1849, or #- 2 154 Surds and Impossible Quantities. [268. which is satisfied if x-y = 43 ; but x+y = 83. Therefore, adding, 2 #=126 or #=63, and subtracting the upper equals from the lower, 2y = 40, or y = 20. Hence the square root required is \/63+A/20, which can be simplified by the method of Art. 253 into the form 3-/7 + 2-V/5. 268. The student can easily verify, by comparison of Arts. 114, 115 and 263, that if the square of Vx + Vy is a + VT, that of Vx Vy will be a Vb. 269. The cube of a binomial surd of which one term is rational is also a binomial surd. This follows from a com- parison of Art. 261 and equation (a) of Art. 263, but it can also be seen by means of example (1) of Art. 122. Thus 2 J + 3 x Occasionally it is possible, by means of this formula, to find values of x and y which shall make (x + Vyf equal to a given binomial surd. 270. It has been seen that the formal laws of algebra however interpreted require the assumption that ( a) (-1} = +ab (Arts. 58, 105). 'Consequently (-a) 2 = + a 2 ; that is, the square of a negative quantity as well as of a positive quantity is positive. Hence such a symbol as ( a 2 )^ involves quite a different sort of difficulty from that which was met with in the symbol 3*. In the latter case there is a number, repre- senting a perfectly definite length, whose square is equal to 3, and an approximation to the value of this number in 272.] Surds and Impossible Quantities. 155 fractions of unity can be obtained, by a process explained hereafter (Arts. 397, 398), to any required degree of accuracy. But no number whatever can have a negative square. Hence such a symbol as ( 2 )* cannot denote a mere number, and it is therefore frequently called an impossible or imaginary or unreal quantity. A few articles must be devoted to the consideration of its possible interpretation. 271. Since the formula (ab) n = a n l n has now been established for all indices (Arts. 238, 244), such a symbol as (-a 2 )* may be written as (- 1)* . ( 2 )*, or (- 1)* . a. Similarly ( afi may be written ( 1)*0*; and as the symbols a and a? have received an interpretation, it only remains to give to (1)* an interpretation consistent with the laws of operation proved to hold in the case of numerical quantities. It will be convenient to denote this symbol by the letter /, where i is determined by the condition ?=_!. (a) 272. It is clear that i cannot represent a number, nor indeed in any case the same kind of quantify as the symbols a, ,V3~ = -1, as ought to be the case. 275. If OC make an angle of two thirds of two right angles with OX, OA is measured to the left, and consequently c is J, while s is still - Hence c + is = The multiplier ought therefore to denote a rotation through one third of four right angles. The multiplier ( ) should thus denote a complete ^ 2 ' revolution. But, as before, 8 8 = 1. In both these cases we see that the interpretation given to i leads to a result consistent with the previous meanings of the signs + and . It is not possible to develope the subject fully without some knowledge of trigonometry : enough has probably 2 7 6.]' Surds and Impossible Quantities. 159 been given to show that the symbol i of the equation (a) of Art. 271 is susceptible of interpretation, and that, with that interpretation, it obeys the laws of other algebraic symbols. No fear need then be entertained of the truth of the results of investigations in which it occurs, provided they are interpretable. 276. If, in the figure of the last article, CA be produced to a distance AD equal to CA, and OD be joined, OD is of equal length with OC; but since AD is measured in an opposite direction to AC, AD is symbolically represented by AC, that is, by -* - - OC. Hence for the line OD, c + is becomes A rotation through two thirds of four right angles, that is, from OX through OC to OD, is thus represented by the multiplier - Bot(=i* 4 2 Here, again, a result is obtained consistent with the interpretation assigned, since a rotation through the angle COX twice repeated is represented by - - ) , and one through twice the angle COX by and 160 Surds and Impossible Quantities. [279. these two symbols ought therefore to be equivalent, as they have just been shown to be. 277. Let the symbol - be denoted by o>, then . l-i\/3 . the symbol - is o> J . 2 It has been shown that &> 3 = 1 . Hence it follows, since (o> 2 ) 3 = (co 3 ) 2 , (24) Art. 72, that (co 2 ) 3 = 1. Each of the three quantities 1, w, o> 2 has therefore the property that its cube (Art. 62) is unity. They are called the three cube roots of unity. Any power of o> as co n is a cube root of unity, since () = ()= 1 ; but the powers of o> beyond co 2 only repeat the same series of values 1, o>, co 2 in virtue of the facts that o> 3 = 1, co 6 = 1, and so on. Thus co 4 = o> 3 o> =co, = a), 278. Since there are three cube roots of unity, that is, three different symbols such that when cubed they give unity as the result, there are three cube roots of any number a. For a can be written as 1 x #, and therefore each of the symbols , a> > j . . . j , - : . . . j 2 9. 9. 2 9. 9. and it is evident that the geometrical positions denoted by these angles up to - or 2 TT are all different, but that the next set of q angles merely repeat these with 2 TT added, and give therefore geometrically the same positions. Hence it follows, as in the last article, that (1) has q and only q different values. A special case of great importance is (1)*, which must denote a rotation through- ,..., that is through TT, 2 ir. . . 2 2 Thus the two values of 1* are 1 and + 1. 162 Surds and Impossible Quantities. [281. Similarly 1 denotes a rotation through any odd mul- i tiple of two right angles. Hence (1)' must denote a rotation through a 3 = 1 ; from which forms it follows that it must con- tain both x + ay + o> 2 and x + eo 2 ^ + o>2 as factors. Since the original expression is only of three dimensions in x, y, z (Arts. 86, 88), it cannot be the product of more than three linear expressions (Arts. 86, 88, 11 2). Hence the only other 283.] Surds and Impossible Quantities. 163 factor possible must be a numerical or symbolical one not in- volving x, y> z. If this factor be denoted by /, we must have #3 _j_^3 ^_ Z 3_ % X yz = k(x +y + z)(x + coj/ + a> 2 2)(# + a> 2 y -f a>z), and by comparing- the term involving 1 # 3 on both sides it is seen that Jc is unity. Hence x 3 +y B \-z z Zxyz= (x +y + z)(x + ay + <*> 2 z) (x + o> 2 ^ + o>z). 283. In the last article it may be noticed that, by exactly similar reasoning, the four expressions co#+y + G> 2 2, a> 2 #+^ + a>2, a># + eo 2 ^ + 2, a> 2 x + a>y + z must also be factors of x 3 -f y 3 + z 3 3 #^2. These factors however are each equal to one of the latter two given in Art. 282 multiplied by some power of o>. Thus o>#+y-fa> 2 2= <># + a> 3 ^ + a> 2 2, since a> 3 = 1, and the resolution in the last article is a complete one, although the factors may be put into different shapes. This first section of the book has been occupied in deducing the laws of algebraical operation and in explaining the various symbolic forms which may arise in applying those laws. The next section will be devoted to the application of these laws to the solution of algebraical equations. EXAMPLES. 1. Reduce to their simplest forms each of the surds V32, 4/16, A/18, Jt/64. 2. Reduce to its simplest form 2 A/f+A/54 -A/24. 3. Given A/2 = 1-4142, calculate the values of to four places of decimals. M 2 164 Examples. 4. Simplify vf v^V 5. Rationalise the denominators of 4 4 6. Simplify ( - --_ Y X (r=- V 4 _y 3 >' V3 + 1 (^2+^/3X^/3 + 7. Simplify 8. Simplify 9. Simplify 10. Simplify 4 59 T^ "4" 7^ i 7^ r 1 1 . Find the difference between the sum and the product of (1 - -s/3) (1 + 2 \/2 - -v/3) ^2-1 2 ^2+^3 12. Simplify 13. Find the square roots of (1) 5 + 2V/6; (2) SS- (3) ^+^2; (4) f+^2. 14. Find the square roots of (1) a+y^ft 2 "; (2) ab + cd+ V(a*- 15. Rationalise the denominator of the fraction 7-2y / lQ N/7-2V10 + 5- Examples. 165 1 6. Find the square root of 1 + n 2 + Vl+ 2 + n 4 . 17. If i be determined by the equation i 2 = l, show that x l+z\/3\ n / 1 *\/3\. ( - ) + ( - 1 is equal to 2, if n be any mul- \ 2 ^ ^ 2 ' tiple of 3, and is equal to 1, for all other values of n. 1 -\-i 8 18. With the same meaning of i show that ( =-\ = 1, and give a geometrical illustration. * 19. If cc, y, 2 be quantities satisfying the equations and if o> have the meaning attached to it in Art. 277, show that SECTION II. EQUATIONS. CHAPTER X. SIMPLE EQUATIONS INVOLVING ONE UNKNOWN QUANTITY. 284. Such, a relation as # 2 a 2 = (x + a)(xa) may be termed an identity ; it is true whatever values may be given to either x or a. On the other hand, suppose that it were given that 2#+5 = 7, (1) it can easily be shown by the use of the axioms of Art. 53 that x must have the value unity. The value unity for x is said to satisfy the equation (1), or to be a root of the equation (1). 285. Equations of condition such as that of the last article occur in all the applications of Algebra to Geometry or Physics. The discovery of the values of the unknown quantity or quantities, which satisfy the given equation or equations, is called the solution of the equation or equations. 286. Equations are divided into classes according to the number of unknown quantities involved. If there be only one unknown quantity to be determined, one equation of condition is sufficient ; if there be two unknown quantities, two equations will be required, and, generally, the number of equations of condition must just equal that of the unknown quantities to be determined. Equations are also divided into classes according to the highest power of the unknown quantity or quantities Simple Equations involving one Unknown Quantity. 167 involved in them. Equations which when reduced to their simplest form only contain terms of one dimension in the unknown quantities are called simple equations ; those in which one term at least is of two dimensions in the unknown quantities are called equations of the second degree or quadratic equations. If the term of highest dimensions involved be of the third degree, the equation in which it occurs is called a cubic equation or an equation of the third degree, and so on. The easiest equations are those of the first degree in- volving only one unknown quantity; and the solution of such equations is the first problem to be examined. 287. For the reduction of all classes of equations the four axioms of Art. 53 are the main bases of operation. They may be repeated here. (1) If equal quantities be added to equal quantities the sums are equal. (2) If equal quantities be subtracted from equal quan- tities, the remainders are equal. (3) If equal quantities be multiplied by equal quantities, the products are equal. (4) If equal quantities be divided by equal quantities, the quotients are equal. These axioms might be all replaced by one to the effect that if equal quantities be similarly operated on, the results are equal ; but it is easier to take the four special cases enunciated separately as above. 288. Take the equation of Art. 284, namely, 2x+5 = 7. By Axiom (2), subtracting 5 from the equal quantities, it follows that 2 # = 7 5 = 2. Hence, by Axiom (4), dividing these equals by 2 we have =1, which gives the only root of the equation. 168 Simple Equations [289. The student should observe that the process of solution is a strict process of syllogistic demonstration, as rigorous as the propositions of Euclid. 289. As another instance take the equation # 2 _ x 3 _ x+ 1 x+5 5 4 ~~2~ 3 The two expressions which are given equal will, by Axioms (3), be still equal when multiplied by any the same quantity. Let them both be multiplied by 60, which is chosen because it is the Least Common Multiple of the denominators of all the fractions occurring in the equation. We thus obtain -2) 60 (#-3) _ 5 4 2 3 or, simplifying the apparently fractional forms by removing common factors from their numerators and denominators, 12 (x- 2) 15 (x- 3)= 30(#+l)-20(#+5), or, multiplying out and removing the brackets, 12#-24-15tf + 45 = 30#+30 20#- 100 ; and collecting terms on each side, The object now to be attained is to get all the terms which involve x on one side of the equation, while those which do not contain #, that is the merely numerical terms, shall go on to the other. This can be done by first adding 70 to both the equal quantities (Axiom 1), which g ives 70 3#+21 = 10#, and then by adding 3# to these equals, whence is obtained 70 + 21 = or 91 = 291.] involving one Unknown Quantity. 169 Then by Axiom (4), if these equals be divided by the coefficient of #, the quotients will be equal, or which of course may be written x = 7. 290. The last example has illustrated the general method of proceeding. The two expressions which are given equal must be multiplied by such a number as shall cause all the fractional forms to disappear. The least number that will effect this object is the L. C. M. of the denominators of all the fractions occurring on both sides of the equation. By Axiom (3) the resulting expressions are equal. When the fractions have thus been got rid of, the expressions on the two sides must be simplified, and the equation will always assume the form ax + b = COD + d ; (a) where 0, b, c, d are numbers. By Axiom (2), subtracting ex and b from both of these equal quantities, it follows that ax COD = db, ((3) or (a o)a = d ~b\ (y) and therefore, by Axiom (4), dividing these equals by (ac), it follows that = ^=*. (8) ac 291. In the deduction of (^3) from (a) it is seen that the terms ex and b which in (a) occur on the right and left hands respectively with a + sign, occur in (/3) on the left and right hand respectively with a sign. It may happen that one of the numbers as b is really negative, so that (a) should be written ax b = cx + d. I/O Simple Equations [292. In this case b must be added to both sides, and the result corresponding to (/3) is ax ex = d + b ; and the term b which was on the left-hand side with a sign , appears on the right hand with a sign + . Hence a general rule arises from the use of Axioms (1) and (2), that any term may be taken from one side of an equation to the other provided Us sign be changed. As a special case a term which occurs on both sides with the same sign may be removed from both. 292. Examples may occur in which the fractional forms involve the unknown quantity in their denominators. The method already given (Art. 290) still applies, provided the L. C. M. of the denominators be taken to include both numerical and literal factors (Art. 193). As an example, take the equation x+l x-l _ 17 -x 2 2(x-l) x+l ~~ 2(x 2 1)' Here the L. C. M. of all the denominators is 2 (a? 2 1), and multiplying the two equal quantities by this L. C. M. we obtain, after reducing the fractional forms to integral ones, (# + l)2_2(#_i)2 = 17 x 2 , or, multiplying out and removing the brackets, x 2 + 2x+l 2x 2 + lx 2 = 17- x 2 , or 6x x 2 1 = 17 x 2 . Adding x 2 + 1 to each of these equal quantities, there results 6x= 17 + 1 = 18, and dividing these equal quantities by 6, x = 3. 293. It is sometimes desirable to simplify the expressions on the two sides of the equation before proceeding to get rid of the fractions by multiplication ; or occasionally, by transposing some of the fractional forms from one side of 294-] involving one Unknown Quantity. 171 the equation to the other and combining two or more of them together, a considerable reduction may be effected and some long operations avoided. 294. As an instance take the equation 1 1 1 1 01 2~0 3 04 The L. C. M. of the four denominators is (0 1) (02) (03) (04), and if the two equal expressions were at once multiplied by this L. C. M., the reduction of the equation would involve the multiplication of three binomial factors for each original fraction. By combining the two fractions on each side into one, before proceeding to multiply, a much simpler process is required. The equation thus becomes (Art. 205) (0-2) -(0-1) _ (0-4)-(0-3) (0-1) (0-2) " (03) (0-4) -1 _ 1 (0-1) (0-2) "(0-3) (0-4)' Now, multiplying these equals by the L. C. M. of their denominators, we obtain (0-1) (02) (0-3)(0 4) or _(0_3)(0_4) =(01) (02); (8) whence multiplying out, -02+ 70- 12 = -0 2 + 30-2; and transposing terms, as before, the two terms 2 disappear and we get 70-30 = 12-2, or 40 = 10 ; and by Axiom (4), 10 5 172 Simple Eqtiations [295. 295. The student will probably be able to pass at once from the form (/3) to the form (8) without writing down the intermediate form (y). It is sometimes also more con- venient to change the sign when both equal quantities are apparently negative. Thus if a = b, it follows, either by multiplying the equals by ( 1) (Axiom 3), or by adding a + b to them both (Axiom 1 ), that a = b. 296. As another example, take the equation 110+12 70 + 4 3(0 + T 6 T ) 11(20+1) "" 2(40+1) ~ 4~(20+l)" Here the L. C. M. of the denominators is 44 (20+1) (40+1). Before multiplying the given equals by this, it is best to combine together the two fractions which have the literal factor in their denominators the same. This is to be done by adding - -- ^ to both sides, or (Art. 291) transposing 4 (20 + 1 j that fraction from the right hand to the left hand, and changing its sign. This gives 110+12 8(0 + A) _ 70+4 11(20+1) 4(20+1) ~~2(40+l)" Combining the two fractions on the left-hand side into one, with the denominator 44(20+1), the expression becomes 44(20+1) 440+48 + 330+18 44(20+1) 770 + 66 70 + 6 of Art " 200 ' Hence the equation becomes 70 + 6 70 + 4 4(20+1) 2(40+1) 298.] involving one Unknown Quantity. 173 Now multiplying these equals by 4(2#+l which is the L. C. M. of these denominators, we obtain or, multiplying out, Whence 31# 30# = 8 6, or x = 2. 297. If the coefficients of the terms be represented by letters instead of definite numbers, the principles of solution are the same. For instance, let the equation be x a x2d # 2c_ 2a ' 3b 4c The L. C. M. of the denominators is 1 2 ale. Multiplying the two equal quantities by this, it follows from Axiom 3 that the products are equal, or 6bc(x a) + 4.ac(x 2b)+3ab(x 2c) = I2a6c, where the fractional forms have been reduced to integral ones. Multiplying out, and collecting all the terms containing x into one term, as in Art. 129, we obtain adding 20abc to both sides, (6bc + 4ac+3ab)x = 32abc ; and dividing these equal quantities by the coefficient of x, 32abc 298. Considerable reductions can occasionally be made in equations by which long operations of multiplication can be avoided. For instance, take the equation 174 Simple Equations [299. If the fractions be at once got rid of by multiplication, it will be found that the coefficients of all the powers of x above the first are the same on both sides, and the equation thus reduces to a simple equation. The process is however long, and the reduction may be more easily effected as follows. By division we obtain 0+4 2x+5 2# 2 +7a?+20 whence Therefore 2x+5 2a? 2 -f- 7x + 20 2x+5 whence x+\ 15 15 and By multiplication 18^ + 45 = 15a?+60, or 30 =15, so that =5. Any simple equation with one unknown quantity can thus, by the use of axioms (l), (2), (3), be reduced to the form ax l . whence, by Axiom (4), it follows that I x = - a 299. A large number of Arithmetical questions can be easily solved by the help of a simple equation. In these cases the chief difficulty is the statement of the question in Algebraical language. A few illustrative examples follow. 302.] involving one Unknown Quantity. 175 300. A sum of j550 is to be divided between three persons A, B and C, so that for every pound sterling" that A gets, B shall have two pounds and C fifty shilling's. In all these so-called problems it is necessary to take a symbol as x to represent some unknown number which is required to be found. Here it will be convenient to assume x = number of pounds sterling that A has. Then 2x number of pounds sterling that B has, 5#? and = number of pounds sterling that C has. 2 5# Hence #-f2#H = total number of pounds =550. Multiplying these equal quantities by 2 to get rid of fractions, it follows that 2#-f 4x+5x = 1100, or \\x 1100. Therefore x = 100. Hence A receives ^100, B receives ^200, and C ^250. 301. Students should, carefully notice that x must represent a definite number, and in their solutions of problems must carefully state of what it is assumed to be the number. Thus in solving the above problem beginners very commonly state, Let x = A's money, or Let x = what A gets, without explaining whether the money is reckoned in pounds or shillings or pence, or, in the latter statement, that it is money of any kind. 302. As another instance take the problem : Two persons A and B working together can do a certain work in 4 days. A working with a different companion C can do it in 3| days, while B and C work- ing together do it in 5| days How many days will A, B and C each take to do it alone ? 176 Simple Equations [302. Let x = the number of days A takes to do the whole work alone. That is, in x days A can do the whole work. Therefore in 1 day A can do - of the whole work. Therefore in 4 days A can do - of the whole work. x But in 4 days A and B tog-ether do the whole. Hence in 4 days B does ( 1 ) of the whole. Therefore in 1 day B does J (l ) of the whole. (a) 3- Again, in 3f days A does - of the whole work. 3 Therefore in 3| days C does (l - ) of the whole. 1 3^ Therefore in 1 day Cdoes (l ^ of the whole ; 1 o or, as it may be reduced, $ (l ) of the whole. (/3) Hence in 5^ days, that is inM days, B does s_e. l(l--) of the whole, i ft and in 5^ days C does ^ $ (l ~j~) f ^ De whole. But in 5} days B and C together do the whole work. 36 I/, 4\|3d 5/, 18\ Hence - T (l--) + . (l- ) = l, whence x can be found. Reducing the compound fractions, the equation can be written 303.] involving one Unknown Quantity. 177 or, multiplying both sides by 7 and getting rid of brackets, x x and transposing terms or 12 = x Therefore 12#=72, whence x = 6. And from (a) it appears that B does T \ of the work in one day ; that is, B will take 1 2 days to do the whole, while from (/3) it similarly appears that C will take 9 days. 303. At what time between one and two o'clock will the hour and minute hands of a clock be exactly in opposite directions to each other ? It is evident that at one o'clock the minute-hand is exactly five minute-spaces behind the hour-hand. Let x be the number of minutes after one o'clock when the required event happens. Then x is the number of minute- spaces over which the end of the minute-hand has moved. Hence, since the minute-hand moves over sixty spaces while the hour-hand moves over five, the hour-hand will have moved over minute-spaces in this time, and consequently be directed at the instant to a mark at the distance (5 + ) minute-spaces from the mark for 12 o'clock. But it is now exactly 30 minute-spaces behind the minute-hand. Hence #=(5H ) + 30 x = 35 + -. N 178 Simple Equations [304. Therefore 1 2 # = 4 20 + # , or 11 x = 420. 420 Therefore x = = 38 T 2 T . 304. Two travellers A and B set out from two places and travel in opposite directions along the road joining them. If c be the number of miles between the places, and a and b the number of miles which A and B respectively walk per hour, find how far each will have travelled when they meet, supposing A to start two hours before B. In problems of this kind it is sometimes useful to assume x to represent not the number actually required but some other number from which the one sought can be easily found. Here, for instance, a good assumption is : Let x the number of hours A has travelled before he meets B. Then x 2 the number of hours B will travel before he meets A, since B starts 2 hours later than A. Hence, since A travels a miles in one hour, he travels xa miles in x hours. Similarly B travels (x 2)b miles in (x2) hours. Altogether A and B have travelled the whole distance c between the two places. Therefore xa + (x 2) b c, or ( 01 ( ' 2b Whence x = r a + b Hence x is the number of hours A travels, and therefore xa, the number of miles A travels, is = = The number a + o of miles which B travels is (x 2)b, that is, 7-^ , which can also be obtained by subtracting from c the number of miles A travels. 305.] involving one Unknown Quantity. 179 305. Examples of such problems can be multiplied ad libitum ; the foregoing- specimens will furnish some hints : for the rest the student must rely on his own ingenuity, sharpened by practice. The first thing necessary is to obtain a very clear understanding of the problem itself, and to recognize the one thing unknown, a knowledge of which is the key to all. The numerical vakte of this quantity is generally the best assumption for x. The next thing is to have a distinct appreciation of the fact that the solution of the problem is a strict process of reasoning, and that the methods of algebra are not a sort of conjuror's box into which symbols may be thrown at random with a certainty of their producing any result which the spectators may desire. EXAMPLES. Solve the equations : 1. 2x= 5 3x. 2. 7# = 4 + 3 involving one Unknown Qtiantity. 181 24. x*+2x-2 x z -2x-2 2x*-Qx+2 - -f- -- = -- x-l x+1 x -3 163 28 x + 2a _ (x + af ' x-2b~ x-b 2 2x'* + 2x+3 x+l ac be 30. ---- = a + b. ox ax 31. From a place A a messenger goes to a place B, 21 miles distant from A, and immediately returns, going at the rate of 4 miles an hour; and simultaneously with the messenger's departure from A, another messenger starts from B at the rate of three miles an hour, goes to A and immediately returns ; find the distance between the two points at which they 'cross one another. 32. A farmer bought equal numbers of two kinds of sheep, one at ,3 each, the other at .4 each. If he had expended his money equally in the two kinds, he would have had two sheep more than he did. How many did he buy ? 33. Two persons A and B divide equally a sum of money consisting of half-crowns, shillings, and sixpences, the values of the several parts being respectively in the ratio of 15, 4, 1. It is found that each has 60 coins. What was the sum ? 34. A cask A contains 10 gallons of wine and' 4 gallons of water. A second cask B contains 12 gallons of wine and 2 gallons of water. How many gallons must be taken out of each so that the mixture may contain 11 gallons of wine and 3 gallons of water ? 182 Simple Equations. 35. At what time between noon and one o'clock are the two hands of a clock exactly opposite to each other 1 36. Find two numbers whose sum is 18, such that when one is divided by 4 and the other by 2 the sum of the quotients may be 6. 37. A certain fraction is equal to f : when its numerator is increased by 5 and its denominator by 9 it becomes |. Find the fraction. 38. The sum of the fourth, third and twelfth parts of a certain number when subtracted from the number leave a remainder which exceeds the fourth part by 5. Find the number. 39. Find a number whose third part exceeds its fourth part by 5. 40. A sum of 10 was distributed in prizes in a mixed school of boys and girls. Three times the sum given to the girls was equal to twice that given to boys. How much was given to each ? 41. A person invests 35000 partly at 4 per cent, and partly at 3 per cent. The income derived from the two parts is equal. How much is invested in each ? CHAPTER XI. SIMPLE EQUATIONS. TWO AND THREE UNKNOWN QUANTITIES. 306. The next problem to be considered is that of the solution of two simple equations involving two unknown quantities. An example will best elucidate the process. 307. Let the unknown quantities be denoted by x and y, and let the equations be 4# 2 4y 5as +y ~3~ '~T~ ~5~ ; 30+1 _ 2a?-y 2y-x 728 The first process is to reduce each equation separately, by methods similar to those of Art. 288 and onwards, so that on one side there shall only be terms involving x and y, and on the other only a numerical term. Multiplying (a) by 30, the L. C. M. of the denominators of the fractions involved, it becomes or 4:0 cc 20 or, transposing terms (Art. 291), 40#+75# 6# 60y y = 20 ; and collecting like terms 109# 66y = 20. (y) Treating (/3) in a similar way, it becomes 8(3#+l)-28(2#- = 7 2-# or 184 Simple Equations. [308. or 24# 56#+7# + 28^ 14y = 8, or 25#+ 14y = 8, or, changing 1 the signs of both sides (Art. 295), 25# 14^ = 8. (8) 308. Any two simple equations in two unknowns x and ^ can always be reduced by the processes of simplification explained in the last chapter to the form (y) and (5). This form may be represented generally by the equations where the suffixes 1 , 2 do not indicate any relation between the magnitudes of the letters . I -- 1 __ .) _ ( __ L 1 1 _ 1 l*T' \ a + b)~ !-:-. y x whence f = - b a Hence from the first equation it follows that x 2x - + = 1, a a or or 3x = a, multiplying the equals by a, or # = > dividing the equals by 3. O And, since j- = - , it is easily shown that y = - d 3 321. Similar principles govern the solution of a set of three simple equations containing three unknown quantities. Each equation must be reduced, by getting rid of fractions and transposing terms so that on one side shall be only terms containing the three unknowns, and on the other 190 Simple Equations. [322. only known numbers or quantities. If the three unknowns be denoted by #, y> 2, the three equations in their reduced forms can be represented by where a lt b l9 c l9 d^ &c. represent known numbers, the notation being in every respect similar to that explained in Art. 308. 322. When the three equations have been reduced to this simplified form, one of the unknowns, as z, must be eliminated between any two of the equations by one of the methods of Arts. 310-315. There will thus result an equation involving only x and y of the form px + qy = r, (a) where jo, q, r are numbers. Let another pair of the original three equations be taken, and z be eliminated between these. There will be obtained another equation involving only x and y of the form where p ', #', r' are known numbers. From (a) and (/3) x and y can be found as in the early part of this chapter. The third unknown, z, can be ascer- tained by substituting these values of x and y in either of the three given equations. 323. As an instance, let the three equations be 5x + 7y+ 3z = 28 ; (l) 7x~ y+llz = 38 ; (2) lla? + 6y- 4* = 11. (3) Multiplying (1) by 11 and (2) by 3, there results 55# +77^+33* = 308, 21 a? -3y+33z = 114 ; subtracting the lower of these pairs of equal quantities each 324.] T^vo and Three Unknown Quantities. 191 from the corresponding- upper quantity, z disappears and we obtain 34 # + 80y = 194 ; and, dividing each, of these equal quantities by 2, it follows from Axiom (4) that 17^+40^ = 97. (4) Again, multiplying 1 (1) by 4 and (3) by 3, = 112, 12z = 33. And, adding- equals to equals, z disappears and there results 53# + 46y = 145. (5) From (4) and (5) a? and^ must be determined. Multi- plying (4) by 23 and (5) by 20 the coefficients of y will become the same, and they give = 2231 ; = 2900. Subtracting the upper pair of equals from the lower, it follows that 669# = 669, thus x 1. Hence, from (4), 17 + 40^ = 97 ; therefore 40y = 80 ; or y=2* Substituting these values of a? andy in (1), that equation becomes 5 + 14 + 3^=28; therefore 3z = 28 5 14 = 9, or z = 3. Hence the equations are completely solved. 324. Arithmetical problems of considerable, intricacy can be solved by the method of this chapter. As an instance, the problem of Art. 302 can be more expeditiously solved by assuming three unknown quantities, instead of one, as was done in that article. 192 Simple Equations. [325 Two persons, A and B, working* together, can do a certain work in 4 days. A working- with a different companion, C, can do it in 3-J days, while B and C working together do it in 5^ days. How many days will A> ', and C each take to do it alone ? Let x = the number of days which A takes to do the whole work alone. y ~~ 33 55 5) 35 -B 53 5) 53 % ~ = ~ 33 33 35 33 ^ 35 33 33 Hence in one day A does - of the whole work, CO H 1 35 33 J3 " 33 33 35 33 7 Therefore in one day A and B together do (- + -) of the v a? y' whole. But as they take 4 days to do the whole, in one day they must do J of the work. Therefore 1 + 1=1. (l) x y 4 By similar reasoning !_i = A. (2) *~3f 18 ( tf*3f 18 1 X - 1 7 y + ^~~5\~ 36* 325. From these three equations #,y, z may be found. Their solution is best obtained without getting rid of fractions, as it may be noticed that #, y, z only occur in the particular forms - > - 3 - , and it is easier to find the values x y z of these latter quantities and then to deduce those of x, y, z. It will be noticed that one of the equations (1) only involves - and - . Between the other two - must be x y z 326.] Two and Three Unknown Quantities. 193 eliminated. The coefficient of - being- the same in both, z this elimination is effected by a simple process of subtrac- tion, from which there results i-I-'-i = m x y 36 12* But by (1) i + i = i- x y 4 Adding these equations, there results 2 _ J_ !_ __ _4_ _ !_ #"12 + 4 ~12 ~ 3~' whence 6 = a?, multiplying by 3#, or a? = 6. Again, subtracting (4) from (l), 2 _ 1 1 2 _ 1 2/ ~~ 4~l2 "~ 12 ~

    -> - And the values of the latter quantities are x y z therefore the best objects for research. 194 Simple Equations. [327. 327. As another example we may take the following problem. There is a number of three digits whose sum is 11. The sum of -the first and last digits is less by unity than twice the middle digit. If the first and second digits be reversed the number is diminished by 90. Find the three digits. If #, y, z represent the digits in the units, tens and hundreds place respectively, the number is represented by 100r+10^ + #. The three given conditions expressed in terms of #, y, z become ll, (1) 2y-l, (2) ) = (1000 +10^ + x) 90. The last of these can be reduced by transposition of terms to 90y 90z = 90, whence y z= 1. (3) Replacing x + z in (l) by its value 2^1 derived from (2), it follows that or 3y = 1 2 ; thus y = 4. Then from (3) 4 z = 1. Therefore z = 5. And then, from (2), x = 2ylz= 8 1 5 = 2. The number in question is therefore 542. 328. Two persons, A and B, set off together to walk from Cambridge to Ely. When A has gone 4 miles, he is obliged to stop for refreshment. B being in haste goes on, and walks faster that he did when with A. After half an hour A goes on, but walks as much slower than his former pace as B does faster, and arrives at Ely two hours and six minutes after B, and an hour and a half later than they 328.] Two and Three Unknown Quantities. 195 would both have arrived had they gone on together at their first pace. B arrives twelve minutes later than he would have done had he gone at the faster pace all the way. Find the distance from Cambridge to Ely and the rates at which A and B walked. Let x = the number of miles walked per hour at the first pace. x +y = the number of miles walked per hour by B alone. Inen x y = ,, A Let z = the number of miles between Cambridge and Ely. -p., z ( the number of hours in which A and B would x \ have walked the whole distance together. 4 _ ( number of hours which they take to walk x \ the first four miles. z 4 ( number of hours which B takes to walk the x +y ~ ( rest of the way. z 4 _ f number of hours which A takes to walk the x y ~~ ( rest of the way. Hence the number of hours actually occupied by B in 4 z 4 the whole journey is H > and the number of hours J x x+y he would have taken had he gone at the faster pace all the way is -. The last condition then gives, +y 4 2-4 z \ x x+y ~~ x+y 5 since twelve minutes make a fifth part of an hour. By subtracting - - from these two equal quantities this equation becomes, x x+y O 2 196 Simple Equations. [329. Again, the whole number of hours occupied by A is 4 j /. _ 4. - H --- H - . Hence the first condition gives # 2 # y 4 1, z- _ 4 z-4 x x 2 xy~x x+y TTy ' 4 1 or, subtracting \- - from these equal quantities, X i -4 -4 8 ,, ' x-yx+y 5 Again, the second condition gives 4 \^ z-4 _z x 2 x y ~~ x T * 4 1 or, subtracting f- - from both sides, ^/ <2 - 4 =^ 4 + l. (3) a? y x 329. The three equations (l), (2), (3) are sufficient to determine z, x, y. Strictly speaking they are not simple equations according to the definition of Art. 286, but, like the equations of Art. 324, they can be solved by similar methods. z _ 4 Equating the two values of - - given in (2) and (3), it follows that z4 _z 8 x ~ x+y 5' or, transposing terms, z 4 __ z 4 __ 3 x x+y 5 Dividing these equal quantities by (z 4), we get 3 _ x x+y 5(^ 4) 330.] Two and Three Unknown Quantities. 197 But from (l), dividing both sides by 4, 1 1 1 Hence x x + y 20 3 1 5(*-4) 20 and multiplying these equals by 20 (2 4), 12 =*-4. Therefore z = 16. 330. To determine x andy there are now left (1) or 1 1 J_ 20' and the result of substituting for z its value 16 in (3), which easily gives 1 1 1 x y x 12 * Getting rid of fractions, the first of these becomes 20^ = x(x+y), (a) and the second gives whence by Axiom (4) 20^ _x(x+y) 12 y x(xy} 3 , . 5 x+ y or reducing - = - . 3 x-y Therefore 5 (?/ A -^ ^ 1 Q *- . K . O f ~f" / U ^^ \J & A */ I 16x+3y-7z= 1. 23. 5 + f.J, f + *=l, S + f.l. a 24. c2/ + 6^=a, a^ + co;=6, bx-\-ay c. 25. ^ + -*=2, ^ + ^=2, ^ + ^ = 2. a 2/ y * 2a; 3y 4z_ 4cc 2y 3^_ Sec 4?/_2^_ r a6c *ct6c ct6c 27. - H =0, H =4. 1 =5. x y z x y z x y z 28. Find z from the following equations in terms of a, 6, c, cy=p-s, d(z-x) = q-r, ez=rs, f( x +y} = qp+g- 29. A person walks from A to .Z?, a distance of 7 J miles, in 2 hrs. 17 J minutes, and returns in 2 hrs. 20 min. His rate of walking up-hill, down-hill, and on a level road being 3, 3J, and 3^ miles respectively. Find the length of level road between A and B. 30. There are two fractions such that the fraction formed with the sum of their numerators for numerator and the sum of their denominators for denominator is f of the greater; and the fraction similarly formed with the difference of the numerators and denominators is J ; also the sum of the numerators is twice the difference of the denominators. Find the fractions. 31. Three persons, A, B, and C, can do a certain piece of work in 8 T \ days when all work together. After two days' working Two and Three Unknown Quantities. 201 all together C goes away, and A and B finish the work in 8 1 days more. If B had stayed for four days, A and C together could have finished the rest in 8 days more. Find the number of days in which each could do the work alone. 32. A certain number consists of two digits ; by the addition of 27 the digits are reversed: find the number. 33. A fraction becomes -| by the addition of 2 to its numer- ator and 3 to its denominator. If 2 be subtracted from the numerator and unity from the denominator it becomes f . Find the fraction. 34. A certain number when divided by a second has a quotient 7 and a remainder 4. If three times the first number be divided by twice the second number, the quotient is 1 1 and the remainder 4. Find the numbers. 35. A cistern can be filled in 5 hours by two pipes A and B. Both are left open for 3 hrs. and 45 min. and then A is shut, and B takes 3 hrs. and 45 min. longer to fill the cistern. Find in how many hours each pipe would fill the cistern alone. 36. There is a number consisting of three digits. If the digits be reversed the new number exceeds the old by 99. If the number be divided by 3 the quotient is a number of two digits the sum of which is equal to the sum of those of the first number, and the digit in the tens' place is double that in the tens' place of the first number. The sum of the first and last digit exceeds the middle one by unity. Find the number. CHAPTER XII. ON GENERAL FORMULAE FOR THE SOLUTION OF SIMULTANEOUS SIMPLE EQUATIONS. 333. THE last chapter has explained the methods by which simultaneous simple equations can be solved. In the present chapter we have to consider certain general formulae by which the results of that solution can be expressed. For this purpose the typical forms given in Arts. 308 and 321 will be employed. 334. Taking the equations of Art. 308, namely c^ (1) c 2 , (2) the elimination of y can be best effected by the first method. Multiplying (l) by b 2 and (2) by b lt it follows that And subtracting the lower pair of equals, each from the corresponding quantity in the upper row, y disappears and we get (a^-a^x = Vi- V* , &> a A c i- The first of these has a sign + . The second is obtained from the first by one interchange, namely of the suffixes 2 and 3. Hence the second term has the sign . The fourth term tf^i^s can ^ e a ^ so obtained by one interchange and therefore has the sign . The third term aj) z c^ can be obtained from the fourth by one interchange, namely of 1 and 3*; hence it is obtained from the first by two interchanges, and therefore it "has the sign + . 208 General Formulae for the Solution of [345- The fifth term can be derived from the first by first interchanging 2 and 3, thus giving a^b z c. 2 ^ and then interchanging 1 and 3. Hence the term a^b^ has a sign + , being obtained from a l b 2 c^ by two interchanges; while aJj.>Ci requires another interchange of the 1 and 2, and has thus a sign . Hence, on the whole, the determinant is the algebraic sum of the six terms and we may therefore write i #1, 2 , # 2 , , 3 , 3 , 345. In Art. 341 the sign of each term has been deter- mined by the number of interchanges of consecutive suffixes. The word 'consecutive' may be omitted; for any two suffixes can be interchanged by an odd number of inter- changes of consecutive suffixes. Thus, let the order of the suffixes in the first term of a determinant of the 9 th order be 1, 2, 3, 4, 5, 0, 7, 8, 9. (a) By interchanging the 1 with its successor 6 times, the order becomes 2, 3, 4, 5, 6, 7, 1, 8, 9; (/3) and by interchanging the 7 with its predecessor 5 times the order becomes 7, 2, 3, 4, 5, 6, 1, 8, 9, (y) which is the same as (a) with 1 and 7 interchanged. The whole number of consecutive interchanges is 6 + 5 = 11, an odd number. 346. To interchange any two suffixes we need only, as in the last article, interchange that one which comes first with its successor till it is just one place in front of the other. Let this require p interchanges. It will clearly require one less than this number, or p 1 interchanges of the second suffix with its predecessor to bring it into the position which the first did occupy. The number of consecutive inter- 347-] Simultaneous Simple Equations. 209 changes required is therefore p +p 1 or 2p 1 , which is always an odd number. Hence an odd number of interchanges of any kind can always be effected by an odd number of consecutive interchanges, and similarly an even number of the one by an even number of the other. Hence the word 'consecutive' in^ Art. 341, although useful in giving a clear idea at first, may be omitted without altering the results. 347. From Art. 344 it follows that 2 b 2 C 2 a t This result, and a corresponding one for determinants of any order, can be obtained otherwise, thus : The terms which involve a^ cannot, by the definition of a determinant, contain any other factor a or any other factor with 1 suffix unity. Hence the multiple^ of a^ must be obtained by taking all possible products of one out of each column and one out of each row of that part of the determinant (which is formed by omitting the first row and column. Hence the terms which multiply a^ are the same numerically as the terms of the determinant Also, since in all terms which involve a^ the interchanges of suffixes are merely those of the ^'s and c's, their signs will be obtained from that of the first, by the same rule as those of the determinant are obtained from its first term. And since the first term aj) 2 c^ is positive in both cases, the whole set of terms involving 1 is 0j L 2 210 General Formulae for the Solution of [348. Similarly the coefficient of a. 2 can be shown to be and that of 0o to be 348. The determinants L 2 2 L and similar ones formed by cancelling any row and column of the original deter- minant, are called minors of the original determinant. 349. It is easily seen that a. For by working out each of these determinants the same expression is obtained as in Art. 344. Or it may be deduced from the fact that the first term of each is a^b^s, and that any other term, as ajb^c^ can be obtained from this either by two interchanges of suffixes so as to obtain in succession o-J^^c^ # 3 #i<^; or by two inter- changes of the letters, leaving the suffixes unaltered, so as to obtain in succession b l a 2 c^^ li^c^a^. Thus the value of a determinant is not altered if rows be changed into columns, and vice versa (Art. 340). 35 O. If the position of any two columns be interchanged, the determinant retains the same numerical value but changes its sign. Thus a 2 3 3 C 3 For any term in the second, as cjb^a^ can be obtained from the first term c^b 2 a z ^ by one more or one less inter- change than is required to bring it from a^b^c^ the first term of the first determinant ; since %^ 2 ^i * s derived from i ^2 c z > by one interchange. Hence every 352-] Simultaneous Simple Equations. 211 term which can be obtained from a^ b 2 c 3 , by an odd number of interchanges will require an even number to come from a 3 b. 2 c L . Thus each term in either determinant is numerically equal, but of opposite sign, to a term in the other. Hence the two determinants are equal in magnitude but of opposite sign. 351. From the last article it follows that a determinant of which any two columns are identical must vanish. For if # ! = Co = #o b n Co + &< a. 354. Another easy deduction from Art. 350 and (/3) of Art. 352 is that rc (y) For, by (/3) of Art. 352, the given determinant a. 2 qb 2 rc. 2 = -p = -pqr ^&2 ^2 ^2 ?