^ liversity of Califorr. FROM THK I.HiKAkV n}- DR. FRANCIS LIEBER, pv.,,f.,c;,., of irit^trry and Law in Columbia College. New THE GU'T 01' MICHAEL REESE Of San Francisci 1873. i Digitized by tlie Internet Arcliive in 2008 witli funding from IVIicrosoft Corporation littp://www.archive.org/details/elementsofalgebrOOdaviricli 77 ?8^ ELEMENTS ALGEBRA TRANSLATED FROM THE FRENCH OF M. BOURDON. REVISED AND ADATTED TO THE COURSES OP MATHEMATICAL INSTRUCTION IN THE UNITED STATES ; BY CHARI^ES DAYIES, PROFESSOR OF MATHEMATICS IN THE MILITARY ACADEMY AND AUTHOR OF THE COMMON SCHOOL ARITHMETIC, ELEMENTS OF DESCRIPTIVE GEOMETRY, 8VRVEYING, AND A TREATISE ON SHADOWS AND PERSPECTIVE. NEW. YORK : PUBLISHED BY WILEY & LONG, No. 161 Broadway. •ITSr.EOTVPCD BY A. CIIA* is called the sign of multiplication; and when placed between two quantities, it denotes that they are to be multi- plied together. The multiplication of two quantities is also fre- quently indicated by simply placing a point between them. Thus, 10 ALGEBRA. 33x25, or 36.25, is read, 36 multiplied by 25, or the product of 36 by 25. 7. The multiplication of quantities, which are represented by let- ters, is indicated by simply writing them one after the other, without interposing any sign. Thus, a6 signifies the same thing as axh, or as a.& ; and ahc the same as aX^Xc, or as a.h.c. It is plain that the notation db, or ahc, which is more simple than axh, or aXbXc, cannot be employed when the quantities are represented by figures. For example, if it were required to express the product of 5 by 6, and we were to write 5 6, the notation would confound the product with the number 56. 8. In the product of several letters, as abc, the single letters, a, b and c, are called /actor-s of the product. Thus, in the product ab, there are two factors, a and b ; in the product acd, there are three, a, c and d 9. There are three signs used to denote division. Thus, a-^b denotes that a is to be divided by b, — denotes that a is to be divided by b, a\b denotes that a is to be divided by b. r 10. The sign =, is called the sign of equality, and is read, is equal to. When placed between two quantities, it denotes that they are equal to each other. Thus, 9 — 5=4 : that is, 9 minus 5 is equal to 4 : Also, a-{-b=c, denotes that the sum of the quantities a and b is equal to c. "^ 11. The sign >, is called the sign of inequality, and is used to express that one quantity is greater or less than another. Thus, ayb is read, a greater than b; and a<6 is read, a less than b ; that is, the opening of the sign is turned towards the greater quantity. 12. If a quantity is added to itself several times, as a+a+a+a +a, we ' —orally write it but once, and then nlaco a nnmber before DEFIMTIONS A>D REMARKS. 11 it to express how many times it is taken. Thus, a-^a-\-a-\-a-{-a=5a. The number 5 is called the co-efficieni of a, and denotes that a is taken 5 times. Hence, a co-efficient is a number prefixed to a quantity, denotmg the number of times which the quantity is taken; and it also indi- cates the number of times plus one, that the quantity is added to itself. When no co-efRcient is written, the co-efficient 1 is always understood. 13. If a quantity be multiplied continually by itself, as ax^Xa XaXfl) we generally express the product by writing the letter once, and placing a number to the right of, and a little above it : thus, aX«X«X«Xa=«^ The number 5 is called the exponent of a, and denotes the number of times which a enters into the product as a factor. Hence, the exponent of a quantity shows how inany times the quantity is a factor ; and it also indicates the number of times, phis one, that the quantity is to be multiplied by itself. When no expo- nent is written, the exponent 1 is always understood. 14. The product resulting from the multiplication of a quantity by itself any number of times, is calledthe power of that quantity j acd the exponent, which always exceeds by onethe number of mul- tiplications to be made, denotes the degree of the power. Thus, a' is the fifth power of a. The exponent 5 denotes the degree of the power ; and the power itself is formed by multiplying a four times by itself. 15. In order to show the importance of the exponent in algebra, suppose that we wish to express that a number a is to be multiplied three times by itself, that this product is to be multiplied three times by h, and that this new product is to be multiplied twice by c, we would write simply a^ IP (?. If, then, we wish to expess that this last result is to be added to itself six times, or is to be multiplied by 7, we would write, 7a'6V. 12 ALGEBRA. This gives an idea of the brevity of algebraic language. 