^3 ^3 a 3 b^ ^ ! c 3 a 3 2 c 2 b 2 a 3 c 3 b 3 a, b, c, b 3 c 3 by Art. 350, (/3) of Art. 352, Art. 350, (0) of Art. 352, Art. 350, Art. 350. 355-] Simultaneous Simple Equations. 213 355. The theorems of the last few articles can now be used for the formal solution of the three typical equations of Art. 321, namely = ^i, (a) Multiplying (a) by (y) and b 2 c 2 , they become xa 2 The sum of the three expressions on the left-hand sides of these equations must, by Axiom I, be equal to the sum of the three expressions on the right-hand side. If the addition be performed, the coefficient of x on the left-hand side becomes -*3 + a. but by Art. 347 this is The coefficient of y becomes \ ; -*, j *i *i c i which by Art. 347 is ^2 *2 C 2 351. ^3 ^3 ^3 H 1 1 which vanishes by Art. 214 General Formulae for the Solution of [356. Similarly the coefficient of z is 'I *l j which also vanishes. The sum of the expressions on the right-hand side is -cL which by Art. 347 is Hence the addition finally gives 'i L i b.. , 2 2 X o 6 d 2 b 2 c 2 356. Similar formulae can be obtained for the values of y and z. The three equations (a), (/3), (y) can be written Whence, by a process identical in form with that of the last article, d 2 a 2 c. 2 b 2 a 2 3 3 358.] Simultaneous Simple Equations. 215 But, by Art. 350, if the first and second columns of the determinants in the numerator and denominator be inter- changed, the determinants will have the same numerical values with opposite signs. Hence the interchange will not affect the value of y. Thus, finally, h d L c, cL 2 *a 3 ^3 a l &1 C l # 2 # 2 C. 2 #o b~ c o 3 o By an exactly similar process it can be shown that a i i < W z = i *i c i a 2 b 2 c. (0 357. The formulae (5), (e), (C) are easy to remember. The denominator in the value of each of the three quantities #, y^ z is the same, and is the determinant of the nine coefficients of the unknown quantities written in the order in which they occur. The numerator in each case is obtained from the denominator by replacing each coefficient of the particular unknown whose value is to be given by the term which stands on the right-hand of the corre- sponding equation. Thus, in the value of x, each a must be replaced by the corresponding d, while in the value of y it is the b's for which d's are substituted ; and in the value of z, cs are similarly exchanged. 358. The student can always solve any set of simul- taneous equations with three unknowns by the formulae of Arts. 355, 356. In the case of equations with numerical or unsymmetrical coefficients there will often be little 216 General Formulae for the Solution of [359. advantage gained by this method over that suggested in Art. 322 and exemplified in Art. 323. When the coeffi- cients of the unknowns are reasonably symmetrical with reference to the letters involved, the determinant method often presents great facilities. 359. The following equations will serve as an instance : = A, a?x + ffly + c z z = A 3 . By (8) of Art. 355, A b c A 2 b 2 c 2 7* ~~ A* b* c 3 v abc a 2 b 2 c 2 a? b* c 3 a # c 111 The denominator a 2 b 2 c 2 abc a be a 3 b 3 c 3 a 2 b 2 c 2 I 1 1 Now if, in a b c , a were given the a 2 b 2 c 2 Art. 354. determinant would vanish by Art. 351. Hence a b must be a factor of the determinant by the theorem of Art. 163. Similarly a c and b c must be factors, and therefore the product of these three, or (a b) (a c) (b c). Now the determinant is a homogeneous function of three dimensions in a, b, c, since each term is the product of one factor out of each row. Hence it can contain no other factors involving the letters a, b, c. Hence 1 1 1 abc = k(a-l)(a-c)(b-c\ a 2 I 2 c 2 where k is some numerical factor independent of a, b, c. 360.] Simultaneous Simple Equations. Hence the denominator of the value of x is 217 But the numerator is obtained irom the denominator by replacing a by A. Hence the numerator is kAbc(A-b)(A-c)(b-c}. _kAbc(A-b}(A-c}(b-c} _ A(A-b)(A-c) -i-Lciiut <& j -. -, i\ / \ / / \" 7 7\~7 \ kabc(a l)(a c](b c) a(a b)(a c) The value of k is easily seen to be 1 , by comparing- the first term in the determinant with the known value of the product of (a b), (a c), (b c); but, as is shown above, its discovery is not necessary for the evaluation of x. The values of y and z can be written down by symmetry. 360. As another example take the equations x+y -f- z = a +b -fc, Ix + cy + az = ad + Ic + ca, cx + ay + bz = ab + Ic + ca. Hence the value of x is given by the equation a+l+c I 1 ab + bc + ca c a ac + ab + bc a b x = 1 1 c a a b By an extension of (a) of Art. 352 the numerator of x can be written as the sum of the three determinants, and = = a = a a 1 1 b 1 1 c 1 I ab c a -f be c a + ca c a ac a b ab a b cb a b 1 1 1 1 1 1 1 1 1 b c a + b c c a + c a c a cab a a b b a b 1 1 1 b c a > since the last two determinants vanish, cab by Art. 351. 218 General Formulae for tJic Solution of [361. Hence x = 1 1 1 b c a c a I = a. 1 1 1 b c a cab Similarly y = b, z = c. 361. If three unknown quantities x,y,z be connected by two relations of the form =0, (1) = 0, (2) it is not possible to determine from these two the values of x, y, z. It is however possible to determine the values of /V> A I _ * the fractions - and - z z The two equations can be written x y 1 z 1 z *' x , y These forms are obtained from the original ones by dividing by z and transposing the terms ^ and c. 2 . From these latter equations, by Art. 337, it follows that ; by Art. 343 (ft) and (6) 362.] Simultaneous Simple Equations. 219 From the first of these equations, multiplying- by the product of the denominators, it follows that x z '1 L l y From the equation giving - it similarly follows that y z c l a l a l b l a L And these two relations can be symmetrically written x y _ z c i a i\~ or, working out the determinants, x y (a) 362. If #, y, z be further connected by a third relation, a z x + bzy + CzZ 0, (3) and if each of the fractions in (a) be assumed equal to a quantity u, the substitution of #, y, z from (a) in this new equation gives ^ q and assuming that u is not zero, that is, that the three equations (1), (2), (3) are satisfied by values of #, y, r, different from zero, it follows that a + 6, -i "i = 0. (ft) By working out the determinants it is easily seen from Art. 344 that this is equivalent to #0 #0 O 6 = 0- 220 General Formulae for the Solution of [363. a l 2 I c c a \ a 2 C l C 2 which by Art. 347 gives a 3 a i a 2 1 b, b. 1 2 C 3 3x 2 +3x- + bx*- + 6sc*-lBa? + - The square root of the first term is evidently # 3 . The first trial divisor is therefore 2 # 3 and the quotient is 3 x 2 . The complete divisor is therefore 2 a? 3 3 a? 2 . When the product of this into 3 # 2 has been subtracted, there is still a remainder, 6 # 4 2 # 3 + 1 5 # 2 6 # + 1 . At this stage the square of # 3 3 a? 2 has been subtracted, and # 3 3# 2 occupies therefore the position of a in Art. 367 after the first step. Hence 2 (a? 3 3 x 2 ) must be taken as the next trial divisor, and the process carried on by similar steps until there is no "remainder. 375. The student can easily verify that at the end of each complete step in the last article the square of that part of the root which has been ascertained has been sub- tracted. This is obviously true after the first step. Suppose it to be true after any step, and that a is the part of the root already ascertained, so that a 2 has on the whole been subtracted ; then 2 a is the trial divisor, and if b be the quotient 2 db -f I 2 is now subtracted. That is, on the whole, a 2 + 2 ab + b 2 , or (a + b) 2 , is the total subtrahend when a + b is the part of the root discovered. That is, if the statement in italics be true after any one step in a process conducted 377-] Square Root. 229 according to the law explained, it will also be true after the next step. But the statement is true after the first and second step, therefore it will be true after the third step, and so on, till the process comes to an end. 376. It follows that, if at last there is no remainder, the process proves that the expression obtained is the square root of the given expression. For the square of the root obtained has been subtracted from the given expression, and there is no remainder, that is, the square of the deduced expression is equal to the given one, which is, by definition (Art. 64), the condition that the former is the square root of the latter. 377. Let x* 10 x 3 +14 x 2 1 1 # + 1 3 be an expression of which it is required to ascertain whether it have an exact square root, and if there be one, to find its value. The process may be conducted as in the previous examples, and is represented below : 2x 2 -5x IQx 11 llx 2 llx+13 It is evident that, if there be a square root, x 2 must be the first term. Reasoning as in previous articles, it is equally evident that the next term must be 5 x, and the next term H. At the end of the indicated process, the square of x 2 5 CD 11 has on the whole been subtracted, and there is a remainder, 66# ^, which is of lower degree in x 230 Square Root. [378. than the next trial divisor. Hence the working- cannot be continued, and it is evident, first that the given expression is not an exact square, and secondly that the square of x~ 5x 1-1 differs from the given expression by a quantity of lower degree in x than that of any other expression. Hence in a sense x 2 5 x 11 may be called the nearest approximation to the square root of the given expression. 378. In a manner similar to that of the last article the approximate square root of any algebraical expression of which x is the dominant letter can be ascertained ; subject to the two conditions, (1) that the index of the highest power of x involved is even, and (2) that the numerical or literal coefficient of this highest power is an exact square. These two conditions are obviously necessary that the first term in the expression shall be an exact square of some quantity a. If, when the process is carried as far as possible, there is no remainder, it shows that the original expression is an exact square ; and in that case the exact square root has been arrived at. If there be a remainder the expression given is not an exact square and the deduced expression is only an approximate root. 379. The investigation of the square root of a number depends on that of an algebraical expression and is con- tained in the following articles. 380. In the ordinary system of decimal notation any number is expressed in the form (Art. 164) p Q x n +? l x n ~ l +p 2 x n - 2 +...+p n -iX+p n , (a) where x has the value 10. The quantities^, p l . Pz'-Pn-nPn i n this case are called the digits, p n being the digit in the units' place, p n -i that in the tens' place and so on. All these numbers are nu- merically less than 1 0, they cannot be negative although any 382.] Square Root. 231 of them may be zero. If the highest power of x involved be the n ih , the number of digits is (n + 1). 381. The restriction that the digits must always be less than 10 introduces the chief difference between the mani- pulation of a number decimally expressed and an algebraical expression. Suppose for instance that in multiplying two numbers together a term qx n is arrived at, where q is greater than 10 or x. Let q contain x p times with remainder r, so that q = px + r, where we may suppose p and r each less than x. Then n and the term qx n is replaced by two terms, one of which, jjx n+l , will have to be added to the next higher term in the number, if there be one, or otherwise set down as the highest term by itself. This is the well-known process of * carrying ' in arithmetical addition and multiplication, the part p which gives the number of #'s, or tens, in q (px + r) being ' carried ' one place to the left so as to be a multiplier of c n+1 instead of x n . 382. The square of any number as (a) in Art. 380 will thus probably assume a different form from that which it would primarily take as an algebraical expression. Treating it algebraically, the terms involving the highest powers of x are easily found by multiplication in the ordinary method to be Now PQ may have any of the values 1, 2, 3 up to 9 inclusive. Hence jt? 2 may be only unity or may bo as great as 81, and part of j? 2 may have to be carried to a higher place. Thus the square of (a) may involve x 2n+l . It cannot involve a higher power of x than this, because (a) which contains no 232 Square Root. [383. higher power than x n is certainly less than # n+1 * , and therefore the square of (a) is certainly less than ft 2 "" 4 " 2 . Hence the square of (a), a number involving n + 1 digits, may contain 2 n or 2n+l digits. Thus the square of a number of one digit will contain either 1 or 2 digits ; that of a number of 2 digits will have 3 or 4 digits, and so on. 383. The second term in (/3), 2j9 /? 1 # 2n - 1 , may similarly give something to be carried to the term involving a? 2 ", and thus the terms, which in the Algebraical process of finding the square and the inverse process of finding the square root were distinct, become in the arithmetical square of a number mixed up and indistinguishable. This fact, as we shall see, sometimes gives rise to a little uncertainty in ascertaining the second figure of the square root. 384. The usual method of ascertaining the number of figures in the square root of a given number, is to put a dot over eveiy other digit beginning with the units' digit. These dots will be over the digits which multiply the unit, 10 2 , 10 4 , 10 6 ,...and every even power of 10. If there be n + 1 dots the last of them will stand over 1 2 *, and the given number contains either 10 2n or io 2ll+1 as the multiplier of its highest digit. Hence its square root must contain 10 n as multiplier of its highest digit. The square root therefore contains as many digits as there are dots placed over the number. Also if the first digit of the square root be J9 , the multi- plier of 1 2n in the square will be at least /> 2 and may exceed this value (Art. 383). Thus p Q will be the integer whose square is next below the multiplier of 1 2n .in the given number ; that is, next * The greatest number containing six digits, for instance, is 999999, and this is less by unity than 10,000,000 or io 7 . Thus the greatest number containing n + i digits is less than the least number containing n + 2 digits ; that is, is less than 386.] Square Root. 233 below the number formed by the figure under the left hand dot and the one to the left of it, if there be one. The first term of the square root is therefore p Q I O n . The first trial divisor will be 2/J 10 n . 385. At this point a difficulty comes in. In the alge- braical process, the trial divisor being 2 a, the terms in the remainder 2 ab + b 2 are quite distinct and the quotient b is obtainable by dividing the first of them by 2 a. In the number the terms are mixed together, and the whole of the remainder must be taken as dividend. Occasionally the value of b thus deduced is too large, and when the complete divisor 2 a + b is multiplied by b, a number is ob- tained greater than the remainder of the given square. If this happen, a value of the second digit smaller by unity must be taken, and the process repeated until the correct digit has been found. After the second digit has been found there is, as will be shown (Art. 391), no further chance of error. 386. As an example let it be required to find the square root of 1522756. 1522756 ( 1000 + 200 + 30 + 4 = 1234 1000000 2a = 2000 b= 200 2a + b 2200 522756 440000 2a = 2400 b= 30 2a + b = 2430 82756 72900 2a = 2460 b= 4 2a + b = 2464 9856 9856 Placing dots over each alternate digit, beginning with 234 Square Root. [387. the units' figure, the last dot falls on the left-hand figure which 'multiplies 10 6 . Hence the first digit in the square root is 10 3 . The trial divisor is 2000, and the quotient must lie between 200 and 300. Hence 200 is the next part of the root. The process in this case, since there is finally no remainder, shows that the square of 1234 is the given number, or that the square root of the latter is 1234. 387. As another example let it be required to find the square root of 31371201. 3i3712oi (5000 + 600 + 1 = 5601 25000000 2a = 10000 I- 600 2a + b 10600 6371201 6360000 2a = 11200 6= 1 11201 11201 11201 Here the last dot stands over the last digit but one. The highest even power of 10 involved is 10 6 . Hence the highest power of 1 in the square root is 1 3 , and the digit multiplying this power in the square root must have its square less than 31, the multiplier of 10 6 in the given number. Hence the first term in the square root is 5000. The rest of the process is similar to the former examples. 388. In working out such examples as those of the last two articles, it is not usual, nor is it necessary, to write out all the noughts. Neither is it necessary at each stage to bring down more than two figures of the remainder, except in the last stage of the second example, where if only the figures 12 were brought down the trial divisor would not go into the part brought down. This is indicated by the ' nought' occurring in the result 5601*. The operations of the last two articles are concisely represented below. 39-] Square Root. 235 1522756 (1234 1 31371201 ( 5601 25 22 52 44 106 637 636 243 827 729 11201 11201 11201 2464 9856 9856 The student should carefully compare these conciser forms with those previously given. 389. Suppose that the square root required is represented by an expression such as (a) in Art. 380, so that 10 n is the highest power of 10 involved. Let us further suppose that the first $ + 1 digits have been found by the process of the last few articles. The whole number of digits being n + 1 there remain n s undiscovered. The highest power of 10 involved in these remaining digits is therefore 1 o- 8 " 1 (Art. 380). And the number represented by the un- discovered digits is less than 10 n ~ 8 (Art. 382, note). 390. If the part of the root already ascertained be de- noted by a and the unknown part by b, the whole square root is a + b. The original number is therefore a 2 + 2 ab + b 2 from which at the supposed stage of the process a 2 has been subtracted (Arts. 369, 372, 375). The remainder is therefore 2 ab + b 2 and the trial divisor is 2 a. The total quotient of the remainder by the trial divisor is. thus , b 2 It has been shown that b is less than 10 n ~ 8 , while a is greater than 10 n . Hence is, on both accounts, less than - , or less 2 a 2 x 1 than | x 10"- 28 , and a fortiori less than lO"" 28 . 36 Square Root. [391. l<2. That is, contains n 2s digits at most (Art. 380), 2a I 2 while b contains n ?, that is, at least s more than . b* Hence the .highest s digits in I and b + - - will be the ' '/ same. Thus when s + 1 figures have been obtained by the ordinary process, the trial divisor will give the next s figures accurately. 391. Hence, for instance, when two figures have been ascertained by the regular process, one more can be ac- curately obtained by division by the trial divisor. If one figure only has been obtained, it is uncertain whether the trial divisor will give even one more figure with accuracy. Trial must be made of the first figure of the quotient, and if this turns out to be too large, a number smaller by unity must next be tried until the correct one is reached. 392. As an example in which this difficulty occurs let the given square be 346921. 346921 250000 1090 96921 98100 Pointing it in the usual way, it is seen that there wall be three figures in the root, and that the first of these is 5, that is, the first part of the square root is 500. Subtracting the square of this from the given number, the remainder is 96921. The trial divisor 2 a is 1000, and the quotient is 90. Adding this to the trial divisor, the complete divisor becomes 1090, and when multiplied by 90 thus gives 98100, a quantity greater than the remainder of the given number. Hence 9 is too large for the second figure. Replacing the 9 by 8 the square root is found, as in former 393-J Square Root. 237 examples, to be 589. The process, omitting needless figures, is represented below : 34692i (589 25 2a= 100 1= 8 = 108 20= 1160 b = 9 20 + 6 = 1169 969 864 10521 10521 393. Let it be required to find the square root of 200000000. Pointing in the usual way, it is clear that there are five figures in the square root and that the first of them is unity. 206060606 (14142 1 20 = b = 20 4 = 24 100 96 20 = 280 1 2a + b= 281 400 281 20= 2820 b= 4 2a + b = 2824 11900 11296 20 = 28280 b= 2 20 + 6 = 28282 60400 56564 3836 238 Square Root. [394. In this process, it may be noticed that in the second step the quotient of the remainder 100 by the trial divisor 20 is 5. If 5 were taken as second figure in the square root the product of the complete divisor unto 5 would be too large. This example thus gives another illustration of the possibility of error in the second figure. It will be further noticed that there is, finally, a re- mainder. Thus the given number is not an exact square. The number 14142 is the largest number whose square does not exceed the given number 200000000, and is frequently called the approximate value of the square root of the latter. 394. The square root of a fraction can be obtained by taking the square roots of the numerator and denominator separately; thus = ^U (Art 252 (0)) and this is true whether a and I are algebraical expressions or numbers. When both the numerator and denominator are exact squares, an exact square root of the fraction can be found. When the numerator or denominator are, either or both of them, not exact squares, only an approximate square root can be found. It is always possible to multiply the numerator and denominator by such quantities as to make the denomin- ator an exact square. When this is done, the fraction whose numerator is the approximate square root (Arts. 378, 393) of the numerator and whose denominator is the square root of the altered denominator of the given fraction, is called the approximate square root of the given fraction. Thus A /- - - ^ V k ~ b 39 6.] Square Root. 239 And if x be the approximate value of Vab, -r is called the approximate value of A / - . 395. The most important class of fractions are those which are expressed as decimals, that is, fractions having some power of 10 for their denominators. The denominator, in the case of a fraction or mixed number expressed as a decimal, is the power of ten whose index (Art. 62) is the number of digits in the decimal part of the given number. In order that the denominator may be an exact square, there must therefore be an even number of digits to the right of the decimal point. If there be originally an odd number, a nought can be added at the end to make the number of decimal places even. The square root of the denominator will then be a power of ten whose index is half the number of decimal places in the given number. To find the square root, exact or approximate, of the fraction or mixed number, all that has now to be done is to find, by previous methods, the square root of the numerator, and from that to mark off half as many decimal places as there were in the original number. 396. Let it for instance be required to find the square root of 3137-1201. This really represents a fraction whose numerator is 31371201 and whose denominator is 10000. The square root of the former is (Art. 387) 5601, and that of the latter is 100. Hence the square root of 3137-1201 is- , or 56-01. It thus appears that the square root of 3137-1201 can be obtained by the same process as that of 31371201. It is clear that since there is an even number of decimal places 240 Square Root. [397. the dots used in pointing (Art. 384) will fall over the same digits whether we begin over the unit figure 7 in 3137-1201 and place a dot over each alternate figure both to right and left, or place a dot over the unit figure 1 in 31371201 and point backwards, as in Art. 387. The former method is usually more convenient, and the rule for finding the square root of a mixed number becomes . Place a dot over the unit figure of the mixed number and over every alternate figure both to the right and the left. If the last dot to the right does not come over the last decimal figure, add a nought, which will make the number of decimal places even, and a dot will fall over this nought. Then find the square root of the whole, disregarding the decimal point, and mark off in the result as many decimal places as there are dots to the right of the decimal point. 397. The approximate square root of a number can now be found to any number of decimal places. Thus 2 can be written as 2-00, 2-0000, or in an infinite number of such forms, adding noughts ad libitum. The approximate square root of 200 has been found (Art. 393) to be 14. Hence an approximation to the root of 2 is 1-4. The first three steps of the working in Art. 393 give the approximate square root of 20000 as 141. Hence a second approximation to the square root of 2 is 1-41. Thus the whole of the work done in that article gives 1-4142 as an approximate square root of 2. 398. The successive approximations 1-4, 1-41, 1-414, 1-4142 are numbers whose squares differ from 2 by succes- sively smaller quantities. For by Art. 375 it follows from the working in Art. 393 that the squares of 14000, 14100, 14140, and 14142 differ from 200000000 by the numbers 4000000, 1190000, 60400, and 3836 respectively. And dividing all 400.] Square Root. 241 the numbers by the square of 10000, that is 100000000, it follows that the squares of 1-4, 1-41, 1-414, 1-4142 differ from 2 by the respective fractions -04, -0119, -000604, and 00003836. 399. The example in Art. 393 has thus given what is called the approximate square root of 2 to four places of decimals. If eight more noughts had been added to the number 2 at the beginning of the operation, the whole square root would have contained nine figures. Of these, five have been found, and therefore, in virtue of Art. 390,/0wr more can be obtained merely by dividing the remainder by the trial divisor. The part already found is 141420000; the trial divisor is 282840000, and the remainder is 383600000000. Hence the four figures required are to be obtained by dividing 38360000 by 28284. 28284 ) 38360000 ( 1356 28284 100760 84852 159080 141420 176600 169704 6896 Hence 1356 are the last four figures, and the approx- imate square root of 2 x 10 16 is therefore 141421356. Hence the square root of 2 approximately to eight places of decimals is 1-41421356. 40O. The examples of numerical square roots in this chapter have not been in all points worked out in the briefest manner possible. The object has been rather to 242 Square Root. exhibit the reason of the process than to give rules for expeditious operation for which the student can refer to any treatise on Arithmetic : although a little practice will soon lead to all useful abbreviations when once the prin- ciple of the method is understood. EXAMPLES. Find the square roots of 2. 9a 2 3. a 4 - 4. 49ce 2 5. a; 4 6. 4a 4 - 7. 8. 4a 6 9. a 6 225 45 111 33 121 256 13. 14. 174-5041, 2732409, 90643-1449. 9 12 15. a 2 -4a+^ -- + 10. or x 16. Sqtiare Root. 243 1 7. a 2 b~ 2 + We- 2 + c 2 a- 2 + 2 ac" 1 + 2 cZr 1 + 2 Sa" 1 . 18. 19. 20. 21. l- 23. Find the value of \/3 to eight places of decimals. 24. Find the value of >/24 to six places of decimals. CHAPTER XIV. QUADRATIC EQUATIONS. 401. An equation which involves the unknown quantity to the second and no higher degree is called a quadratic equation (Art. 286). We shall at present deal only with equations involving one unknown quantity. 402. It is evident that by processes of reduction similar to those employed in solving simple equations (Arts. 290, 291) any quadratic equation can be reduced to consist of three terms at most, two of them involving respectively # 2 and %, and the third being independent of x. The general type of a quadratic equation is therefore ax 2 -\-bx-\- c = 0, (1) where a, b,c are scalar quantities (Art. 226) independent of x. 403. The following preliminary theorem is useful in the solution of such equations. From Arts. 114 and 115 it follows that = (x-af. If in these identities \k be written instead of a, they become ^ 2 ; (a) Hence, if (|^) 2 be added to either of the expressions or x 2 kx, the sum will be an exact square. 404. Let the quadratic equation be = 0. 405.] Quadratic Equations. 245 Subtracting 3 from both sides of this equation, it follows that x 2 - 4# = -3. On the left-hand side there is now an expression of the form x 2 kx, k having the value 4. Hence if (|/) 2 , that is, in this case, (2) 2 , or 4, be added to both sides, the expression on the left-hand side will be the square of x \k> or a? 2. Thus # 2 -4# + 4 =4-3 = 1, or (#-2) 2 = l. Hence x 2 is a quantity such that its square is 1, or, by Art. 280, x 2 must be either +1 or 1. Thus the given quadratic equation resolves itself into the tivo simple equations x~2 = +1, x-2 = -1. The former gives x = 3, the latter x = 1. Hence the quadratic equation has two roots (Art. 284) ; that is, two values of x, namely 3 and 1, satisfy it. 405. Let the equation be x x+l 13 x+ 1 x 6 Multiplying the equal quantities by 6 #(#4-1) so as to get rid of fractions, it follows by Axiom 3 (Arts. 53, 287) that or, multiplying out, whence transposing and collecting terms, x 2 # + 6 = 0, which is of the form (l) of Art. 402. It follows, by transposition, that x 2 -}- x= 6. Here the left-hand expression is of the form k having the value unity. The addition of (J) 2 to both 246 Quadratic Equations. [406. sides will, by Art. 403, (a), make the left-hand member the square of x + }, . Is 2 1 25 (*+ 5 )= 6+ - = - 25 Hence x -f J is a quantity whose square is ; that is, must (Art. 280) be either +- or - The quad- 2 2 ratic equation gives therefore the two simple equations 1 5 and {B+ 2 = ~2' From the former we obtain 5 1 4 ff _ _ 9 . ~ 2~2~2~ and from the latter _5_ I- 5- ~2 ~ 2~ ~ 2~ The two roots are therefore 2 and 3. 406. The solution of the general type equation of Art. 402, namely ax 2 + bx + c = 0, can be conducted in a similar manner. Subtracting c from each of the two quantities given equal, it follows that ax 2 + Ix = c. Dividing these equal quantities by a, the quotients are equal by Axiom 4, or I c x 2 + - x = -- a a 4 o6.] Quadratic Equations. 247 If the square of - - be added to these two equal quan- 2i a tities, the sums will be equal by Axiom 1, or 402 By Art. 403 the left-hand member of this equation is the square of (x + - -) Hence the last equation can be > & &' written b , 2 b 2 That is, x + is a quantity whose square is whence x -\ must be either 2a or 2a 2a Thus the quadratic equation is satisfied by a value of x which satisfies either of the two simple equations or These equations give respectively * = -/(l) 2 -4xl x(-6) ~~ 4io.] Quadratic Equations. 249 1+5 4 -6 = - or 222 = 2 or 3 ; agreeing with the result of Art. 405. 409. The formula (/3) of Art. 406 thus contains implicitly the solution of every quadratic equation when reduced to its simplest form. The student is recommended to make himself thoroughly familiar with this formula so as to be able to apply it at once to any given equation. The solution of any particular equation may of course be conducted on the lines of the working in Art. 406 and the roots investigated independently, as has been done in Arts. 404, 405, but the quotation of the formula (/3) is usually the easier way. The general method of pro- ceeding may be verbally stated thus : Reduce the equation by getting rid of fractions and transposing so that the terms involving # 2 and x shall be on one side, and the term or terms independent of x on the other side of the equation. Then divide both sides by the coefficient of # 2 , so that its coefficient shall become + 1. To both sides of this reduced equation add the square of half the coefficient of x. The left-hand side then becomes, by Art. 403, an exact square, and the square root of this expression must be equal to one or other of the numbers whose square is the number or expression on the right- hand side. Thus two values of x are obtained. 410. In the process of solution of Art. 406 the following equation is deduced from the given quadratic, The sign of the number (d 2 4 ac) comes here to be of great importance in discriminating between the character of the roots in different cases. 250 Quadratic Equations. [411. First let l 2 4ac be positive. Then it has been already explained (Arts. 220-227) that a length can be geome- trically constructed such that the square on it is represented by I 2 4 ac whether the latter number be an exact square or not. Thus Vb 2 4ac in this case represents a real or scalar quantity (Art. 226), and if this quantity be denoted by 7e, the two values of #, namely - - and - are 2a 2a also real or scalar quantities. 411. Secondly, let 2 4ac have the value zero. Then 7 o 7 (x + ) is also zero. Hence x-\ is zero, and there- fore x can only have one value, namely In this case the two values of x in the general solution, namely _ and 2a 2a each reduce to the same value, -- Jj a Hence if b z 4 ac is zero, the two roots of the original quadratic equation become coincident and equal. 412. Thirdly, let b 2 4ac be negative. Then 4acb 2 , which = (I 2 4 ac), is positive. Let 4ac b 2 be denoted by the symbol & 2 , where k is a scalar quantity. It follows that b 2 4ac = k 2 = i 2 & 2 , where i is the operational or imaginary quantity introduced in Art. 271. Hence in this case And thus x + must have one of the values -- . or ik . A , l + ik -l-ik - that is x must be either - -- or - --- 2a 2a 2a In this case the two values of x are operational or im- aginary. 4i 4-] Qiiadratic Equations. 251 413. The roots of the equation ax 2 + bx + c = are thus real and unequal if I* lac be positive, they are real and equal if b 2 4ac be zero, and operational or imaginary if I 2 4 ac be negative. The expression & 2 4ac is often called the discriminant of the expression ax 2 + lx + c. 414. Whatever may be the character of the roots, the formula (ft) of Art. 406 holds good. In fact the results of Arts. 410-412 can be readily deduced from that formula. Let the two values of x in (a) or (/3) of Art. 406 be denoted by as 19 # 2 , so that 2a By addition + \/6' 2 4ac 6 Vl z 4ac *,+*,= - -^- -26 b 2a a By multiplication (i) (Art .n 6 ) Since the ordinary laws of multiplication and addition hold for operational or imaginary quantities as well as for 252 Quadratic Equations. [415. real or scalar quantities, the results (l) and (2) hold in all cases whatever be the quality of the roots of the original equation. 415. From these results it follows that the expression ax* + l>x + c can be resolved into factors in terms of #j and x 2 . Thus, by ( 1 ), b = - a (^ + * 2 ), by (2), c = ax l x. 2 . Therefore ax 2 + lx + c = ax- ax x + = a (xxj (x-x 2 ). (Art. 130) (y) It may be noticed that this last result follows from the theorem of Art. 163, on the assumption that the quadratic equation ax 2 + lx + c = has two roots, a^ and # 2 . Because x is a root of the equation it follows that the expression ax 2 -\-lx-\-c vanishes when x^ is put for x. Hence x x-i is a factor of the expression. Similarly x x t is a factor, and as the product of (x x^) (x x 2 ) is of the second degree in #, there can be no other factor involving x. The numerical factor a is evidently necessary to make the coef- ficient of x 2 the same in both. Hence 416. The whole subject may now be treated from a dif- ferent starting-point. The solution of the equation ax 2 + Ix + c = means the discovery of the values of x which make the ex- pression ax 2 + bx + c equal to zero, these values being called the roots of the equation. By the theorem of Art. 163 it follows that this dis- covery will be effected if the factors of the expression 417.] Quadratic Equations. 253 ax 2 + bx + c can be found. Because if a? x be a root of the equation, the theorem referred to shows that x x l is a factor of the expression ; and conversely, if x x l be a factor of the expression, # x is a root of the equation. We must proceed to investigate the factors of the given quadratic expression. 417. It will not limit the usefulness of the investigation if a be supposed a positive scalar. With this restriction it can be shown that ax 2 + bx + c can be always reduced to one of the forms u 2 + v 2 or u 2 v 2 , where u is an expression involving x to the first degree and i> is a real quantity independent of x. For ax 2 + Ix 4- c = a (x 2 + - x -f -) \ n n f a a / * \ 2 / * \ 2 C l ( ) - ( ) + - \2a' \2a' a) First, let ^ 2 4 2 4.ac is negative, =a - where .^ and a? 2 are the values obtained for x in Art. 419. If b 2 lac is zero, the two values of x are equal, and each is --- It has been already shown that in this case / \ ax 2 + bx+c = a(x + j 2 where os 1 is the value of the single root. But if # 2 be equal to as ly (as a? x ) (a? # 2 ) becomes (x a^) 2 . Hence in all cases, if a?j , a? 2 be the values of the roots of the equation ax 2 + bx + c = 0, the expression ax 2 + bx + c is identically equal to (x x^ (xx 2 ). 423. The converse proposition, that if aaP + fa + c be identically equal to a(x x l )(x # 2 ) then # 15 # 2 are the roots of the equation ax 2 + bx + c = 0, follows at once from Arts. 51, 167. 424. By means of the last article it is possible to write down a quadratic equation whose roots are any two given numbers. For if these numbers be represented by a^ , # 2 the quadratic equation required is (a? <&!)(# a? 2 ) = 0, s 258 Quadratic Equations. [425. or, multiplying out, 2 = 0. Thus the quadratic equation whose roots are 1 and 3 is (0-l)(a_3) = 0, or a? 2 4 a? +3 = 0. This last result can be compared with Art. 404. 425. The relations (l) and (2) of Art. 414, namely, -, which can also be deduced from the identity of Art. 422, sometimes simplify the solution of a given quadratic equa- tion. Thus the equation # 2 + 2001#-2002 = is obviously satisfied by the value x = 1. Hence one root being unity and the product of the two roots being 2002, it follows that the other root must be 2002. The two roots are thus 1, 2002. 426. The same relations can be used to express any symmetrical function of the two roots in terms of the J coefficients of the given equation. A symmetrical function means an expression involving the two quantities in such a manner that its value is not altered when they are interchanged. Thus x l + x 2 , x-f -f # 2 2 , x*^x. 2 + x^x are instances of symmetrical functions of x l and # 2 . 427. Since # 1 + # 2 = -- > squaring these equal quan- tities it follows that 427.] Quadratic Equations. 259 or a?! 2 + # 2 2 = -5 2 # x Also, cubing tlie same quantities, it follows {Art. 122 (1)} that -f- u #i X< ~T 6 X^ X% Whence ^ + # 2 3 = -- 3 x^x^ (x^ a 3 Again, by squaring the two equals in (l), n , - Therefore xf + a? 2 4 = - ^ ~ ~ 2 2~ In a similar manner the sum of any equal powers of the two roots can be obtained, and by processes not more difficult the values of any symmetrical functions of the roots. 428. Any equation involving only two powers of the unknown quantity, the index of one of which is double that of the other, can be solved in the same manner as a quadratic equation. 260 Quadratic Equations. [428. Thus if the equation be the assumption x* y reduces it to / 9^ + 8 = 0, the roots of which by the processes previously explained are found to be y = 1 and y = 8. Whence x z = 1 or # 3 = 8. The scalar values of x are therefore x = 1, x = 2, and the operational ones are o>, co 2 , 2o), 2o> 2 , where co has the meaning given to it in Art. 277. 429. Again, let the equation be t -I 3 . f x% ~~ 6 " Multiplying both equals by x% and transposing, it all becomes .....+,=,. Assume x^ = #, then x = y 2 , and it follows that 2 3 whence ^ = - or - 3 2t rt O Hence a?* = - or - > 3 2 and * = (D or (2) ' 4 9 = 9 r 4" 430. Equations of a more complicated nature sometimes occur, the solution of which can be reduced to that of two quadratic equations. For instance, the equation -3#-2 = 43I-] Quadratic Equations. 261 can be written or # 2 _3#) 2 + (a? 2 -3#) 2 = 0. Thus a? 2 3 a? is the only formal combination in which x occurs in the equation. Assuming- x 2 3 so = y^ the equation becomes /+^-2=0; whence it easily follows that y = 2 or 1. Hence x must satisfy one or other of the equations # 2 -3# = -2, or a? 2 3#= 1. The values of x derivable from these equations are re- spectively x _ i or # = 2 3 WT3 and a? = - - This method of simplification can often be adopted in cases when the unknown quantity occurs in some one formal combination only. 431. One or two instances of the reduction of equations in which apparently surd forms occur will be useful to the student. As a first example let the equation be x + Vx+l = 5. This may be advantageously solved in 'either of two ways. (1) Assume x+1 = y 2 . Then, since x =y 2 1, the equation becomes 262 Quadratic Equations. [432. or whence y = 2 or 3. Hence a?+l = 4 or 9, therefore x = 3 or 8. (2) Transpose the term x to the other side so as to have the surd form VHc + 1 alone on the left hand. Thus Vx+1 = 5 a?. Squaring these equal quantities, a process which is merely an application of Axiom 3 of Arts. 53, 287, it follows that a? + 1 = (5 - a?) 2 = 25 IQx + x 2 . Whence, transposing, a? 2 - 11 #+24 = 0, the solution of which gives x = 3 or 8. 432. It may be noticed that only one of the values of x obtained in the last article, namely x= 3, is really a solu- tion of the original equation. The other solution x = 8 satisfies the equation x \/x+ 1 = 5, a fact which is indicated in the first method of solution by the negative sign of one value of y. It sometimes happens in similar cases that neither of the roots obtained satisfies the given equation. Then if it be required to solve the equation = 0, the solution conducted by either method gives the values x = 0, x = 3. Neither of these values satisfies the given equation ; both of them satisfy the relation x + 3 3 it follows without any further work that the two roots must be 1, / / __ /Z\2 t A ( v a + V or + 4 and. = = 6 c 434. The relations x-, + a? = ? a?, a? 9 = - j and the a - 1 a results of Art. 427, can be often used to discover equa- tions whose roots are related in some given manner to those of the original equation. Thus let it be required to find the equation whose roots are ^ > 11 1 a* Since -yg- = 2 j = - ' 1 1 x^ + x & 2 2ac and ^-g + ^~2 = ^ 2 2 = 2 ' (Art. 427) it follows that the equation required is or Cx-2acx + a = 0. Quadratic Equations. 265 EXAMPLES. Solve the equations : 1. OJ 2 +16 = I7x. 2. x 2 -l2x+27 = 0. 3. l-x*-3x+4: = 0. 4. x*x-2 = 0. 5. 5a 2 -4a-l = 0. 6. 7x*-8x+l = 0. 7. 20j s -41a;+20 = 0. 8. 12x*-23x = 77. 2 V - ' 2 ' 3 2 11. abx*-(a 2 + 6 2 )a + a& = 0. 12 x 2 13 13 2X ~ 1 1 3 2 x~~ 6 3 2*-l- 14. 1 1 8 x 4"~ 3(x 3)' u.'^. 55 ~6~* x-2 + X i a 1 1 1 1 7 ^"T J- a + 2 13 x a x b x c ' ar+2 ' C+1 6 18. xa 1 x-b a 2 + 6 2 19. 4- 1 11 . a o as a ao x a xb a b _ (xma)(xb)~~(xa)(mxb)' 21. a < c ~ d )^ d(a-b) = b(c-d) c(a-b x + a x + d ' aj + 6 x+c 22. a; 4 -5a 2 +4= 0. 23. >v/0 + a;+ Va x = 2 v^. 24. 2 2 -/ar 2 -2a;-3 = 25. jct+ a t 26. >/a; 3+ v^+4 = \/7. 27. (aj" + 8a*+16aj- !)*- = 3. 28. (x*-l)*+(x*-l)* = x s . 29. 2 (x + a) { x -(x*-a*)*} = a 2 . 30. e^-ss^+e^-ssoj+e = 0. 31. aj-3aj-4-5-6 = 2 266 Quadratic Eqiiations. 32- 33 a ^ = 1 a 34. 35. 37. If the equations a? 2 7a + c = and cc 2 9#+2c = have a root in common, solve both of them. 38. Find the condition that the equations ax z + bx + c = and px" 2 + qx + r = may have a common root. 39. If a?y = z(x + yzf, prove that y =. z or 2/2 = (xz)*. 40. If a, ft he the two roots of the equation x 2 +px + q = 0, form the equations whose roots are respectively i,i; a',/3 2 ; and a 2 + /3, /3 2 + a. 41. If a?j, a? 2 be the two roots of the equation ax bx cx where a + /3 + y = 1 ; and if y lt y 2 be the corresponding values of y derived from the equation ax bx cx then x l + y 2 = a + b + c = # 2 + y x . 42. If the roots of the equation x*+px+q = be a show that the equation whose roots are - + is ( 2 Quadratic Equations. 267 43. If the roots of the equations = 0, x 2 +^ 2 a + ?=0, x*+p 3 x + q = be respectively a, b ; ma, ; j mb ; m m prove that P?(p? 44. If one root of the equation x*+px + q = be a root of the equation a? 2 + ax + b = 0, its other root is a root of x* + (2p d)x+p* ap + b = 0. 45. Find a number which shall exceed its square root by 110. 46. Two kinds of pears are sold in the market, two more of one kind being given for a shilling than of the other ; a score of the inferior sort costs sixpence more than a dozen of the superior sort. Find the price of the pears. 47. There is a number of two digits whose sum is 11 ; the square of the smaller digit exceeds the larger digit by 9 ; find the digits. 48. The united ages of a father and son amount to 64. Twice the father's age exceeds the square of the son's age by 8. Find their ages. 49. The length of a rectangular field exceeds its breadth by 33 yards. Its area is an acre : find the dimensions of its sides. 50. A messenger starts from A at 8 a.m. to go to B ; after waiting an hour at B, he returns walking at one mile per hour less than his pace in going, and arrives again at A at 4 p.m. Had he walked half a mile an hour faster at first he would have reached B at 20 min. to 11 a.m. Find his pace going and returning, the time of reaching B, and the distance from A ioB. CHAPTER XV. CUBE BOOT AND CUBIC EQUATIONS. 435. THE extraction of the cube root (Arts. 65, 67) of an algebraical expression or number is not such a frequently occurring or important process as that of finding the square root which was explained in Chapter XIII. Some ex- position of the method is however desirable. 436. The cube of a + b is known (Art. 122) to be # 3 +3fl 2 -f 3ab z + b 3 . The object of the present investi- gation is to discover a method by which a + l may be recovered from the latter expression supposed given. 437. The first term, fl, is evidently the cube root of the first term of the given expression. If the cube of a be subtracted from this ktter, the remainder is 3a 2 b+3ab 2 + b 3 . The trial divisor which will give b is evidently 3 a 2 , that is three times the square of the term already discovered. It only remains to verify that #, the quotient obtained by dividing the first term of the remainder by the trial divisor, is a quantity such that (a + b) 3 is equal to the given expression. 438. To do this, various methods of building up the expression 3a 2 b + 3ab 2 + 3 from the discovered values of a and b are adopted. The most usual and on the whole the easiest is to calculate separately the term 3 a 2 , which is the trial divisor, 440.] Cube Root and Cubic Equations. 269 and the term 3a + b. Multiplying the latter by I and adding to the former, the expression 3a 2 +(3a + b)b or 3a 2 +3ab + b 2 is obtained. This may be regarded as the complete divisor, and the product of this into 6, which is 3a 2 b + 3ab 2 -\-b 3 , ought to be equal to the remainder of the original expression. 439. The operation is usually written in the following manner. (a + b 3 a 2 (3a+b)b 3a 2 b+3ab 2 + b* The student who has carefully followed the method of the square root (Arts. 367-369) will have no difficulty in under- standing the meaning and use of the present method. 440. As an example let it be required to find the cube root of 8^ 3 36# 2 y + 54#/ 2 -27y 3 . The cube root of the first term is 2x. The trial divisor is consequently 3 (2#) 2 , that is, 12 x 2 . The remainder after subtracting (2#) 3 being 36# 2 t ^+54#/ 2 27^ 3 , the quotient of the first term of this by the trial divisor is 3y. Hence the cube root is 2x 3y. The process is indicated on p. 270 (*). The values of a and b having been determined, the re- mainder of the process consists in the calculation of When this is done and the result is subtracted from the remainder of the given expression, it is found that nothing is left. Thus the process proves that the cube of 2 a 3y is equal to the given expression, that is, that 2# 3y is the cube root of the latter. 270 Cube Root and Cubic Equations. [440. "0 (M + 1 + + + + tt 1 1 14 II L 1, v!L ^ ^5 55 % % % 12 22 CM (M O3 1 1 1 1* 15 % O GO 1 s ** *^>> ?&N o *c * *o CO CO CO CO CO CO 1 1 1 J . 1 1 "* "- 1 15 Ii CO 00 ?5s> C$ C5 55 55 CO CD CO 1 1 1 II t*4 Oq 6_ 6 05_j_ 15a ,4_20# 3 -f 15a? 2 -6#+l. The process is indicated on p. 270 (f). The cube root of the first term is x 2 . The first trial divisor is consequently 3 a? 4 and the quotient of the first term in the remainder by this is 2 a?, which must con- sequently be taken as b. When the expression has been formed and multiplied by b, and the product subtracted, there is still a remainder. At this stage, on the whole, the cube of % 2 2x has been subtracted from the given expression. Consequently # 2 2 a? occupies the position of a in the general investi- gation of Arts. 437-439. In the above working it is denoted by a' . The second trial divisor or 3 a' 2 is easily formed, and the next quotient is unity. Calling this b' and forming the expression 3 a' 2 + 3a'b' + b' 2 , multiplying this by b' and subtracting the product, there is no re- mainder. Hence on the whole a'* + 3a' 2 b' + 3a'b' 2 + b'* or (a' + bj has been subtracted, and there being no remainder, the process proves that the given expression is (a' + # / ) 3 , or (x 2 2#+l) 3 . Hence # 2 2#+l is the cube root re- quired. 272 Cube Root and Cubic Equations. [443. 443. The second trial divisor is 3(a + b) 2 or 3a 2 + 6^+33 2 . In the operation of forming the first complete divisor it will be noticed that the expressions (3a + b)b and 3a 2 +3ab + l 2 are formed in succession. The sum of these is 3a 2 +6ab + 2b 2 . If to this sum b 2 be added, the result is the second trial divisor. This is usually as easy a method as any of deducing the trial divisor in any stage from the results obtained in the previous one. 444. The application of this process to the calculation of cube roots of numbers is easy after the full explanations given in relation to the process of the square root in the articles from 379 onwards. The cube of 1 being 1 and that of 10 being 1000, it follows that the cube root of any number between 1 and 1000 must lie between 1 and 10, that is, must have one digit only. Since the cube of 100 is 1000000, it follows similarly that the cube root of any number between 1000 and 1000000 must lie between 10 and 100, that is, must have two digits. Hence it follows that the cube roots of all numbers with one, two or three digits have one digit, those of numbers with four, five or six have two digits, and so on. 445. Again, if any number be represented in the form /? a? B +/? 1 a? n " 1 + ... +p n -iX+Pn, (a), (Art. 380), where x has the value 10, the cube of this number will assume the Now /? 3 may be greater than 10 but must be less than 10 3 , since p Q is less than 10. Hence if the highest power of 10 in any number be 10 n , the highest power of 10 in its cube may be either 10 3n , 10 3n+1 , or 10 3n+2 . If then in any number whose cube root is required a dot be placed over the unit figure, and in succession over 447-] Cube Root and Ciibic Equations. 273 each third figure going to the left-hand, these dots will lie over digits which multiply 10 3 , 10 6 , 10 9 ..., that is, powers of 10 of the form 10 3m . The number of these dots will thus indicate the number of digits in the cube root re- quired, since the n ih of them will lie over the digit which multiplies lO 3 ^- 1 ). 446. Again, jt? 3 must be the principal part of the coefficient of # 3n in the cube. Owing to the process of carrying it will probably not constitute the whole of this coefficient. When the dots have been placed as suggested in the last article, the number formed by the figure under the left-hand dot and the figure or figures to the left of it, is the coefficient of # 3n . Hence p Q will be the greatest number whose cube is less than this coefficient. Thus PQ", the first term of the cube root required, is easily found. The remaining terms must be discovered by a process exactly equivalent to the algebraical one, the only difference being that there is some little doubt as to the second and third figures of the root, for a similar reason to that which made the second figure in the square root uncertain. 447. One example in illustration of this not very im- portant subject will suffice. Let the number be 259694072. Placing dots over the unit and every third figure, there are seen to be three figures in the cube root. The first digit must be the number whose cube is nearest below 259, that is 6, and the principal part of the cube root is therefore 600. The first trial divisor is 3 x (600) 2 or 1080000. The first digit in the quotient of the re- mainder, after subtracting (600) 3 , by this trial divisor is 4, but it will be found on forming the complete divisor with this number that 4 is too large, and accordingly 3 must be taken. The different steps in the process are indicated pretty clearly in the following diagram : 274 Cube Root and Cubic Equations. [447- a 3 3a 2 = 1080000 = 30 = 1830 = 54900 259694072(600 + 30 + 8 = 638 216000000 = 1134900 43694072 34047000 = = 1189800 b 2 = 900 3a' 2 = 1190700 V =8 3a' + 6' = 1898 ' = 15184 = 1205884 9647072 9647072 = 3a' 2 6' + 3a'6' 2 + 6" A great number of unnecessary noughts have been written down in order to make the principle of the process clearer. With a little practice the student will discover for himself how to omit needless figures. 448. Suppose that the whole number of digits in the cube root is #+1, and let s+2 of these have been found by the above process. Let a represent the number formed by these digits, and let b represent the remaining part of the cube root. Then, since the whole cube root contains n + 1 digits, the first of its digits is the* coefficient of 10 n ; while the first digit in the part denoted by b is the coefficient of 10 n ~ 8 " 2 , and b contains nsI digits. Hence a must be greater than 10 n , and b must be less than lO"" 8 " 1 * It is here assumed that any number with p digits, which consequently contains no higher power than 10 P ~ 1 , must be less than 10*. The Arithmetical student who is familiar with the fact that 9999 is less by unity than 10000, will have no difficulty in allowing 449-] Cube Root and Cubic Equations. 275 the general truth of the above statement (see note, Article 382). 449. The remainder of the given number, after a has been found and a 3 subtracted, is 3a 2 b + 3ab 2 + b 3 . The trial divisor is 3 a 2 . The complete quotient of the re- b 2 b 3 mainder by the trial divisor is b -\ --- f- 5 a 3a* Now b is less than lO"" 8 - 1 . Hence b 2 is less than io 2 "- 28 - 2 . Also a is greater than 10 H . *2 in2n-28-2 Thus is less than or 10 n - 2 '- 2 . Similarly, b 3 is less than 10 3 *- 38 - 3 , and a 2 is greater than 10 2w . 3 103H-38-3 Therefore -% is less than ^ or 10 n ~ 3 '_ 3 Hence + -^ is less than 10 n - 28 - 2 + . lO"- 3 *- 3 , that a 3 a is, less than lO"- 28 -^ + 2 )> which is evidently less than 10"" 28 " 1 , since + - +2 is a proper fraction. b 2 b 3 Hence -- + VT' "being less than 10 n ~ '\- contains at most ft 2s 1 digits, while b contains n 51, or (n 2s 1) + $, digits. Thus the highest s digits of the complete quotient, b + - - + -r-$ > are the same as those d 3d ofb. Thus when (s+2) digits have been found, the trial divisor will give the next s digits accurately. For instance, when three figures have been obtained the trial divisor will give with certain accuracy the fourth. T 2 276 Cube Root and Cubic Equations. [450. There will be a little necessity for guessing in ascertaining the values of the second and third digits. 450. All the statements (Arts. 378, 393, 394, 397) made about the approximate square roots of numbers apply, mutatis mutandis, to approximations to cube roots. The cube root of a fraction is obtained by finding the cube root of the numerator and dividing by that of the denominator. If the fraction be not an exact cube it is more convenient to multiply its numerator and denomin- ator by such a number that its denominator shall become an exact cube. The approximate cube root of the numer- ator can then be found, and this, divided by the cube root of the denominator, is called the approximate cube root of the fraction. 451. Any number can be expressed in the form of an improper fraction with any power of 10 as denominator, by the simple process of adding noughts to the number and taking it so altered for the numerator. If the index of the power of 10 be a multiple of 3 the denominator will be an exact cube. The approximate value of the cube root will thus be a fraction with a power of 10 as denominator, and in this way an approximate value of the cube root of a number can be obtained to any required number of decimal places. __ 2000 _ 2000000 us> " 1000 ~ i oooooo ' and if the approximate cube roots of 2000, 2000000 be obtained, these values, divided by 10 and 100 respectively, will give the approximation to the cube root of 2 to one and two places of decimals respectively (compare Arts. 397-399). 452. The solution of a cubic equation that is, an equa- tion with one unknown quantity, in which the unknown quantity occurs to the third and no higher degree can 454-] Cube Root and Cubic Equations. 277 be effected by the help of the previous articles and Art. 282. ^ The most general form of a cubic equation is # 3 + ax 2 + bx + c = ; since by getting rid of fractions, transpositions, and division of the whole equation by the coefficient of # 3 , every equation involving the third and no higher power can be reduced to this form. 453. The expression x* + ax 2 + bx + c can be reduced to the form u 3 + qu + r where u is equal to x -\ and qr and r are constants. +!, Art. 122, (1) Hence 3 db 2a 3 _ + _ = u , a 2 ab 2 a 3 where q = b and r c --- 1 -- . 454. The solution of the general cubic equation will therefore be effected if that of = (1) is effected. 278 Cube Root and C^tbic Equations. [454. The latter problem depends on the resolution of the ex- pression n* + qu + r into factors, since (Art. 163) if a be any value of u which satisfies (l), u a is a factor of u* + qu + r ; and conversely if u a be a factor of this last expression, u = a is a solution of (l). In Art. 282 it is shown that where y a?z) (u uPy a>z). If then by a proper choice of y and z, the expression tfi + qu + r can be made identical with u 3 3yzuy 3 z 3 , the resolution of the former into its factors and, conse- quently, the solution of (1) will have been completely effected. 455. The conditions of identity of the two expressions are Zyz = -q, y* g*= r; 3 whence *z z = , = r. Hence (Art. 414), it follows that y z and 2 3 are the two values of t in the quadratic equation But these values are r (Art. 406) (a) 457-] Cube Root and Cubic Equations. 279 Hence the values of y and z are 456. The three factors of the expression being, with the above values of y and z, uyz, u <*> 2 y z, u toyuPZ) it follows that the three roots of the equation (l) are y + z, uPy + vz and wy-f a> 2 . Hence the equation is completely solved. r 2 a 3 457. If + ^ be a positive quantity, the values of y 4 27 and z are scalar quantities. Hence, in this case, one root of (1) is scalar and two are operational. r 2 a 3 If + |- be negative, y and z are both operational. Nevertheless all three values of u in this case are scalar. For y* and z 3 assume (Art. 271) the respective forms a + bi and a bi, which, by Art. 273, can be written in the forms r(c + is) and r(c is). Hence y and 2, or (a + bi)* and ( bi)* can be written as r*(c + is)* and r*(c is)* respectively. Now c + is denotes (Art. 273) a rotation through some angle which we may call a, c is denotes a rotation through an equal angle in the opposite direction. The multipliers (c + is)* and (cis)* must denote rotations through angles of one third the amount, or - > also 3 in opposite directions. Hence (c + is)* can be replaced by same multiplier c' + is', where 2 y + o>0 is really scalar, although like the other two roots it assumes an operational form. 458. The actual deduction of the scalar values of the r 2 3 roots of the cubic in the case when -- 1- is negative, is beyond the powers of elementary Algebra, and must be left to the more advanced subjects of Trigonometry and Theory of Equations. 459. It is scarcely necessary to remark that the solution of any equation in which only two powers of the unknown quantity occur, the index of one of which is three times that of the other, can be made to depend on that of a cubic equation. Thus the equation 4 x + 3#* 7 = becomes, by the assumption x% = M, u 7 = 0, 3 7 or 3 + - u - = 0. 4 4 Comparing this latter equation with (l) of Art. 454, 3 7 r r = -- 459-] Cube Root and Cubic Equations. 281 Hence 7 + 5v/2 r / r 2 whence y + z 1, which is the only scalar value of x : the two operational roots can easily be written down. It may be noticed that, in this case, the deduction of the value of y and z from those of y* and z 3 is as difficult an operation as that of the solution of the original equation. It can in fact be reduced to depend on the latter. In any case in which - + ~- is not an exact square a similar difficulty is met with, so that for practical purposes the general formula is of very little value. EXAMPLES. Find the cube roots of : 2. 3. 8ar ) -12c 8 + 6a? 7 -37a7 6 + 36^-9^ + 54ar 5 -27z 2 -27. 4. 8ic 9 -36ie 8 + 54^ + 21^-144^+108^ + 96ce 3 -144a 2 +64. 5. 8z 9 - 12^- 6. tt 3 - + 12y 2 z-6yz* + z*- \2xyz. 7. 1879080904, 66775173193, 258474853. 282 Cube Root and Cubic Equations. Solve the equations : 8. ar 5 -2a+4 = 0. 9. x*-3x z +5x-3 = 0. 10. x 3 +2x 2 -3x-G = 0. 11. If each of the equations have two equal roots and the third common to the other equation, the unequal roots not being the same in the two equations, show that ' and that = 12. If a, b, c, be the roots of the equation or 5 + 3px* + 3 qx + r = 0, prove that those of caco + ab co 2 be + ca co 2 + ab co are ^ - ; 5 - > where co is one of the imaginary cube roots of unity. 1 3. If the equations ace 3 + 3 bx 2 + d = 0, and for 5 + 3 cfo + e = have a common root, prove that 14. Prove that if = cx + by + az, Z = foe + ay + cz, CHAPTER XVI. SIMULTANEOUS EQUATIONS OF A HIGHER DEGREE THAN THE FIRST. 460. THE solution of two equations containing two unknown quantities to a higher degree than the first, involves, as a general process, the solution of a single equation, with one unknown, of a higher degree than any within the province of elementary Algebra. There are, however, many pairs of equations in which, by particular artifices, the solution can be made to depend on equations similar to those which have been already considered. The present chapter will be devoted to the consideration of some of the most commonly occurring forms. 461. The primary object to be attained is usually the elimination of one of the unknown quantities and the deduction of a single equation involving only the other. Sometimes it is advantageous to express one or both of the given unknowns in terms of some other quantities, the values of which may be more easily determinable in the first instance, from which those of the original unknowns can be deduced. 462. The first class of equations for which a general method of elimination can be suggested, consists of those in which one of the unknowns occurs, in one of the equations, only to the first degree. In such a case, the value of this unknown can be determined from this equation in terms of the other ; this 284 Simultaneous Equations higher than the First. [463. value being substituted in the other equation, there results an equation with only one unknown. If it so happen that this final equation is either a quadratic, or reducible to one, the solution can be completed. 463. As an example, suppose the pair of given equations to be 3^ + ^=18, (1) 2xy-y* = 3, (2) Here y occurs in (1) and x in (2) only to the first degree. The method may be applied either by finding the value of y from (1) and substituting in (2) ; or the value of x from (2) and substituting in (1). Taking the latter plan ; it follows from (2) that # whence x =- - (3) 2y Hence from (I), or, multiplying these equals by 4^ 2 , so as to get rid of fractional terms, 3(/ + 6/ + 9) + 2/ (/ + 3) = 72 f ; whence, transposing and collecting terms, 5y 4 -48/ + 27 = 0. This equation, involving only y 2 and y 4 , can be solved as a quadratic as y 2 (Art. 428). By the formulae of Art. 406 it follows that 2 _ 48+ \/(48) 2 -4x5x27 _ 48 + 42 J 1 A 1 A 10 10 = 9 or . Hence y 3 or A/|. Corresponding to each value of y, (3) gives one value of x. 465.] Simultaneous Equations higher than the First. 285 The four pairs of values are therefore 9 + 3 y = 3, x = = 2, 9 + 3 y=-3, a? = = -2, 8 i q 464. The second -class of equations for which a general method can be suggested are those in which the terms containing the unknown quantities are homogeneous (Art. 88), or nearly so, as regards those quantities. The equations solved in the last article are a v/121-72 _ 11 + 7 _ 18 4 12 12 = T2 01 12 18 18 * * 2 = ^= whence x = +2, and y = vx = |x2 = +3. Tf i 2_ 18 18 27 " 3+v ~~ 3+ ~" ~5~ ' whence x = 3 \/f ; and y = vx = % x 3 \/f = + \/|. The solutions obtained by this method are of course identical with those given by the former. 466. A third class of equations consists of such as are symmetrical in form with respect to the two unknown quantities; that is, which remain unaltered when x is changed into y and y into x. 468.] Simultaneous Equations higher than the First. 287 In this class it is often useful to make the substitutions By means of processes such as those given in Art. 427 the given equations, being symmetrical with respect to x and y can be expressed in terms of u and v. It is frequently much easier to solve the resulting equations with respect to u and v than the given ones in x and y. Suppose that from these equations the values u = a, v = /3 are deduced. Then, since x+y = a, and xy = /3, it follows from Art. 414 that x and y are the two values of t in the quadratic equation tf 2 a* + = 0. Thus x and y can be completely determined. 467. As an example of this class of equations let us take the pair = 133, * 7. Making the assumptions x+y = u, xy = v, it follows that whence Squaring these equals whence, since x 2 y 2 = v 2 , subtracting these equals from the former two, it follows that Hence the original equations become 4 -4 a i; + 3t; 2 = 133, , u 2 3v= 7. 468. In the latter of these equations v occurs to the first degree only. Hence the method of Art. 462 can be used. 288 Simultaneous Equations higJicr than the First. [468. u 2 7 Substituting for v in the first equation its value - 9 derived from the second, we obtain 4 = - = 6. o o Taking u = 5, v = 6, x and y are the two roots of the equation that is, x and y have the values 2 and 3, a? being equal to 2 and y to 3, or a? to 3 and y to 2. If ^ = 5 and v = 6, x and y are the roots of the equation ^ + 5^ + 6 = 0, where x and y are 2 and 3. On the whole there are four pairs of values of x and y^ namely x = 2, y = 3 ; #=3,^=2 ; 0=-2,jf=-3;a=-3,^ = -2. 469. The equations of the last article can also be solved by the method of Art. 464. They can be solved perhaps even more simply by noticing that whence the first equation gives, by the help of the second, x't+ocy+f 1 = 19. Also, since a? xy+y 2 = 7, it follows by subtraction that 2xy= 12, or xy = G. 47^] Simidtane otis Equations higher than the First. 289 Hence, adding xy to the first equation, x 2 + 2xy+f = 25, whence x+y = 5, and the rest of the solution can be conducted as in the last article. 470. The solution of simultaneous equations involving- more than two unknown quantities must be effected by similar methods to those which have been indicated in the case of two unknowns. General rules are less to be relied on than special artifices, which can only be learnt by practice and close attention to the results of algebraical transforma- tions. 471. The elimination of one unknown between two equations each of the second degree leads in general to an equation of the fourth degree. The most general form of an equation in x and y of the second degree may be represented by ax 2 + If + 2hxy + 2gx + 2fy + c = 0. The second equation may be taken as 0V + 6y + 2//^+2/#+2/> + c'= 0. These equations may be written in the forms lf + 2(7ix+f)y + ax*+2(/x + c= 0, Vf + 2 (h'x +/') y + a'a? + 2g'x + c' = 0. or ly* + Py+Q = 0, (1) 0y+p> +# = <), (2) where P = 2(hx+f), Q = ax 2 + 2gx + c and P f and Q' represent similar expressions in the second equation. Multiplying (l) by V and (2) by b, and subtracting these results (tfP-6P^y + b'Q-bQ' = 0. (3) 290 Simultaneous Equations higher than the First. [471. Multiplying (1) by Q' and (2) by Q and subtracting, it follows that or dividing by y, (bQ'-VQ)y + (PQ'-P'Q) = 0. (4) Equating the values of y obtained from (3) and (4), we obtain VQ-W __PQ'-P'Q. b'P-bP' '' '' bQ'-b'Q ' whence, getting rid of fractions and transposing, (PQ'-P'Q) (b'P-bP') + (b'Q-tQJ = 0. (5) Now P, P / are expressions of the first degree in x, and Q, Q' are of the second. Hence PQ' P'Q is of the third degree, VP-bP* of the first, and b'Q-bQ' of the second degree. Thus the equation (5) will be of the fourth degree. The student will easily verify for himself that equation (5) can be written in the determinant notation as and he will recognise that each of the constituents of this determinant is itself a determinant. Further acquaintance with the theory of determinants will show him that the whole process might be effected by the investigation of a determinant of the fourth order, which can easily be reduced to the above determinant of the second order. 472. In one particular case the result has a special interest, namely, when the equations are of the simplified forms = 0, = 0. 473-] Simultaneous Equations higher than the First. 291 Here P = 2 fa, Q = ax 2 , P' = 2//#, Q' = aV, and the equation (5) reduces to whence, dividing by # 4 , we obtain 4 (a'b - ah'} (b'h - bh') + (aV - a'b}* = 0. (a) This is therefore the condition that the two given equa- tions may be satisfied by any common values of x and y differing from zero. Each of the two equations may be written as a quadratic equation in -, and the equation (a) is the condition that x V they shall be satisfied by a common value of - 473. It is theoretically possible to eliminate any number of quantities, as ft, from a set of equations, provided the number of the latter exceed by unity the number of the quantities to be eliminated. The general method is to find the values of the n quantities from n of the equations and substitute these values in the remaining one. There thus results a relation independent of the n quantities, For example, let it be required to eliminate x and y from the equations x+y = a, The first two equations will give by the method of Art. 466 whence x and y can be determined and .their values substituted in the third equation. It happens, however, that x 5 +y 5 can be readily expressed (Art. 427), in terms of xy and #+y, and the further investigation of the values of x and y is unnecessary. U 2 292 Simultaneous Equations Jiighcr than the First. [473. Thus, a 30 Thus the final result is 3a 90 or For the solution of problems of this class the student must learn to rely on special artifices, only to be acquired by great practice and considerable familiarity with differ- ent algebraical expressions and their transformations and relations to each other. 474. Problems are frequently proposed similar to those which have been solved in Arts. 299-304 and 324-330, the solution of which depends on that of equations of a higher order than the first. There is no difference in the principle of the methods to be adopted from those formerly described. The conditions of the question must be carefully stated in algebraical language, one or more of the unknown numbers required being represented by the letters #, y, and so on. The resulting equations must then be solved by such of the previous methods, or such special artifices as may be most applicable. EXAMPLES. Solve the equations : 1. x*+3xy = 34, #?/+4y 2 = 110. 2. x + y= 7, x* + if = 25. 3. 3x 4y = 2, 4ar 2 ajy=14. 4. x^ + xy = 12, xy + y* = 4. Simultaneous Equations higher than the First. 293 5. x*-xy= 6, xy-y z = 2. 6. aj 2 +37/ = 7, 2/ 2 +3cc=7. 7. 4# 2 +7#=78, 5o*/+92/ 2 = 66. 9. 10. 11. ^. 18, x + y= 12. 12. x-y- 2, cc?/ + 5a7-62/= 120. 13. 14. So;-- = 4, 9aJ 2 +\ =40. 6 a _ 15. aa? + 62/ = c, - + - = d. x y 16. 17. ,-,= 18. x 4 + 2/ 4 = a 4 = ay (a? + if). (x*y + xif=l80, \ a?H s =189. 91 22. xy = c(x + y\ aa? = 6 23. a 2 - 24. / cc+ 32M-_5 3a; + y+4 _ 4 J x + y+1 " x + y-I ' ((x+2yY+(y+2xY = 5(x+y) z 294 Simultaneous Equations higher than the First 25 f 27. 28. iry=8, y = 28, aa? = 14. + ^1_ = 29. 'I 9+ * + y= C - tax+cybz by + azcx cz + bxay 30. | -^ = ~P- -7- 31. - + f + "= 3 ' \a o c 33. 4 [c(x + y z) = 35. 36. Eliminate x, y, z from the equations = a 3 + z* = Sabc, xyz = a 3 . Simultaneous Equations higher than the First. 295 Solve the equations : / x y z y x z 0-7 ! ~H 1 = 3?r = f- I > 37. y,z from the equations 41. A person spends five shillings in the purchase of eggs. He spends part of the money on one kind and part on another. The number of pence he spends on the cheaper kind added to the number of eggs of the dearer kind is 52. He gets f as many for a shilling of the dearer as he does of the cheaper, and he has twenty more of the cheaper than of the dearer eggs. How many does he buy of each kind ? 42. A number consists of three digits whose sum is 6. If the digits be reversed the number is increased by 99. The sum of the squares of the digits is 14. Find the number. 43. A regiment is drawn up in marching order. If the first ten ranks be taken off and the remaining men be arranged so that there is one fewer in each rank, but the same number of ranks as before, there will be ten men left out. The number of ranks exceeds the square of the number of men in each rank by 19. Find the number of men and their arrangement. 44. Two persons, A and B, run a race to go five times round a certain course. "When A has gone three laps, B is 150 yards behind him. A then slackens speed and goes at J5's rate, while 296 Simultaneous Equations higher than the First. B quickens his rate and goes at A's first rate. A wins by 30 yards. Find the length round the course, and compare the original speeds of A and B. 45. The sum of three numbers is 6, the sum of their squares is 14, and the sum of their cubes is 36. Find the numbers. 46. Three persons, A, B, (7, are engaged to do a certain work. When working together they can do it in 6 days. If A only work as hard as B, while C works as before, it will take them 8J working days to finish. The sum of the number of days which A and G would respectively take to do it alone is double the number which B would take by himself. Find in how many days each of them will separately do the work. 47. The owner of 20,000 divides it into three portions, which he puts out at different rates of interest. The total interest he receives is 950. Three times the interest on the first por- tion is equal to five times that on the second and also to ten times that on the third. The sum of the rates per cent on the last two portions is double that on the first. Twice the first rate added to the third is equal to four times the rate on the second portion. Find the sums of money and the rates per cent. 48. There are three numbers whose sum is 54. The square of the middle one exceeds the product of the other two by 4, and the sum of the first and last is double the middle one. Find the numbers. 49. A cistern can be filled by three pipes A, B, C. Twice as much water passes through B per minute as through A. The three when open together fill the cistern in one hour, and B alone will take one hour longer to fill it than C alone. Find the time in which each alone will fill the cistern. SECTION III. SERIES. CHAPTER XVII. PEKMUTATIONS AND COMBINATIONS. 475. THE third section of this book will be devoted to the subject of Series, that is, to the consideration of the properties of algebraical expressions consisting of a number of terms formed according to some law, the value of each term depending in some manner on its position in the series. In some of the series which will have to be considered, the coefficients depend on the number of arrangements which can be formed out of the different letters involved. It is therefore necessary in the first place to investigate the arithmetical formulae which determine this number in different cases. 476. The number of possible arrangements of three quantities among themselves has already been incidentally shown to be 6 (Arts. 121, 344), that is, 3x2x1. The number of arrangements of n things among themselves can similarly be shown to be the continued product of all integers, beginning with n and going down to unity, or, as it may be written, n(nl)(n 2) ... 3.2.1.- 477. This product occurs so frequently in mathematical investigations that it has received both a name and an appropriate symbol. It is usually called factorial n, and denoted by the symbol n. 298 Per imitations and Combinations. [477. Thus factorial 6, or [6, means 6.5.4.3.2.1, or 720; factorial 5, or |5, means 5.4.3.2.1 or 120. From the law of formation it is clear that |J5 = 6 [5. More generally, \n = n(n l)( 2) ... 3.2.1, \n +1 = (+ l)n(n l)(ft-2)... 3.2.1. |ft +1 thus contains all the factors of [ and one factor, ft + 1, additional. Hence |M +1 =(+!) [ft. If in this last result n have the value zero given to it, there appears the anomalous result \l = LO. Now the value of 1 1 is obviously unity, while the sym- bol | has at present no meaning. It will be convenient to give it such a meaning as to make the relation | # +1 = (ft+ 1) |^ft as general as possible, and it is there- fore usual to consider | as merely a symbol for unity. It will be seen later on (Arts. 546, 550), that this assump- tion renders slightly more general certain forms obtained in the multiplication of binomials and polynomials. 478. In recent years the notation [ft has by some writers been replaced by n !. The latter symbol has the disad- vantage that the second part of it is in most minds already associated with other ideas, and a page in which such signs frequently occur gives at first sight an impression of astonishment beyond even that due to the most brilliant achievements of the mathematicians who introduced it. The notation In will accordingly be exclusively adopted here. 479. Suppose that there are n pupils in a class among whom a number of prizes have to be distributed. As the first prize may be awarded to any one of the pupils, there are n different ways of giving the first prize. 48 1.] Permutations and Combinations. 299 When the first prize has been assigned to a particular pupil A) there are n \ pupils left to contend for the second. Thus, A having 1 the first prize, there are (n 1) different ways of giving the second prize. Similarly if B^ another pupil, have the first prize, there are n 1 ways of giving the second, and so on for each of the n pupils. Hence the total number of different selections of a first and a second prizeman is n times n I or n (?i 1 ) ; different, that is to say, in one at least of the two prizemen, each from every other. Again, when the first two prizes have been assigned in any particular manner theie are n2 pupils left as competitors for the third prize, and consequently (n2) different ways of awarding it. Hence, since the number of different ways of awarding the first two prizes is n(n 1), and with each of these ways there are n 2 different ways of awarding the third, there must be n(n 1) times (n 2) ways of awarding the first three prizes ; or the number of choices of three out of n is n(ii !)(# 2). 480. The student will notice that this method of reckon- ing counts ABC as a different choice from BAC, or CBA, or any other arrangement of the three pupils denoted by ABC. The number of different choices when this is the case is called the number of permutations of n things, in this case 3 together, or if r be the number chosen, r together. 481. It has been shown that the number of choices of three things out of n, the order of choice being a matter of importance, as in the case of choosing a first, a second, and a third prizeman out of a class of n, is n (n 1 ) (n 2 ). When any set of these have been chosen, there are ( 3) left, and consequently (nS) ways of choosing a fourth. Hence, under the condition stated in italics, the number of different 300 Permutations and Combinations. [481. \vays of choosing 4 thing's out of n is n(ti \)()i 2) times (//-3), or n(nl)(n 2)(n3). Thus the number of choices is so far represented by the product of as many factors as there are things to be chosen, the first factor being n and each factor being less by unity than the previous one. 482. It is obvious that this law will continue to hold for the number of choices r together, r being any integer less than n. For let JP r denote the number of distinct choices of r things out of n. When any particular set of r things have been taken, there are (n r) left, and the number of ways in which one more thing can be chosen and placed at the end of this set of r things is (n r). Hence the total number of ways of choosing (r+l) things, that is, one more than r, is n P r times (n r), or n P r (n r). Hence n P r+l = n P r . (n-r). (1) This shows that in deriving the number of permutations of n things (r+l) together from the number r together, one additional factor is introduced. Further, as r increases by unity, this additional factor (n r) decreases by unity. Thus for each additional thing chosen the number of per- mutations has a new factor less by unity than the last factor in the previous number. Thus the number of factors being at first the same as the number of things chosen, must continue to be so, and the successive factors diminish by unity. Also (1) shows that the last and least factor in n P r+1 is (n r). Hence the last and least factor in n P r must exceed this by unity or be (n r + 1 ). Thus, finally, it follows that n P r = n(n-l)(H-2)...(n-r+l), (2) where the dots represent the intermediate factors, which v 485.] Permutations and Combinations. 301 cannot be written down without specialising the value ofr. 483. The formula (l) of the last article can be used in a slightly different manner to deduce (2). In Art. 481 it has been shown that the number of permu- tations of n things 4 together is n(nl)(n 2)(n 3), or n P, = n(n-l)(n-2)(n-3). Hence by (l), n P 5 = n P 4 (rc-4) = n(n - 1) (n - 2) (n - 3) (n - 4). Similarly, n P 6 = n P 5 (n-5)=n(n-l)(n-2)(n-3)(n-4)(n-5), and so on. Thus each additional thing chosen introduces a new factor less by unity than the last factor of the former pro- duct. The number of factors and the number of things chosen increase therefore parj.. passu, and being equal at first musti remain equal. Also the second factor is n 1 , the third^/fc 2, the fourth 3, and so on. Hence the r th factor, which is the last in ? P r , must be n r + 1 . Thus, as before, n P r = n(n-l)(n-2)...(n-r+l). 484. A particular and very important case is that of the total number of distinct arrangements of the n things. This is evidently the number of permutations of n things, n together, and will therefore be obtained from n P r by giving to r the value n. The number required is therefore the product of n factors beginning with n and diminishing from factor to factor by unity. The last or n ih factor must therefore be unity, and the required number of arrange- ments is n(n\)(n 2) ... 3.2.1, or the -product which has been denoted by the symbol \n. s/ 485. Suppose that from a body, such as a Town Council consisting of n persons, it is required to select a committee for a special purpose consisting of r persons. The number 302 Permutations and Combinations. [485. of distinct ways in which this can be effected is evidently quite a different thing 1 from the number of distinct choices on the supposition hitherto adopted, that the order of choice is an important element. For instance, if three persons have to be chosen, the number of permutations is by the previous investigation n(n !)(>* 2). This reckoning counts all the different arrangements of any one set of three persons, A, B, C, as distinct. In whatever order these three may be chosen they will, however, still form the same committee, and this set of three must therefore in the new problem only be counted once. The number of orders in which this same set of three persons would be chosen is by the last article 3.2.1, or | 3. Hence each distinct committee of three persons fur- nishes | 3 different permutations. The whole number of permutations, three together, must be therefore j 3 times the number of distinct committees. The number of the latter must therefore be - of the number of permutations, or must be _ 486. The number of distinct choices of r out of n things, when the order of choice is not considered, is called the number of combinations of n things r together. It is often denoted by the symbol n C r . 487. As a particular illustration, let there be five letters 0, b, c, d, e, and let it be required to find the number of per- mutations and combinations of these three together. The combinations are easily seen to be abc, abd, abe, acd, (ice, ade, bed, fice, bde, cde. This comprises all that can be found, the first six being 488.] Permutations and Combinations. 303 all that contain a and two of the other four letters, the next three all that contain b and two of the letters which follow I, and the last being the only remaining possibility. The whole number of permutations can be found by arranging each of these combinations in all possible ways. The permutations will thus be abc, acb, bca, bac, cab, cba, abd, adb, bad, bda, dab, dba, abe, aeb, bea, bae, eab, eba, acd, adc, cad, cda, dac, dca, ace, aec, cea, cae, eac, eca, ade, aed, dae, dea, ead, eda, bed, bdc, cbd, cdb, dbc, deb, bee, becy cbe, ceb, ebc, ecb, bde, bed, dbe, deb, ebd, edb, cde, ced, dee, dec, ecd, edc. There are obviously | 3, or six, times as many permu- tations as combinations. The number of permutations being 5.4.3, that of the combinations must be one-sixth 5.4.3 of this, or I! 488. Suppose now that any one combination of r out of n letters is represented by a be . . . Jc. The letters which form this combination can be arranged among themselves in I r ways (Art. 484), each of which arrangements will count as a different permutation. If all the n C r different combinations be similarly treated, each of them will yield |^r permutations. All the permu- tations given by one combination will be different from all those given by any other combination, since they must differ in at least one letter. Hence the n C r combinations produce | r times that number of different permutations. But since every possible permutation of n things r together does contain some particular combination of r letters out of the n, the above process must produce all the possible permutations. Thus \r n C r must be the number 304 Permutations and Combinations. [488. of permutations of n things r together. By Art. 483 this number is (# 1) (n 2) ... (# r + 1). Hence [f ..C r *(- l){-2)...(-f+l); n(n\)(n 2) ...(% r-f 1) whence n C r = - y * 489. The proof of the last article may be put in rather a different point of view. Supposing all the possible permutations of n letters r together to be written down, it is clear that any one per- mutation being taken, a large number of others can be found which contain the same combination of r letters. The number of permutations containing this particular set of r letters is evidently the number of different ways in which these r letters can be arranged, or | r. Let the I r permutations containing the particular combination be set aside.\ Any other permutation being taken, which must contain a different combination from all of the former ones, [_r permutations will be found containing this second com- bination. In this manner the permutations may be arranged in sets, every permutation of each set containing the same combination of r letters, and those in one set having a different combination from those in all the other sets. Thus the whole number of permutations may be arranged in as many sets as there are different combinations, each set containing |r permutations. Hence the number of permutations is | r times that of combinations, or .P r =[r..e r ; whence, as before, ^*') (Art . 482 ). 49^ ] Permutations and Combinations. 305 490. The formula for n C r can be put into a more con- cise form. Multiplying both numerator and denominator of the fraction by the product of all integers beginning with n r, that is, a number less by unity than the least factor in the numerator, and going down to unity, the value of the fraction is unaltered, and _n(n-l)(n2)...(n-r+l). (n r)... 3.2.1 n r ~ |V.(-r)(-r-l)...3.2. 1 The numerator is now the product of all integers from n down to unity inclusive, or is [% (Art. 477). The product of the factors introduced may be written as n r } and the value of n C r becomes = p . [r \n r 491. The result of the last article shows that the number of combinations of n things r together is the same as the number n r together. For n C n _ r = - - = r - = n C r . r \n (nr) \nr This can however be even more easily seen from the consideration that for every different set of r letters taken away from n, there is a different set of n r letters left. Hence the number of distinct sets of r letters out of n must be the same as the number of distinct sets of (nr) letters. 492. The fractional form which represents the value of n C r contains an equal number, r, of factors in numerator and denominator. Each additional letter taken in the combination introduces an additional factor in both numer- ator and denominator. 306 Permutations and Combinations. [493. there being r1 factors in both numerator and denom- inator. _ (- !)...(-*+ 2). (tt-r+1) " r= 1 .2.3 ... (r-l).r an additional factor (n r + 1 ) being introduced in the numerator, and an additional factor r in the denominator. Hence .C r = ^i . /?,_, . (1) 493. The result of the last article is of considerable importance. An independent proof of it follows. Let abed ... Ji represent any combination of r1 letters out of n. There are consequently { n (r 1 ) } , or (n r + 1 ), letters which are not included in this combination. By placing each of these letters in succession along with the given combination of (r1) letters, (n- r + 1) combinations of r letters will be formed. From each of the different combinations of r1 things the same number of combinations of r could be formed. On the whole there would thus be produced (n r + 1). n C r ^ l such combinations. These would not all however be different. Each com- bination of r letters, as abc ...kk, would appear r times, by the combination of each letter with the other r1. Thus, on the whole, the process would produce r times the number of combinations of n things r together ; or it would follow that (n-r+l). n C r . l =r. n C r r whence n C r = - - . n C r _ r 494. This formula, obtained as in the last article, will give another proof of the number of combinations of n things r together. 495-] Permutations and Combinations. 307 For replacing r by r 1, r 2, &c., in succession, it follows that _ n r+2 . r ~ 1 ~ r-l n r - 2 ' .C, = *- r+3 i| C'_. -2 2 and n C l evidently is n. Hence, by successive substitution, r+3 n 2 n\ r r-l r-2 3 2 which is the same as the formula of Arts. 488, 489. the factors being written in an opposite order. 