16. The root of a quantity, is a quantity which being multiplied by itself a certain number of times will produce the given quantity. The sign y/ , is called the radical sign, and when prefixed to a quantity, indicates that its root is to be extracted. Thus, Vfl or simply ■v^a denotes the square root of a. "^ a denotes the cube root of a. ^ a, denotes the fourth root of a. The number placed over the radical sign is called the index of the root. Thus, 2 is the index of the square root, 3 of the cube root, 4 of the fourth root, &c. 17. Every quantity written in algebraic language ; that is, with the aid of letters and signs, is called an algebraic quantity, or the algebraic expression of a quantity. Thus, C is the algebraic expression of three times the i number a ; C is the algebraic expression of five times the ( square of a ; c is the algebraic expression of seven times the i product of the cube of a by the square of b ; ( is the algebraic expression of the difference be- 3a — 5o < , „ . , ( tween three times a and five tunes b ; ^ is the algebraic expression of twice the square 2a^—Sab+4:b^7 of a, diminished by three times the product of « ( by b, augmented by four times the square of b. 18. When an algebraic quantity is not connected with any other by the sign of addition or subtraction, it is called a monomial, or a quantity composed of a single term, or simply, a term. Thus, 3a, 5a-, laW, are monomials, or single terms. 19. An algebraic expression composed of two or more parts, separated by the sign + or — , is called a polynomial, or quantity involving two or more terms. DEFINITIONS AND REMARKS. 13 For example, 3a — 5 J and 2a-—2cb-{-'il/ are polynominls. 20. A polynomial composed of two terms, is called a hinomial ; and a polynomial of three terms is called a trinomial. 21. The numerical value of an algebraic expression, is the number which would be obtained by giving particular values to the letters which enter it, and performing the arithmetical operations indicated. This numerical value evidently depends upon the particular values attributed to the letters, and will generally vary with them. For example, the numerical value of 2a^=54 when we make a=3 ; for, the cube of 3=-27, and 2x27=54. The numerical value of the same expression is 250 when we make a=5; for, 53=I25,and 2x125=250. 22. We have said, that the numerical value of an algebraic ex- pression generally varies with the values of the letters which enter it : it does not, however, always do so. Thus, in the expression a — h, so long as a and b increase by the same number, the value of the expression will not be changed. For example, make a=7 and S=4 : there results a — Z»=3. Now make a=7+5=12, and ^=4 + 5=9, and there results a — ^=12 — 9=3, as before. 23. The numerical value of a polynomial is not affected by changing the order of its terms, provided the signs of all the terms be preserved. For example, the polynomial 4a'' — 3a-i+5ac'^= 5af'-— 3a''^o+4a^=— 3a-Z'+5ac^+4a^. This is evident, from the nature of arithmetical addition and subtraction. 24. Of the different terms which compose a polynomial, some are preceded by the sign +, and the others by the sign — . The first are called additive terms, the others, suhtractive terma. The first term of a polynomial is commonly not preceded by any sign, but then, it is understood to be afTected with the sign +. 25. Each of the literal factors which compose a term is called a dimension of this term ; and the degree, of a term is the i umber of 2 14 ALGEBRA. these factors or dimensions. Thus, 3a is a term of one dimension, or of the first degree. hah is a term of two dimensions, or of the second degree. la^hr=laaabcc is of six dimensions, or of the sixth degree. In genei-al, the degree, or the number of dimensions of a term, is estimated by taking the stitn of the exponents of the letters ivhich enter this term. For example, the term Sa^bcd^ is of the seventh degree, since the sum of the exponents, 2+1 + 1+3=7. 26, A polynomial is said to be homogeneous, when all its terms are of the same degree. The polynomial 3a— 2&+C is of the first degree and homogeneous. — ^ab-\-¥ is of the second degree and homogeneous. 5a^c— 4c''+2c^(Z is of the third degree and homogeneous. 8a^— 4aS+c is not homogeneous. 