495. The numbers of the combinations of n things taken one, two, three, and so on, together, form a series (Art. 475), n(n l) n(nl)(n 2\ the terms of which are n, \ 9 - *- ' and 1.2 1.2.3 so on. The r th term of this series n C r is derived from the r 1 th term n C r _ 1 by multiplying the latter by the factor - (Arts. 492, 493). Hence for all values of r for which this factor is greater than unity, the r th term of the series is greater than the r 1 th ; on the other hand, for all values of r for which this factor is less than unity, the r th term is less than the rl th . If for any value of r, as _p, this factor has the value unity, the jo tb term will be equal to the p 1 th . x 2 308 Permutations and Combinations. [496. If r have the values 1, 2, 3, 4 ... a, given to it in sue- A * ^4 I 1 cession, the numerator of the fraction - - continually decreases from n to 1 ; the denominator, on the other hand, increases from 1 to n. In virtue of both these facts the value of the fraction decreases, having-, to begin with, the ., i value -, and ending with the value -. Thus for the smaller values of r the multiplier by which n O r is derived from n C r _i is greater than unity, and n C r is consequently greater than n r -i , while for the latter and larger values of r the reverse is the case. The terms in the series under con- sideration increase in the first part and decrease in the latter part of the series. 496. Let p be a quantity such that the fraction - = 1 ; it follows that n p +1 = p, or n + 1 = 2 p^ whence p = Thus, if n be an odd number, and n + 1 consequently even, p is an integer and, as above explained, the jo th term is equal to the p 1 th . Also when r is less than^?, the considerations of the last article show that the nr+1. , n p-\-l ., factor - is greater than - , or unity, and thus r p for all values of r less than p, n C r is greater than M C r r _ 1 ; while similar reasoning shows that for values of r greater than j9, n C r is less than n C r - r Hence, when n is odd, the number of combinations of n things r together increases as r increases from 1 to \ (n + 1 ), remains unchanged as r passes to ^ (n+ 1), and decreases as r assumes successively greater values. 497. If, on the other hand, n be even, no integral value of r makes - - equal to unity. Hence no two terms 498-] Permutations and Combinations. 309 of the whole series are in this case equal. By the last two articles it follows that if r be any integer less than />, that is, - - or - + - 1 this multiplier is greater than unity. 2 22 Hence the terms of the series increase, until r is -. After that point they will decrease, since the next and all suc- ceeding integral values of r, being greater than - - or jt?, 2 must make the multiplier - - less than unity. Hence AJ the terms of the series increase until r = - , for which value the term is greatest, and then decrease. 498. Whether n be even or odd the terms during the latter or decreasing part of the series are equal respectively to those in the former or increasing part in a reverse order. (Art. 491.) As an instance of the last article, suppose n to be 6. The terms of the series are 6.5 6.5.4 6.5.4.3 6.5.4.3.2 6.5.4.3.2.1 ' T72"' 1.2.3' 1.2.3.4' 1.2.3.4.5' 1.2.3.4.5.6' or 6, 15, 20, 15, 6, 1. Here the third term is the greatest, and the terms increase up to that point and then decrease. Again, if n be 7, the terms of the series are 7.6 7.6.5 7.6.5.4 7.6.5.4.3 7.6.5.4.3.2 7 ' 1T2' Y.2.3 ' 1.2.3.4 J 1.2.3.4.5' 1.2.3.4.5.6' 7.6.5.4.3.2.1 . 1.2.3.4.5.6.7 ' or 7, 21, 35, 35, 21, 7, 1. Here the third and fourth terms are equal, the terms of the series increasing up to the third and decreasing from 310 Permutations and Combinations. [499. the fourth onward. This result agrees with that of Art. 496. 499. A problem of great importance in many applica- tions of this subject is that of the number of distinguishable permutations, all together, of n things which are not all unlike one another. Suppose that there are n letters, of which p are as, q are #'s, r are c's, and the rest are distinct from each other and from these. Let any permutations of these letters be represented by aabbaabccabcdaef. (1) Suppose that each of the different letters represented by a has a distinguishing number affixed to it. This permu- tation will then be written a l a 2 bba z a 1 bcca 6 bcd^ef t (2) where the a's are now distinguishable. By altering the positions of the p as among themselves without changing those of the other letters, [/? distinguishable permutations can be found out of this single one. Thus if there were x distinct permutations, when the as were all alike, there will be \p times that number, or x . | p when the as are made distinguishable. Suppose that (2) represents any one of this latter set of permutations, and let q letters b be made distinguishable- by numerical suffixes : the permutation (2) can then be written a^a^b^a^cca^cda^ef. (3) By interchanging the qb's among themselves without altering the positions of the other letters, this one permu- tation will give \q permutations. Thus the x . J p permu- tation will give |j? terms that number, or, when both the a's and &'s are distinguishable among themselves, there will be x . | ^ \q_ distinct permutations. 5oo.] Permutations and Combinations. 311 Similarly, if the r c's be made distinguishable by suffixes, each of these will give \r^ permutations. Thus, on the whole, there will be x \ p \q \r permutations when all the letters are distinct from each other. But on this latter supposition the number is known to be \n (Art. 484). Hence x [P Li IT = 1^ ; \n whence x \P [? \ L 500. The result of the last article may be obtained in a somewhat different manner. The number of arrangements required is the same as the number of distinguishable ways in which n things whereof p are #'s, q are b's, r are cs, and the rest unlike, can be arranged in n places. The number of different sets of p places that can be selected for the p #'s is the number of combinations of n things p together, or (Art. 490) is' \p_ \n-p There are now (n p) places left unoccupied, and q of I n p these can be selected for the q b'a in - == - ways. 1* -p-1 Hence the number of different ways in which the p a's and q b's can be placed in p + q of the places is the product of \n \n p these two numbers, or r - x : > that is, \p \n -p \q_ \n -p-q is There are now (np q) places left vacant, and r of n-p-q these can be taken for the r cs in \r_ \n -p-q-r ways. 312 Permutations and Combinations. [501. Hence the total number of distinguishable ways of assigning (jp + q+r} places to the p a's, q 6's and r cs is the product of this into the former number, or L? |-jp-g \T\T that is, IS. \ n -P-V \L \ n -p-q- [n | [0 [r | -p-q-r If all the remaining letters be unlike, they can be distributed in the remaining (np q r) places in I n p qr different ways. Hence the total number of distinguishable arrangements is X \n p qr, or [ H If \JP-q-r [p \q_ \r_ 501. The student will easily see that, by either method of proof, if there had been s quantities of another kind alike, an additional factor | s would be introduced in the de- nominator of the fraction, and the number of permutations would be \p_ \q \r_\t_" 502. There is an almost infinite number of problems of a similar nature to those which have been discussed in this chapter. The preceding articles contain those which are most important for algebraical purposes, and the principles which have been employed will serve as a guide to the methods to be employed in any similar questions of some- what greater arithmetical difficulty. Permutations and Combinations. 313 EXAMPLES. 1. Find the number of words of three letters that can be formed out of the English Alphabet. 2. In how many different ways can three prizes be awarded to a class of fourteen scholars 1 3. How many committees of four can be formed from a council consisting of 1 3 members ?,, 4. A Board of Education consists of nine members. In how many different ways can a chairman and vice-chairman be selected ? 5. In how many ways can a party of twelve be selected from a company of 100 soldiers! In how many of these will a particular soldier be found ? 6. How many words each containing one vowel and two consonants can be made out of the letters of the word number % How many will there be if the vowel is to occupy the middle place ? 7. Prove that the greatest number of combinations that can be formed with 2n things, each containing the same number, is always double the greatest number that can be formed with 2n l things. 8. If the number of combinations of 2n things taken n 1 together be the number of combinations of 2(n 1) things taken n together as 132 is to 35, find n. 9. If the number of permutations of 2n things 3 together is equal to twice the number of permutations of n things 4 together, find n. 10. If the greatest number of combinations of n things r together be -J of the greatest number of combinations of n I things, find n. 314 Permutations and Combinations. 11. If m denote the number of combinations of n things taken 2 together, prove that the number of combinations of in things taken 2 together is equal to three times the number of combinations of n+ 1 things taken 4 together. 12. In how many different ways can a party of six people form a ring ] 1 3. Find the number of arrangements of n people at a round table ; one of them being supposed to occupy a particular chair. 14. A gentleman invites a party of m + n friends to dinner, and places m at one table and n at another, both tables being round. Find the number of ways in which he can arrange them among themselves. 15. Out of six ladies and eight gentlemen, how many different parties can be formed, each consisting of three ladies and four gentlemen ? 16. Find the number of committees that can be formed out of a House of Representatives containing 45 Liberals and 50 Conservatives, each committee to have 9 Liberals and 10 Con- servatives. 17. What is the number of distinct arrangements of the letters in the word precipitate ? 18. Find the same thing for the words Mississippi, Papa- toitoi, Ngahauranga. 19. How many words containing two vowels and three con- sonants can be formed out of 21 consonants and 5 vowels? How many will there be if the vowels are to occupy the even places ? 20. Prove that \2n = 1 . 3 . 5 . 7. ..(2 n-l) . 2 n \ n . Hence show that the number of combinations of 2n things n together is 1.3.5...(2n-l).2 n 21. Prove that the number of ways in which p positive signs and n negative signs can be placed in a row so that no two Permutations and Combinations. 315 negative signs shall be together is equal to the number of com- binations of (j) -f 1 ) things taken n together. 22. If n (7 r denote the number of combinations of n things r together, prove independently that n i r-i From 6 ladies and 5 gentlemen in how many ways could you arrange sides for a game of croquet, so that there should be two ladies and one gentleman on each side 1 23. If the permutations of the things a 15 a 2 , 3 , ..., a n be taken all together, and Q n be the number of ways in which no one of the suffixes indicates the place which the corresponding thing holds in the permutations, show that 24. If ,<7 r+1 = . +1 r = ~ _, C r , find n and r. CHAPTER, XVIII. ARITHMETICAL AND HARMONICAL PROGRESSION. 503. A SERIES of terms such that each is greater, or less, than the preceding" by a constant difference, is called an Arithmetical Progression. The first term and the magnitude and sign of the common difference between each term and the preceding being given, it is obviously possible to determine by repeated additions or subtractions the value of any succeeding term. It will be shown that it is also possible to express the sum of any number of terms in terms of the same two quantities and the number of terms. 504. The series 1 + 2 + 3 + 4 + .. . is an arithmetical progression, the first term being unity and the common difference also unity. The series is also an arithmetical progression, the first term being 3 and the common difference i. The common difference is always taken to be the algebraical quantity obtained by subtracting any term from the next succeeding term. 505. If the first term be denoted by a, and the common difference by d, the second term is a + d, the third a + 2 d, and so on. The series of terms is represented as below. 1, 2, 3, 4, 5, 6 ...... a, a+d, a + 2d, a+3d, a-\-4d, 506.] Arithmetical and Harmonical Progression. 317 The number in the upper of these two lines is the number of that term in the series which is written below. The coefficient of d increases by unity in passing from term to term, since each term is obtained by adding d to the previous one. The number which denotes the term also similarly increases by unity from term to term. In the terms written down, the coefficient of d is less by unity than the number which denotes the position of the term in the series. As these two numbers increase at the same rate, the difference between them must remain always the same. Hence in the n ib term the coefficient of d must be n1, and the n ih term must be a + (n l)d. If this n ih term be denoted by /, we have l = a + (n-l)d; (l) whence it also follows that a = l(n-l}d. 506. If s denote the sum of the first n terms of the series, s = a + (a + d) + (ai-2d) + ... + {a + (n l)d}. Writing the terms in the reverse order, and remembering that as any term is obtained by adding d to the one before, so it can also be obtained by subtracting d from the following one, Adding the equal quantities 011 both sides of these two equations, it follows that 2s = a + l + a + l + a + l+ ... to n terms Hence * = ( + *); ( 2 ) or, since I =. a + (n l)d, (3) 318 Arithmetical and Harmonical Progression. [507. 507. The formulae (l) and (3) suffice for the solution of all problems relating to arithmetical progression. Such problems are of two classes. In the first class, the values of a and d are given, and by means of (l) and (3) every other required result about the progression can be deter- mined. In the second class, some two conditions which the progression is to satisfy are stated : from these conditions, by means of (1) and (3), the values of a and d can always be obtained, and then any further result can be investigated as in the former class. 508. An important problem of the first class is that of the determination of the number of terms of a given progression whose sum is a given quantity. Here a, d are known since the progression is given, and s is also known ; n is required. From (3), multiplying out, (-!) 7 s = na + - ' d n 2 d nd From this equation two values of n can be determined by the ordinary process of solving a quadratic equation. If either value of n be a positive integer this gives the required number of terms. A fractional or negative value of n is irrelevant, regarded as a solution of the problem considered, though an interpretation may be found for such a value by seeking some other problem to which it applies. 509. Let it, for instance, be required to find how many terms of the series 1 1 + 9 + 7 + ... amount to 32. If n be 5io.] Arithmetical and Harmonical Progression. 319 the required number, the formula (3) of Art. 506 gives, since a = 11, d = 2, s = 32, = 1 In n (a I) whence n z 1 2 4- 32 = ; which gives n = 4 or 8. Here both values are admissible. The first four terms are 1 1, 9, 7, 5, the sum of which is 32 ; while the first eight terms are 11, 9, 7, 5, 3, 1, 1, 3, the sum of which is also 32. 510. As an example of the second class of problems, let it be required to insert n arithmetical means between two given quantities a and It. The problem means, to find n quantities # 15 a? 2 ,...# n , such that the whole set, a, sc l9 # 2 , 3 ,... # n , b shall be in arithmetical progression. Here a being the first term, 1) is evidently the (n -f 2) th . Hence if d be the common difference b = a + (n + 2-1) d, by (1), , , b a . whence d = -- - (a) n -j- 1 Hence d is known, and the value of a? t , # 2 can be determined by (l). Thus # r being the r+ 1 th term of the series, # r = a + rd , . n+l 320 Arithmetical and Harmonical Progression. [511. 511. A special case of the last article is the discovery of a single arithmetic mean between a and , that is, a quantity x such that #, #, I are in arithmetic progression. The formula (a) and (/3) of the last article, putting b a a + b n = 1 , give a = and x = -~ 2 2 This latter result can be obtained more easily from the consideration that if a, .r, b be in arithmetic progression, it follows from the definition of Art. 503 that x a = bx\ whence transposing, 2x = a + b, a + b or *=-g- 512. As another problem of the same class, let it be given that the p ih term of an arithmetic progression is x and the # th term is ^, required the ft th term and the sum of n terms. Here p, q> x, y are supposed to be known quantities. Let a be the first term and d the common difference of the progression. The / th term will be a + (p \}d, and the th term a + (q 1 ) d. Hence, by the given conditions, From these two equations the two unknown quantities a and d can be found. Subtracting the second from the first, 7 whence 0= - p-q 513.] Arithmetical and Harmonical Progression. 321 Substituting in the first equation whence a = x (p 1) p-q xy Hence the n ih term, which, by (1), is a + (n 1) d, = (p-i)y-(s- 1 )* , fa ^x-y p-q and, by (3), the sum of n terms can similarly be deter- mined. 513. The n ih term of any series is conveniently denoted by a symbol such as u n , the suffix indicating- the position of the particular term in the series. In all cases of series proceeding 1 according to any law, u n is a function of n (Arts. 165, 475). In the case of an arithmetical progression, the n ih term has been shown to be a + n 1 d. Hence in this case, u n = a + (n 1 ) d (ad) + nd. Thus u n) regarded as a function of n, is an expression of the first degree, or a linear expression in n (Arts. 88, 139). Conversely, an arithmetic series can always be discovered whose # th term shall be represented by any expression of 322 Arithmetical and Harmonical Progression. [514. the first degree in n. For let the given expression be p + qn. Then if d be taken equal to q and a d to J9, this becomes identical with (a d) + nd. Hence a-^dp d q. Therefore a = p + q. The required series has therefore p + q for its first term, and q for the common difference between consecutive terms. 514. The sum of n terms of a series is conveniently denoted by s n . Hence in the case of an arithmetic progression ,. = (a _|) +|. (Art. 508) Thus the sum of n terms of an arithmetic progression is an expression of the second degree in n (Art. 139). Conversely, an arithmetic series can always be found such that the sum of n terms shall be represented by any expression of the form pn + q?i 2 . For let a and d be taken so as to satisfy the two conditions d 5-* whence a = p + q, d = 2q. Then pn + qn* = (a - ^) n + ^ 2 , or _pn + qn 2 is the sum of n terms of an arithmetic progression whose first term is a or p -f q and common difference d or 2 q. 517.] Arithmetical and Harmonical Progression. 323 515. The subject of Harmonical Progression is closely connected with, that of Arithmetical Progression. Before entering on it a preliminary definition must be given of the term * reciprocal/ Two numbers whose product is unity are called reci- procal numbers and each is said to be the reciprocal of the other. Thus if ab = 1, a is the reciprocal of b, and b is that of a. From the relations db 1, it follows that a = T and b = . Thus the reciprocal of a may also be denned as 1 being - 1 45 As examples the reciprocal of 2 is -, and that of - is - 516. A series of numbers is said to be in harmonical progression when their reciprocals are in arithmetical progression. Thus all problems relating to quantities in harmonical progression can be solved by taking the reciprocals of these quantities and using the formulae relating to arith- metical progression. 517. The discovery of n harmonical means between two given quantities a and b depends on the investigation of n arithmetical means between - and T a b If the harmonic means be called z lt z 2 , z z ,...z n , so that the (n + 2) numbers mm-** ? I a ) ^v z <2? * *n) are in harmonical progression, the (n + 2) numbers 1111 11 > , > > ... , 7- a ^ z 2 z s z n b will be in arithmetical progression. 324 Arithmetical and Harmonical Progression. [518. If cl be the common difference of this latter, by (l) of Art. 505, j j 1 ,1 K whence d = - - ( _ -- ) ra + 1 v 0' Hence, since - is the r + 1 th term of the series, *r 1 1 L Hence 518. The particular case of inserting one harmonical mean z between two quantities a and b may be either de- duced from the last article by putting n 1, r = 1, whence z = -- - ; or may be independently investigated thus. . 1 1 1 If a, z, b are in harmonic progression, - , - > -? are in a z u arithmetic progression. Hence by the definition of Art. 503 1111 z a b z 2 1 1 whence - = - + , ; z a fj and, getting rid of fractions, 2ab = z(a + l\ 2ab or z=. -- 7 a+b 519. There is no general formula for the sum of n terms of a harmonical progression. .Arithmetical and Harmonical Progression. 325 Find the w th following series : 1. l+f + i+.... 3. 2 + 2i + 3 + ... 5. 11+9 + 7 + ... 7. 4 + 3 + 2 + .... 9. + 1 + 2 + .... 10. (a + &)' + (a a + 11. (n-l) + (n 2) + (n- EXAMPLES. term and the sum of n terms in each of the 2. 1 + 3 + 5+.... 4. I + J+2 + .... 6. 15+^+16+.... 8. 1 + J + 0+.... ' 26 a -' 1 1 15. -5-31-2-.... a-b tfb* 14. 13 + 9 + 5 + .... 16. -2i + lf+6+.... 17. Find how many terms of a series, whose first term is unity and third term is 5, amount to 169. 18. The fourth term of an arithmetical progression is 7, the eighth term is 15; find the n^ term and the sum of n terms. ' 19. The third term of an arithmetical progression is e, and the fifth term /; find the thirteenth term and the sum of n terms. Find the w th term in each of the harmonical series : 20. 1 + 1 + 1 + .... 21. l + * + 2+'.... 22. i + t + t+...- 23. 1 + 4-2. 24. 1, 2, oo,-2. 25. The first term of a harmonical progfession is unity, the thkd term is ; find the tenth term. Arithmetical and Harmonical Progression. 26. The first term of a harmonical progression is unity, the Him of the first three terms is ~, find the progression. 27. How many terms of the series 11 + 9 + 7+... amount to 20? 28. If the wth and n^ terms of an arithmetic progression be M and N, find the ^ term. 29. If = - , then z is a harmonic mean between x and y. 30. Find 17 harmonic means between 1 and T ^. 31. If = = , and p, a. r be in arithmetical px qy rz progression, then x, y, z are in harmomcal progression. 32. If s be the sum of any number of term's of the series 1 + 2 + 3+..., prove that 8s+l is always a square. How many terms of the series must be taken to make 21? 33. Find the arithmetic series whose fourth term is 3, and the sum of seven terms is 21. 34. S lt 2 , S s are the sums of three series of n terms each in arithmetical progression, the first term of each being unity and their common differences in haimonical progression ; prove tha t _ 35. A and B go round the world; A goes east one mile the first day, two the second, and so on in arithmetical pro- gression. B goes west at the uniform rate of twenty miles a day. Find when they will meet, and interpret the negative answer. The circuit of the world being 23661 miles. CHAPTER XIX. GEOMETRICAL PROGRESSION. 520. A SERIES of numbers such that each is equal to the preceding multiplied by a constant factor is called a series of numbers in Geometrical Progression. Anticipating here the definition of the ratio of a number a to another number b (Art. 644) as being measured by the fraction j , the common factor is the ratio of each term to- the preceding one. Hence this factor is usually called the common ratio of the series. 521. Let a represent the first term, and r the common ratio. Then since each term is obtained by multiplying the preceding term by r, the second term is ar, the third ar 2 , the fourth ar 3 , and so on. The index of r increases by unity from term to term and is always less by unity than the number of the term. Hence the n ih term is ar n ~ l ; or, denoting the n ih term by w n , n n = ar n ~ l . (l) 522. Let the sum of the first n terms be denoted by s n . Hence s n = a + ar + ar 2 + ar B + ... + ar n ~ l . Then rs n = ar + ar 2 + ar B + ... +ar n ~ l + ar n . Subtracting the second of these equations from the first, all the terms on the right hand except the first in the first row and the last in the second row disappear, and *(!>) = *", l-r n or , n = 0__. (2) If r be less than unity, rs n is less than s n , and the sub- traction gives a positive result on each side. If r be greater 328 Geometrical Progression. [523. than unity it would be more convenient to subtract the upper equation from the lower, and the result would be *( r - 1 ) = ar n -a, or ^ = a 7rr' ( 3 ) The form (2) is more convenient when r is less, and the r n 1 form (3) when r is greater than unity, but since is 1 r n algebraically equal to - - > either form may be safely used in either case. 523. The formulae of the last article give the means of solving any problem involving a finite number of quantities in geometrical progression. (Compare Art. 507.) Let it be required, for instance, to insert n geometrical means between a and d ; that is, to find n quantities y\ 5 2/2 ' ^3 ' & su h th^ the whole series may be in geometrical progression. If r be the common ratio, b is the n + 2 th term of the series, and must therefore, by (l), be equal to ar n+1 . Hence I = ar n+l ; therefore r n+l = -, a = (-} whence y p , the p ih of the means which is the (p+ l) tb term of the series, is given by the equation np+l (a) 526.] Geometrical Progression. 329 524. The particular case of the insertion of one geome- trical mean y between a and b, may be deduced from (a) of the last article by putting n and p each equal to unity. We thus et y = (ab)*. (j8) The same result may be obtained independently from the consideration that if , y, b are in geometrical progression the quotient of each term by the preceding must be the same (Art. 520). Whence in this case or, multiplying these equals by ay, ./ = at, whence y = (ab)%. 525. In Arts. 511 and 518 it has been shown that if x and z be the arithmetical and harmonical means be- tween a and b, a + b x = 2 2ab Hence xz = ----- x - y = ab y^ (Art. 524) y being the geometrical mean between a and b. Hence, since xz = ^ 2 , or x, y, z are in geometrical progression. 526. The connection between x,y, z, a, b may be geome- trically illustrated. Let the line OA represent a, and OB represent b. ' If A3 be bisected in (7, since AC = CB, it follows that OC-OA = OB-OC-, whence OA, 0(7, OB are in arith- 330 Geometrical Progression. metical progression, and OC is the arithmetical mean between OA and OB. AYith C as centre and CA as radius describe a semi-circle on AB as diameter ; and draw OP from to touch this circle. Join PA, PB, and draw PN perpen- dicular on AB. Since OP touches the circle and OAB cuts it, the square on OP is equal to the rectangle contained by OA, OB, or, replacing the lines by the numbers which represent them, OP 2 = OA . OB, OA OP whence OP = OB' or OP is the geometrical mean between OA and OB. Again, the angle OP A is equal to the angle PBA in the alternate segment, which again is equal to the angle APN, because the triangles ANP, APB are similar. Hence PA bisects the angle OPN ; and PB, which is perpendicular to PA, since the angle in a semicircle is a right angle, must bisect the angle between PN and OP produced. NA NP Hence, by Euclid VI. 3, -^ = -^- ; ^L (J Jr (J and, by Euclid VI. A, whence or or NA_NB AO~BO' ON-OA OB -ON OA OA OB ON ~ OB' 527.] Geometrical Progression. 331 and, dividing these equals by ON, JL JL J_ .L OA ~~ ON ~ ON ~~ OB ; whence > , are in arithmetical progression, UA. UJM UJj and therefore OA, ON, OB are in harmonical progression (Art. 516), or ON is the harmonic mean between OA and OB. If CP be joined, since the angle OPC is a right-angle, the triangle 0JVP is similar to OPC, and therefore jyp = TT-^; whence 0A r , OP, 0(7 are in geometrical pro- gression. 527. The following is another method of comparing the conditions that three quantities shall be in arithmetical, geometrical, and harmonical progression respectively. If a, I, c be in arithmetical progression, by the definition of Art.. 503, a b = 6 c, which may be written . b c a If , I, c be in geometrical progression, by the definition of Art. 520, - = - j or ac = l/ 2 -, whence, subtracting each of these equals from ab, ab b 2 = ab ac, or b (a b) = a (bc], or, dividing these equals by (b c) b, = _. b-c b If a, b, c be in harmonical progression, by the definition _ _ b a ~~ c b' 332 Geometrical Progression. [528. a I bc or 7- = . ; ab be whence, multiplying these equals by ab and dividing by b-c, a-b_ab_a .. b-c~ bc~ c' V ' Hence in the three cases the fraction -= is equal to bc a a a - > j- , - respectively. The condition (3) is sometimes given as the definition of harmonical progression. 528. The sum of n terms of a geometrical progression has been shown to be a (Art. 522) Supposing r to be less than unity, the second term in the numerator, r n , diminishes as n increases. Hence the sum of the series to n terms continually approaches nearer to the value Since s n = \r lr the difference between s n and is , or r n s n . The 1 T 1 r difference continually diminishes as n increases, as has been already observed. It may be further asserted that by making n sufficiently large the value ofr n s n can be made less than any number how- ever small. If this be granted, it follows that the sum of the geometrical series carried on absolutely to infinity must be . The symbol oo, read infinity, is frequently used 530.] Geometrical Progression. 333 to denote a number to the magnitude of which no limit can be assigned, and thus in this case we may write a. = ~. (4) 1 r 529. The truth of this equation depends upon the two statements in italics. For if s be not equal to it must differ from it by some finite quantity x. Then, in virtue of preceding statements, by taking n sufficiently large, s n can be made to differ from by less than #, however small x may be. Also, as n is increased s n keeps approaching nearer to . Hence s& must be nearer to - - than s n 1 *> Ir *" % * is, that is, s^ cannot differ^from - - by so much as x. . Hence the supposition that s^ differs from by any JTJ l _ r J J finite quantit^ is false, and s*> must therefore be absolutely equal to 530. The theorem of the last article is so important that it is perhaps worth while to illustrate the second statement in italics by a few particular examples. Let, for instance, r = - - Then the first ten powers of r 1111111 1 1 1 are > > ) ? 3 - > > -T--T > and 2 4 8 16 32 64 128 256 512 1024 Hence r 10 < < < -001, the symbol < being used as an abbreviation for the words ' is less than.' Con- sequently r 10 " < ; and by taking n sufficiently large, this latter quantity, which is decimally written as 3 n 1 334 Geometrical Progression. [531. ciphers and a unit to follow, can evidently be made less than any given measurable or conceivable quantity. Suppose, again, that r = - , the successive powers of r are 5 25 125 625 1 6 ' 36 ' 2l6 ' 1296 ' 1S 2 ' r 4 < - ; whence by the previous calculation Hence, as in the former case, by taking m sufficiently large, r m can be made less than any conceivable quantity however small. The nearer r approaches to unity, the higher power of it will have to be taken before its value comes below any given small quantity, but by taking a sufficiently high power, the object can always be obtained provided r is at all less than unity. The symbol < has been used for the words * is less than/ The symbol > is similarly frequently used as an abbrevia- tion for the words ' is ' or ' are greater than.' 531. The formulae (l) of Art. 521, (2) and (3) of Art. 522, and (4) of Art. 528 are the fundamental formulae of the subject of Geometrical Progression. They may be repeated here = ar"- 1 , (1) t n =.a l ~=a^ i, (2) and (3) = - > where r< 1. (4) lr 532. A repeating or circulating decimal is an instance of a geometrical progression carried on to infinity. Thus the decimal *43527 is a concise expression for the series 43 527 527 527 - ad ' mf ' 533-] Geometrical Progression. 335 The terms of this after the first form a geometrical series 527 1 whose first term a is -^ and whose common ratio r is ^ . Hence the value of the repeating decimal is 527 43 To" 5 10 2 J_ " 10 3 43 527 " 10 2 T 10 2 (10 3 -1) 43(l0 3 -l) + 527 10 2 (10 3 -1) _ 43000-43+527 100x999 43527-43 99900 a result which agrees with the ordinary rule given in treatises on Arithmetic. 533. In the general case we may suppose that the non- repeating part 1 contains p figures and the repeating part Q contains q figures. p The value of the non-repeating part is therefore - > that of the first period g , and, since each of the repeating periods begins q places after the first figure of the previous one, the successive periods really represent Q Q and so on. Hence the whole value of the fraction is Z_ that A of the second payment is -^ > and so on. Hence, if l\ n denote the present value of the annuity, the latter being supposed to continue for n years, n -- A Thus n n is the sum of a geometrical series whose first A 1 term is -~ and common ratio -^ JK H Therefore H n = (\ ~R( R .- A The present value of an annuity A to continue for ever is given by the formula 340 Geometrical Progression. [540. This last result is obvious, for it is merely the statement that A is equal to rU ao , the interest to be annually paid for the use of I I ? common ratio is r is - / of the sum of the first n terms, and the sum of the first, third, and fifth terms is |J of the sum of the second and fourth terms : find r and n. 20. If a,b,c,d,...l be n terms in geometrical progression, prove that %/ abed... I = Geometrical Progression. 343 21. Prove that the arithmetic mean of the arithmetic and geometric means between any two quantities is equal to the arithmetic mean between their square roots. 22. , S' and 2 denote the sum of a geometrical series to n terms, 2n terms and infinity respectively. Prove that o _ y ,, tS is independent of n. o o 23. If s lt s 2 , 5 3 be the sums of a geometrical progression to n terms, 2 n terms and 3 n terms respectively ; shew that 24. If a series of terms in arithmetical progression be collected in order into groups of n terms and the terms of each group be added together, the results form an arithmetical progression whose common difference is n 2 x original common difference. If this process be repeated on the second series to form a third, and so on, and if a r , b r be the first term and common difference of the r th series, then a^~ $b lt a 2 ^b 2 , a 3 1-6 3 ,... form a geometrical progression whose common ratio is n. 25. The sum of six terms of the series 1 xV 1 o?+ ... is 65 times the sum to infinity. Find x. 26. The arithmetic mean between two numbers exceeds the harmonic by 1, and twice the square of the arithmetic mean exceeds the sum of the squares of the geometric and harmonic means by 1 1 . Find the numbers. ! L ! 27. If a, b, c be positive integers and a b , b ac , c*> be in 1 i 2. geometrical progression, shew that a 6 *, 6 a " c ", c l * are also in geometrical progression. 28. If an arithmetical, geometrical and harmonical progression have the same first and second terms, and their third terms be a, b, c respectively, prove that 344 Geometrical Progression. 29. A man lays by annually *k of his income which he invests. Prove that if his income for any one year be 1, his r n income n years later will be 7 (1 H ) , r being the interest of XI for one year and compound interest being assumed. 30. Interest is payable m times a year. Assuming that the interest is invested at once, find the amount of P in n years. 31. What is the real rate per cent, when the interest is nominally 5 per cent, per annum paid quarterly, assuming that the interest is invested immediately ] 32. Find the present value of an annuity of 100 a year paid quarterly to last for 10 years, interest being reckoned at 4 per cent, per annum. 33. Find the present value of an annuity of A per annum payable m times a year to last for n years, r being the interest on 1 per annum. 34. If the arithmetic means between the pairs of quantities (a, x) (6, y) (c, z) be in arithmetical progression, the geometric means in geometrical progression, and the harmonic means in harmonical progression, prove that 112 l>y( j T) = 2b a c, or that by = ac. CHAPTER XX. ON THE SERIES FORMED BY THE PRODUCTS OF BINOMIAL FACTORS. 543. IN Arts. 129 and 131 the following relations have been proved, (x + a) (x + b) x 2 -f (a + 1) x + ab, (x + a) (x -\-b}(x + c)=x* + (a + b-\- c)x 2 + (be + ca + ab)x + abc. Here the product of two or three binomial factors, the first term in each of which is #, have been expressed in series arranged according to descending powers of x. The product of any number of binomial factors of the same form can be also expressed in a similar series. In Arts. 104 and 106 it is shewn that the product of any two algebraical expressions is the algebraical sum (Art. 93) of all the products which can be formed by mul- tiplying any term in the first expression by any term in the second. Hence it follows that the product of three or more ex- pressions is the algebraical sum of all the products that can be formed by multiplying together one term out of each of the factors. Let there then be n binomial factors, the first term in each of which is x, and let the factors be represented by Each term in the product will be the product of one term out of each of these. The highest power of x will occur when all the #'s are multiplied together, and, since there 346 On the Scries formed by the [543. are n of these, it will be x n . A number of terms can be obtained involving x n ~ l , namely, the product of n 1 a?'s, out of any set of (n 1 ) of the binomials, into the second term of the remaining 1 binomial. Thus, on the whole, the coefficient of x n ~ l must be x 4 a 2 + a z + + a n which may be concisely denoted (Art. 119) by the symbol 2(). The terms involving x n ~ 2 must similarly contain the product of two of the terms a l9 a 2 , a z , a^...a n , and the coefficient of x n ~ 2 may be denoted by the symbol ^(a^a^ since a term will certainly occur containing the product of any two of the quantities a t , a 2 , 3 , ... a n . The terms involving x n ~ r must contain r of the second terms as factors : and by the same reasoning, the coefficient of x n ~ T must be 2 (a^ a 2 . . . a r ), the symbol meaning the sum of all possible products of r out of the n quantities 1> # 2 > fl 3J ' a n' The only term independent of x will evidently be a l .a 2 .a 3 ...a n . Thus, on the whole, 544. If all the quantities a 19 a 2 , a 3 ,...a n be equal, and each be denoted by a, the left-hand member of the above identity becomes (x + a) n . The number of terms in 2 (a) is n, and, since the value of each term is a, the value of 2 (a) is na. The number of terms in 2(a 1 2 ) is the number of distinct pairs that can be formed out of the n quantities ' 488), - As each of these terms will have the value a 2 , it follows that 545-] Products of Binomial Factors. 347 The number of terms in 2 (a 1 a 2 . . . a r ] is the number of distinct sets of r out of the n quantities a^ , a 2 , # 3 , ...#, nn-l)...(n-r+\) that is (Art. 488), -...- ^ ^^ ^ rf ^ terms, being the product of r factors equal to 0, is # r . Hence The last term, a 1 a 2 ... a n , becomes a* Hence, on the whole, it follows that ( ( 2 ) the r + 1 th term of the series being If the value unity be also given to a, the formula gives or, as it may be written, using the notation of Art. 486, 2=l+ n must be 3. All the sets of values allowable are given in the accompanying table, the method of obtaining them being by giving to , s, successively, lower values until all the possibilities are exhausted, no value of any letter being greater than 5. There are thus eight terms which contain a? 7 , and the values of these terms can be obtained by putting in (a) the sets of values in each horizontal line of the table. They thus become 2 3 x5 [2 |J_ [2 2x4 2 + which reduces to 400 + 1800 + 800 + 960 + 1080 + 320 + 1080 + 720, or 7160. The term in the expansion involving x* is 7 160 a? 7 . 553. The expansion of a polynomial can be completely determined by means of the last three articles. The different terms can be arranged in classes as in Arts. 119 and 121, but the terms cannot be easily arranged in any definite order as in the expansion of a binomial in Art. 545. 554. The general formula for (x+d) n can be arrived at by what is known as the method of Mathematical Induction. A a 354 On the Series formed by the [555. This method consists of three steps. The first step is the proof that the theorem holds in some simple case. The second step is the proof that if the theorem holds in any one case it holds in the next case. The third step is the deduction that the theorem holds in all cases following- that for which it has been shown to be true in the first step. 555. The theorem for the expansion of a binomial given in (l) of Art. 545 holds when n has the values 2 or 3. For (Arts. 114, 122) (x + a) 2 = x 2 + 2xa + a 2 = x 2 + 2 C l xa + 2 C 2 a 2 , (x + af = This gives the first step in the induction. 556. Again, if it be true that ra r +...+ n C n a, (1) the expansion of (x + a) n+1 must be obtained by multiply- ing the right-hand member by x + a. In the product a term involving a r as a factor will arise from the multiplication of n C r x n ~ r a r by a? and also from that of n C r _ l x n - r+l a r ~ l by a. Hence the term in the product wiU be ( c * C }x n ~ r+l a r {n^r + n^r-lj a > but n C r + n C r _i = H+1 r . For n+ iC r means the number of combinations of (n+ 1) things r together. The number of these in which one particular thing occurs, is the number of combinations of the remaining n things r 1 together, or n C r _ l . The number in which that same thing does not occur is the number of sets of r that can be obtained from the remaining n things, or n C r . Since the particular thing either does or does not occur in every combination, it follows that n n _ r 558.] Proditcts of Binomial Factors. 355 Hence the term involving a r in (x + a) n is on the given supposition C r x n+l ~ r a r . n+l Thus, on the supposition that the expansion of (x + a) n is correctly given by the form (l), that of (x + a) n+l will be given by the precisely similar form x n+l + n+l C 1 x n a + n+1 C. 2 x n - l a*+.... 557. From the last two articles it follows that since the theorem (1) is true when n has the value 3 it must be true when n is 4 ; and thence its truth follows for the value 5, and so on for any integral value of n. 558. The r th term of the expansion of (x + a) n is f 1 The r+ 1 th term is similarly The latter of these can be obtained from the former by multiplying by the expression nr+l a t r x If this factor be greater than unity the r+ 1 th term will be greater than the r th , if it be less than unity the r+ 1 th will be less than the r th . Obviously, as in Art. 495, the factor - - decreases as r increases. The greatest term in any given expansion will therefore be ascertained by finding for what value of r the expression first becomes equal to or less than unity. If rx there be an integral value of r, as jo, which makes this factor equal to unity the jo th and p + 1 th terms will be equal, A a 2 356 On the Series formed by the [559. and greater than all that precede or that follow. If there be no integral value of / for which the factor becomes unity and p be the least value of r which makes it less than unity the jt? th term will be the greatest. If p be greater than n + 1 the terms continue to increase to the end, so that the last term is the greatest. 559. The Binomial Theorem can be often used to prove relations similar to those of Art. 548. Suppose, for in- stance, that the coefficients in the expansion of (1 +#) n be denoted by a , a lt 2 ,..., so that (l+#) n = a + a l x + a 2 3? + ...+a r x r +...+a n sc n . (l) It follows, replacing x by #, that (1 xf = a a l x + a 2 x 2 +...+( l) r a r x r + ... ...+(-l)" H a?". (2) Hence (1 -f x} n x(l -x) n or (l-# 2 ) n is the product of the two series on the right-hand side. But replacing x in (l) by a? 2 , we have ...+(-l)*a n a?. (3) Hence (!)'' must be the coefficient of # 2 '' in the product of the two series on the right-hand sides of (1) and (2). But this coefficient must be obtained by multiplying together all terms, one in each series, the sum of the indices of x in which is 2 r, and will be seen to be a a 2r -a 1 a 2r _ 1 + a 2 a 2r _ 2 -...a 2r _ l a l + a 2r a () . (4) Hence ( l) r a r = a a 2r a l a 2r - 1 +...a 2r - 1 a 1 + a 2r a the last reduction being obtained by noticing that there is an odd number of terms, 2r+ 1, in (4), and that the last r are the same as the first r in the reverse order. Products of Binomial Factors. 357 EXAMPLES. 1. Expand (a -cc) 7 ; (2a x 7 v 3 20 2. Expand to five terms (2-) ; (ce 2 ^-) . 3. Expand (x + -) ; (a-6) 8 . ^3 20 4. Find the term involving 2/ 15 in (a? 2 ) - 95 _ i 5. Find the r+ l*h term in (#+ -) x' j 2n+l 6. Find the terms in (cc + -) which involve x* and x respectively. 1 2W 1 7. Find the terms in (x -\ ) which involve oP and or x 2 respectively. 8. Four consecutive terms of a binomial expansion are 14, 84, 280, 560, respectively : find the expansion. 9. Find the greatest term in (a + &) w when a = 2, b = 3 and n = 7. 1 0. Find the coefficient of cc 4 in the expansion of 11. Find the coefficient of x 5 in (2 x + x 2 -3cc 3 ) 3 . 12. If A be the sum of the odd terms and B that of the even terms in (x + a) n , prove that . (xt-a^^A 2 -^ 2 . 13. If the whole number of combinations of (mn) things is of the whole number that can be made out of (m + n) things and - of the number that can be made out of 128 2 (m n) things, taking an odd number together in each com- bination ; find m and n. 14. If f(ri) be the (r-f l)tb term of (a + 6) n , prove that {/(*)}' _n(n-r+l) 358 On the Series formed by the 15. If the (m+l) th and (m + 2)th coefficients in(a + 6) n be equal, and also the (m + 3)^ and (m + 4) th terms, then (w + 3) a = (m 1)6. 1 6. If (x + a) n = p # n +Pi a"" 1 a +^ 2 a;n ~ 2 2 + , find the value of p q r +p l q r+1 +p. 2 q r+t + .... prove that (1) a r =a n _ r : \2n { ) a a r ^a,.^ a z r+2 a n _ r a n ^ n _ r |^ + r > (3) a a. 2r a^a^. 2 \r_ \n- (4) a a r - ttj a r+1 + a 2 a r +2 -...+(- 1 ) n ~ r a n _ r a n I !("-) |*( + r) In (4) examine the nature of n and r and find the value of the expression on the left-hand side when this condition is not satisfied. 18. Prove that if S r denote the sum of the products r together of the n quantities # 15 x 2 , x 3 , ... x n , 19. If p + q + r = p' + q + /= p" + q" + r"= a positive in- teger , and thea Show also that if p + q + 2r and also p' + q'+2r be in- variable and equal to two numbers whereof the former exceeds the latter by unity, then 2 9 '.9 r ' 2*.9 r |/|/|2/-H=^ 2 |p \q |2r' Products of Binomial Factors. 359 20. In the expansion of (a+bx + cx*) n if the coefficient of x n vanish that of x zn ~ m will also vanish. If the coefficient of as 2 vanish, prove that 2ac= (n 1)6 2 . 21. Show that if for any positive integral value of n the value of (2 + \/3) n is calculated, its integral part will be an odd number. 22. Prove that the integral part of (Va*+ 1 +a) n is even if n be odd and odd if n be even. 23. If a r be the coefficient of x r in (1 +cc) n , prove that * \n or 0, n. z o) according as n is an even or odd integer. CHAPTER XXI. n(n-l) _ ON THE INFINITE SERIES 1+HX + -+ ' X? + . . . . 560. The expansions of the last chapter, and the con- verse summations of series in Chapters XVIII and XIX, with one exception, have all contained the expression of the arithmetical identity of two different forms, each of which might be employed to calculate the value of either, though one form might be sometimes more convenient than the other. Thus the value of the sum of an arithmetical progression may be obtained in any given case either by adding all the separate terms, or by calculating the value of the ex- pression to which this sum is proved to be equal (Art. 506). The latter processes usually the easier. Similarly the value of (x + #) n for any given values of a and #, may be obtained either by multiplying n factors x + a together, or by computing the values of the separate terms in the equivalent expansion of Art. 545. 561. The one exception alluded to has been the theo- rem of Art. 528, in which it is proved that the sum of a certain series carried on for ever is equal to Here it is impossible to go through the processes of calculating the separate terms absolutely to infinity and adding the result. The identity arithmetically of the series a + ar + ar 2 + . . . ad inf. and the quantity - - is therefore one which cannot be * 1 r 563.] On the Infinite Series l+nx+ ^ 2 ^ x' 2 + . . . . 361 verified by actual computation of their equivalence. The proof of the equality of the two expressions depends on the fact that the supposition of their inequality leads to an absurdity. There are many other series which when carried on absolutely for ever can be shown to be equiva- lent to finite algebraical forms. The numerical equivalence of the finite and infinite forms in cases where it holds good must be established by considerations analogous to those which have been employed in the treatment of the infinite geometrical series. 562. The series of which the r+ 1 th term is where r is, from the nature of the case, integral, is for general values of n an infinite series. It is true that if n have any positive integral value all the terms after the u 4- 1 th will contain a factor n n or zero. The value of the series will in that case reduce to that of its first n+1 terms, but regarded as a general formal function of n (Art. 165) the series still contains an infinite number of terms. The next few articles will be devoted to an investigation of the properties of the series in question, namely , x 2 ... j- ... ad inf. 563. The series considered depends for its value on two quantities n and x. It may therefore be regarded as a function of either n or x or both (Art. 165). For present purposes it will be treated as a function of. n to which different values will be given while the value of x remains unaltered. For shortness the series will be denoted by (n) : and the quantity n will be called the parameter of the series. 362 On the Infinite Series [564. 564. By the theorem (2) of Art. 548, it follows that the value of $ (n) when n has any positive integral value whatever, is (l-f x}". It will have been observed by the student that two algebraical forms which are equivalent for any general series of values of the letters involved are also equivalent for all other values. It will probably seem hardly conceivable that the two forms $ (n) and ( 1 + x) n should be equivalent for all in- tegral values of n and that the equivalence should fail for values intermediate to the integers. The equation, when n is any quantity whatever, may be perhaps accepted at once as a particular instance of a general principle of the permanence of equivalent algebraical forms. The assumption involved in this method, an assumption which really lies at the bottom of all proofs of the theorem, is perhaps rendered more easily acceptable in the develop- ments of the following articles. 565. Let it be required to multiply a series 1 -f a^x -f a 2 x' 2 + a 3 x 3 + ... ad inf. by a second series, By conducting the multiplication in the ordinary method of Arts. 108, 109, or by that of Art. 136, the product can be ascertained to be that is to say, it is a series of the form l+c l x + c 2 x 2 + c 3 x* + ..., where c 19 c 2 , 3 ,... are definite functions of the coefficients S 66.] l+*+=.... 363 1 . The point of chief importance to notice is that the quantities c l9 c 2 , c 3 ... involve the coefficients 15 2 , fl 3 ... ? ^i> ^2 ^3-*' 2 ^ exactly the same manner whatever be the values of these coefficients. t 566. Thus, if the series (n) or 1 +nx + "" 'a? 2 + ... ad inf. be multiplied by the series $ (m), or . ... ad inf., the product will be a series of ascending- powers of a?, the coefficients of the different terms in which must involve m and n in the same manner whatever be the values of m and n. The coefficients of each power of x cannot be a different formal function of m and n when there" are positive integers from what it is when they are fractional or negative. The first few terms of the product are easily ascertained to be ml) n(n 2 } +mn+ \ 2 The coefficient of x 2 in this series _ m(m l)-f 2mn + n(n I) 1.2 1.2 _ (m + n) 2 (m + n) 1.2 + n I) 1.2 Hence for the first three terms the product is 1) }2 364 On the Infinite Series [567. which agrees with the first three terms of the series < (m + w). Any number of coefficients in the product could be cal- culated and the law would be found to be followed as far as the calculation went. The universality of the result can be proved by the following considerations. 567. When m and n are any positive integers whatever the two series (j)(m) and $(n) are equivalent to (!+#)"* and (l+#) n respectively (Art. 548). Hence their product is equivalent to (l + #) m x (1 +#)" or (l+#) TO+n , which again (Art. 548) is equivalent to the series $(m + n). Thus, when m and n are any positive integers (f> (m) x $ (n) = $ (m + n). But the forms of (f)(m) and <$>(n) are the same whether m and n be positive integers or not ; hence the coefficients of all the terms in the product must be of the same form, regarded as functions of m and n, whatever values these letters may have (Art. 565). Hence, since the product assumes the form of the series (fr(m + n), when m and n are positive integers it must be always of the same form ; that is, for all values of m and n *0)x (n) must be expressible in the form k n where k is some quantity independent of n. In the par- ticular case before us it is evident that, since when n is a positive integer, the value of k must be 1 + x. Another familiar instance of a function satisfying the general equation (1) is afforded by the Demoivre Function cos 6 -f i sin 0, which is shewn in treatises on Trigonometry 5 6 9 .] l + nx+~ & + .... 365 to be equivalent to z ' d , e being the base of hyperbolic logarithms which will be investigated in a future chapter, and i being a quantity such that i 2 = 1. 569. Returning to equation (l), it follows that $(m)x (n) can be represented by a series of the same form, the value of the parameter in the product being the sum of the parameters in the several factors. Thus let s series, the parameter in each of which is , be multiplied together, the product will be a similar series whose parameter is sn, or If n be any positive quantity commensurable with unity, s can be so chosen that ns is a positive integer. Let this T integer be r. Then ns = r, whence n - {* (J)}' = *W- Hence But since r is a positive integer the series $ (r) is equal to (l+#) r , (Art. 548). Hence U( -H = (l+#) r ; that is, <#>(-) is a quantity which when raised to the s ih power is equal to (l+a?) r ; that is (Art. 66), (-) can be- - s otherwise expressed as (l +x) 8 . Hence when n is a positive fraction, the series < (n) or n(nl) 1 + a? H J ; - x 2 + ... ad inf. 1 . 2 is one of the values of (l +#) n . 366 On the Infinite Series [570. 570. In Art. 279 it has been shewn that any such r expression as (1 + x)* has s different equivalents. It is obvious that if the value of the series (-) be calculable, it can only give the value of one of these equivalents. It is usual however to reverse the equation proved in the last article, and to write it (1+*)-= i+ + ^"rW..., 1 . > in which, when n is a fraction, it must be borne in mind that the meaning is that the series on the left hand is one of the values of (1 +#) n . 571. The equivalence of the last article may be ex- tended to negative values of n. For, let m be any quantity numerically greater than n. From the equation (l) of Art. 567, replacing n by ft, it follows that (n) = <})(mn). Also by the same law (m). Hence (^(m) x <$>(n) x $(ft) = (m) ; or, dividing these equals by ^(m), <( n)x (t>(n) = 1. Therefore *(-) = -J- } = ( -^ (Art. 570) = (1 +)-. Thus the equivalence of the series (#) and (1 -f #) n holds when n has a negative value given to it. 572. The series $(n) regarded in the light of its equiva- lence to (1 +#)" is frequently called the expansion of (!+#)", which may be now enunciated in the following 573-] l+~ + - 2 + .... 367 general form. If n be any number, integral or fractional, positive or negative, the series n(n I)(n 2) - -T27JT is one of the values of (!+#)". And the fact of their equivalence is known as the Binomial Theorem. 573. The equivalence which has been proved to exist between the form (l+#) n and the series (f)(n) does not always lead to any verifiable result of numerical equality. An instance of failure to do this will be afforded by the expansion of (1 x)~ l . This can be deduced from the general formula by the substitutions of x for x, and of 1 for n. The r+ 1 th term in the series will be In the numerator of the coefficient there are r negative factors ; each negative factor being taken as the product of 1 into the numerical value of the factor, the numerator becomes ( l) r . 1 .2.3 ...r, or ( l) r . \r, while ( x) r is ( l) T .x r (Art. 127). Hence the whole term becomes (-l) r \r //- (-l} r x r =(-\)* r .x r = x r . (Art. 126.) Hence the series for (1- x)~ l becomes 1 + x + x 2 + as 3 + . . . ad inf. Now in Art. 528 it has been shewn that the sum of this series is arithmetically equal to or(l x)~ l on 1 x condition that x is less than unity. 368 On the Infinite Series [574. The sum of r terms of the series can in all cases be I _ # 1 written as - j and therefore differs from by lx lx J l-x This latter quantity continually diminishes, as has been shewn (Arts. 528-530), as r increases if x be less than unity, but increases without limit with r, if x be greater than unity. Hence the equivalence given by the Bino- mial Theorem in the latter case cannot in any ordinary sense include the notion of arithmetical equality. 574. The equation . ad inf. is thus only true as an arithmetical equation when x is less than unity. The equation (1 a?)-^ \+x + a?+. ..+&-* + ~ A ~~~ X x r is true for all values of x. The term - - may be called 1 x the remainder after r terms of the expansion of (lx)~ l . In treatises on the Differential Calculus methods are given by which ' the remainder after r terms ' of the ex- pansion of any function can be ascertained. If in any case this remainder diminishes indefinitely as r increases, the expansion to infinity may be regarded as an arithmetical equivalent of the function. If the contrary be true, the equivalence of the function and its expansion must be regarded as a merely formal truth without arithmetical significance. 575. The following theorem is useful to distinguish between the cases in which the binomial expansion gives an arithmetical equality from those in which it fails to do so. 576.] i + nx+~- 2 x 2 +.... 369 If, from and after a fixed term in a series whose r ih term is denoted by u r (Art. 513), the value of the fraction - ?1 i- is always less than some quantity which is itself less than unity, then the sum of the terms of the series carried on for ever is a finite quantity. For let u p be the fixed term after which the given condition holds good, and let y be the quantity less than unity. Then (Arfc. 530), using the symbol < as an abbreviation for the words ' is less than,' Jl "p+1 whence u p+l 2r ll since ( I) 2 **- 1 = ~ 1 (Art. 126). B b 2, 372 On the Infinite Series [579. In either of these cases by giving to r the values 1,2,3 etc. in succession all the terms of the expansion can be obtained. Thus, in the last example, . x , x 2 3# 3 3.5 3.5.7 = 1 -*--3-* 4 - --* 5 - 580. The expansion of the Binomial Theorem can be used in the case of a binomial whose first term is not unity. Thus As many terms as are required can be found of the 7 fl 7 expansion of ( 1 H ) by substituting -- for x in the general formula. Multiplying the series thus obtained by 0", the expansion of (a + )" is obtained. 581. As an example of the numerical application of the Binomial Theorem to purposes of practical calculation, let it be required to compute the value of the square root of 20. This square root can be written as (25 5)*, which by the last article = (25;* (l - ^) = 5.(l - i)*- Also the expansion of ( 1 -) can be obtained from the general formula of Art. 570 by writing - for n and --- for x, or it 2 5 may be deduced from the expansion of the last article by putting for 2x the value -i and consequently the I value for x. 5 82.1 \+ nx + - i/aj+.... 373 "JO Thus, x K*_ * J * * * 5 l 7 l "5' ~ 10 "" 2 10* "~ 2 10 3 ~" 8 10 4 "~ 8 1^ 5 " The terms in this series after the first are all negative and rapidly diminish in numerical value. The sum of a small number of them will therefore differ but slightly from that of the whole series. The calculation may be conducted in the following manner: ^r= -100000 = -005000 --. - 3 = -000500 = =: X I -k = -0000625 = -00000875 10557125 The succeeding terms will not affect the first four or five places of decimals. Hence to four or five places of decimals (l- i) =-894429. And therefore (20)* = 5 (l if = 4.472145. The value is really true to four places of decimals, as can be verified by the process of Art. 397. By taking in a few more terms of the series a still closer approximation can be obtained. 582. A formula has been given (Art. 550) for the general term of the expansion of any positive integral power of any polynomial. The theorem of Art. 572 can be used to give a similar formula for the general term when the 1 10 1 1 1 1 __ ^s 2 10- " 20 10 i i Therefore X 2 ' To 3 5 1 1 1 1 8 10*"" 8 > C 2 ' 10 3 7 1 7 5 1 8 ' 10~ 5 ~ 50 > : 8 " 10* 374 On the Infinite Series [582. index is fractional or negative. The most important case is when the polynomial is itself a series, finite or infinite, arranged in ascending power of some letter #, in which case it may be denoted by a + a^ + a 2 x 2 + Then Treating - - as one quantity, thejo+ 1 th term in the expansion will be ^.(.-iM.-j+i) f (M , + y...)' t which may be written as n(n l)...(n p+1) J a Q n ~ p fax + a 2 x* + ...)*. [ Now p being a positive integer the general type of the terms in the expansion of fax + a 2 x 2 + ...) p is by Art. 550, ^ r sxY(si *)', I3 *Y ll \L li- Lg f r . "llllll-^^^' where q + r + s+ ... = p. Hence the general type of all the terms in the expansion of the polynomial will be, [ [it Li \q_ \r_ [*... 23.. 583. In the particular case of ^ being a positive integer, jo is also a positive integer. By multiplying the 584.] l+nx+ \ o '& + .... 375 idc 3 IS numerator and denominator of the coefficient by \n p the term becomes which agrees with the formula of Art. 551, with the change of (np) for p ; for since q + r + s+. t .=p, it follows that ( p) + q + r + s+ ... = #. 584. The calculation of the coefficients of particular terms or particular powers of x in the expansion can be conducted exactly in the manner indicated in Art. 552. EXAMPLES. 1. Expand to four terms (1 xfi and (1 +x)~% and find the eighth term of each. 2. Expand (1 ar 2 )^ to five terms. 3. Expand (1 + ) 4 and find the first negative coefficient. o 4. Find the first four terms and the r+ l^ term of each of the following expansions, (1-2*)*, (1-2*)-*, (I-*)-*, (I-*)"*- 5. What term of (1 x)% is equal to fg of the same term of(l+)T*1 1+x 6. Find the general term of rp (i x) (1xY 7. Find the coefficient of as* in the expansion of j- 4, 8. Find the values of >/24 and ^/ 31 each to five places of decimals. 9. Find the value of */\7 to four places of decimals. 10. Expand, to the second power of a;, 376 On tJte Infinite Series 1 -f- nx + n ^ ' x* + . . . . 11. By ordinary multiplication find the square of the series for (1 x)*, and shew that the coefficients of x 2 , a; 3 , x 4 , x 5 , ac 6 all vanish. 12. From the identity i -\-x prove that if r be any integer greater than unity the series being carried on until a term vanishes. 13. If . t (r--2) 1 2 |r-2 |^- 1 J prove that 2 {0(0) + <(!)+... 0(w-l)} + 0(0 = 3 n . 1 4. Expand, to the 4^ power of a, (1 2 ax + a 2 )" *. 15. Find the coefficient of cc 4 in the expansion of (l+a-^-Sor'-x 4 )- 3 . 16. Find the coefficient of as* in the expansion of 17. Show that the coefficient of a? 3r in the expansion of (9a 2 + 6acc+4cc 2 )- 1 is 2 3r (3a)~ 3r - 2 and that every third term vanishes. 18. Find the coefficient of a 6 in (1 + 2#-3z 2 -a; 4 )i 19. Prove that 2x(n-x}(\nY 20 Tf /"M ; VL_/ "n'^njln-xl^' prove that /(0)+/(l) + /(2)-f ... +f(n) = 2n-l 21-Provethat ^ = 1 + I + l.| + i .| .^ CHAPTER XXII. UNDETERMINED COEFFICIENTS AND RECURRING SERIES. 585. FOE the treatment of several problems connected with the series which are next to be considered the following proposition, Usually known as the principle of indeterminate or undetermined coefficients, is useful. If two expressions of the n ih degree in #, of the forms AQ + A I X + A^x z + ... + A n x n and a + a 1 x + a 2 x 2 + ...-f n #", have equal values for more than n different values of x they must also be equal for all values of #, and the coefficients of the one expression must be severally equal to those of the other, namely, A = a , A^ = a lt ... A n =: a*. From the given conditions it follows that (l) must have the value zero for at least n -f- 1 different values of a?. Let these be a l5 a 2 , a 3 , ... a n , a n+1 . Then arranging the expression (l) in descending powers of a?, it becomes (A n - a n ) x 4- K-! - a^) x-* + ... ... + (^ 1 -a 1 )aj + J -ff ; (2) and since this expression vanishes when x is a l it follows (Art. 163) that x a x is a factor of the expression. Simi- larly x a 2 , . . . , x a n are factors of the expression ; and this latter being of n dimensions only in x cannot contain any more factors involving x. 378 Undetermined Coefficients [585. Hence the expression must identically equal (4, - ,) (x -a,) (x - a 2 ) . . . (0 - a,). But, by hypothesis, this expression also vanishes when x = a n+1 , which is impossible unless A n a n vanishes, since a n+1 is not equal to any of the quantities a lf o 2 , ... o n . Hence A n must equal a n , and thus the expression (2) becomes really only of the n 1 th degree. Applying- the same reasoning to this, it follows that A n _ 1 = # n _ l5 and so on, until we arrive at A Q = . Thus the coefficients of the one series are severally equal to those of the other, and the two expressions are identical with one another for all values of x. 586. If the two expressions be infinite series, n is indefinitely large, and n + 1 , the number of values for which the expressions are given equal becomes also infinite. It is usual to assume, as a limiting consequence of the last article, that if two infinite series, A Q + A l x + A 2 x 2 + ... ad inf. and a + a 1 x + a 2 x 2 -f- ... ad. inf., be equal in value for an infinite number of values of #, then A Q = -P a i where b and c can be 1 px qar chosen so as to give any required values of a and a v . The sum of any such series to infinity can be therefore always represented by such a fraction as the above. The method of undetermined coefficients can be employed however to give expressions in a general form for the value of a n and also for the sum of n terms of the aeries. 591. A fraction whose denominator consists of any number of factors of the form 1 ##, \bx, I ex, and whose numerator is of lower degree than the denominator, can be replaced by a number of fractions whose denominators 382 Undetermined Coefficients [591. are the factors of the given denominator and whose numerators are de terminable constants. Let the fraction be p + qx + rx 2 ^ (I ax) (I bx) (I ex)' then constants A, B> C can be found such that this fraction shall be equal to the sum of the fractions ABC I ax \ bx I ex For the Litter sum can be expressed as a fraction with the denominator (1 ax) (lbx) (lcx) and with an ex- pression of the second degree in x for its numerator, namely If, by properly choosing the values of A, B, C, this latter expression can be made equal in value to p 4- qx + rx 2 for three values of x, it must (Art. 585) be equal to it for all values. Let the three values of x chosen be - > T j - Putting j a b c - for x in the two expressions and equating their values, the terms involving B and C disappear, as they involve a factor zero (Art. 51) ; and there results Similarly the equality of the two expressions for the values -7 and - of a? is ensured by taking - (c-a)(c-by 593.] and Recurring Series. 383 With these values of A, B, Cthe sum of the three fractions will therefore be equal to the given fraction. 592. The process of the preceding article is known as that of the resolution of a fraction into its partial fractions. For fuller developments the student is referred to treatises on the Integral Calculus. It will be now used to investigate a general formula for the coefficient of the n ih term in a recurring series. It may be noticed that it has been assumed that the numerator of the given fraction is of lower degree than the denominator. If this be not the case the denominator can be divided into the numerator, giving a formally integral quotient and a remainder of lower degree than the de- nominator (Art. 158). 593. A recurring series + !# + 2 # 2 + ... + a n x n + ..., where the relation between successive coefficients is a n -pa n . l qa n _ 2 = 0, can, as we have seen (Art. 590), be produced by the expansion of a fraction of the form 1 px qx 2 The values of b and c determine the first two coefficients of the series and the general relation determines the succeed- ing ones. If the factors of 1jpx qa? be 1 ax and 1 /3#, T\ !_ st/Y* the fraction - ^ can be replaced by the sum of two 1 px qx* A 7? fractions - - + - ? where A and B are constants to 1 ax I fix be determined as in Art. 591. But if these two fractions be separately expanded in series, they give (Art. 588), 384 Undetermined Coefficients [593. But this series is identically equal to a Hence (Art. 587), a Q = A + B, The last result gives a general form for a n . The values of a and (3 are the two roots of the quadratic equation in t, fipt q = 0, for (1 ax) (I fix) = \pxqx z by assumption. And writing - for #, it follows that 0-7)0-!)='- l-fr or, multiplying- hoth sides by ^ 2 , (t-a)(t-{t),= t*-ft-t. The values of A and B can be determined from those of the first two coefficients , a l , since Hence in any recurring series whose first two terms are given, as well as the scale of relation between successive coefficients, an expression can be obtained for the n ih term. 594. A recurring series in which four successive coefficients are connected by a relation of the form a n -pa n _ l -qa n _ 2 -ra n _ 3 = 0, can similarly be derived from the expansion of a fraction 1 px qx* rx z and the value of a n will be of the form Aa n + B (3 where a, /3, y are the roots of the cubic equation 596.] and Recurring Series. 385 The values of A, JB, C will depend on those of the first three terms of the series. 595. The sum of the first n terms of a given recurring series can be obtained when the value of a n is known. Let the relation between coefficients be n-^-l-^n- 2 = > 0) and let S n be the sum required. Then Thus px S n = pa^x Adding these different results, it follows that S n (1 -px - qx 2 ) = a + (# ! -pa ) x since the coefficients of a? 2 , a? 3 . . . , x n ~ l in the sum of the three expressions on the right-hand side vanish in virtue of the general relation (1). Hence n 9 1 px qx 2 If a? be a quantity less than unity, so that x n and # n+1 diminish indefinitely when n is increased indefinitely (Art. 530), the series can be summed to infinity, and we O _ a result agreeing with the assumption of Art. 590. 596. It may be observed that a geometric series is merely a particular case of a recurring series, the relation being a n -pa n _^ = 0. The w th term, a^as*- 1 , is therefore a Q p n ~ l x n ~ l , and the c c 386 Undetermined Coefficients [597. /7 Tin ff& ft (1 ^_ & T^\ sum of n terms is - !=i or -^- - ' > results with 1 px 1 /?# which the student may compare those of Arts. 521, 522. 697. The expansion of Art. 588 can be used to determine the sum, or at any rate the number, of all the homogeneous products of any dimensions (Art. 88) of any number of letters 0, b, c, . . . . For = I ax l-bx 1 1 ex Hence On the right-hand side the index of x in any term on the product will evidently be equal to the sum of the indices of the letters 0, #, or (1 #)~ r , 7 , w r (1 ax) (I bx)(l ex}... if there be r of the letters 0, #, c. . . . Hence the number required is the coefficient of x n in the expansion of (1 x)~ r . But by the formula of Art. 572 the n -f 1 th term in this expansion is (- f )f-r-l)...(-r-+l) b which reduces, as in Art. 578, to t And since ( l) 2n is always + 1, the coefficient of x* iix the expansion is which is the number of homogeneous products required. c c 388 Undetermined Coefficients and Recurring Series. EXAMPLES. 1. Resolve into partial fractions - " 1 3x+2x 2 2. Expand to five terms ^ by the method of 1 3 x -j- ^33 Art. 589, and also by the help of the last example and the binomial theorem. Find the ri^ term of the series. 3. Resolve into partial fractions -)(1 -2*) (1-3*) 4. Resolve into partial fractions - - = 1 x 3 5. Find the n th term and the sum of n terms of the re- curring series 3 + 7a+ 15x z + 31x 3 + 63a; 4 + ... . 6. Find the scale of relation in the series and hence sum the series to n terms. 7. Find the scale of relation in the series and sum the series to n terms. Deduce the sum of the squares of the numbers 1, 2, 3,...w. CHAPTER XXIII. SUMMATION OF SOME SPECIAL SERIES. 599. IF the general, or ft th , term be represented by n , and the sum of the first n terms by S n , both u n and S n , will be functions of n (Art. 165). Also, u : being the first term, Therefore -#-! = . (a) Hence if w n can in any given case be resolved into the difference of two terms one of which is the same function of n that the other is of n 1 , that is to say, one of which involves n in the same manner as the other does l, the former term will be a possible value for S n . 600. There are two large classes of forms of u n for which this resolution is possible. The first of these is when u n is the product of a fixed number of factors in arithmetical progression, the first factor of each term being the same as the second of the previous term. The general form of u n will thus be (an + l) {a (n+l) + b} {a (n + 2) + b}...{a (n + m-l) + b], where m is the fixed number of factors, a is the common difference of successive factors, and b is a constant. The term may be denoted by the symbol n> m , the letter m being introduced to mark the number of factors. 390 Summation of some Special Series. [600. The symbol #, m+1 will consequently denote the product of m 4- 1 factors of which the first is an + b, or m+l ...(a .n + m 1 -f b) (a,. (i) lin w n _ 1 ^2 - g ' and as in Art. 606 Again, if ^ = 0, as before, and A , ^3 , ^ 2 , ^ are determined by the equations, l = 4 J 4 , 3 = 6 3 = 4 J 4 6 jo.] Summation of some Special Series. 399 whence A = - > A = - , A = - > A l = ; = whence 1 3 + 2 3 + 3 3 + ...+ 3 = *-+ + ! the result previously obtained. 610. Any series of which the general term is a rational integral algebraical function of n can be summed by the method of Art. 607, or the sum required may be deduced by separating the given series into a number of series one for each power of n contained in the general term. Thus if the general term of a series be n 2 + 3n+ 2, the series itself to n terms is {l2 H _3. 1 + 2} + {2 2 + 3.2 + 2} + ...+ {tt 2 +3tt + 2}, which can be otherwise written as + 2 (1 + 1 + 1... to n terms) ; which therefore is equal to {2fl 2 + 1272 + 22} 400 Summation of some Special Series. [6ro. This can be written in the form and in this form the result will be recognised as a particular case of Art. 601, since the given general term n 2 + 3 n + 2 can be also written as (n+ l) (n + 2). A series whose general term is a rational, but fractional function of n can sometimes be summed by special artifices similar to that of Arts. 603 or 604. Thus, if the general term be . - -r-, - 7 - N , the ' series can be summed by taking the general term as the sum of two terms, 2 _ 3 _ + n(n+l)(n+2)(n+3) ' and applying to each of the terms the result of Art. 603. The general term can also be resolved into either of the two forms 1 1 1C1 1 1 1 and the sum can be obtained from these forms by the principle of Art. 599. 611. Series are sometimes given for summation of which the law of formation is not obvious, and the general term of which requires some skill for its discovery. For instance, let the given series be 2, 7, 14, 25, 44, 79,.... (l) By subtracting each term from its successor a new series is obtained, 5, 7, 11, 19, 35,..., (2) 6 1 1.] Summation of some Special Series. 40 1 of which the law is still not evident. Repeating the process, a third series is found, 2, 4, 8, 16,..., (3) which is evidently a geometrical series of which the n ih term is 2 n . The n ih term of (2) is evidently obtained by taking the sum of n 1 terms of (3) and adding 5, the first term of (2). Hence the n ih term of (2) is 5 + 2 n 2 or 2 n +3. The n ih term of (1) is similarly obtained by adding the sum of (# 1) terms of (2) to 2, the first term of (l), and is therefore 2 + (2 n -2) + 3(- l) or 2 M + 3(rc-l). The sum of n terms of (l) can now be obtained by the help of Arts. 506 and 522. The series (2) is called the series of first differences of (l) ; and (3) the series of second differences. In all cases where the n ib term of the original series can be expressed algebrai- cally as an integral function of n, a series of differences will be at last reached which assumes the form either of a geometrical or arithmetical progression. EXAMPLES. 1. Sum to n terms 2.3 + 3.4 + 4.5+.... 2. Sum to terms 3.5.7+5.7.9 + .... 3. Sum to n terms 2.2.3 + 4.3.4 + 6.4.5 + 8.5.6+ 4. Sum tor, terms - +- +- + .... 5. Sum to n terms 402 Summation of some Special Series. 6. Sum to n terms + ~^ + = + 1.3 3.5 o.7 7. Sum to w terms ^ ^ + - I -_ i + _ r ^ + .... 8. Find the w*k terms and the sum to n terms of 1.2 + 2.3 + 4.5 + 7.8 + 11.12 + .... 9. Sum to n terms w + 2 (n 1) + 3 (n 2)+ ... . 1 0. Find the w th term and the sum of n terms of the series 2 + 3 + 5 + 9 + 17 + 33+.... 11. If every term of a series be subtracted from the following one, find the nature of the resulting series (1) when the original one is a geometrical series, (2) when its rfo- term Find the w^ 1 term and the sum of n terms of the series 6, 16, 34, 68, 142, 328, 842,.... 1 2. Find the rift- term, and the sum to n terms, of the series 1, 2, 4, 5, 7, 8, 10, 11. 1 3. Find the nfa term and the sum of n terms of the series 4, 6, 13, 18, 31, 48, 85,.... 14. Sum tow terms + ^-+ + .... 15. Sum to n terms 2.3 + 4.5 + 6.7+.... 16. Sum ton terms 1.2.3.4 + 2.3.4.5+.... 17. Sum tow terms ... 18. Sum tow terms -^+ ^__+ -^L^ + ... . 19. If Sj be the sum of w terms of the arithmetical pro- gression a, a + b, a + 2 b, . . . , and s 2 that of w terms of c, c + d, c + '2d, ..., then, if 8 be the sum of w terms formed by mul- Summation of some Special Series. 403 tiplying together the corresponding terms in each of the former two, prove that 20. Find the nfl* term and the sum of n terms of the series 2 5 8 ~l~ r A ft* ' A t~ f* "l * * * * 2.3.4 3.4.5 ' 4.5.6 21. A person discharges a debt in n years by paying at the end of each year sums in the ratio of 1, 2, 3, ... ; what was the first payment ? D d 2 CHAPTER XXIV. EXPONENTIAL AND LOGARITHMIC SERIES. 612. THE word 'limit' has been introduced in Art. 608. As it will be necessary to make some use of this term in the following- articles it will be as well to repeat more formally the definition. The limit of a function (Art. 165) of any variable quantity, for any assigned value of that variable, is a quantity towards which the value of the function con- tinually approaches as the variable approaches its assigned value and from which the value of the function may be made to differ by less than any definite magnitude, by making the variable approach sufficiently near to its assigned value. 613. Thus when r is less than unity the limit of the sum of the geometrical series a + ar-{-ar 2 + ... to n terms is when n is indefinitely increased (Arts. 528, 529), p -\-n and the limit of - p '^ - when n is indefinitely in- creased is - (Art. 608). p+l V 614. We have now to investigate the value of a very 1 n important limit, namely, that of (l + -) when n is in- definitely increased. By the Binomial Theorem, (-l)...(-r+l) ,1 y If HK Exponential and Logarithmic Series. 405 This is true whether n be integral or fractional. In the latter case the series on the right-hand side contains an infinite number of terms, but if n be greater than unity, and consequently - less than unity, the series is % convergent (Art. 576), and the equation therefore implies an arithmetical equivalence for all large values of n. The series on the right-hand side may be somewhat differently written, thus 1.2 (j-iUl-?)...^-?^- 1 ) n' v n' ^ n ' ___ . + ... ? ( a ) the r + 1 th term being reduced to this new form by dividing each of the r factors in the numerator of its coefficient by one of the factors n in the denominator, n r . The series on the right-hand side of (a) is always less than the series Further, the right-hand member of (a) approaches nearer to (/3) the larger n becomes ; and by making n sufficiently large the difference between the right-hand member of (a) and the series (/3) can be made less than any assignable quantity. Hence the limit of (l + -) when n is in- definitely increased is the series (/3). 615. The numerical value of the series (/3) is usually denoted by the letter e. It follows from Art. 576 that () must be convergent, since the right-hand member of (a) 406 Exponential and Logarithmic Series. [616. is so for all values of n greater than unity ; or the follow- ing- special proof may be given since each term, after the third, of the first series, is less than the corresponding term of the second, and the first three terms are identical. Hence the series (0) <1 + T (Art. 528), 55 L-L m = -0000003 10 2-7182819 617.] Exponential and Logarithmic Series. 407 The last figures in the several rows have been increased by unity when the first figure omitted is greater than 4. The sum of the remaining terms of the series 1 1 JTT ' h jliz + '" is less than 1(11 ) TTTr h iT H ~TT* + -j or 36288000 <-00000003, so that the above result is probably accurate to seven places, certainly to six places, of decimals. 617. Since the limit of ( 1 + -) when n is indefinitely n' j * increased is e, it follows that the limit of ( 1 + -) , which r J n^x n' is equal to if 1 H ) > , is e x under the same conditions. But, by the Binomial Theorem, [3 " V " nx(nxl) ... (nx ~~ x(x -- ) x(x -- ) (x -- ) ^ _ !L + J: _ 21: _ ^-i-... 1.2 [3 / IN / 2x / r K l -- 1 I* -- )...(# -- -I v #/ v ^/ ^ / - ;+ - IE - 408 Exponential and Logarithmic Series. [618. And, as before, the limit of ^he series on the right-hand side, when n is indefinitely increased, is ar # 3 # r 1+ * + ^ + l + - + t + - Hence, finally, X 2 X* X r , . - ,* =1+ * +| ^+__ + . .. + _+... ad inf. This equation is frequently known as the exponential theorem. 618. A somewhat more general relation can be deduced from the theorem of the last article. Writing ex for x, it becomes ... Hence, if e c = a, (1) this gives c 2 x 2 c*x* c r x r fl . = l +caM ._ + _ + ...+ _+.... (2) This relation gives an expansion of a x in ascending powers of x, the only new quantity involved being the quantity c, which is connected with the given quantity a and the known number e by the relation (l). This quantity c is of so much importance for many pur- poses, both theoretical and practical, that it has a name given to it. It is called the logarithm of a to the base e. 619. The general definition of a logarithm may be thus given. The logarithm of a number to a given base is the index of the power to which the base must be raised to be equal to the number. Thus in (l), e c = a. Hence c is the index of the power (Art. 62) to which e, the base, must be raised in order 62i.] Exponential and Logarithmic Series. 409 that the result may be a ; ^ c is therefore the logarithm of a to the base e, or, as it is concisely written, c = \og e a. In the general way if a x = m, x is the logarithm of m to the base , or x log a m. The two equations a x = m, x = Iog ^, must be regarded as merely two different ways of stating one relation between the three quantities a, m, x. 620. The equation (2) of Art. 618 can now be written II II 621. The relation of the last article can be obtained by another method which leads to some other important results. By the Binomial theorem, The coefficients of the successive powers of (a 1) in- volve different powers of x. The whole expansion can therefore be arranged in powers of x. The only term in- dependent of x is the first term, while the first power of x occurs in every succeeding term. The coefficient of this first power is thus an infinite series, the successive terms of which involve successively higher powers of (a 1). The term of this series arising from the r+l th term o ( W>* (-!)... (-r+l) ~\L~ is <-i)(-')-(-+ y iy or (-^(-'r. \r r 410 Exponential and Logarithmic Series. [621. Hence the coefficient of a? in (1) when rearranged is ..... Let this be called c, and let the coefficients of # 2 , # 3 , . . . be denoted by c 2 , c 3 ,..., these quantities being functions, at present undetermined, of a and independent of tf. Hence a x = 1 -f ex + c It follows that and that since c, which makes ex equal to unity, we get This is the quantity previously called e. Hence i a = *, or a = e c , whence c = log e #, and equation (3) becomes a* = the relation of Art. 620. 622. Since c is equal to the expression (2) of the last article, it follows that or replacing a by 1 +z, log e (l+,) = ,-i^+Ie3_... + (_l)-^+... > ( a ) a series from which the value of log e (l +z) may in some cases be calculated. 623. The series for c is only certainly convergent when a 1 is less than unity, or a is less than 2. Thus the proof of Art. 621 is strictly only applicable when a is less than 2. The proof for cases when a is equal to or greater than 2 can be easily deduced. For if a be any number and A any number less than 2 and greater than unity, it is always possible to find a number y such that 412 Exponential and Logarithmic Series. [623. Hence a* = A yx , and since the theorem of Art. 621 has been proved for all numbers less than 2, Now if e c = A, c = \og e A. Also e cy = A v , and cy = log e A v , Hence a* = A*= I+(y\og e A)x+ + ... a? 2 (log- a) 2 = l + alog e a+- v *' ' + ..., which proves the theorem for all values of a. The proof contained in Arts. 614-620 is independent of any limitation to the value of a. 624. Other series can be derived from (a) of Art. 622 which are often more convenient for purposes of calculation. Before entering on the discussion of these, it is necessary to examine more fully the nature and properties of a logarithm to any base. Let m and n be any two numbers, and a any base. Let x and y be such numbers as to satisfy the equations a x = m, a v = n, so that, by definition (Art. 619), x = log a m, y = log a n. By multiplication mn = a x . a v = a x+v . Hence by the definition of a logarithm (Art. 619), \og a (?nn) = x+y It easily follows that log a (mnp) = \og a (mn.p) = loga (*) + log a (p) = log a m + log a n + log a p. 629 ] Exponential and Logarithmic Series. 413 Thus it follows that the logarithm of a product is equal to the sum of the logarithms of the factors. 625. Similarly by division whence by definition . m Joga - = -y> 71 =: Iog a ^-log tt, (2) or the logarithm of a quotient is equal to the logarithm of the dividend diminished by that of the divisor. 626. Again, with the same notation, if m = a x , it follows that m p = a px ; whence again by definition (Art. 619), log P = P, = p\o<* a m. (3) This formula holds whether p be integral or fractional. 627. The formulae (1), (2), (3) of the last three articles are those on which the practical utility of logarithms depends. If tables be calculated giving the logarithms of all numbers to any base these will enable the operations of division and multiplication to be performed by means of addition and subtraction, and formula (3) will give the means of raising to powers or taking roots by means of multiplication or division. 628. The base to which tables of logarithms are usually calculated is 10, the radix of the ordinary arithmetical notation. This has some important practical advantages in simplifying both the construction and the use of such tables. 629. With this number as base, the only numbers whose logarithms will be integers are 10, 10 2 , 10 3 , 10 4 ,.,., or 414 Exponential and Logarithmic Series. [629. 10, 100, 1000, 10000,.... The logarithms of all other numbers will be fractions or mixed numbers. Thus, since the number 7834 is greater than 1000 and less than 10,000 its logarithm must lie between the logarithms of these two numbers, that is between 3 and 4. The integral part of its logarithm is consequently 3. The integral part <9/*the logarithm of a number is called the characteristic of the logarithm : the fractional part when expressed as a decimal, is called the mantissa of the logarithm. Thus the characteristic of the logarithm of 7834 to the base 10 is 3. Since any number containing p + 1 digits in its integral part is greater than 1 O p and is less than 10 P+1 , it follows that the logarithm of any such number lies between p and p+1. Hence the characteristic of the logarithm to the base 10 of any number containing p+ 1 digits in its integral part is p, one less than the number of digits. The use of the base 1 thus enables the characteristic of the logarithm of any number greater than unity to be determined by inspection. 630. The logarithm of any number to the base 10 being ascertained, those of all numbers obtained from the first by merely changing the position of the decimal point can be written down by the help of Arts. 624, 625. Thus the logarithm of 52185 to the base 10 is 4-7175457. The numbers 5218-5, 521-85, 52-185, 5-2185, -52185, -052185... are obtained from 52185 by dividing successively by 10. Hence the logarithms of these numbers will be derived from 4.7175457 by subtracting unity, the logarithm of 10, and will all therefore have the same decimal part, namely 7175457, and for their characteristics the integers 3,2,1,0, -1, -2. Thus, since any two numbers which are formed by the 632.] Exponential and Logarithmic Series. 415 same digits in the same order, and only differ in the position of the decimal point, can be obtained from each other by multiplying' or dividing by some integral power of 10, it follows (Arts. 624, 625) that the logarithms of any two such numbers to the base 10 must differ by some integer and must therefore have the same decimal part or mantissa. 631. In tables of logarithms to the base 10 it is only therefore necessary to give the mantissa of the logarithm corresponding to any set of digits for the number. The characteristic can be determined by the rule of Art. 629 when the number is greater than unity. When the number is a proper fraction expressed as a decimal, as '52185, the logarithm is a negative number. In the last article it has been seen that this logarithm is expressible as 1+-7175457, which is more usually written 1-7175457, the sign , written over the characteristic, indicating that it affects the characteristic only, and that the mantissa is a positive quantity. The logarithm of -052185 is similarly 2-7175457, and the general rule is easily deduced that, the characteristic of the logarithm of a decimal fraction is negative and numeri- cally greater by unity than the numbers of ciphers before the first significant digit. The convention that the logarithms of a proper fraction shall be always written in the form of a negative integer with a positive fraction makes the rule of Art 630 abso- lutely general. 632. The last three articles exhibit the great advantage of 1 as a base for practical purposes. Most of the published tables of logarithms contain full directions for their use, and to these the student is referred for further exemplification of details. One of the most convenient collections of tables is the volume of ' Mathematical Tables' in Chambers 7 416 Exponential and Logarithmic Scries. [633. Educational Course, published by W. and R. Chambers, Edinburgh. 633. The fundamental series for the calculation of logarithms is (a) of Art. 622. Other more advantageous series will be deduced presently : but as the logarithms found from these series will all correspond to the base , it is necessary to find a means of deriving the logarithms to the base 10. 634. Let a and I be two different bases, and let x and y be the logarithms of the same number N, corresponding to these two bases respectively. Thus ,. Consequently, by the definition of a logarithm (Art. 619), a x = N, i* = N t X Hence* a x = b y , and therefore ai* = I, y_ and b* = a. Hence, by definition of a logarithm, (2) Thus x = y log a #, or, log a ^= Iog b ^xlog a 3 by(l). (3) Similarly, y x Iog 6 a, or, log 6 -#= Iog a ^xlog 6 fl. (4) Incidentally it follows from (2) that log a xlog b a = 1. (5) 635. As a particular example of (3) or (4) we have whence Iog 10 N = j x log e N. 638.] Exponential and Logarithmic Series. 417 Hence if the logarithms of all integers, including 10, to the base tf, be calculated, those to the base 10 can be deduced by multiplying by -. * to 636. This multiplier, -, , is often called the modulus of the system of logarithms to the base 10. 637. The series (a) of Art. 622 is log e (l+*) = *-^ 2 + ^ 3 -.... (a) If for z we write z, the formula becomes Iog.(l-*)=-*-^-J*"-..., (ft) since the signs of the odd powers only of z are changed (Art. 126). Hence, subtracting (/3) from (a), log. a +)-!<*. a -*) = But by Art. 625 the left-hand side of this equation is 1+2 equal to log e Hence, log e ^ = 2 j* + i^ + i^ + .. .J. ( y ) The series inside the bracket will be rapidly convergent if z is small, and thus, if log e (1 z) be known, log e (1 +z) can be easily computed. 638. Let a quantity n be taken so as to satisfy the equation n+l l+z n ~ I-/ Whence, by solution as a simple equation with regard to z> it easily follows that 1 z = 6 418 Exponential and Logarithmic Scries. [639. Hence (y) gives, ,.2*4- whence From this series the logarithms of different integers can be successively calculated. 639. Since # = 1 (Arts. 70, 246) whatever a may be, it follows that the logarithm of unity to any base is zero. Hence if in (6) of the last article we put n equal to unity, - The calculation to a few places of decimals can be easily effected. Thus, -=333333333; .'. - = -333333333; 3 - 3 dividing by 9, =-037037037; - (-) =-012345679 3 3 ^3 - =-004115226; - (-) =-000823045 3 5 * 3 7 = -000457247; - (-) =-000065321 3 7 ^3 9 = -000050805 ; - (-) = -000005645 -^ = 000005645; (-) =-000000513 -^ = -000000627; (-) =-000000048 1 1 /1\ 15 -^ = -000000069 ; (-) =-000000005 .346573589 2 693147178 640.] Exponential and Logarithmic Series. 419 Hence with probable accuracy to eight places of decimals, log,. 2 = -693147178. Hence, since log 4 = 2 (log e 2) (Art. 626), it follows that log^ = 1-386294356. 64O. Again, putting 4 for n in (5) of Art. 638, than !1 1 /1\ 1 /1\ I h ( ) + - ( ) + ... The series in the bracket is more rapidly convergent than the former one, and the calculation can be conducted in the following way. Thus, =-111111111; =-111111111 y y p= -012345679; 1 11** = -001371742 ; -(-) = -000457247 y o y i= -000152416; 1 II 5 _ = -000016935 ; -(-) = -000003387 i= -000001882; = -000000209; - (-) = -000000030 -g = -000000023; ^ = -000000003 ; 111571775 2 223143550 log e 4 = 1-386294356 Hence log e 5 = 1-609437906 Again, Iog tf 2= .693147178 2-302585084 420 Exponential and Logarithmic Series. [641. But (Art. 624) log e 1 = log e 2 + log a 5. Hence Iog fl 10 = 2-302585084. 641. By Art, 635, Iog e 2 _ -693147178 gl ~ log a 1 ~~ 2-302585084 ' and effecting the division there results Iog 10 2 = -30102999...; or, to seven places of decimals, Iog 10 2 = -3010300. The logarithm of 2 being determined, that of 3 can be deduced by (8) of Art. 638, and so on for all succeeding integers. It will not however be necessary to calculate independently the logarithms of any but prime numbers since the logarithm of a number composed of two or more factors can be deduced from those of the factors by Art. 624. 642. The first figure in the value of the logarithm of 2 to the base 10 can be deduced by means of simple Arith- metic. Thus, 2 10 = 1024, and therefore 2 10 is very slightly greater than 10 3 . Thus 2 is very slightly greater than 10 1 ", and Iog 10 2 exceeds -3 by a very small amount. 643. It has been seen that for important practical reasons it is desirable always to keep the mantissa of a logarithm a positive quantity. Logarithms of proper fractions are thus always the algebraical sum of a nega- tive integer and a positive fraction. In any case where such a logarithm has to be divided by an integer it is convenient to add to both negative and positive part the smallest number which will make the negative part exactly divisible by the divisor. Thus the quotient comes in the desired form of a negative integer and a positive fraction. 643.] Exponential and Logarithmic Series. 421 For instance, let it be required to divide the logarithm 2-8450980 by 3. The process adopted is really the fol- lowing : i(2-8450980) = ^(-2 + -8450980) 3 3 1-8450980) = -1+-6150327 = 1-6150327. EXAMPLES. 1. Find the logarithm of 32 to the base 2. 2. Find the logarithm of 625 to the base 5. 3. Find the characteristics of Iog 5 497 and Iog 7 400. 4. Given Iog 10 2 = -3010300 and Iog 10 3 = -4771213 : find the logarithms of 6, 8, 9, 24 and 36. 5. Given Iog 10 5 = -6989700 : find Iog 10 Vl . 25. 6. Given Iog 10 12 = 1-0791812 and Iog 10 15 = 1-1760913 : find the values of Iog 10 2, Iog 10 3, and Iog 10 5. 7. From the logarithms given in question 4, find the logarithms of A/2, and ^-0125. 8. Given Iog 10 7 = -8450980, log J0 2 = -3010300: find the 64 A/1715 9. Find the values of the logarithm of 125 to the base 25, and of the logarithm of 343 to the base 7*. 10. From the logarithms in question 8, find the value of 1 1 . Given a = 7, 6 = 6, c = 5, s = J (a + b + c), and the logarithms in question (4) : find the logarithms of values of Iog 10 7 = =,log 9-8, log 350. S ( S -a)(s-b)(s-c) and 422 Exponential and Logarithmic Series. 12. Shew that 13. Sum the series - -f - - -f - + - + ... to infinity. 1 . 3 ^ 5 3.7 4.9 14. Prove that *!1 = * + ^ + j + '" . e 2 1 1 1 1 + ^ + +- 15. Prove that 16. Prove that g,(a+l) = 2 Iog e *-log 6 (*-!)- jl + -^ 17. If a, 6, c be consecutive integers, prove that 18. Prove that log. | = j-f-^ + J-^ + -J- 7 + ... ad. inf. 19. A person has a capital of A at interest. He spends 2 annually - the interest on A, taking the difference between his expenditure and income, at the end of each year, from his capital. At the end of five years he reduces his expenditure in the ratio of 2 to 15, and invests at the end of each year his surplus income. If at the end of 5 years more his capital is g increased on the whole by - A, find the rate per cent. Given Iog 10 13 = 1-1139434, log 1053 = 3-0224284, Iogl054= 3-0228406. 20. If y q = z p , prove that p log y a = q Iog 2 a. 21. Prove that if log a a, log^ 6, log v rb and < (r + 1) b. If c and d represent two other incommensurable quan- tities, the ratio of which is equal to that, of a to b, according to the test of Art. 647, it follows that since na > rb, nc > rd, and since na<(r+\) b, nc<(r + \)d. Hence a r r+l _ c . r r+1 T > - and < - i and - 7 is also > - and < - on n a n n 428 Ratio, Proportion^ and Variation. [649. This is true whatever be the magnitude of n. Thus whatever meaning le attached to the forms 7- and , they must represent quantities whose difference is less than - however large n may be, that is, they must represent equal quantities. The equation a_ _ c b~~d is therefore in all cases a necessary and sufficient condition for the equality of the ratios of a to b and c to d. The fractional form T is thus a suitable measure of the b ratio of the two quantities represented by a and b, whether commensurable or incommensurable. 650. The equality of two ratios is termed proportion, and the four quantities involved are called proportionals. The ratio of a to b is often denoted by the symbol a : b. Thus the equality of the two ratios of a to b and of c to d may be denoted a : b = c:d. In this particular case the symbol = for equality is often replaced by the equivalent symbol : : ; the propor- tion may be also written a : b : : c : d. 651. A number of consequences of a simple proportion are deduced by Euclid from the definition quoted in Art. 647. They follow easily from the equation For multiplying each of these equal quantities by Id it follows that ad = ^ 655.] Ratio, Proportion, and Variation. 429 Again, dividing each, of these by cd, we get a 6 or a : c : : I : d ; (3) that is, , d, c, d are also proportionals when taken al- ternately 652. Again, dividing the equals in (2) by ac, d_b_ c ~ a' or d : c : : I : a ; (4) that is, b d or a + b:b::c + d:d. (6) This is quoted by Euclid as ' componendo.' a+b c+d 655. Since and also d ab cd 430 Ratio, Proportion, and Variation. [656. dividing 1 the equal quantities in the first equation by those in the second, it follows that a + b __ c + d a b "~ c d' or a + b: a b : : c + d:c d. (7) 656. Dividing each of the equals in (2) by , we get .S. a Hence, if four quantities be proportional the fourth is equal to the product of the second and third divided by the first. This is the basis of the Single Rule of Three in Arithmetic. 657. The ratio compounded of two or more given ratios is defined to be the algebraical product of the fraction which represent the ratios. Thus, if j , -j represent two ratios, the etc ratio compounded of these is represented by the fraction =-j Euclid's definition leads to this same result. Let a, b be any two quantities ; then x being any third quantity, a fourth quantity y can be obtained such that a:b::x:y. (l) Similarly from the quantities c, d, y another quantity z can be found such that c: d :y : z. (2) Euclid defines the ratio of x to z to be that compounded of the ratios of a to b and c to d. From (1) it follows that a x 1 ~~ y - . z Similarly -, d Therefore j x ^ = - x ^ b d y z ac x or YJ - Id z 659-] Ratio > Proportion, and Variation. 431 658. If the two ratios be equal, the ratio compounded of them is called the duplicate of either ratio. Thus a 2 : b 2 is the duplicate of the ratio a : b. If c be a quantity such that a : b : : b : c y in which case c is called a third proportional to a and I and b a mean proportional between a and c, a b b = ~c' a a a b a 2 a therefore r x T = T x ~ J or 7^ = o o b c cr c Hence the ratio of a to c is the duplicate of that of a to b. Similarly the ratio compounded of three equal ratios is called the triplicate of the given ratio. If the ratio x to y be the duplicate of the ratio a to b ; the latter ratio, a to , is called the subduplicate of the ratio x toy. 659. If any number of ratios a : b, c : d, e :f t . . . be equal, each of them is equal to the ratio of Let k denote the value of each of the equal ratios, ,, a , c e 7 then ? = *, 3:=*, ^ -*...-. Hence a = bk> c = dk, e = f /,...; and a + c + e+... = With the same conditions it follows that p> q, r, . . . being- any multipliers, each of the ratios is equal to that of 432 Ratio, Proportion, and Variation. [659. For, as before, a = Ik, c = dk, e = fk, ____ Whence pa pbk, qc = qdk, re = rfk,...-, and pa + qc + re+... = pa + qc + re + . . . _;___ _ . > - = -- A still more general result can be deduced. For, since a = Ik, c = dk, e = //,... , it follows that n = n /fc n , c n = d n k n , e n =f n k n ,..., and j n =pb n k n , qc n = qd n k n , re n = rf n k n ,... ; whence a n + c n + re* + ... = b n k* + d n k n + r n & n +... whence ... _-,____ / QX -~~~ 660. When one quantity depends on another in such a manner that if the value of the second be changed, that of the first is changed in the same proportion, that is, so that the two values of the first quantity and the two values of the second form a proportion, the first quantity is said to vary as the second. Thus the wages of a man will change with the time during which he works, so that the wages for two different numbers of days will have the same ratio as the corre- sponding numbers of days. The wages are then said to vary as the number of days he works. 661. If A denote the value of one quantity correspond- ing to B the value of another, the fact that the first quantity varies as the second is often denoted by the symbol a, and thus A&.>B. 663.] Ratio > Proportion, and Variation. 433 This appears to express a proportion by means of only two quantities. The statement really however involves four, namely, two values of the second, and two correspond- ing* values of the first quantity, and is equivalent to the proportion where B , B 2 are any two values of the second quantity, and A^ , A 2 the two corresponding- values of the first. 662. Thus if A l , B 1 be any given pair of corresponding values of the two quantities and A, B represent any cor- responding values whatever, if the first quantity A vary as B, we have = .. 9\ ^ But is a definite given quantity and does not change "i when. B, and consequently A, is changed. Thus, denoting - by a symbol c, the relation B\ Ax B may be replaced by A = cB, where c does not change with A and B. Thus if W denote the wages of a man for D days, we know that Wx 2), which may be replaced by W = cD, where c is evidently the wages of a man for one day. 663. One quantity is said to vary as a second inversely when, if the second is changed, the first is increased in the same ratio as the second is decreased, and vice versa. Thus, if A varies inversely as B, and B l , B 2 be two values if 434 Ratio, Proportion, and Variation. [663. of the second, and A^ A 2 the corresponding values of the first, we have A . A . . T> . 7? AI . A 2 . . j). 2 . j^ l ; whence A l B l = A 2 B 2 , (Art. 651, (2)) ^2 -#2 and ^ l = _22. D \ If then A 2 , B 2 be any given corresponding values of A and B, while A l ,B 1 are any other variable pair of corresponding values, denoting A 2 B 2 by the symbol c, we have Thus, if A varies inversely as B, we may write, either ^~. or -. In this latter form the result can be deduced directly from the proportion A,: A, ::,:,. For, dividing the terms of the second ratio by B^^B^, which will not affect the value of the ratio, it gives 1 1 ^1^::^-:^, or (Art. 660) A Proportion, and Variation. [666. values of the three quantities. Hence (Art. 664) A varies jointly as B and C. 666. A good illustration of the last article is afforded by the case of wages earned by different gangs of men, working for different numbers of days. Evidently if the number of men be the same in two cases, the wages earned will vary as the number of days they are employed, while if the number of days be the same for two gangs the wages earned will vary as the number of men employed. Hence, if W be the wages earned by M men working for D days and w the wages earned by m men working for d days, it follows that W:w:: HD : md. This and the last article contain the theory of what is called the Double Rule of Three in Arithmetic. It is evident that if five of the quantities A, B, (?, a, #, c be given, the sixth can be determined from the equation A EC ~ = ~T~ a be and will in all cases be obtained by multiplying three of the given quantities together, and dividing the product so formed by the product of the remaining two. EXAMPLES. 1 . If a : b : : c : d, prove that a? + ab : ab-b* : : c^ + cd : cd-d\ 2. If a : b : : c : d, prove that bade 3. Find the duplicate of the ratio 4 : 5, and the ratio com pounded of 2 : 3, 6 : 7, 14 : 5, and 5 : 8. 4. Find a third proportional to 4 and 6, and a mean pro portional between 9 and 16. Ratio, Proportion, and Variation. 437 5. Two numbers are in the ratio of 3:4, and if they be increased by 6, they are in the ratio 4 : 5. Find the numbers. 6. If a : b : : c : d, then 7. If a + b:a-b::b + c:2(b-c)::c + a:3(c-a), prove that 8a + 96+5c = 0. 8 If y + z = z + x x + y , 36 c 3c a 3a 6* then x + y + z _ax + by + cz T 2x y 2yz 2z x 9. If f = -f -- = - } shew that 2a + b 2b+c 2c + a c)(x+2y+3z) = f = - > prove tha 6 c (x* + y* + s 2 ) (a 2 + 6 2 + c 2 ) = (a* + by + 10. If - = f = - > prove that a 6 c T 11. It a b c prove that (la 2 + m6 2 + we 2 ) (Za 3 -f mb* + nc 3 ) = (a 2 + 6 2 + c 2 ) (Pa 3 + m 2 6 3 + n 2 c 3 ). c^ cy + bz-ax az+cx-by 12. If - = 7- - ^ = 5-; 2 J prove that a 2 + 6 2 6 2 + c 2 c 2 + a 2 a; + y 4- 2 _ ax + by + cz a + b + c ab + bc + ca 13. There are three numbers, such that the arithmetic means of the two least and of the two greatest are in the same ratio as the corresponding mean proportionals. Moreover the sum of these mean proportionals is double the sum of the two least numbers, while the difference of the mean proportionals is 12. Find the numbers. 1 4. If < ^y = 5Z? = o 2 / +a \ , shew that each fraction =y x+3y x-3y 2(x 1) and solve the equations. 438 Ratio, Proportion, and Variation. 15. If y + z + u 6 z+u + x 1 shew that eacli of these fractions = J- or 5 ; and find the values of x, y, z, u. 16 if x (y+*~ x } _ y(*+x-y) _z(x+y-z] ^ logos logy log z prove that y z z v = z x x z = x v y x . 17. Taxes are laid upon a lodging-house in proportion to the rent of the rooms ; they were one-seventh of the rent and are raised to one-sixth. In what proportion must the rent be raised to meet this additional charge ? 18. If n arithmetic means A lt A z , ... A n be taken between a and b, n geometric means G lt 6r 2 ,...6r n ; n harmonic means ff lt ff 2 ,H a , ...#; prove that 19. Itx + y + zxx + y z and cc 2 + f + z 2 oc o? 2 + 7/ 2 z 2 , shew that each of x and y varies as z. 20. A is walking along a road and passes JB, when finding he has lost something, he turns back and meets B t hours afterwards; having found what he had lost he overtakes B again t' hours after he met him, and arrives T hours too late at his destination. Compare their rates of walking. 21. Given that the volume of a sphere varies as the cube of its diameter and that a sphere of 14 feet diameter has a volume of 1433-74 cubic feet : find the number of cubic feet in a sphere 2 feet in diameter. 22. If 8 men working 9 hrs. a day dig out 400 cubic yards in 5 days, how much can 15 men working 7J hours a day dig out in 1 1 days ] 23. If the sum of two quantities vary jointly as the difference of their cubes and a third quantity ; and the difference vary jointly as the sum of the cubes and the third quantity: shew that the one of the two quantities varies as the other. Ratio ', Proportion, and Variation. 439 24. Supposing that the velocity of a steamer varies inversely as the area of its greatest section when the tonnage is constant, and inversely as the tonnage when the area is constant, and that a steamer whose section is 200 sq. ft. and tonnage 1000 goes 15 miles per hour ; find the velocity of a steamer whose section is 250 square feet and tonnage 1200. 25. Divide 111 into three parts such that the products of each pair may be in the ratios 4:5:6. 26. Two coins of the same bulk, whose values are as 25 : 4 and whose weights are as 9:8, are each composed of silver and copper. Bulk for bulk silver is \ as heavy again as copper : weight for weight silver is 42 times as valuable as copper. Find the proportions of silver and copper in each coin. 27. A certain number is added to each term of the ratio 3:10, and the same number is also subtracted from each term, and it is found that the resulting ratio in one case is the duplicate of the resulting ratio in the other. "What is the number ? 28. There are two vessels each containing mixtures of two liquids in the ratios of r : 1 and s : 1 respectively. A mixture consisting of certain quantities from the two vessels is com- posed of the two liquids in the ratio x : 1, and one consisting of the same quantities each taken out of the other vessel is composed of the two liquids in the ratio y : 1 : shew that (+l)(y+ 1) xy-l (r+ !)(*+!) rs-l ' 29. The sum of four quantities in proportion is 55, the sum of the extremes exceeds that of the means by -f- of the first term, and the sum of the squares of the extremes exceeds the sum of the squares of the means by ^- of the square of the first term : find the four terms. CHAPTER XXVI. CONTINUED FRACTIONS. 667. THE term ' continued fraction ' will be employed to denote such forms as _1 1 1 1 1 1 + 4 1 where all the numerators of the partial fractions are unity and all the letters a,b,c 9 ... denote positive integers. For economy of space the continued fraction is usually written in the form 1 1 1 1 b + c + d + 7' 668. Any positive rational fraction in its lowest terms can be expressed as a continued fraction. N For let the fraction be -^ > where N and D are positive integers having no common factor, D)N(q, r... 669.] Continued Fractions. 441 Let the process of finding- the Greatest Common Measure (Arts. 179, 180) of the two numbers N and D be carried on, and let the successive quotients be q^ q^ ^3, ... and r 19 r z , ... the corresponding remainders. The last of these must obviously be unity, since N and D have no common measure. Then, by the nature of division (Art. 157), the following identities hold N a JD + r , Whence D r 2 - - Whence, substituting in succession, N 1 1 355 669. Let the fraction be > 113 113) 355(3 339 16) 113 (7 112 1 ) 16(16 16 442 Continued Fractions. [669. 355 1 1 Hence - 323 As another example take > 117)323 (2 234 89) 117 (l 89 28) 89 (3 84 5)28(5 25 8)5(1 3 2)3(1 2 1)2(2 2 323 111111 Hence =2-1 117 h 1+ 3+ 5+ 1+ 1+ 2 N 670. If a fraction -j- be reduced to the form 1 1 it can be shewn that the quantities obtained by taking one, two, three, ... of the quotients will be alternately less JV~ and greater than the value of -=r For q l is evidently less than -^ since the fractional part is omitted. 6 7 1.] Continued Fractions. 443 1 . N On the other hand, q l -\ -- is greater than -=- > since the 2 V latter exceeds q^ by a fraction whose denominator is greater 11 7V than q 2 . Similarly q 1 + - -- is less than -=- The fractions obtained by taking one, two, three, ... of the quotients are called convergent* to the value of the continued fraction. 671. The successive quotients being q lt q z , -l - n P d *>-l~ n P-l d P _ (~ 1 ) P . J-llUOj J ' " 7 *"~ 7 7 ~~ 1 7 From this we see that if p be even -f > -f^, while if p be ( 'p "P-I odd the reverse is the case. Hence the convergents are alternately greater and less than the previous one. ft 675. The convergent -f- is produced from the proceeding a p one by the introduction of a new quotient q p . The whole fraction -jr will be produced from -jp^ by introducing JJ d p _i 1 N q p 4- - - instead of q p . Thus -=- can be deduced from 2Wi + '" -^ -^ by the same rule as -*-> if for q p we substitute q p +f when /is a proper fraction denoting 1 446 Continued Fractions. [675. Her - f ~ Hence AltJIILtJ ^ *' - <* r ~ ,' ~ (1) Hence if p be odd -=- ~- is positive, and j^- is less than -=- > while if p be even, the reverse is the case. Thus, as was seen before, the first, third, fifth . . . convergents are less than the original fraction, while the second, fourth . . . are greater. Also since as p increases, d p obviously increases, the A7 fy* difference between -=r and -j- diminishes. That is, each convergent is nearer in value to the given fraction than the previous one. It is evident from (1) that the difference between -=- and n 1 -T 5 - is numerically less than -^ 676. A slightly closer limit can be assigned to the error j. i n P c N in taking ~ for for / = 1_ < -- or -j > q p+l 678.] Continued Fractions. 447 N n v 1 1 Hence -^ < ~ - - < ' Again, - which = q p+1 + - - < q p+l + 1 . J KM-I "*" JV Hence ^^ d p {d p .^(q p+l+ l)d p } 1 n N Thus the error involved in taking - instead of -^? lies 1 1 d p V between -7= and 7 , 7 y-r. ^P^+i ^(^+1 + ^) 677. The successive convergents to a given fraction are thus fractions, expressed by smaller numbers than the original one, whose values are continually closer and closer approximations to that of the original one. For instance, the successive convergents to the fraction 355 22 - in Art. 669 are 3 and -> the value of the latter 113 7 exceeds very slightly that of the given fraction. 678. The method of continued fractions thus gives a means of approximating to the values of any commen- surable fractions. It may also be used to approximate to the value of a quadratic surd. Thus A/ 18 can be expressed as a continued fraction in the following manner, 4) = 4+ = - = 4+ J - , V18 + 4 V18 + 4 2 _ A/18-4_ _1 ' ~~ ~ = 8+ and the process will repeat itself indefinitely. 448 Continued Fractions. [679. Thus V 18 = 4 + -L JL J_ 4+ 8+ 4+ 8 + .. . ad inf. The quadratic surd is thus expressed in the form of an infinitely repeating continued fraction. It is proved in treatises on the subject that if a be the integral part of A/5 is slightly greater than 2, hence the integral part of is unity. Thus, 3 + ^/5 _ A/5-1 5-1 : A + - JL + A/5+1 A/5 + 1 = 3 + >/5-2 = 3 +~ -=3 + J - A/5 + 2 >/5 + 2 -s/5 + 2 = 4+ A/5-2 = 4 and the process will now repeat itself indefinitely. Hence, finally, 3+x/S 11 1 681. Conversely, any infinitely repeating continued fraction may be expressed in terms of a quadratic surd. For let the fraction be 111111 1 a+ - y - --- -, + c+_p+ q+ r+ s+j}+... where the quotients fiomp to $ repeat indefinitely. Let the value of the whole fraction be denoted by x, and let the infinite repeating fraction p -\ -- q+ r+ s+ be denoted by y. Hence 1 1 1 x = a+ I - -, o+ From (1), f^y =1 or y = 2/5. Evidently the value of y must be positive and a little greater than 4, hence the sign + must be taken, and y 2 + \/5. 4?/+l 9 + 4N/5 It follows that x = - - = - = ; 3-V/5 and rationalising the denominator by multiplying numer- ator and denominator by 7 3\/5, the fraction becomes Continued Fractions. 451 EXAMPLES. Reduce each of the following fractions to the form of a con- tinued fraction and find the successive convergents : 13 ]_ 355 31459 ' 23 ' 17* * 113* ' lOOOO' 2723 617 _ 148 ' 1799* ' 1839* ' 263* 8. Find the convergents to 1 H -- - --- 9. Two bells begin to ring together; the one rings 12 times in 7 minutes, the other 17 times in 9 minutes. What strokes most nearly coincide in the first half-hour ? 10. Express 27-321661 days, the average period of the moon's revolution, by the nearest equivalent fraction whose denominator is less than 1000. 11. If the arithmetic means of each two consecutive con- vergents be formed, prove that they will be alternately greater and less than the continued fraction. N N' N" 1 2. If j: > -jy > -jyf be three consecutive odd (or even) con- vergents to a continued fraction, and p be the product of the quotients corresponding to the two convergents preceding the *. ^ N"-X' (p+l)N'-N- last of these, then - ]yr -^ = (p+1)J) ,_ D * 13. If the quotients q 2 , q 3 ,...q r+1 corresponding to the con- N N N vergents ^j jr>~> jjr^* ^ e a ^ equal, then *' * ^ G g V.VJ Continued Fractions. 14. If be the w th convergent to the continued fraction prove that p n = b n p n \~^~ a nPn 5 5 5 5 Hence must be integral and therefore - must o o be integral. If the integral value of this be denoted by t, or y= 2 5t. (a) Indeterminate Equations. 455 Substituting in the original equation, 5#+7(2-5) = 24, or 5x = 10 + 35^; whence x = 2+ It. (/3) The only integral value of t which will allow both x and y to be positive integers is zero. Hence x = 2, y = 2 are the only values of x and y which satisfy the given conditions, and the problem is in this case perfectly determinate. 685. The general form of the problem is the discovery of values of x and y which satisfy an equation of one of the forms ax + 6y = c, (l) ax by = c, (2) and which are at the same time restricted to be positive integers. We may premise that the equation is supposed to have been previously freed from fractions, so that a, 6 and c are all integers. There will obviously be no limitation in supposing that a and b are positive integers. The solution will obviously be impossible if a and I have any common factor which is not a factor of c. For any measure of a and b must measure axby if x and y be integers (Art. 181). Since any factor common to a, b and c may be removed, we may assume that a and b are positive integers which have no common factor. In the case of form (l) no solu- tion is possible unless c be also a positive integer. 686. A very useful practical way of solving such equa- tions is exemplified in Art. 684. Suppose that a is the smaller of the two numbers a and b, and let the whole equation be divided by a ; we thus get l c -*=- (a) 456 Indeterminate Equations. [686. Now each of the numbers b and c when divided by a must give a quotient and a remainder, if these be q, %', r, /, respectively, (a) becomes r' ' or i f _ fy Hence - must be an integer, not necessarily posi- tive. Let this be called z. We then have r'=ry+az. (/3) In this equation y and z are integers, and r is obviously less than a. Divide by r, and by a similar process a new equation of the form ru + /"z = i" (y) is obtained, where u and z must be integers. Now from the process of formation the coefficients of the successive equations continually diminish. Hence at last the coefficient of one of the unknowns will become unity. Suppose that in this last equation /"= 1. Hence z = r"-ru. And therefore from (/3), ry = / az = / ar" + aru. (6) But by the law of formation r'" or unity is the re- mainder after dividing a by r. Hence a 1 is divisible by r. Also / r" is for a similar reason divisible by r. Thus /_/'_(0-l)/', or r -ar", is also divisible by r, and (8) gives r ar" (1) where /3 = - - 688.] Indeterminate Equations. 457 Substituting in (a), we get cbfibau -a r But ft = - -- Hence /-r/3 = a/', and finally, # = /-2/3 + /' bu, = a fa. (2) The value of u is only restricted to be integral. Hence the solutions of the original equation will be obtained by giving to u all integral values including zero, which make both a bu and (S + au positive integers. 687. The solution of the equation axly c can be conducted on the same plan as that of the last article. "688. Sometimes one solution can be discovered by in- spection. When that is the case the general solution can be deduced. Thus, let x = a, y = ft be one solution of the equation ax + by = c. It follows that and if as = sc lt y = y^ be any other solution whatever, we have Subtracting the former from the latter, or whence 458 Indeterminate Equations. [689. Now, by supposition, -7- is a fraction in its lowest terms. Hence, since j3 y l and x^ a are both integers, it follows, as will be strictly proved hereafter (Art. 727), that j3 y and x l a must be equimultiples of a and I. Thus we must have, if u be some integer, or and the whole assemblage of solutions will be obtained by giving u all integral values consistent with making x^ and y positive integers. 689. In a similar way, if x = a, y = (3 be one solution of the equation , ax by = Cj the general solution will be x = a + du, y= fi + au. It is obvious that while in the equation in the last article, u can only have some integral value lying between T and + - > in the present case, u may have any positive integral value whatever, as well as possibly some negative ones, and therefore the number of solutions of the equation ax by = c is unlimited. 690. The discovery of one solution can always^ be effected by means of one of the theorems of the last chapter. Thus let - be converted into a continued fraction, and a , let -j- be the convergent next preceding 691.] Indeterminate Equations. 459 Then it has been shewn (Art. 672) that an-bd = 1, the + or sign being taken according as - is even or odd in the order of convergents. Multiplying by c, anc bed = + c. If the sign be + , this can be written a (nc bu) + b (au cd) = c-, and evidently x = nc~ bu, y = au cd, give a solution if u be so taken that nc bu and au ccl are both positive integers. These results in fact give the general solution, as may be seen by comparison with Art. 688. If, on the other hand, the sign be , we shall have a(bu nc) + b(cdau) = c: whence a solution is given by x = bunc, y = cd au, u again being taken so that these shall both be positive integers. 691. The solution of the equation ax by = c can be conducted in the same way and will lead to the results, x _ nofa, y cd au ; or x = bu nc, y =. au cd; according to the order in which - comes in the con- vergents to itself expressed as a continued fraction. 460 Indeterminate Equations. [692. 692. The student will notice that, in the method of Art. 686, the process is really gone through of finding the greatest common measure of a and b leading at last to a remainder unity. The same process has to be gone through in expressing - as a continued fraction. The two methods, though apparently unconnected, do thus depend essentially on the same fundamental process. The method of Art. 686 generally gives the solution in a more con- venient form than that of Arts. 690, 691. 693. It has been seen (Art. 689) that the number of solutions of the equation ax by = c is unlimited. It has been also shewn that if a, /3 be any values of x and y which satisfy the equation ax + by = c the general solution is given by a; = a + bu, y = p au. Hence u may have any integral positive value such that an, is not greater than /3, and any integral negative value such that lit is not numerically greater than a ; u may also have the value zero. rt Let then = m +f, where f is a proper fraction, and m an integer ; and similarly let j = n -\-f, where n is an integer and f a proper fraction. The number of values admissible for #, or the number of distinct solutions will be m + n+1. Now -+j = m + n+f+f. a o Hence m+n+l=- + __+i But x = a, y = {3 is one solution of the original equation. 696.] Indeterminate Equations. 461 Hence aa + bft = c, and therefore If therefore f+f exceeds unity, the number of solutions /> is the integral part of r > but if f+f is less than unity, Q the number of solutions is the integral part of r + 1 . 694. If either of the quantities f or f is zero, as/; it /? /3 follows that - is integral, and when u - , y = 0. If the nature of the question . render a zero value for y inad- missible, this solution is excluded, and the number of admissible solutions is the integral part of . If/ 7 also vanishes the value u = j- makes x = 0, and if this be excluded by the nature of the question the / y number of solutions will be r 1, r being in this case ab ab easily seen to be an integer. 695. Problems sometimes occur requiring the disco very of all the positive integral values of three quantities #, y, z which satisfy a single equation of the form ax + ly + cz = d. The method of solving such a problem depends on the methods already elucidated, and will be best illustrated by one or two examples. 696. Let the equation be = 36. It is obvious that z, whose coefficient is the largest, has the least range of possible values. The greatest possible value of z is evidently 9, and if z have this value x and y must be zero. 462 Indeterminate Equations. This one solution is [696. x = 0, y 0, z = 9. Let s have given in succession the values, 8, 7, 6, ... 2, 1 ; thus we get the several pairs of equations 2=8, 2=7, z = 6, 2 = 4, 2 = 3, 2=2, 2=1, 2=0, = 4 = 8 = 12 = 16 20+3^ = 20 = 24 = 28 = 32 = 36. Then each of the equations with two unknowns can he solved by previous methods, and the whole series of values ascertained. As another example take 2#+3^ 42 = 4. Here 2 may be zero or any positive integer. Con- sequently there is an infinite number of sets of values of #, y, 2, derived from the series of equations 2=0, 2#+3^ = 4 ; 2=1, 2#+3y=8; where n is any positive integer. In this case the complete solution can easily be shewn to be contained in the formula x = 2 (+!) 3^, y=2^, 2 = fl, and being both positive integers, and the only restric- tion being that 2 (n -f l) must be greater than 3 1. 697.] Indeterminate Equations. 463 Problems of this kind have been met with in Arts. 551, 552. 697. Indeterminate equations of higher degree than the first present too many difficulties for an elementary treatise. One class of such admits however of easy treatment. Let the equation be axy + Ix + cy + d = 0. Multiplying by a and transposing, this becomes a 2 xy + abx + acy = ad, or, by adding be to both sides, (ax + c) (ay -f b) = be ad. The number Ic ad can ordinarily be resolved into two factors in several ways. If a, /3 be one pair of these we may have ax+c = a, a y + b = p, and if these give positive integral values of x and y, there will be one solution. By taking all the possible pairs of factors of bo ad, all possible solutions can be obtained. Thus the equation xy 2x y + 14 = can be written The pairs of factors of 12 are 1, 12 ; 2, 6 ; 3, 4 ; 4, 3 ; 6, 2; 12, 1 ; of these only 0-1= 6, 2-y=2; a? 1 = 12, 2-y=l are admissible ; thus a? = 7, ^ = ; # = 13;y=lare the only solutions. EXAMPLES. 1. Solve in positive integers the equation 7a?+5y= 31. 2. Find the general integral solutions of 2lx13y = 95. 3. Find the general solutions of8#+7y= 56. 4. Find all the integral solutions of 5x+7y = 62. 464 Indeterminate Equations. 5. Solve the equation 13x+l7y = 116. 6. Find the number of positive integral solutions of 5x+3y= 74. 7. "What is the least value of ra that 5x+ 3y = m may have six solutions in positive integers ? 8. Find the number of solutions in positive integers of llx+I5y = 1031. 9. Find the general solution of IQx+lly = 1001. 1 0. Find the simplest way in which a person who has only guineas can pay 10s. 6d. to a person who has only half-crowns. 11. A has 15 florins and 8 half-crowns, and B has five shillings and 4 half-crowns, and A owes 12s. 6d. to B. In how many ways can A pay B 1 12. The mint price of standard gold is 31. 17s. IQ^d. per ounce; find the least integral number of ounces that can be coined into an integral number of sovereigns and guineas ; and how many of each sort may be taken. 13. A man was born in a certain century and died in the next, the last two digits in the years of his birth and death being the same. His age when he died consisted of two digits, the first of which equals the second digit in the year of his birth, and the other equals the first digit in the sum of the years of his birth and death, which ends with three digits the same. When was he born ? 14. A certain number consisting of two digits is multiplied, and thus becomes greater by one than the number formed by inverting the digits : what is the number and the multiplier ? 15. A and B begin to read simultaneously two works each consisting of a great number of volumes. A' 8 volumes contain 89 pages each, and B's 100, and A reads 7 pages while B reads 11. "Will they ever finish the same page at the same instant, and if so, when ? 16. In how many different ways can a volunteer shooting at a target score 16 in 9 shots, a bull's-eye counting 3, a centre 2, an outer 1, and a miss 1 Indeterminate Equations. 465 1 7. There is a number of two digits, which if its digits be re- versed, becomes less by unity than its half. Find the number. 18. Find three numbers such that twice their sum is a number whose digits are the greatest and least of the numbers, and three times their sum, one whose digits are the least two of the numbers. Shew that there are two such sets of numbers and that in each case six times their sum is a number whose digits are the greatest two of the numbers. 19. Find the greatest integer which can be formed in 9 different ways and no more by adding together a positive integral multiple of 5 and a positive integral multiple of 7. 20. Find the general form of numbers which when divided by 7 and 9 leave remainders 5 and 7 respectively. 21. Find a number of two digits such that the product of the digits is less by 100 than twice the number. H h CHAPTER XXVIII. ON INEQUALITIES AND ARITHMETICAL MAXIMA AND MINIMA. 698. IN the chapter on Permutations and Combinations, Arts. 495-497, an instance has been given of the deter- mination of the greatest value of an algebraical expression, n C r , where r was restricted to have only integral values. Another instance is given in Art. 558. There are a few general propositions which enable the maxima and minima values of some algebraical expressions to be determined, without any restriction as to the integral character of the values of the letters involved. 699. An algebraical quantity a is said to be greater than another 6 when the expression a b is a positive quantity. Thus a > b, and a b is positive, are interchangeable and equivalent assertions. 700. The product (a b) (ab) is always positive whether ab be positive or negative (Art. 59), con- sequently a 2 2a6 + b 2 is positive, or or > ab. This is true whatever scalar quantities a and b may represent. Hence for a we may substitute Vx, and for (j we may write Vy. Thus we get - > Vxy. Or (Arts. Inequalities and Arithmetical Maxima and Minima. 467 511, 524) the arithmetic mean of two quantities, x and y, is greater than their geometric mean. The only exception is when a b is zero, or when a is equal to It, and consequently x is equal to y. In this case the arithmetic mean is equal to the geometric mean. 701. It is not infrequently required to find the least value of an expression which can be put into the form A where x is the only variable part. By the last article, A x+ A , or x + - - > 2 V A, sc except when x = > in which case x -\ becomes equal to 2 yz A A Hence x-\ has its least value when x = , or x 2 = A, x x or x = VA, and this least value is 2 \/A. 702. The result of the last article can be obtained in another way. Let x -\ = y, x whence x~ xy + A = 0. Solving as a quadratic in a?, 2 Hence if x be real or scalar, y 2 4A must be positive, or at any rate cannot be negative, thus y 2 must not be less nh 2 468 On Inequalities and [703. than 4 A, or y cannot be less than 2 -2*) (2*- 3) 2 unless 6 2x = 2# 3. Hence the greatest value of (6 2#) (2 x 3) is given when 6 2# = 2# 3 or when x , and this greatest value is -|. Hence the greatest value of 9# 2x 2 9 is f, and is obtained by given to x the value . 704. The general principle underlying the result of the last article can be stated that the product of two expressions whose sum is constant is greatest when the expressions are equal. The corresponding principle for Art. 701 is that the sum of two quantities whose product is constant is least ivhen the quantities are equal. 705. If there be n quantities a lt 2 , a 3 , ... a n equal or unequal, the value of - " is called their Arith- n I metic mean, and the value of (ff 1 agO s ...a a ) n is called their Geometric mean. 706.] Arithmetical Maxima and Minima. 469 These terms are derived by analogy from the case of two quantities. It can be proved as an extension of Art. 700, that the arithmetic mean of n quantities, which are not all equal, is greater than their geometric mean. For it is evident that when the quantities are all equal their arithmetic mean is equal to their geometric mean. If any two, as a r and a 8 , are unequal, let them be replaced by two equal quantities each equal to half their sum or J (a r + a 8 ). This will not affect the value of the arithmetic mean of the n quantities, but, by Art. 704, will increase the value of the geometric mean. Thus by repeated substitutions of this kind, as long as any two remain unequal, the arithmetic mean is unaltered while the geometric mean is increased. In the final result the two means are equal: hence previously the geometric mean must be the less of the two. 706. This principle can occasionally be used to determine the greatest value of an algebraic expression which is resolvable into factors. For instance, let it be required to determine the values of x and y which give its greatest value to the expression (c-x)(c-y)(x+y-c). (a) By the theorem of the last article, N-.I (c x) + (c v\ + (x + y c\ c -c}}*< v -- > V ^ ^ < 3' unless the three quantities c #, c y and x+y c are all equal. Thus (a) will have its greatest value when x and y satisfy the conditions c x = c y = x+y c 2c whence x = y = . 3 470 On Inequalities and [707 707. There are two propositions with respect to ratios or fractions which are sometimes useful. The first is, that a fraction greater than unity, or as it may be called, a ratio of greater inequality, is diminished by adding the same quantity to its numerator and denominator. On the other hand, a fraction less than unity is increased by the same process. Thus > or < b + x b as (a + x) b > or < a (b + #), as ab + lx > or or or therefore *,=$,. a 2 a l T < T b., ^ a n a, a, f n 709.] Arithmetical Maxima and Minima. 471 Adding these results, when ce l+a + 3 + +. 1 fl 1 + fl 2 +...+fl z~' ^ + ^ 2 +...+^ n 709. In some of tlie foregoing articles the following principles have been assumed. If one quantity is greater than a second, any positive multiple of the first quantity is greater than the same multiple of the second. If one quantity be greater than a second the sum of the first and any third quantity is greater than the sum of the second and the same third quantity. If one quantity be greater than a second the remainder after taking away any quantity from the first, is greater than the remainder after taking the same quantity from the second. If one quantity be greater than a second, any positive even power of the first is greater than the same power of the second. These may be assumed as axioms, extensions of those in Art. 53. They may also be proved, if the student prefer, by means of the definition of Art. 699. Thus a > t>, when a b is positive. But in this case p(a b) or pa pb is also positive ; therefore pa > pb, which proves the first principle, and the others can be treated similarly. 472 On Inequalities and EXAMPLES. 1. Prove that -+- + -+->4 unless x = y = z = u. y z u x 2. Prove that ab + (a b)x x 2 is never greater than 3. Find the greatest value of ax x*. 4. Find the sum to n terms of the series + + ... , and 1 ,t t . O hence shew that (n-}- l) n ~ 1 < ([*>>)* 3m(3m+l) 2 OT /jg- 5. Prove that - - > v }6m ' 4 6. Prove that (1 + x) n (1 + x n ) > 2 n+1 x n , n being positive. 7. Prove that xyz > (y + z x)(z + x y) (x + y z), all the factors being positive. 8. Prove that (1) r 4n H-2nr 2n - J > l + 2wr" +1 . W-l +J_ M + 1 ~ 7ir 2 +1. 9. Prove that os 2 + 2/ 2 + 2 2 > yz 10. Which i. the greater "',, or s - 11. Prove that > 2 Z, the first fraction yz + ssx + xy a* + b 2 + c 2 being positive. 12. Prove that |2n-l < | w_(2n) llr - 1 . 13. Prove that (2 x + a) A/a as has its greatest real value when 14. Shew thatpo; 9 - r -f qx r ~ p + rx p ~ 9 >^ + j + r unless cc = 1 or p =. q=. r. 1 5. Shew that the limit of f/(x + a) (a? + ar) . . . (a? + ar w ~ l ) when n is indefinitely increased, r being less than unity, is x. Arithmetical Maxima and Minima. 473 16. Supposing that the consumption of tea is forty millions of pounds when the duty is 2s. Qd. per lb., and that the increase of consumption varies as the decrease of duty, so that if there were no duty, the consumption would be sixty millions : find the consumption when the duty is 2s. per lb., and what would be the most productive duty. 1 7. Prove that if a;, y, 2, ... , a, 6, <*,... be all positive quantities (x v z ) (a b c ) ]- +f + - + ... > <- H h- -f ... f can never be less thanw 2 . (a b c ) (x y z } 1 8. Prove that (x 2 + y 2 + z 2 ) (a 2 + b~ + c 2 ) > (ax + ly + cz) 2 unless x y z a b c cc + 4 1 9. Find the values of x which give the fraction -. , . . (x + 5)(x + Q) its maximum and minimum values. jp 2 _i_ 3 x -4- 2 20. If x be real prove that the expression '-^ can have no real value between - and 1. f 21. Prove that CHAPTER XXIX. ON NOTATION AND NUMBERS. 710. IF a number N be divided by a smaller number m, with a quotient q and a remainder r , the following identity holds (Art. 157), N = qm + r . The number m being called the modulus and r the residue, any two other numbers are said to be congruent with respect to the modulus m when they have the same residue. Thus 12, 17, 22, 27... all leave the same remainder when divided by 5 and are therefore congruent to each other with respect to that number as modulus. The word number throughout this chapter will be always understood to mean positive integer. 711. If q be a number greater than m, we can similarly obtain an identity of the form q = Whence, by substitution, If q^ be still greater than m, we may write 2i whence again = q 2 m -f r 2 m + This process may be repeated until a quotient ^n-i is arrived at which is less than m, and finally we obtain or, writing r n for q n -^ since q n _ l will be also the remainder after dividing q n _ l by m, (l) On Notation and Numbers. 475 Thus any number can be expressed in a series of powers of any modulus which is less than itself. The coefficients of these powers, beginning- with the lowest, are the successive remainders after dividing N and the successive quotients by m. 712. When a number N is expressed in terms of a second number m in the manner of the last article, m is called the radix, and the numbers r n , r n _ 15 ...r 2 , r l9 r , are called the digits. These digits are all, from the nature of the process by which they are found, less than m. Any of them, with the exception of r n , may have the value zero. 713. In the ordinary system of Arithmetical notation, the number 10 is employed as radix, and the coefficients only of the different powers of the radix are written down, the power of 1 belonging to each digit being indicated by the position of the digit. Thus 416732 is an abbreviated way of writing 4xl0 5 +l x 10 4 + 6x 10 3 +7x 10 2 + 3xlO + 2. Numbers expressed in this manner are said to be expressed in the scale of the radix 10. Similarly the number TV in (l) of Art. 711 is said to be expressed in the scale of the radix m. 714. A number expressed in any scale can be expressed in any other scale by the help of 7 ^673521 the foregoing articles. Thus, let _ 195217 2 673521 be a number expressed in 13745 2 the ordinary decimal scale, and let it be required to transfer it to the scale 7. The process is represented by the diagram : the given number when divided by 7 leaves a remainder 2. The quotient divided by 7 again leaves a remainder 2, and so on. By 476 On Notation and Numbers. [715. the theory of Art. 711 it follows that the given number is equal to 5x7 6 + 5x7 5 + 0x7 4 + 3x7 3 + 4x7 2 + 2x7 + 2; or expressed in the ordinary \vay may be written as 5503422, it being remembered that the shifting of a figure one place to the left multiplies its value by 7 and not by 10. 715. To transfer a number from any other scale to the scale 10, the student will probably find it easier to work out the several powers of the radix, multiply by the successive digits, and add the results. Thus the number 5503422 in the scale of seven means 5x7 6 + 5x7 5 +3x7 3 + 4x7 2 + 2x7 + 2, and 2=2 2x7= 14 4x7 2 = 196 3x7 3 = 1029 5x7 5 = 84035 5x7 6 = 588245 673521 Hence the number 5503422 in the scale 7 is represented by 673521 in the scale 10. 716. The transformation in the last article can be also effected in the manner of the previous one, the only difficulty . being that the student will not be so familiar with the formal id 400zdo 1 ., ., ,. ,.. ,. .. '- results ol the multiplication 13125431 2 . ' table in the scale 7 as in the I scale 10, and thus the division will present greater difficulties : the process is indicated by the diagram. In the scale 7 the number 10 is evidently denoted by the symbol 13. 7 1 8.] On Notation and Numbers. 477 Thus the number is represented in the scale 10 by the figures 673521. The student will at first find the above process facilitated by turning the portion of the dividend operated on into the scale 10. Thus 55 in the scale 7 means 5 x 7 + 5, or 40 in the scale 10, whence the quotient by 10 is 4 and the remainder zero. 717. A number may thus be expressed in any scale, and transferred from that to any other. The digits in the case of scales whose radix is not greater than 1 may be expressed by the ordinaiy symbols. For scales whose radix is greater than 10, additional symbols will be required. Thus for the scale with radix 12, two symbols will be needed for the digits 1 and 1 1 which may occur. The letters t and e are usually employed for this purpose. Thus the number in the scale 12 means 5x 12 6 + 6x 12 5 + 10x 12 4 +7x 12 3 + Ilxl2 2 + 4x 12 + 3. It is hardly necessary to add that the processes of multi- plication, division, square and cube root, &c., can be conducted in any other scale on the same principles as in the scale of 1 ; the only requisite is a familiarity with the expression of the multiplication table in the scale in which the opera- tions are earned on. 718. If N be a number expressed in the scale m with digits r , T-J, r 2 , ... so that N= r + r 1 m + r 2 m 2 + ..., we may write N in either of the forms 3 ( 3 - 1)+ from the first of which it follows that N will be divisible by m 1 if r -f r 1 + r. 2 + . . . be so divisible ; and from the second that ^V is divisible by r + 1 if r r l + r 2 r 3 + . . . be divisible by r+l. 478 On Notation and N limbers. [719. In the scale 1 it follows that any number is divisible by 9 if the sum of its digits be so divisible ; and by 1 1 if the sum of the even digits and the sum of the odd digits leave the same remainder when divided by 11. 719. It has been previously assumed that in the scale of 10 any number with n digits is less than the smallest number with n + 1 . This is true in any scale. For if N be the number with n digits in the scale of m, and r denote the greatest digit, it follows that N cannot be greater than rm n ~ l + rm*~ 2 4- rm n ~ z + . . . + rm + r ; that is, cannot be greater than r(m n -\) m 1 Now r is not greater than m 1 . Hence N is not greater than m n 1 and is therefore less than m n , which is the least number represented in the scale of m by n + 1 digits. 720. Taking 2 as the modulus (Art. 710) any number must have a remainder or 1. Thus any number can be expressed in one of the forms 2 q or 2q+l. All numbers of the first form are called even numbers^ those of the second are called odd numbers. Thus all even numbers are con- gruent with regard to the modulus 2, and similarly all odd numbers. 721. Any two numbers are congruent with respect to a given modulus when their difference is divisible by that modulus. Let N and N* be the numbers and m the modulus, and let Ni-^N' be divisible by m. Let N = mq + r,N'= mq' + / where both r and / are less than m. Then N-N'= m(q-q') + r-r'. But NN'= mp by hypothesis where p is an integer. Hence r r'= m(p q + q'). 724.] On Notation and Numbers. 479 But rand /, and therefore/ 4 /, are numerically less than m, and the only way in which this equation can hold is that both sides shall be identically zero. Hence r must equal /', or N and N' are congruent with respect to the modulus m. The converse of this proposition is obviously true. 722. The problem sometimes occurs to find an expression for all numbers which are congruent with one number with respect to one modulus and to a second number with respect to another modulus. Thus, let it be required to find an expression for all numbers which are congruent with 5 the modulus being 7, and with 8 the modulus being 9. The numbers in question must evidently be equally represented by the symbols 1 p + 5 and 9^ + 8. Thus the problem is to find all possible values of p and q to satisfy the equation or 7p 9q = 3. The solution of which by the principles of Chapter XXVII is given by p = 3 + 9*, q = '2 + 7t, where t is any integer. Consequently the numbers required, which are represented by Ip + 5, can be expressed in the general form 26+63. 723. The series of numbers 1, 2, 3, 4, 5,... ad inf. is called the series of natural numbers. With reference to any modulus m the natural numbers may be divided into sets, each containing m consecutive numbers, which have the numbers 1, 2, 3, 4, ... m 1, as residues in the same order. Thus if 5 be the modulus, the natural numbers may be arranged in sets, thus 1,2,3,4,5; 6, 7,8, 9, 10; 11, 12, 13, 14, 15; and each set has as residues with respect to the modulus 5 the numbers 1, 2, 3, 4, 0. 724. A number which has no factors except itself and 482 On Notation and Numbers. [728. 728. It follows from the last article that if a number// which is prime to q divide aq it must divide a. Another proof can be given of this result. Let the ordinary process of finding- the G. C. M. of q and p be carried out, and let r 19 r 2 , r 3 , ... be the successive re- mainders. Since q and p have no common measure higher than unity, the last of these remainders must be unity. Let a? 1} # 2 , # 3 ,... be the corresponding quotients. Then we know (compare Art. 180), q r-, aa ar-, i- = # + i , whence - = ax-, + -- ; (1) P P P P ar, ar* p = r^ + r^ = i ^ 2 +y; (2) ar-, ar ar* fi = r 2 a 3 + r 3 , -^ = 2 # 3 + 3 ; (3) Since is an integer it follows from (l) that - is so fl /" also, and from (2) that , and so on. Since the last of the quantities r lt r^ r 3 , ... is unity it finally follows that - is integral. From this the result of the last article can be deduced. 729. A number prime to each of two others is prime to their product. Let p be prime to each of q and r ; then p is also prime to qr. For, if possible, let p and qr have a common factor *, so that p = st and qr = su, where u and t are prime to each other. Hence = - But q and r are each prime * r l to p or st and therefore to s, so that - is a fraction in its , s lowest terms. Thus (Art. 727) u and r are equimultiples of q and g ; or / is both prime to s and a multiple of *, which is absurd. Thus p must be prime to ab. 733-] On Notation and Numbers. 483 730. Any number must obviously be either a prime number, or resolvable into factors which are prime. Let N be the number, and let abed ... be its prime factors, of which some may be equal. If possible let N also equal a/3y8 . . . , where a, /3, y, 8 are also prime numbers. Then a divides N or abed.... Hence a must divide one of the factors 0, b, c, d. For if not, being a prime number, it must be prime to each of them and therefore to their product (Art. 729). Let a divide a. Since a is a prime number it has no factors except unity and itself. Hence a must equal a. Similarly 0, y, . . . must each be equal to one of the factors b,c,d, . . . , that is, the two resolutions into factors are identical. 731. The most general form of resolution of a number is a x b v (f ... where a, 6, c, ... are different prime numbers and sc,y, z, ... are any integers including unity. 732. When a number is expressed as in the last article it is evident that all the possible divisors of it will be the different terms of the product, and the sum of all these divisors is therefore, -... (Art. 522 . a I bl cl The number of these divisors is evidently the number of terms in the above product, or (a+l) or, if unity be excluded, 733. From Art. 723 it follows, that any set of m successive natural numbers will leave the same residues in the same cyclical order with respect to the modulus m, these residues being the integers 1, 2, 3 ... (m 1), 0. li 2 484 On Notation and Numbers. [733. If any m equidistant integers be taken the same pro- position holds good provided the interval between the integers be a number which is prime to m. Thus if any five integers, increasing in succession by 4, be taken, as 3, 7, 11, 15, 19, the residues of these relatively to the modulus 5 are 3, 2, 1, 0, 4. The cyclical order of the residues is different to what it is when the interval between the numbers chosen is unity. The general proposition is, that if n be any number prime to m the residues of the numbers #, a + n, a+2n, ... a + (m l)n, with reference to the modulus m will be in some order, the m numbers 0, 1, 2, 3, ... (m 1) : and the cyclical order of the residues is independent of the starting point a. In the first place the residues must be all different. For if two of them, as those of a+pn and a + qn, were the same, then (Art. 721), (a+pn)^'(a+q*\ or (p~*q)n, must be divisible by m. But m is prime to n, consequently (Art. 728) m divides p ^ q, which is impossible, since p and q. and a fortiori p r^ q, are less than m. Hence, since there are m different residues, these must be the m integers 0, 1, 2, ... (m\). For the second part it is evident that whatever a may be the difference between any two consecutive residues must be n m or m n according as n or m is the greater. Hence if any two particular residues are consecutive in the cyclical order when a has one value they will also be con- secutive for any other value of #, that is, the cyclical order is independent of the value of a. 734. Among the numbers which are less than any given number which is not a prime, some will be prime to it and others not. A very simple formula gives the number of the former. 735-] On Notation and Numbers. 485 Let the number N = a x where is a prime. This is the simplest case of the general formula of Art. 731. Then the numbers not greater than N and having a common measure with it, are 0, 20, 30, 40, ... a x ~ l .0. The number of these is 0*" 1 or Hence the number of numbers less than N and prime to it is ^-*or jr(l-2). ^ 0' 735. Let now $(n) represent the number of numbers less than n and prime to it ; and similarly let < (m) represent the similar quantity for another number m. Further, suppose that n is prime to m. Then, (mn) representing similarly the number of numbers less than mn and prime to it, it can be shewn that < (mn) $ (m) . $ (n). For the mn integers from 1 to mn inclusive can be divided into m sets of n consecutive integers. Taking the first number out of each set we get the series 1, #+1, 2n+ 1, ... (m 1) n+ 1 (a). The residues in this series with regard to the modulus m will by the last article be the same as those of the series 1, 2, 3, ...(i-l), *(/3), only in a different order. The same number of the numbers in (a) will therefore be prime to m as of the numbers in (/3). That is, the number of numbers in (a) which are prime to m, must be $ (m). Similarly in the series formed by taking the second number out of each set, namely 2, n + 2, 2n + 2, ... (ml)n + 2, there will be $ (m) which are prime to m, and so on for the 486 On Notation and Numbers. [736. different series formed by taking the third, fourth, &c. out of each set. Now all the numbers in each such series will be prime to n or not according as the first terms, 1, 2, 3, ... n 1 are prime to n or not. I'he number of series in which all the numbers are prime to n is therefore (n). Hence the number of those numbers which are prime both to m and to n is (n) . (m), that is, $ (mn) = < (n) . (m). 736. Let now N=a x b v (f.... Then with the notation of the last article, = (a*). = *(). = (!_ !).(! - 1)^(1 - -)... (Art. 734) which is the formula required. 737. It follows from Art. 723 that any fraction whose denominator is m, whether improper or proper, can always be reduced so that its fractional part shall have for its numerator one of the m 1 integers 1, 2, 3, 4, ..., (m 1) ; unless the denominator be a factor of the numerator, in which case the fraction is really an integer. If m be a prime number, all the fractions 1 2 3 m 1 will be in their lowest terms. 739] On Notation and Numbers. 487 738. The student of Arithmetic is familiar with the process of reducing a vulgar fraction to a decimal one, that is to a fraction whose denominator is some power of 10. The process really depends on the principle that the value of a fraction is not altered by multiplying both numerator and denominator by any number. Thus any fraction = ? where n is any number whatever. If, by taking n sufficiently large, 1 O n can be made to contain q as a factor, and the quotient of 10 n divided by q be r, the fraction = ~ = n which is the required decimal form. It is evident that this process is only possible when q contain no factors except those of 10, namely 2 and 5. The only fractions then which can be reduced exactly to decimals are such as have denominators of the form 2 s . 5 e , where s and t are positive integers. The denominator of the resulting decimal will be 10 s or 10*, according as s or t is the greater. 739. Any fraction, > can however be reduced approxi- mately to a decimal fraction by the following process, 10 p r, fl i + p _ IQp _ q ~q~^0q~'~~ 10 where 0j is the quotient and ^ the remainder after divid- ing I Op by q. Again, 1 r. IQq I0 2 q 10 2 10 2 and r 2 being the quotient and remainder after dividing r t by q. -+ 488 On Notation and Numbers. [740. The numbers a l9 a 2 are thus the first two figures in the decimal fraction equivalent to This process can be carried on indefinitely. If any of the quantities r lt r 2 ,r 3 , ... vanish, the series will terminate and the fraction will reduce to a finite decimal. This, as in the last article, can only be the case when q is of the form 2 8 .5 t . 740. If the quantities r lt r 2 , ... none of them vanish they must after a time recur in the same order. All the numbers r l9 r 2 , . . . are less than q. Hence they have only q 1 different possible values. If at any stage of the process any remainder, as r 2 , occurs again, the whole series of operations will recur from that point, and thus the decimal fraction obtained will be in some part at least a repeating decimal (Arts. 532, 533). 741. The process by which a fraction can be expressed as a series of fractions with the successive powers of 10 as denominators may be applied to any other radix. Thus, if m be the radix and the fraction, mq and so on ; a^ , a 2 , a z , . . . being the quotients and r l , r 2 , r 3 ,. . . the remainders after dividing mp> mr : , mr 2 , . . . by q. 743-] On Notation and Numbers. 489 Thus, finally, p will reduce to a recurring decimal repeating from the first place, and the number of figures in the period will be either ( r 2> r s' w iU obviously also be the remainders after dividing 10j, 10 2 p, 10 3 j, ... by q. Assuming then that q is prime to 10 and also a prime number, each of the 10?% 10/-0 tractions -, , ... is in its lowest terms, since the ? 9 quantities r lt r 2 , . . . are all less than ^, and there is con- sequently no factor common to numerator and denom- inator. Thus, if it be required to reduce the fraction -, for instance, whose denominator is q and whose .numerator is any one of the series of residues r l , r 2 , . . . , the process will be the same as that of reducing only beginning at a later point, and in this case the figures of the equivalent decimal will be 3 , 4 , 490 On Notation and Numbers. [744. Suppose then that a 3 is the first figure that recurs in the decimal equivalent to Let n be the number of figures in the repeating part, the figures themselves being there- fore #3, 4 , ... a n+2 . Thus in reducing - to a decimal the figures will recur from the beginning and will be 3 , 4 , ... fl n+2 . In reducing , they will also recur from the first and be 4 , 5 , ... fl n+2 , 3 , and so on. Thus in reducing - to a decimal, n fractions whose denominator are q, and whose numerators are different integers r 2 , r 3 , ... /+! have been reduced, the period of each being the same, only the figures beginning at a different point. 744. By Art. 533 it follows that the fraction -^ can be . N % expressed in the form -, where N is the number formed by the repeating figures. By Art. 727 it follows that N is a multiple of r 2 . Thus if JV= r 2 N / 9 we have 1 N' 1 - = - , that is (Art. 533), - can be expressed as a repeating decimal of n places recurring from the first. Multiplying both sides \>j p, it follows that L _ P N' q ~ 10 n -l Hence can also be expressed as a similar repeating decimal. Thus, if q be any prime number, all the fractions - will be convertible into repeating - -, -9 decimals of the same number of places recurring from the commencement. 745. Again, if the number of places in the period be n, we have seen (Art. 733) that in working out the period for any one of the above fractions we really work out the 748.] On Notation and Numbers. 491 period for n of them. Thus # 1, the whole number of fractions, must either be n or a multiple of n, since the fractions must either all have the same figures in their repeating decimal, or must be divisible into a number of sets of n, all of eacli set having the same figures. Thus n, the number of places, must be either q1 or a factor of q-\. 746. It follows that the process of reducing to a decimal must always repeat itself after q 1 places. Hence by the observation in Art. 743 it follows that the re- mainder after dividing I0 q ~ l .p by q is p. Thus where M is some integer ; or (I0 q ~ l l)p = Mq. But since p divides Mq and is prime to q it follows that p divides M. Let Q be the quotient. Hence 10'- 1 -! = Qq. (a) 747. The conditions under which the equation (a) holds good, are that q is a prime number and also prime to 10. If any other number N be taken as radix the reasoning of Arts. 742-746 holds good, provided q be a prime number and prime to N. Hence under these conditions, it follows that 'N*- l -l = Qq, (ft) where Q is some integer. This result (ft) is known as Fermat's Theorem. Another proof can be given depending on the proposition in the next article. 748. The product of any n consecutive integers is divisible by [#. This is evident from the fact that the fraction (m+l)(m + 2) ... (w-f n l)(m represents the number of combinations, of (m + n) things 492 On Notation and Numbers. [748. n together (Art. 488) and must therefore be integral, that is, the product (m + 1 ) (m + 2) . . . (m + n) of any n con- secutive integers is exactly divisible by In. A more direct proof can be given. Let the above fraction be denoted by the symbol I(m,n). The fraction can also be written as = (Art. 490), \m_ \n ^ and is thus symmetrical with respect to m and n. Then j = (+ 1 )( + 8)--- ( + -*) (m+ l)(m + 2) ... (m + n l).n : .te ... (m + n 1) (m _ \nl or I(m,n) = I(m-l, n) + I(m,nl). (I) rf \ (m+l)(m + 2) .Now I(m, 2) = } and this is always If integral since one of the two numbers m -f- 1 , m + 2 must be even (Art. 720). Consequently I(2,n) is also integral for all values of n. By the above relation (l), we have I(m, 3) = I(ml, 3)+I(m, 2). Hence, since I(m, 2) is integral, I(m^ 3) will be integral if I(m 1, 3) is so. But /(2, 3) is integral, consequently 7(3, 3) is so, and therefore 7(4, 3) and 7(5, 3), and so on, to any value of m or I(m, 3) is an integer. Again by (l), I(m, 4) = I(m 1, 4) + I(m, 3), and since 7(m, 3) is integral it will follow that I(m y 4) is so if 7 (m 1, 4) is an integer. Hence, as above, I(m, 4) is seen to be always integral : and in a similar way 7(?#, 5) and so on to I(m, n), where n is any integer. 75o.] On Notation and Numbers. 493 749. It follows that all the coefficients in the expansion of the binomial (a + x) n , where n is an integer, are integral. The coefficient of the r+ 1 th term is K (!) ...(nr+1) - or - \r \n r \r If n be a prime number, since none of the factors in the denominator can divide n, this coefficient will be divisible by n. Hence all the terms of the binomial expansion (a + x) n , when n is a prime number, are divisible by n except the first and the last, namely a n and x n . 750. The coefficient of any term in the expansion of ...) n can be written as - -, where \p_ \q_ \r_... ... n (Art. 550). This coefficient is obviously an integer from the method of its formation. Hence, if n be a prime number, this coefficient will, as before, be always divisible by n except in the cases when one of the quantities j, q, r, ... is equal to n and the others zero. The coefficient in this case is unity and the corresponding terms are a n , l n , c n , ____ Hence (a + b + c + d+ ...)* = a n + & n + c n + ...+M.n, where M is some integer. Now let each of the numbers a, 5, e ... be unity, and let the number of them be N. Hence N n = N+Mn, or N n -N = Mn, or N(N n ~ l -l) = Mn. If then .ZV be prime to n it follows that the other factor N n ~ l l is divisible by n, or JV"- 1 -! = Q.n, where Q is some integer. Thus we again arrive at the theorem of Art. 747. 494 On Notation and Numbers. EXAMPLES. 1. Express 2345 in the scales of 5 and 9. 2. Express 174-26 in the scale of 5. 3. Transform 1234-56 from the denary to the septenary scale. 4. Express in the common scale and in the scale of 8 the number denoted in the scale of 9 by 723. 5. Transform 119716 into the scale 7 and extract its square root in that scale. Transform the square root back into the scale of 1 0. 6. Multiply 12923 by 215 in the scale 12 ; divide the product by 3 and extract the square root, and transform the result to the scale 10. 7. Extract the square root of 112123-0213 in the scale 6, and transform the result to the scale 8. 8. Shew that any number of six digits in the decimal scale formed by the repetition of three digits in the same order is divisible by 7, 11 and 13. 9. Find a number which is expressed by the same two digits in the scales of 7 and 9. 10. The difference of two numbers having the same digits is 35453221, in what scale are they expressed? 11. If ,, $ 2 , S s be the sums of every third digit in a number, beginning at the units', tens', hundreds' place respectively, and in each series making the alternate digits negative ; then the number is divisible by 7 if S l + 3 2 + 2 S 3 is so. 12. The successive digits of a number expressed in the decimal scale beginning with the units' place are , 15 a 2 , a 3 ,... ; shew that the number will be divisible by 77 if 13. The /number 1865 when expressed in a certain scale becomes 12345 ; find the radix of the scale. On Notation and Numbers. 495 14. Find the scale in which a certain number is expressed by 1111 when were the scale doubled it would be expressed by 125. 15. A certain odd number is expressed by 6 digits in the scale of 3 and by the last three of those digits in the scale of 12. Find the number. tf>. A number is denoted by 4-440 in the quinary scale arid by 5-54 in a certain other scale. What is the radix of the latter ? 17. Shew that a number in the decimal scale is divisible by 3 if the sum of its digits is so divisible. 18. Shew that a number is divisible by 8 if the number formed by its last three digits is so divisible. 19. The square of every number is of the form 3m + 1 or 3m; and the square of every prime number greater than 2, diminished by unity, is divisible by 8 without remainder. 20. Every cube is of one of the forms 4m or 4w + 1. 21. If n be a prime number different from 2, then in the scale of notation whose radix is 2w, any number ends with the same digit as its ri^ power. 22. If (r 1 ) be a prime number, then in the scale of r, of the numbers 121, 12321, ... none but the last is divisible by r 1, and that is divisible by (r I) 2 , the last digit in the quotient being unity. 23. If n be an integer, shew that 7n 3 3n 2 4n is divisible by 6. 24. If m be any even number, m 7 m 5 m? + m is divisible by 90. \2n 25. Prove that if m be a prime number -; -^=- is an integer. 26. If m be a multiple of 5, m (m 2 + 89) is divisible by 30. 27. If - be converted into a circulating decimal with pl re- curring figures !, a 2 , a 33 . . . ; shew (1 ) that p is a prime number ; 496 On Notation and Numbers. (2) that the recurring period being multiplied by 2, 3, ... p 1, will reproduce its own digits in their own order ; (3) that I + P+I = 2 + ap+3 = ... = Op_i + a^_i = 9. 22 2 28. How many numbers will divide 800 exactly ? 29. What is the number of factors of 720 1 30. How many numbers are there less than 720 and prime to it? 31. If p be a prime number, not a factor of a and not equal to (a 1), then the sum of the remainders when a, a 2 , a 3 ,... a 1 *" 2 are divided by p is less by unity than a multiple of p. Also shew that the coefficients of (1 +x) p ~ l will differ from multiples of p by unity in excess or defect alternately. Shew also that if A Q , A l} A 2 , ... be the coefficients in (1 +)~ 2 , then A 1, ^ + 2, A 2 3, ^ 3 + 4,...will be multiples of;?. 32. If n be a prime number and p any integer, prove that (n 2 p* ly^+l and (rap i)-i + (wp + i)-i have the same remainder when divided by n. 33. If m and n be any two prime numbers, prove that (m n ~ l + n m ~~ l ) divided by mn leaves a remainder unity. 34. If n be a prime and m not divisible by n, shew that |2n-l (m 1) is divisible by n. In In 1 35. The difference of the squares of any two prime numbers greater than 3 is divisible by 24. 36. Shew that three prime numbers, each greater than 3, cannot be in arithmetical progression unless the common difference is a multiple of 6. 37. If in the scale of 12 a square number ends with a single cipher the preceding digit must be 3, and the cube of the square root ends with 60. CHAPTER XXX. PROBABILITIES OK CHANCES. 751. ALL persons are conscious that their expectation of any undecided event may vary from an almost certain anticipation that the event will happen to an almost equal certainty that it will not. Thus expectation in such cases is susceptible of being greater or less, and therefore must be capable of quantitative measurement. It can therefore also be represented by a numerical symbol. The number which represents the expectation of a well-informed person in regard to a given undecided event, is called 'the chance' of that event happening. The unit is taken to represent that degree of expectation which is called certainty, and any less expectation than this must therefore be represented by a proper fraction. Thus the numerical measure of ' the chance * of any uncertain event is always a proper fraction. 752. Suppose a halfpenny is tossed up. As it falls it must have either ' head ' or ' tail ' uppermost. The expec- tations of these two events are equal. As one of them must happen, the sum of the expectations is certainty. The sum of the two equal chances of ' head ' or ' tail ' is therefore unity. Hence each of these chances is represented by . 753. Similarly, if a six-faced die be thrown, the chances of the different numbers being uppermost are all equal. The sum of the chances is unity, hence each of them, as for Kk 498 Probabilities or Chances. [754. instance the chance that the number 2 shall be uppermost, fa*. 754. The chance that either 1 or 2 shall be uppermost is similarly or J, since of the six ways all equally likely two are favourable. 755. The general rule is that, if any event can happen in a ways and fail in I ways, the a + b ways being the only possible cases and all being equally likely, the chance of the event happening is ^ and the chance $f its failing is y tt-\- u a + b For, evidently, if the chances of the event happening and failing respectively be x and y, we have the two conditions x : y : : a : b, x+y = 1; the former equation being obvious from the data, and the latter expressing the fact that the event must either happen or fail. From these equations we easily obtain a b X T > #= !- a+b a+b 756. The result of the last article is the foundation of the mathematical treatment of chance. The student who desires further elucidation of the logical basis of this result can consult with great advantage the ' Essay on Probabilities' of the late Professor De Morgan. For full information on the earlier history of the subject, Dr. Todhunter's ' History of Probabilities' may be referred to. 757. Let p represent the chance that the event A will happen ; then 1 p represents the chance that A does not happen. Similarly, if q be the chance that another inde- pendent event B happens, 1 q will be the chance that B fails. 758.] Probabilities or Chances. 499 With these assumptions the chance that (1) A happens and B fails is 7? (1 q) ; (2) A fails and B happens is (1 p) q ; (3) A happens and B happens is pq ; (4) A and B both fail is (1 p) (1 -q). For, reverting to the notation of Art. 755, let a be the number of ways in which A can happen and b the number of ways in which it can fail, while a' and I' represent the number of ways in which B can happen or fail respectively. Then each of the ways of A happening may be combined with each of the ways of B failing, and there are thus ab' '. ways altogether in which A may happen and B fail ; that is, ab' is the number of ways in which the compound event described in (1) may happen. But the total number of possibilities is (a-\-b) (a' + b'}. Hence the chance of (l) must be -if a r - * - 77 J a 4-0 that is, Similarly, the chances of (2), (3) and (4) may be shewn to be as stated. Hence the chance of the concurrence of two independent events is the product of the chances of those events happening separately. 758. As an instance of the last article, suppose that there are two bags, one containing three black and four white balls, the other containing five black and three white^ balls, and let it be required to find the chance of drawing a black ball out of each at the first drawing. There are eight possible draws out of the second bag, and seven out of the first ; any one of the eight may be com- K k 2 /lx f ; 500 Probabilities or Chances.- 1 [759- bined with any one of the seven ; that is, on the whole there are 7 x 8 distinct possible results of drawing one ball out of each bag. All these possibilities are equally probable. There are three different ways of drawing a black ball from the first, any one of which may concur with either of the five possible ways of drawing a black ball from the second ; that is, there are 3 x 5 different ways of drawing a black ball _out of each bag. Thus the chance of drawing a black ball 3x5 out of each bag at the first trial must (Art. 755) be , or 35 7x8 as it may be written - x - ; that is, the product of the 7 o chances of drawing a black ball at the first trial out of each bag separately. 759. By reasoning similar to that of Art. 757 it will appear that if p l9 p 2 , p%, ... be the chances of any number of independent events, the chance of the concurrence of all these events will be the product of these separate chances, 760. The investigation of the measure of the chance of any event simple or compound is thus reduced to depend on the calculation of the number of ways in which the event may happen or fail. In the case of a compound event, this is again reduced to the consideration of the number of ways in which each of the separate events can happen or fail. 761. Let p be the chance of a given event happening and q that of its failing in any one trial ; then p + q = 1, since the event must either happen or fail. It is required to find the chance of the event happening exactly r times in n trials. The chance that it will happen in one particular set of r trials out of the n and fail in all the others is evidently p r q n ~ r . The number of different sets of r out of the n trials I* is (Art. 490) equal to - ~ - Hence the chance that 762.] Probabilities or Chances. 501 the event will happen in any set of r trials out of , and fail in all the rest, is - == p f ' q f ' " This is evidently the r+ 1 th term in the expansion of (q + p)* (Art. 547). It follows that the chance of the event happening at least r times in n trials is the sum of all the terms in the above Binomial expansion beginning 1 with the r+ 1 th , while the chance that it will not happen so often as r times is the sum of all the terms preceding the r + 1 th . Since q+p = 1, we have (q +p) n = 1, whence it may be noticed that the sum of the chances of the event happening at least r times, and not so often as r times, is unity ; as of course it ought to be, since one or other of these events must happen. 762. As an example, let it be required to find the chance of throwing head exactly seven times in ten tosses of a coin. j Here the chance of throwing head being -> that of j failing is also - : thus the chance required is equal to The chance of throwing head at least seven times is the A lv 10 sum of the last four terms of the expansion (- + -) or is ^J 6' I ^Wj I ^VWa ^ 3~]^(~2) W h j~?V2^ V2/ h |T~[9 ^ix^cio.g.s 10.9 = (2) T^y + lT2- ,1 = (-) r 2 502 Probabilities or Chances. [763. The chance of throwing head less than seven times is the sum of the first seven terms of the expansion, or 10 ^ ^ 10.9,lN 8 xl 2 10-9.8 2 /1\ 10 t^\ f^\ 10.9,lNxl\ (2) + ^ (2) + T^ (2) (2) 10.9.8.7 AJ* xlx 4 10.9.8.7.6 ,K 6 1.2.3.4 2 1.2.3.4.5 ' 10. 9.3.7.6. 5,lx 4 ,l\ 6 1 (-) 1.2.3.4.5.62 10 10 + 45 + 120 + 210 + 252 + 210} 848 _ 53 = "2 15 " ""64* 11 ^ ^? The sum of the two chances and is of course unity. 64 64 763. The preceding articles have all referred to problems of what may be called direct chances. We have next to consider briefly the subject of inverse chances. Suppose that a given event has happened. It may have been produced by any one of a certain number of causes. It is required to find the chance that it shall actually have arisen from any particular one of these causes. Let P t , P 2 , . . . P n be the probabilities of the existence of each of the several causes, estimated before the given event has been observed to happen. Let p l , p 2 , p 3 , ...p n be the probabilities that the event would happen on the supposition of the existence of each of the possible causes. Then the chance of the compound event that the first cause should exist and the event happen as a consequence of it is, by Art. 757, P^p-j. Similarly, the chances that it will happen as a consequence of each of the other causes are Now it is in accordance with the ordinary behaviour of 764.] Probabilities or Chances. 503 reasonable beings to assume that phe chance of the existence of any one cause as the efficient agent in producing the given event is proportional to the chance, estimated before the event has been observed, that tlie event would happen as a consequence of the given cause. | Hence, if # 15 # 2 , # 3 , ... # be the chances required that the event actually was produced by the 1st, 2nd, ... % th causes respectively, we have P lPl (Art. 657). But the event has happened, and must have been produced by one of the causes. Hence and therefore P lPl . ~ and so on ; where 2 (Pp) is an abbreviation for the sum of all the products P 1 p , P 2 p. 2 , . . . . 764. Suppose, for instance, that a bag is known to con- tain six balls which may be either black or white. A ball is drawn and proves to be black, it is required to find the chance that it is the only black ball. A priori there are seven possible and equally likely cases, namely, that the bag may contain 6 black; 5 black, 1 white ; 4 black, 2 white ; 3 black, 3 white ; 2 black, 4 white ; 1 black, 5 white ; or 6 white balls. The chances of the existence of each of these causes, or the values of the quantities P 1 , P^ P s , P^ P 5 , P Q9 P^ are therefore -each - . The probability of the event happening if the first state exist, is evidently certainty or 1, so that jp 1 = 1. Similarly, 54321 504 Probabilities or Chances. [765. Hence 1 3 P -* 1 ) 2 _ 11 - x -> < 6 ' ^' G= 7 X e'^^ 7 "" * r* /> 6 6 (3 d '1 ^~ c.i P; i 4 + 34 -2 + 1 21 5 4 3 2 1 The chance therefore that the ball drawn is the only black ball is 765. One very important application of the theory of chances is to the calculation of the value of Ihejzxpectatiw of a sum of money, the obtaining of which is contingent on some as yet undecided event. The general rule will be best elucidated by considering a particular case. Suppose that there is a lottery with a + b tickets altogether, a of which give a prize of ^c, while the remaining b are blanks. The whole sum of money to be divided among the purchasers of tickets is thus ac. Any person who chose to purchase all the (a -f b) tickets might therefore pay this sum with a certainty of not losing. As all the tickets have an equal chance of winning the prize, and as the fair price for the (a -f b) tickets is seen to be and -- , is the chance of his winning a 768.] Probabilities or Chances. 505 prize. Hence the value of his expectation is measured by the product of the chance of his getting a prize into the value of the prize he may get. The general principle to which this example leads is : The mathematical value of the expectation of a con- tingent gain is represented by the product of the sum of money which is possibly attainable into the chance of getting it. 766. The proviso that the loss of the sum of money paid for the ticket is a matter of no importance to the speculator is a very important one, as distinguishing between the moral and mathematical values of expectation. In the book ' Choice and Chance,' by W. A. Whitworth, will be found a very full and clear discussion of the difficulties of this subject. 767. One of the most important practical applications of the foregoing theories is afforded by the subject of Life Insurance. A Company for the Insurance or Assurance of Lives is a society which, in consideration of regular annual payments during the life of the assured person, undertakes to pay over to his representatives a certain sum after his death. The primary basis of the operations of such societies is the Life Table. From a considerable number of observations of the mortality among persons of the class who will be clients of the society it is found that of a given number 1 Q born at any time, a definite number l x will be alive at the end of x years. The series of numbers , l lt 1 2 , / 3 , ... l x ... to the last year at which it is found that any persons survive, constitutes what is called the Life Table. It is assumed that such a table will represent the actual mortality in the future among persons similarly situated to those who were the subjects of the observations from which the Life Table was formed. 768. It follows that of l x persons alive at the age ef x 506 Probabilities or Chances. [769. only 4+i w iU survive to the age of #+1 years. Hence the chance of any one person aged x years living for one year more, is estimated by the fraction -^ The *z chance that he will die within the year is therefore Similarly the chance that a person aged x years will live for n years is j^ > while the chance that he will die before ^x the expiration of n years is 1 ^~ or -' , x+ -* The latter result may also be seen from the consideration that l x l x+n is the number of persons out of the l x who die within the n years considered. 769. The first problem is to find the formula giving the mathematical value of the expectation of a sum of A to be paid at the end of the year in which death takes place to the representatives of a person aged x years. The present value of a sum of A certainly to be paid after n years is, by Art. 537, -n^* The chance of the payment being made at the end of n years exactly is 4+n-i-^+n. HenC6} by Art 765j the present value of fr the expectation of the payment of A at the end of n years exactly is ^-^-.j,. 770. The above expression can be put into another more convenient shape. Let p x t denote the chance that a person aged x will live for x years ; then^ n = -y^ Let also v be substituted for -^ - Then the value of the li 77i.] Probabilities or Chances. 507 expectation of A being paid at the end of n years, which is l ^-~ l ' +n ' Becomes ^- VA. The sum of the values of this expression for all values of n, from unity to the last age in the table, is the present value of the assurance. Thus the present value of an assurance of ^1, which is the sum usually taken for con- venience of calculation, may be denoted by where the symbol of summation refers to n. 771. As the payment to the assurance company is not usually made in one lump sum but by a series of annual payments during the life of the assured, it is necessary to investigate also the present value of such a series of pay- ments, that is, of an annuity of a fixed sum payable during the life of the assured. The present value of the payment of ,! *& a 27. -r-. 28. ? 29. 1. x 2 x + * a 4as ~~' q 2 32. -r-7 - i - ,- 33. ^r abc(a + b + c) a; 3 ^, _ 34. - (see examples 17 and 18). 35. 6 36.0. . . . 7 CHAPTER VIII. 1. a 2 . 2. 1. 3. a^b^c^. 5. 6. a;- 6 +7 a~ 4 -6 4. 7. 8. a + a*6^ + 6; a 2 + a^ + 6 2 . 9. 10. 16a;-t-12a;-^-f+9/-i 11. as 2 a?S + a? a 518 Answers to Examples. 12. *+?/* si 13. 2+2ce*2/* a y. 14. .r*y*+l. 15. a'-a 2 . 16. (- !)*(- 2)* (a?- 3)*. 17. --' C 18. - 20. (a + &)*. CHAPTER IX. 1.4/2; 2 =2,y= 1; (2) a; = 3, y = 2. 3. wubv, bcu 2 , cwuv. 4. a&c+2?m0 aw 2 fo 2 cw 2 . 5. 3abc-a s -b*-c 3 . 6. 7. Use Article 352 (a) : the result is a, 6," c a or 6a&c-2a 3 -26 8 -2c 3 . c, a, b 8. a= 3, ?/ = 1, z= 2. 9. The third equation is derivable from the other two. 10. x = y = z = a 2 + 6 2 + c 2 6c ca at. 11. a? = a, y = 6, = c. 12. cc = y and z. 13. * = (c-a)(a-b) m + n n + l and symmetrical values for ' 1. 4. 7a;- 6. 2a 2 - 8. 2 3 - CHAPTER XIII. 2. 3a-26. 5. x z - 7. 9. # 3 - 3. a; 2 - 2. 522 Answers to Examples. 13. whence the result will easily follow. 43. q = db y p l = m Hence p 2 +#, = - 1 2 -I and since (mH ") =w 2 H -- s + 2, the result follows. V m' m 2 44. See examples 37, 38. 45. 121. 46. 6 and 8 for a shilling ; or 8 and 10 for a shilling. 47. 4, 7. 48. 54 yrs. 10 yrs. 49. 55 by 88 yards. 60. He goes at 4 miles per hour, reaches B at 1 1 a.m. ; returns at 3 miles per hour : distance is 12 miles. CHAPTER XV. 1. 3x+5y. 2. 4/3. Answers to Examples. 525 11. The expression on the left-hand side of the first equation must be identical with (x af(x ft), while that in the other equation must be identical with (x a) (x(3f. 12. The expression ar 5 + 3px z + 3 qx + r must be identical with (x-a)(x-b)(x-c); whence a + b + c = 3p, be + ca + db = 3 q, abc = r ; applying the formulae of Art. 414, and referring to Art. 282 and the value of o> given in Art. 277, the result is easily arrived at. 13. Work out the highest common divisor of the two ex- pressions and equate the last remainder to zero. If the equa- tions have a common root, the expressions must have a common factor (Art. 454), or see Art. 473. 14. It is easily seen that JT+ 7+^= ( a + 6 + c) (x + y + z) and then use the result of Art. 282. CHAPTER XVI. 1. a; = 2, y = 5; x = -2, y = -5; x = 34, y = - 11 ; x= -34, y = 11. 2. x 3, y = 4 ; x = 4, y = 3. 28 55 O O 1 . . ~13' 26 4. 3, 1; -3, -1. 5. 3, 1; -3, -1. 6. a? = 2/ = ^ii -, or x= 1, y = 2; as = 2, y= 1. 4 tf? 7. 85=3, y = 2; x = -3, y = -2; x = 22, y = -; a; =-22, y = y 8. a; = 2, y = 6 ; aj = 5, y = 3. 9. x = 7, y = 4; = 4, y = 7. 526 Answers to Examples. 11. 12. 13. 14. x = 8, y = 4 ; x = 4, y 8. 12, y= 10; a = -9, y = 11. 11 43 -,y = -;x = ~--, ,,= --. _ a A/26 2 - a 2 _a+A/26 2 -a 2 2 ~~' ^~ 2 17. x 16, y= 4; a? = -4, y = -16; a? = 6 + 3vT3, 18. x = = 19. cc=4, y= 5; a? = 5, y= 4. 20. 21. 3, y=4; aj=4, y= 3, and imaginary values. 9, y= 3; x = 3, y=9. 22. = ab + ac-bc 23. x - , &c. 24. 25. 26. 27. 28. 19 3 -2, y=; aj =~ , y = 0. = 147, y= 140. , &c. 2abc ab, y a~ l b 3 , z = p-- 2, 2/= 4, = 7. x _ y _ z _ ' ~~~ ~~ ca + ab 31 a(b c) b(ca) c(ab) a 2 (c =0 Answers to Examples. 527 o t 32. * = 1, y= 1; *=, y = ! put y = we. 33. 2 and each of these factions = or x y b + ca c + ab 18(bc + ca + ab) nt i/ 9. 34. y or 0. 35. x, y, z must have the values (b c), (c a), +(a b). 36. 7a 3 = 3a(6-c) 2 . 37. The values of x, y, z are 16, 16, 8 or 10, 10, 20 in any orders. 38. 6 4 (a 4 -6 4 Y 2 =2a 4 (& 8 -c 8 X 39. 2 (6 4 V3b*-c 8 ) = (a 2 + 40 - = 41. 36, 16; 24, 4. 42. 132; 213. 43. 900 men in 100 ranks, or 20 men walking in single file. 44. 300 yards ; A runs six yards for B'& five. 45. 1, 2, 3. 46. A takes 11 days; B 22; C 33. 47. 10,000 at 5 per cent.; 7500 at 4 per cent.; 2500 at 6 per cent. 48. 16, 18, 20. 49. A in 6 hrs., -B in 3 hrs., C in 2 hrs. CHAPTER XVII. 1. 26.25.24. 2. 14.13.12. 3. . J . 4. 72. 528 Answers to Examples. 1100 199 's> 88 6 ' 72 ' 24 ' 7. See Arts. 496, 497. 8. n = 6. 9. n = 8. 10. n = 5. 12. [5, or if right-handed and left-handed arrangements are |5 counted as identical, -==- 2 13. 1 7i 1 or I 1 7i 1 according to the interpretation of the word different arrangements. 14. . , | ra 1 \nl, or one quarter of this number. 15. [m \n 6 |8 |45|50 1111 II fi 17. 11 110 111 '(ll) 2 '^' >. J^^-r^r-15; 21.20.19.5.4. CHAPTER XVIII. 1. ; - g 2. 2-l, w 2 . 3 + n ~~ ~~~ 5. 13-271, 12n + 7i 2 . 6. 2 Answers to Examples. 529 7. 5 w, 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 19. 20. 23. 26. 28. 30. 31. 1, n i n 0, ^ { CT (3-^) + (w 1)6} " 2(a-6) a- (n 2) 6 w{2a-(w- 17-4n, 37^-13 3 or THE J 530 Answers to Examples. 32. (2?i+l) 2 , 6. 33. 0+1 + 2 + 3 + .... 35. 280 days or 229. The negative answer implies that 229 days previous to starting they would have been together had the arithmetic progression been continued backwards and A'B motion consequently been westward. CHAPTER XIX. 1 1023 2"^ ' "512" 3-V/30/3 + 1) , -i- -^ 4 + 2-/3 10. {1 -(-)}. 14. 1+1+1 + .... O 6O 18. 6. 22. The expression = 2. 26. 2, 6. (-I) H762 19683 11. l 2 |l_JLj,i2. 15. 10 terms. 19. r - 2, 7i = 3. 25. re = 2. Answers to Examples. 531 27. The condition gives (ac) ac , = l b , and ac must = b since a, b, c are positive integers, whence the result follows. 3O. P (l H ) , lOOr being the nominal rate per cent, per annum. 3. CHAPTER XX. 672 a; 2 560a; 3 ~ 120 2n+l jit |2w |2n 35 21 7 1 + ~ + - + - ? ; 6 4 56a 3 6 6 + 8. (l + 2) r . 9. 35x2 3 x3 4 , the 5* h ' term. 10. 3. 11. 48. 12. (aj + a)" = 4 + J? and (aj-o)" = -4-A 13. Use Art. 548 (3) and (4) ; prove 2"-" = 16, m = 6, n = 2. M m 2 532 Answers to Examples. \n 14. By Art. 558 /(n) = - - a n ~ r b r 7* I 97* ^~ 7* Similarly /(n+l) = r -.a" /(n+1) ^+1 /(w) n whence J v = - . a ; similarly -^p -r = -- a, f(n) n r+1 r /(^ !) w r whence the result can be easily deduced. 15. Use Arts. 492 (1), and 558. 16. See Art. 559 : the coefficient of x r in |r I m + n r 17. Use Art. 559. In (4) n and r must be both even or both odd, otherwise the series = 0. 18. Compare Art. 543 and Example 12. 19. (l +a? + 8 ) = 2. , a g+2r , (i) i Also ( 1 + cc 2 + a? 4 ) = (1 + x + 2 ) (1 x + 2 ) ; and equate co- efficients of any power of x in (3) to coefficient of same power in product of (1) and (2). For second part take (1 and put x = 1. Compare Art. 548 (2). 20. _ / i / ' Vac if y = whence the result follows. Answers to Examples. 533 21. Let (2+ >/3) n =7+7 T where 7 is an integer and F a proper fraction. Then, since 2 Vs = ~ j=* (2 A/3) n must 2+ v 3 be a proper fraction. Call this F. Hence (2 + A/3) n + (2 - >/3) n = 7+ JP+ 7". But it is easily seen that the sum of the expansions on the left is integral and even. Hence F+ F' must = 1 and 7 must be an odd integer. 22. Va 2 + 1 a = - and solve as 21. CHAPTER XXI. 1 l^^,^ __ ~3. 1 _ yJ. 2* 8* 16^ 2 a?f 8 33 7 429 7 -- a; : -- x" 2 11 2 11 2 - ^r 2 - 4 -^-^ 21-17.13.9.5.1.3 3 . 1 + 2 r 24 4 7 1_7 4. The r+ 1th terms are x r ; x r ; 1.2 [4_ 5. The 8th term. 6. 7. z = (i-3x+3a?-x*)(l + x)-\ Hence the coeffi- ( 1 -f a;) 2 cient required = 1 x coefficient of x* in (1 +a?)~ 2 3 x coefficient of x 3 in (l+cc)- 2 + 3 x coefficient of or* in (1+a)- 2 1 x coeffi- cient of ain (1+x)' 2 = 5 + 12 + 9 + 2 = 28. 534 Answers to Examples. 8. ^24 = 5 (l-^)*= 4-89898; 4/31=2(1-^7=1-9873. 9. 4-1231. 10. 12. The series is the coefficient of x r obtained by picking out the term involving x r in each of the binomials on the right- hand. 13. Use the identity then (/> (r) is the coefficient of x r : the final result is obtained by putting 2=1. 14. 15. 33 a; 4 . 16. 0. 17. the given expression = (3a 2a?)(27a 3 _ 875630 6561 19. lw-1 20. Coefficient of y*- 1 in (1 +2/) n ~ J is -==- ; coefficient ~~~ L of y n x * is T . ; the sum of the products of these for | a; \n x 1 all values of a? from 1 to T& 1 is the coefficient of y n ~* in \2n 2 n _\ \2n (1 + y)2- or = ' - = ^- !_.-!=-; whence \n-2\n 2(2n-l) [n\n 9 1 n-i | a? \n-x-l \xl \n-x 2 (2n 1) [n [n whence the result follows. Answers to Examples. 535 CHAPTER XXII. 4 _1 2. 3 21 4 I$_L_-i. - + - [ where o> is one of the cube - 3\l- x 1-ax l-a> 3 x) roots of unity (Art. 277). The last two fractions can be joined so as to give a real form, and the result becomes 1 2 + x 4(1 2 n cc n ) lx n K /on+l 1\a;-l. ^ } _ ' l-2co l-x 6. a n 2 a n _ 1 + a n _ 2 = 0, For the second part put a = b = I and x = 1 A, and expand all the terms in the numerator by the Binomial theorem in powers of h. The terms involving lower powers of h than A 3 will be found to vanish, while that involving 7i 3 reduces to n(n+l)(2n+l) hs 11 The denominator becomes h s ; hence the fraction reduces to II and making x = 1, and therefore h = 0, the required sum is obtained. 536 Answers to Examples. CHAPTER XXIII. (2n+l)(2yi+3)(2?i+5)(2n+7)-1.3.5.7 4 MJ ___ 1 | ' 2(2.3 (+*)(+*))" 1(J ___ 1 ) '' 2(3.5 (2n+3)(2w+5)} 7 ' use Art. 610. 10. 1 +2-'; 11. (i) A geometrical series; (2) a series whose w th term is of the form A'*?'* + B V~ 2 + . . . ; (3) "= use Art, 611 13. 2^ + 3^1-(-ir; 2 n_ 1 + 3^fn + l-(-ir 14 n n( + l)(4n+5) ' n+l 3 Answers to Examples. 537 16. 18. o 3(+2)(*+3) 21. 7^" v ~ -' , D being the debt and R the amount of l in one year. CHAPTER XXIV. 1. 5. 2. 4. 3. 3, 3. 4. -7781513, -9030900, -9542426, 1-3802113, 1-5563026. 5. -0484550. 6. Iog 10 12 = 2 Iog 10 2 + log la 3, Iog 10 1 5 = Iog 10 3 + Iog 10 5 and Iog 10 5 + Iog 10 2 = Iog 10 10 = 1, whence the values of Iog 10 2, Iog 10 3 and Iog 10 5 can be found: the values are given in examples 4 and 5. 7. 0-1505150; 1-3656367. 8. 0-1890480, -9912260; 2-5440680. 9. ?; - 10. 1-8186440. 11. 1-1672269; 1-9119544. 12. In the series for log e (1 +z) put z=l. 13. 2(1 -log. 2). . 15. Put - for z in Art. 637. x 538 Answers to Examples. 16. Put -5 for z, in (/3) Art. 637. as 17. ae = &* 1 : use the series for log c - in Arts. 6, 38, n putting n = ac. 18. The general term 231 (2n l).2n.( the sum of which gives 3 log e 2 1. 19. The capital at end of 5 years can be shown to be A -i that at the end f 10 y ears 209 Equating this last to cA, a quadratic equation in (1 -fa?) 5 is obtained which gives (1 +cc) 5 = , whence log (!+*) = ^ > and 1 + r can be obtained from the given logarithms. 20. Take logarithms and use Art. 634 (5). 21. Art. 634 (5). 22. If log a a = logpft =...= x; a = a*, b = (3 X , ... whence 23. The harmonic mean of n quantities is the reciprocal of their arithmetic mean (Art. 705). 25. Assume a == m x , b = m y , c = m*: the two conditions x z z y .... give xy = yz and --- = --- , whence eliminating x z y y x' we obtain ly 3 3y*z 3yz*+2z? = 0, whence y = 2 or 4y* + yz 2z* = 0; from the latter 2y z 1 . 2y s s 3 as a 3 - -- = - and - --- = - > or --- = - z y 2 z y 2 2y2 Answers to Examples. 539 CHAPTER XXV. 1. and 2. Assume ? = 4 = # 3. 16 : 25 ; 1 : 1. 6 a 4. 9, 12. 6. 18, 24. 6. As (1) and (2). For examples (7) to (12) use Art. 659 (1) and (2) : thus in (8), y + z __ z+x __ x + y p(y + z) + q(z + x) + r(x + y) 3b-c ~~ 3c-a~~ 3a-b ~ p(3b-c) + q(3c-a) + r (3a-b)' cc I *// -4- % and by taking p, q, r each unity this becomes - ~ - ? while by taking p = b + c a, q = c + a b, r = a + b c, it becomes after reduction, ^ ^ -^ ; the difficulty consists in the proper choice of the quantities, p, q, r in each case. 13. 2, 8, 32. (2x y) + (5y x) (2y + x) 14. Each fract.cn = = V, 3 3 x= 4, y =!, = -, y= --. 15. Use Art. 659(1); 2,3,5,6 or -^, -^, 5, ^ l 16. Each of the fractions _ yx(x + y-z) + xy(z + x-y) __ 2xyz ylogx + xlogy ~ ylogx + xlogy' and so on with the other possible pairs. 17. In ratio 36 : 35. 18. It is easy to prove that A r H n _ r = db. 19. The conditions are equivalent to x + y = kg and as 2 + 1/ 2 = &V ; whence a? = Az, y = \LZ; k, k', A, ju being constants. 20. Rate of^i irate of ::t':f + 2t- T. 540 Answers to Examples. 21. 7223.04. 22. 1375 cubic yards. 24. 10 miles an hour. 25. 30, 36, 45. 26. vol. of silver in first : to vol. of copper : : 57 : 7; in the second : : 1 : 7. 27.|. 29.8,12,14,21; orO, 0, ~^~- O A 2 CHAPTER XXVI. J_J_ J_I ill 2 - 'TTr+3T3' '2*7' '2"+2T3 4 z+J- JL JL JLJ__LJLJL. r 6+ 1+ 5+ 1+ 5+ 1+ 1+ 1+ 10 5 14 J-l-J II. 1+ 1+ 17+ 1+ 6 e J L 1 1 J_I ' 2+ 1+ 50+ 2+ 2+ 2 7 J_ J__L_L JL ' 1+ i + 3+ 2+ 16 3 10 43 225 1393 '2' T' 30' 157' 972 " 9. The 49* of the first with the 54* of the 2nd : the first bell rings 108 times in 63 minutes, the second 119 times. 248 10. 27, days. 11. Use Art. 675. 12. Use Art. 671 (1): it can be shown that N* N" -jy, Answers to Examples. 541 13. If q 6e the value of each quotient, it follows that N r+l N r -N r N r ^ = qN r \ whence the result follows. 14. Prove by induction as in Art. 671. 15 3 +' 16 - 4 + 4T T iT sT^' 4th conver g ent ri . and the I 1 error lies between - ____ and 136x1121 136x1257 J_ _1_ J_ J_ J_ J_ _1_ J f "" 18 - " 19 ^ 37 - 4 18. 1. / o 20. The positive root of (6 + l)aj2_ ( a b + a + b-l)x- (a+l) = 0. 21. The second fraction is the positive root of = 0, the roots of which are the reciprocals with the signs changed of those of the former equation (see Arts. 414, 424), whence the result follows. M.J--LJL-LJL }1 ,?,_L,^,^,^L. 1 1 h . 322391 10+ 10+ ...' 104030' 111111 , 4389 . Qth convergent 1 + 9+ 2+ 10+ 2+ 10+...' 4830 24. The equation gives x = y-\ , whence y = x y+x x+y 542 Answers to Examples. 25. By means of the relation of Art. 671 it is easily proved that for any three consecutive even convergents and for any three consecutive odd ones ^ 2 n +1 -(&+2)^ 2n _ 1 + ^ 2B _s = 0, similar relations holding for the denominators. Hence, by Art. 593, it follows that each of the quantities -fl^n-n ^m ^ 2 n 1> ^2n can be expressed in the form Ay '* -\-ByJ 1 , where A, B are constants and y lt y^ the roots of the equation y *-(ab + 2)y+l = 0. The constants for each can be determined from the first con- vergents and then the required results will follow. 26. Work as in Art. 678. 27. p n = ^apn-i+Pn-zi whence, by Art. 593, where y l and y 2 are the roots of the equation y*2ayl *=. 0. 28. Use examples 14 and prove by induction. CHAPTER XXVII. I. a?= 3, y = 2. 2. a? = 7 + 13*, y 3. x=7t,y = 8 8. 4. x= 4, y = 6; x= 11, ?/ = 1. 5. x = 5, y = 3. 6. Five solutions. 7. m = 75 if zero solutions are allowed, otherwise m = 83. 8. Seven solutions. 9. x = 99 112, y = 1 + 10 1. 10. He must pay 3 guineas and receive 21 half-crowns. II. If x be the number of florins A pays to B and y the number of half-crowns, while z and u are the numbers of shillings and half-crowns B pays to A, the equation is w 2 = 25. Answers to Examples. 543 Here y u is merely the balance of half-crowns that passes from A to JB, and may be treated as one quantity v : the equation thus becomes 4x+5v2z = 25, which can be treated as in Art. 696, it being borne in mind that the limits of x are and 15, of y u, or v, 34 and +8, and of z, and 5. Then 4 a? 2z = 5 (5 v). Hence 5 v must be even and can only have values between 3 and 9. Hence 5 v may = 2, 0, 2, 4, 6, 8, and we have the series of equations 2xz = 5, 0, 5, 10, 15, 20, the number of admissible solutions of which can be ascertained ; total number 1 6. 12. 24 oz. ; 11 sovs. 79 gs.; 32 sovs. 59 gs.; 53 sovs. 39 gs. ; or 7 4 so vs. 19gs. 13. He was born in 1741. 14. 37x2 or 14x3. 15. Depends on the solution of the equation whence x = 4t : results are, 71 st page of 5 th and 7^ volumes, 38i of 13th and 18th, & c . 16. In 1 4 different ways : it is the number of solutions of 3o3+2y + 2 = 16 consistent with x + y + z not being greater than 9. 17. 52. 18. 4, 2, 1 or 8, 4, 2. 19. 292. 20. 63 -2. 21. 78, 65, 50. CHAPTER XXVIII. 1. Use Art. 705. 2. Art. 703. 3. 4. -^- : use Art. 705. 4 71+1 5. .1 544 Answers to Examples. 7. , or x , 8. (i) 7 ^"" 1 = l+r 2 + r 4 +...+r n - 2 , use Art. 705. (2) = 1+r + r 2 +...+r n - 1 , use Art. 705. 9. cc 2 + 2/ 2 >2a?/, y z + z?>2yz, 10. The former. 11. The first fraction >1, the second is less than unity since 12. 13. (2x + a)a x = J<(2cc + a) (2x + a) (4a 4as). 14. A.M. of ^> quantities = x q ~ r , and ^quantities == a; r ~ p , and r quantities = x p ~ q > G. M. 15. Use Art. 705. 16. 44 millions; 3s. 9d. per Ib. 18. = (ax + by + cz) 2 + (ay - ^) 2 + (az ex) 2 + (cy bz)*. 19. x= 4/2 4 gives a minimum value, x = in which case, as there is no information, A's chance of winning is Hence ^t's whole chance of win- /?o q 70 ning is + = , thus the value of his expectation is 780. 9. ; the condition that a number is divisible by 11 is 21 given in Art. 718. N n 2 548 Answers to Examples. 331 11. -; ; -- The value of the expectation in shillings is the sum of the terms 6543 2/2 K 6 54321 *i"F J8 .5 SVj ' + 3) + S 7'6*5'4*3' 2 ' which reduces to 22f shillings. \mn 12. The total number of distinct distributions is - If the two balls be called A and 5 and A be put into any one box, the number of arrangements which put B into the same \mn box is - ' Hence the chance of their being \n 2\mn n+2 m(\n) m together is Ti 2 \mn n+2 >. i 13. If e lt c 2 , ... c n be coefficients of x, x 2 , ... x n in (1 +x) n the chance required, n being the number of balls of each colour 2 2n -l But l + c 1 a + c 2 2 + ...+c n 2 is easily seen to be the term inde- pendent of a? in (1 + x) n (1 H ) or - -- ~~ Hence ^ x/ x Inln The chance required therefore = - (I39) 2 4(|39) 2 14 /T\ VI - } . / 2 N vl ; ^ J) |26|52' () [26[52' Answers to Examples. 549 ,,i. ,..?. IX (.)ij <>! 18. i - 19. | 20. 9s. 7\d. 3 o 24. This can be obtained by the principles of Arts. 769 and onwards. The chance that after a? years A will be living and B dead is - Hence the value of A's ex- 86 m 86 n pectation of a payment of XI at the end of cc years is (86-m-a?)a? (86-m)(86-w)' By summing up these for all values of a? the required result is obtained. MISCELLANEOUS EXAMPLES. MISCELLANEOUS EXAMPLES. 1. Simplify 2. Prove that a(a x](a 2x) = (a b)(a b x)(a + b(b~x)(3a-2b-2x). 3. Prove that x (x + y) (x + 2y) = ( x -y)(x-2y) (a?-3y) + 9y(a;-y) (x- 2y) + ISy 2 (x- 4. Simplify 5. Multiply 2a + 36 by 3 a 46, and divide a? 4 by 2 -2^ + i/ 2 . 6. Shew that {26c(a-6)-(6 2 + c 2 - 2 )(a-c)} 2 {abc (b+c a)(c + a b)(a + b c)}. 7. Multiply a 3 _7a6 2 -66 3 by a 3 -lla6 2 -66 3 , and divide the product by a s - 6 a 2 6 + 7 a& 2 + 6 6 3 . 8. Multiply 7a 4 -3a'&-2a& 3 -26 4 by a* + and divide the product by a 3 6 3 . 0. Multiply a 4 3a 3 x 9a 2 2 36aa; 3 +14x 4 by 2a 2 and divide the product by a 2 6 a# 4- 2 cc 2 . 10. Multiply 4x 2 5cc 4 raj^ + Bas" 9 by 3x and divide the product by 3x- 10+ lOcc" 1 4aT 2 . 554 Miscellaneous Examples. 11. Prove that 12. Simplify + # / x \ z / /# a 13. Prove that -n)z} {(m-n)z + (ml)x] {nl)x + (nm)y} 14. Simplify c 1 p c 1 -^+ + 15. Simplify (xa)(xb)(xc){bc(x-a)-[(a+b+c)x--a(b+e)]x\. 16. Simplify 17. Shew that (1 +xz) (1 +yz) z - {(1 -a) (1 - = 4 (a + y xy) (xyz 3 + xyz* + z). 18. Simplify 19. Simplify o a a b a b* a b TH ^ T 2 T o a 6 a" o a Miscellaneous Examples. 555 20. Simplify )-f 4c-{2a (6 + 2c a)}]. 21. Divide 6(a 3 -a 3 ) + aa(a 2 -a 2 ) + a s (a;-a) by (a + b)(x-a). 22. Find the Highest Common Divisor of cV + bcx% + c (a + 6) cc + 6 2 x and 5cjci + (6 2 + c 2 )a; + 6(a + c) 23. Find the Highest Common Divisor of 24. Find the H. C. D. of o? 4 + 4o; 2 +16 and x*- 25. Find the H. C. D. of at a; 2 y? + x% y xy* + x^y^y^ and x^ x^y^ x% y + xy% + x* y z y^. 26. Find the H. C. D. of a; 4 +2a5 3 -3ar J -4a;+4 and x* + 2x 2 -x-2 27. Find the L. C. M. of ar J -6 2 +llce-6 and x 3 - 9 x* + 26^-24. 28. Find the H. C. D of and 29. Find the Highest Common Divisor of cc 10 and 4or 5 +7or J 3# 15. 30. Find the H. C. D. and L. C. M. of and 31. Find the H. C. D. of 5x Q and 7a; 6 - 556 32. If Miscellaneous Examples. Z =r 0, find the H. C. D. of and 33. Find the H. C. D. of and 29^- 34. Find the H. C. D. of 190# 5 -235o; 4 -5ar } + 70^- and 121a 4 -214ar J -fll5ce 2 -23a;+l. 35. Find the H. C. D. and L. C. M. of and 6a 4 -5a 3 6-13a 2 6 2 +17a& 3 -5& 4 36. Find the H. C. D. of 5a 3 + 38cc 2 - 195^-600 and 4aj 3 -15.T 2 - 37. Shew that 38. Shew that ~~3 ( 2 39. Simplify 15 40. Shew that 41. Shew that -2i\ Solve the equations , where i 2 = 1 Miscellaneous Examples. 557 43. 44. 2 3 12 ~4 1111 3 o:5 # 2 a; 4 1 171 45. -T-- + - -=- + - 46. C 2 -1 X+l 8 oj+l 5 x 7 x-2 2(x-l) 3 47. x 3 = - 48. (a? l)(a; 2)(aj-3)=2.3.4. 49. (aj-3)(a?-4)(aj-5)(a!-6)= 1.2.3.4. 51. a;- 52 " 53. 54. 55. 56. 57. = x(x 2 +5mn). := ma. 20 = 34. a? y = 558 58 59. Miscellaneous Examples. I A** + 2 ) (2/ ( a 2 ) = (a + 60. 2-10 = a + b. = 0. 12 6 x x2 5 328 61. + 5-fl? 4 x 2 + x _ ox ax !- !- es -V + 3 14-2ax_ W-x 4 8 / fi 64. 65. 66. 67. ax = 2x-7 5 3 (a? 5) 2x-8 2 &V f - -2 12a;+34a;+3 12 7x-l~ 16 70. x 2 + ?/ = 66, x*-y z = 11. 72. Miscellaneous Examples. 559 / 36 18 1 r'*' " + '" , 74. = _= ___ = ___ and 3+w-2-4y = 8. = a^= 6. (b c)Vx + (c a) Vy +(a b) Vz = 0, yz zx xy 76. 77. Find the value of x from the equations = 33, C 78. Solve the equations : cx-by + az = 2 + c 2 , 79. Eliminate x, y, z, u from the equations x +y +z +u = 0, 80. Form the equation whose roots are the squares of the sum and difference of the roots of + (m 2 + 7i 2 )= 0. 81. If ^+px s + qx z + rx+l =0 and a; 4 + rx* + qx* +px + 1 = have a common root, shew that p + r = q + 2, the symbols being supposed essentially positive. 560 Miscellaneous Examples. 82. If each pair of the equations x 2 +p 1 x + q l = 0, a?+p z x + q t = 0, x*+p s x + q 3 have a root in common, shew that 83. Eliminate x, y, z from the equations x v z V z x - 84. If x* + px + q = and x* + qx*+p = have two roots in common, shew that ^ 3 +^ 2 + 7jt? 1 = 0, p and q being positive. 85. If ax* + bx* + c be divisible by x* + hx + h 2 , then will ac = 6 2 . 86. Prove that ; 2 ) (ah'* + 2 &A'&' + c&' 2 ) - { aM r + b (hk' + V 87. From the equations (i) ex 2by + az = 0; (2) h?x + 2hty + t?z= 0; (3) A'*aj + 2VA'y + *"*=0; (4) (5) a/i' 2 + 2ta' deduce that (ac-6 2 ) (Wh'Kf = 1. 88. Eliminate , y, from the equations cc 2 /a V 2 ^^ & -+ y -=l; -+ = m; - y x 2 zx y l xy 89. If the two expressions # 3 -^-px 1 + qx + r, x 3 +p'x* + q'x + r, have the same quadratic factor, prove that r r _p'rpr' __ q'rq 1 / ^ p-p'~ q-q ' r-r Miscellaneous Examples. 561 Shew also that the third factors are x + - r and _ ,/; and that the quadratic factor is P-P P-P 90. If x 3 + ax* + bx + c be divisible by cc 2 +^+3', prove that 01. If a + 5 + c = 0, then (1) (a 2 (2) o 2 (g-ft) (5-<*) + (5-c) (g-a) + (d-a) (a-c) _ a-c (a-c) ( c -d) + (b-d) (d- a ) + (d-b) (b-c) " 6-ci ' shew that either a + d = 6 + c or (a b) (c d) = (bc)\ 93. If a + 6 + c = 0, and a (by + czax) = b (cz + ax by) = c (as + 6y cs), then a?+y + = 0. 94. Find the values of a and 6 in order that may exactly divide ~ xt. 95. If x = - j then x^ (6 a + c) a 96. If (a + 6) 2 = c(a-6), prove that (a + 6 c) = 8a6c 2 . 97. If , y, s be unequal, and if y / v _^\ 2 then will 2a 3a; = ^ - -, and cc + y + z = a. o 562 Miscellaneous Examples. 98. Prove that the result of eliminating a?, y,z from the equations aoj 2 + fy 2 + C2 2 = ax + by + cz = yz + zx + xy = 0, is abc = (6 + c a) (c + a b)(a + b c). 99. Solve the equations : (i (3) a + a 100. If a? 1} cc 2 be the roots of the equation a a cc 6 c find the value of (x 1 a) (x 2 d) without solving the equation. 101. Solve the equation (a + a?)-*-6 (a 2 -a 2 )-* = -5 (a -a)-*. 102. The rational values of a? which satisfy the equation y* = a? are given by the formula cc = (- -- ) , where p is any integer positive or negative. 103. If as 2 + 2 ay 2 is a square, a 2 + ay 2 is the sum of two squares. 104. If a and /3 are the roots of ax* + bx + c = 0, shew that the equation whose roots are z and z is 105. Solve the equations Miscellaneous Examples. 563 27 T x+y 15 Y ; y + z (x z)(x y) (xz)(yz) xy __ __ a ^ *MI^ ^B^M ^ ^^^ X\ ^"^ """ TJ" ' (x-y)(x-*) + (x-y)(*-y) X Z (3) (4) H 106. If a and /3 are the roots of the equation oo? 2 -f fo + c=0, prove that the equation of which the roots are a 4 + /3 4 and - (a + /3) (a 3 + /3 3 ) x+z z+y 107. Prove that 108, Prove that 109. Prove that c) 2 c 2 (a 1+cc 234 1 2+a? 3 4 1 2 3+w 4 1 2 3 4+a? a* b* (a 110. Prove that the determinant 1111 a /3 y 8 a 2 /3 2 y 2 S 2 a* j3 4 y 4 8 4 = (a-8)(^-8)(y-8)(a-y)(^-y)(a-^)(a + /3 + y + 8). 111. Reduce the equation a 8 6 3 c 8 + A) 8 (6 + X) 8 (c + A) 3 (2a + A) 8 to the form of a quadratic in A. 002 =0 564 Miscellaneous Examples. 112. Prove that mn , m 2 -fra7i, nl = (mn + nl + lmf. I 2 + Jra, Im + w 2 , foi 113. Given V+y 8 *^ 2ys 2af gay 55 0, prove that -v^sc -f Vy + Vz = 0. 114. Investigate the necessary independent conditions for the coexistence of the four equations ex -\- az & C and shew that they can be put into the form 115. Two pedestrians start at the same time from two towns and each walks at a uniform rate towards the other town. When they meet, it is found that one has travelled 96 miles more than the other, and that if they proceed at the same rate they will finish the journey in 4 and 9 days respectively. Find the distance between the towns and the rates of walking per day. 116. Two boats A and B row a bumping race against a stream flowing 1 mile per hour; in still water they row re- spectively 13-2 feet per second and 266 yards per minute: B gains 44-3 feet on A but does not bump. A time race is rowed down the stream over A' a course. By how many seconds will B win 1 117. A farmer sold 10 sheep at a certain price, and 5 others at 10s. less per head. The eum he received for each lot was expressed in pounds sterling by the same two digits. Find the price of each sheep. 118. Two persons A, B walk from P to Q and back. A starts 1 hour after B, overtakes him 1 mile from Q, meets him 20 minutes afterwards, and arrives at P when B is -f of the way back. Find the distance from P to and the rates at which they walk. Miscellaneous Examples. 565 119. In a match a person fired 7 shots at each of three ranges. He made an equal number of centres at each of the first two ranges and twice as many at the third, and he made the same number of outers at the second and third ranges ; the number of bullseyes and misses at the first and last ranges were all the same, and he scored as much for bullseyes at the second range as for outers at the first. A bullseye counts 4 points, a centre 3, and an outer 2, and his total score was 56, that at the last range being 20. Find the number of bullseyes, centres, outers and misses at each range. 120. A steamer, whose speed in still water is v : miles per hour, starts at a certain hour up a river which it ascends against the current in \ hours. N hours afterwards another steamer (speed v z ) starts down the stream, and accomplishes the distance in t 2 hours. In how many hours will they meet ? Find also the length of the river and the velocity of the current. 121. On a tidal river a boat's crew can row with the tide half as fast again as they can in still water. They start with the tide at 12 o'clock to row from A to B, but on reaching a certain place, (7, the tide changes and flows with equal velocity in the opposite direction. The rest of their journey occupies half an hour more than the time from A to G. If the distance from A to B had been 10 miles more they would have been three hours longer on the journey ; or if they had gone twice the distance they did before the tide changed, the whole journey would have been completed in two-thirds of the time actually consumed. Required the distance from A to B and the time at which the tide changed. 122. The distance from London to Peterborough is 75 miles, and from Peterborough to Grantham 30 miles. A-n up ordinary train starts from Grantham at the same time that the down express leaves London. The up train is delayed 30 minutes at Peterborough, and then passes the express 10 minutes after leaving Peterborough. Again, a down ordinary train leaves London at the same time that the up express leaves Grautham, 566 Miscellaneous Examples. each train travelling at the same rate as the other ordinary and express respectively; but in this case, the express, delayed only 6 minutes at Peterborough, arrives at the point where the former trains met, when the ordinary train has only got two-fifths of the distance of that point from London. Find the speed of the trains. 123. A market-woman having bought equal quantities of eggs at two different prices, sells them all at one price, giving to her customers for twopence the sum of the numbers she got for a penny at the two prices, and finds that she has lost pence equal to the difference of these numbers. If she had given one less for twopence, she would have gained pence equal to the sum of the same numbers. If she had sold them all for 70 pence, the price of each egg would have exceeded the average price per egg at which she bought the whole by half the price of one of the cheaper sort* Find the number bought and the price. 124. A flock of s sheep is turned into a field of turnips which would last them d days; after df days, ' sheep are added to the flock. In how many days will the remainder be consumed 1 125. A number consisting of two digits is such that when divided by the sum of the digits the result is the second digit ; and if the digits be reversed the square of the number thus formed is four times the cube of the sum of the digits. Find the number. 126. What is the price of eggs per score, when 10 more in half-a-crown's worth lowers the price 3d. per score ? 127. In the astronomical clock where the hours are marked upon the dial from 1 up to 24 ; find the two times between 8 and 9 o'clock when the hands are at right angles. 128. An officer can form the men in his battalion into a hollow square 4 deep and also into a hollow square 8 deep. If the front in the latter formation contain 1 6 men fewer than Miscellaneous Examples. 567 in the former formation; find the number of men in his battalion. Shew that the battalion can be formed into 3 other hollow squares only. 129. Extract the square roots of a 4 + + -x 3 + 4z-2 and 4 C 130. Having given that a? 4 6 cc 3 + 4 as 2 is the difference of two perfect squares neither of which vanishes with a, find them, 131. Prove that 132. Having given that = 0, and /l) aj + ( Caj 2 + ( shew that ax by cz and - + ^- + - - = 0. 2/1 2/2 133. If s = J (a + & + c), prove that 8 3_( 5 _ a )3_( 5 _&)3_( s _ c )S 134. Prove that (y + zy + y(z + xy + z(x + y)*-4xyz = 135. If n be an odd integer, prove that is divisible by (b + c) (c + a) (a + b). 136. Given aX+ b 7+ cZ = 0, a'X+ V 7+ c'Z = ; where -T= ox + aV + a", Y=bx + b'x' + b", Z= shew that c'-b'c)* + (a'c-acj + (a&'-a'fc) 2 568 Miscellaneous Examples. 137. Find the square root of 4738-027, and of 3\/2 4/3 138. Simplify V3 + VG A/6 + 139. If x^+pxt + qx^ + rx+s be a perfect square, shew that =j9 2 s and p* 4pq + 8r = 0. 140. If a+Vb+ /# can have its square root extracted in the form of a polynomial surd, shew that de = cf= bg and that A / must be rational, and find the other condition. Extract the square root of 141. The first term of an arithmetical progression being 2, the fifth being 7, and the sum being 63 ; find the number of terms. 142. In an arithmetical series consisting of an odd number of terms, the sum of the odd terms is 44 and the sum of the even terms is 33. Find the middle term, and the number of terms. 143. Find the sum of 4 + 3 + 2 + ... to nine terms and the number of terms whose sum is 9. 144. Insert 7 arithmetical means between 13 and 3. 145. A complete polynomial of n dimensions in x and y is shew that the number of terms is - - 1 . i 146. The sum of a geometrical series whose greatest term is 12, is 21 ; but when the series is continued through as many more terms the sum is 189 ; determine the series. 147. Find an arithmetic series whose fourth term is 3 and the sum of 7 terms is 21. Miscellaneous Examples. 569 148. If four quantities a, 5, c, d are in harmonical pro- b-d 2d b-d 2b gression, prove that - -. = > . -. = a b a cd c 150. Let a, J, c be any three quantities; if n harmonic means be inserted between a and b, and ra harmonic means between b and c, find the condition that all these quantities may be in harmonic progression. 2 2 151. Sum 18 terms of the series -, ^-1, 2-,.... 3 3 152. If the 5 th and 18 th terms of an arithmetical pro- gression be respectively 7 and 72, find the first four terms. 153. If two geometrical progressions having the same first terms and continued to infinity are in the proportion of their common ratios, shew that these common ratios are equal or have their sum equal to unity. 154. Sum to n terms /tf Prove that - 4. ?! ?! as + ar 1 + a? 8 ~ x n (x-l) 3 Deduce the value of ! 2 + 2 2 + ... +n\ 155. Three persons A, B, C whose ages are in geometric progression, divide amongst them a sum of money in amounts proportional to the ages of each. Five years afterwards when C is double the age of A they similarly divide an equal sum, A now receiving 17 10s. more than before, and B 2 105. more than before. Find the sum divided on each occasion. 156. Find the present value of an infinite series of annual payments, the first payment being 1, the second 2, the third 3, and so on. 570 Miscellaneous Examples. 157. A property now worth A per annum increases every year in the ratio 1 +p to 1 ; find the present value of the enjoyment of it for n years. What does this become if p = r, r being the interest on l for one year 1 158. I borrow 1000 on condition that I repay .10 at the end of every month for 10 years. Find an equation for determining the rate of interest I pay. 159. There are 24 oranges at 7 for a shilling. How many selections can be made in buying three shillings' worth 1 In how many of these will a particular orange occur ? 160. There are fifty different kinds of cakes at a con- fectioners. A person orders two particular kinds to be sent in, and any other three that the confectioner chooses. How many different assortments may be sent ? 161. How many different sums can be made by taking three coins from a purse containing a sixpence, a shilling, a florin, a crown, a half-sovereign and a sovereign 1 162. Four ladies and four gentlemen are arranged to dance a quadrille, two couples standing side by side, at one end of a room, vis & vis to the others at the other end. How many arrangements are possible 1 163. A crew for an eight oar has to be chosen out of eleven men, five of whom can row on the stroke side only, four on the bow side only, and the remaining two on either side. How many different selections can be made 1 164. Fifty-two cards in four suits are dealt out to four people in rounds ; find expressions for the number of hands which one person can have and for the number of possible sets that can be dealt. 165. Find the general term of (a* 166. Find in their simplest forms the coefficients of (i) a 3 in (V^l -fa) 10 ; (2) x r in (1-7*)*. Miscellaneous Examples. 571 167. Find in its simplest form the coefficient of x r in (1 + 3^)^, and the first negative term. \_ 168. Give the general term of (1 nx)~ n . 169. Write down the general terms of i - (i)(l-5*)-*; ()(3+f)~5 (8) ()"". 170. If in (a + b) n the 7 th and 8^ terms are in the same ratio as the 6 th and 7^ in (a + 6) n+1 , find n. 171. If the two middle terms of (a+) 2n + 1 be taken and n = 0, 1, 2, ... in succession, the sum of all will be equal to ( -\ - ), X ^^ - ), expanded in 173. Find the coefficient of a 3 " in (1 a? ) 174. If ^^ n = l+B l x+3 2 x' 2 + B^+..., find the values of B V B,,B,. 175. Shew that 2=? + - + -^.+... ad infinitum. 4 o lo 176. Find the coefficient of a 20 in (x + a 3 + a 5 + x 7 + a; 9 ) 4 . 177. Find the coefficient of sc 5 in (1 + a; + 2 a 2 + 3 3 ) 6 . 178. If S^, S 2 , S 3 , 4 be the sums of every fourth coefficient in (l+rc + ^ + as 3 )" beginning with the 1 st , 2 n ^ 3^ 4th } respectively, prove that $ x = >S 2 = 3 = 4 . 179. If n be any prime number except 2, the integral part of (v / 5 + 2) n -2 n+l is divisible by 2Qn. ISO. If /(w, r) be the number of combinations of n things taken r together, prove that /(, 1) /(", 2)+/(, 2) -/(n, 3) + ... -/(n, 1). 572 Miscellaneous Examples. 181. Prove that the sum of the products of the first n natural numbers three and three together 48 182. Prove that the sum of the products of the n quantities c, c 2 c 3 ,...c n taken m at a time 183. Resolve into its partial fractions 184. Resolve into its partial fractions , - r\2T~a i Y 185. Resolve , -- . 2 into partial fractions. (X 4) (X +1) 186. Prove that __ _ 187. Expand j- . 2 . pr in a series of ascending powers of cc, and find the nfl* term of the series. 188. Shew that the series u + u 1 + M 2 +... is convergent if i (u n ) n is always less than a quantity which is itself less than unity, however much n may be increased. 189. Transform the series r into one ascending by powers of cc, and find the n^ term of the series. Hence sum the series ax / ax \ 2 / ax \ 3 _l +(_ -)+... ad inf. ^ 1+ao; 190. Prove that Miscellaneous Examples. 573 where o> 3 = 1 and JT and Y are rational functions of a? and y. Hence prove that (x^ + xy + y*) 1 " can be put in the form X* + XY+ Y\ n being integral. If* =2, X = x*-y\ r= n = 3, X=a?-3x3f-y 191. * A person has 3 n homogeneous balls of diameters 1,2,3,... inches : the weight of the first n balls is an aliquot part of that of the others ; how many balls are there and what is the aliquot part in question ? 192. If l + 2*+... to are integers ; find p, q, r. 193. The population of a country increases slowly by -& of P itself every year, and after every decennial census an epidemic takes off -*h of the population ; at the end of ^V years it is found that the population is the same as it would have been in half the time had there been no epidemics ; shew that approximately 5 V~ NJ where n is the number of years elapsed since the last census. 194. Shew that n(n+l) ... (n + m-1) _ n(n + l) ... (n + m4) \m is zero if m > 2n and = 1 if m = 2n. Give the last term in the series. 195. Prove that the expression (a x) {+ v^+fc 2 } cannot exceed \ (a 2 + 6 2 ). 196 . If 1+1=4. + - b =---+-^=7' u v u + a vb ua vb / then will f (a&'-a'6) 2 = aa' W (a-a^ (b -V). 574 Miscellaneous Examples. 197. There cannot in any scale be found three different digits such that the three numbers formed from them by placing each digit differently in each number shall be in arith- metical progression unless the radix of the scale exceed by unity a multiple of three. If this condition be satisfied and the radix be 3/> + l, there are then (p 1) such sets of digits; and the common difference of the progressions is in all of them the same. 198. If C 2 + ?/ 2 = z* where x, y, z are integers, prove that one of the three x } y, z is divisible by 5. 199. If a 3 -f b 3 = c 3 where a, b, c are integers, shew that dbc is a multiple of 7. 200. If a n + 6 n = c n , shew that ale is a multiple of (2n+ 1), provided 2n+l be a prime number. ANSWEES TO MISCELLANEOUS EXAMPLES. 2. and 3. Eeduce the expressions on the right. 4. 96. 5. 6a 2 + o&-12& 2 ; ^ + 2 xy + 3 y*. 6. Keduce both sides. 7. a 6 - 8. 7a 8 - 10. a? + axx* xc 12. -- -. -- r- - 14. -rr- - =- 15. 16. 2 (6c + ca + a&). 17. Use formula of Art. 116. 18. 0. 19. 1. 20. 2a-5b + 3c. 21. ce 2 + aos + a 2 . 22. 23. a; 2 -3a+l. 24. a 2 - , , 25. x* y*. 26. 27. (*-l)(a-2)(*-3)(*-4). 576 Answers to Miscellaneous Examples. 28. 3a 2 (2^ + 30;+ 5). 29. 4a;-5. 30. (2a 3 + 3a 2 &-a& 2 + 6 3 )a&; o 2 6 2 (a 4 -6 4 ) (2a 3 31. 32. Putting (# + 2/) for z in each of the expressions the H. C. D. will be found to be x^ + xz + * or 33. x*2xy+y*. 34. x 1. 35. 2a 2 - 36. a 5. 37. A2x-(cc 2 -a 2 ) = +- y (Art. 266). 2=^ = ^1 (Art. 266). 39. 2cc 2 \/l-a; 2 -a; 2 . 40. Use Art. 259. 41. It is easily seen that (1 +if 2i and that (l-iy=-2i, whence v^2z = 1 + 1, \/ 2i = 1 i. 42. a; =11. 43. x = 24. 44. cc=3j. 45. x 3 or - 46. w = 4 or - 47. The equation may be written in the form aj*(a- 1) = 3 (a? + a+ 1). The factor a-t-o+ 1 is obvious, and we have the two equations x + x$+ 1 = 0, or a? x* = 3. 48. The root x = 5 is obvious. Hence the others can be obtained by removing the factor x 5. They are impossible quantities. 49. The two roots 7 and 2 are obvious. 50. x = 2. Answers to Miscellaneous Examples. 577 51. The equation can be written in the form f = 0, whence a factor x m n is obvious: the other values are x = (m + n) ^3 52. x = 3, y = 2. f, ftc. 56. o?= 10 + 4\/6, */= 10-4-/6. 67. x = -, y = 0. 58. # = a, y = 5 or # = 5, y = a : obtain a result 156. .-= TT^J R being the amount of l in 1 year. - 157. -H - T1 - '> if # = 1 +p it becomes r H 1 + l+r 158. 1 r- - r^- = 1 00 r, r being the interest of 1 for one month. 2 48 161. 20. 162. |x[3_. 163. 145. Take the different cases of selecting both the indifferent men, one of them and neither. [52 , % [52 164 ' 5.2.9 ...(7r-12) 166. (i^ -120^/-1; (2) -- -TT- 584 Answers to Miscellaneous Examples. 167. The first negative term is the 7^. If r be greater than 6, the coefficient of a? r is 13.10.7.4.3.1.2.5...(3r-16) [r 7.9.11 ...(2r+5) ,5xS u 100. ill i I-TT ) > . . 170. w= -8. |2n+l 171. Thesum of the two middle term8=a"6 w . l j === 172. (-I) n ~ 1 4w 2 +l. 173. 174. B, . 176. 73. 177. 726. 178. Give to x the values 1 and 8 can be deduced from co by changing the sign of V 3. For second part 191. If the aliquot part be -^ the equation between p and n reduces to 9 (3 n + 1) 2 = (p + 1) (n + I) 2 . Hence p + l must be a square number greater than 9. Also - - - or 9 -- w-l-1 n -f 1 must be an integer, thus n = 1, 2, or 5, and p+l 36, 49 or 64. 192. By the method of undetermined coefficients it is possible to prove that l + 2 + ... + n > = = +++. (Art. 607). By raising this to the r^ power and equating corresponding terms we get (p-}- 1) r = (?+ 1), (p + l) r = q + 1, whence can be deduced that either p = q or ;; = 1. If p q, r = 1, which gives an obvious solution whatever value p and q i may have. If p = 1 we can get r r ~ l 2, which is satisfied by r= 2, and then q = 3. 1- 193. The equation is (l--) *=(!- -)* Expand after taking the fifth root of both sides, keeping terms to the n 1 1 first power of and - - and to the second power of - since N 1+p q that is a larger fraction than the others. 194. (1 -x*) n (1 -)-"= (1 +a + x 2 ) n . Equate coefficients of powers of x on both sides. The last term will be different Answers to Miscellaneous Examples. 587 according to whether m is of the form 3r, 3r+ 1 or 3r + 2. If |n m be of the form 3r the last term is ( l) r . . a; 3r . Ir. \ n ~* 195. Equate the given expression to y and solve as an equation in #. If the values of x be real (a 2 + 6 2 ) y^ Zy* must be positive. 196. From the first equation u(u + a) v(v b] H* v* uv ^ - ' = -- j or -- r = u + v= a b a b f u* uv v* , u* uv v 2 Hence- --- y = 0. Similarly -, - j - ? = ; ,b b\ uv , _ A f a a\ uv . , whence u- (- - -,) = - (b-V), v" ( ^ - ^) = _ (a'-). 197. If r be the radix, x, y, z the digits in descending order of magnitude, we must have x- y = yz- 1 = r x + z, whence r= 3 (cc y) + l. Hence r must be of form 3/? + l. Also y = p = 2/- 1 ; /, aj = 2;; + +l; thus the greatest value of z is ^; 1. 198. The residue of any number with respect to the modulus 5 must be 0,1, 2, 3, or 4. Hence the residue of any square number must be the same as that of 0, 1, 4, 9 or 16, that is, must be 0, 1, 4, 4 or 1. Hence the equation cc 2 + 2/ 2 = s 2 must assume the form = 5r + c, where a, b, c can only have the values 0, 1, 4. Thus residue of a + b must equal c. The only possible cases are a = 0, b = 1, c = 1, o = l, 6 = 0, c = 1 ; o = l, 6 = 4, c = 0; a = 4, 6 = 1, c = 0. In every case one of the quantities x, y, z is a multiple of 5. 199. With respect to the modulus 7 every cube number must have a residue 0, 1 or 6. Hence tlie given equation can be put into the form 7^, + a ' + 7 q + 6'= 7 r + c', where a', 6' c' have each one of the above three values 0, 1 or 6, and the residue of of + b' must be c'. This is only possible when one of the three quantities a', 6' or c' is zero. 588 Answers to Miscellaneous Examples. 200. By Fermat's theorem, since 2n+l is a prime number, if ^Vbe any number less than 2n+ 1, ^ 2n 1 is a multiple of (2 n + 1 ). Hence either ^ n 1 or N n + 1 is a multiple of 2 n + 1 . Thus the residues of the numbers O n , l n , 2 n , ... (2n) n with respect to the modulus 2w + l are either 0, 1 or 2. It easily follows that the residue of the w th power of any number must be 0, 1 or 2n. Hence, if three numbers be connected by the equation a n + b n = c n , the residue of one of the three numbers a, 6, c must be zero. UNIVERSITY THE END. -* r\ K u3* MAR 8 1' 13 1949! SEP MAR 31 1942 P MAR 1943 CD LD 21-50m-8,-32 . 7V^ YC169553 . t .- ; *. .-;