27, A vinculum or bar , or a parenthesis ( ), is used to express that all the terms of a polynomial are to be considered to- gether. Thus, a-\-b-\-cxb, or (a+Jxc)xJ denotes that the trinomial a-\-h-\-c is to be multiplied by b ; also a-{-b-\-cxc+d-{-f or (a+J + c)x(c+(^+y) denotes that the trinomial a-\-b-\-c is to be multiplied by the trinomial c-\-d-\-f When the parenthesis is used, the sign of multiplication is usually omitted. Thus {a-\-b-\-c)xb is the same as {a-\-b-\-c) b. The bar is also sometimes placed vertically. Thus, +a +^ + c is the same as {a-\-b-\-c) x or a+b + cXx 28. The terms of a polynomial which are composed of the same letters, the same letters in each being affected with like exponents, are called similar term^. Thus, in the polynomial lah + Sab—^taW+oa^l^, the terms lab and Sab, are similar ; and so also are the terms— 40^^ and 5a'Zr', the letters and exponents in each being the same. But in the bino- DEFINITIONS AND REMARKS. 15 mial 8a^b-{-7aP, the terms are not similar ; for, although they are composed of the same letters, yet the same letters are not affected with like exponents, 29, When a polynomial contains several shnilar terms it may often be reduced to a simpler form. Take the polynomial 4a^Z>— 3a-c+7a^c — 2a-^. It may be written (Art. 23), 4a"b—2a^b-{-7a^c — 3a'c, But 4:a^b—2a^b reduces to 2arb, and 7 arc— Sac to 4a^c, Hence, 4a^J — Sa'c + 7 ah — 2a'b= 2a^ + Aa^., When we have a polynomial with similar terms, of the form + 2a'br - 4.a^bc'-\-6a'b(r—8a^bc^+ lla'bc^ Find the sum of the additive and subtractive terms separately, and take their difference : thus, Additive terms, Subtractive terms, + 2a''b(? - 4a^c^ + Ga^c" - Qa^b(? -^\Wb(? Sum -\2a%e' Sum +19a='Ac2 Hence, the given polynomial reduces to \^a''bc^-\2d'b(?=7a^baxy, and -2 ^ay— "^x —Qap. Ans, ^/^_^/^-^^7. 7. Add together the polynomials 7a'6 — 3aic — 8Zrc — 9c'+ca^, 8ak-5a-5+3c3-45-c+cd2 and ^a^b-Sc'+^Wc—ZcP. Ans. Qd'b -^babc — Z¥c — 1 4c ' + 2c(? — 3(Z'. SUBTRACTION. 36. Subtraction, in algebra, consists in finding the simplest ex- pression for the difTerence between two algebraic quantities. The result obtained by subtracting 4^ from 5a is expressed by 5a — 4&. In like manner, the difference between '{(Cb and Aa^b is expressed by la^b-A.a^b^2a^b. Let it be required to subtract from . . .4a the binomial . . . . ■ . .2b — Sc In the first place, the result may be written thus, 4a — (2i— 3c) by placing the quantity to be subtracted within the parenthesis, and writuig it after the other quantity with the sign — . But the ques- tion frequently requires the difference to be expressed by a single polynomial ; and it is in this that algebraic subtraction principally consists. To accomplish this object, we will observe, that if a, b, c, were given numerically, the subtraction indicated by 23— 3c, could be performed, and we might then substract this result from 4a ; but as 22 ALGEBRA. this subtraction cannot be effected in the actual condition of the quantities, 2b is subtracted from 4a, which gives 4a— 2i ; but in sub- tracting the number of units contained in 2b, the number taken away is too great by the number of units contained in 3c, and the result is therefore too small by the same quantity ; this result must therefore be corrected by adding 3c to it. Hence, there results from the proposed subtraction 4a— 2J + 3c. 4a 2b-Sc 4a — 25+ 3c Again, from ..... 8a"—2ab subtract dci'—Aab+Shc — P. The difference is expressed by 8ar—2ab—(5ar—4:ab+Sbc—lf^) which is equal to . . . 8a^— 2a5— 5a-— 3ic + 4a5+i^. or by reducing, equal to . . . . Sa^-{-2ab—Sbc-\-l^. The reduction is made by observing, that to subtract 5a^— 4a5 -^Sbc — I^, is to subtract the difference between the sum of the ad- ditive terms 5a^-{-Bbc, and the sum of the substractive terms 4ai+^. We can then first subtract 5a^+3bc, which gives 8a^— 2ai— 5a^ — 3bc; and as this result is necessarily too small by 4ab+P, this last quantity must be added to it, and it becomes 8a^— 2aZi — 5a^ — 3bc + 4:ab-\-P ; and finally, after reducing, 3a^-\-2ab—3bc+P. 37. Hence, for the subtraction of algebraic quantities, we have the following general RULE. I. Write the quantity to be subtracted under that from lohich it is to be taken, placing the similar terms, if there are any, under each other. II. Change the signs of all the terms of the -polynomial to be sub- tracted, or conceive them to be changed, and then reduce the polynomial result to its simplest form. SUBTRACTION. 23 From Take Remainder From Take. . Remainder From . . Take . . Remainder ^d 6ac — bab+<^ ■ Sac—Sab-\-lc Sac—8ab-\-c'+'!c. 6ac—5ah-\-c? Sac + Sab— 1c Sac — 8ab-\-c^+lc. _(2). 6\/2«y— \r^-{-s¥ ■SV2;ry-Vx+x+2U'. byx-Sx'+S + bb- (3). Qyx—S3?-\-U yx—S + a (4). -ba^'—Aa^b+S^c -2a^+Sa^b-Sl^c 7a^-la^b-\-llI^c. (5). 4ab- cd + Sa' 5ab—4:cd+Sa-+5b' ab+Scd—blr 7. From 8abc — l2Pa + 5cx—lxy, take Icx—xy — lSb^a. An^, 8abc-\-Pa — 2cx—6ry. 38. By the rule for subtraction, polynomials may be subjected to certain transformations. For example . . 6a^-Sab+2if'-2bc, becomes . . . 6a''—{Sab—2P + 2bc). In like manner . . '!a'-8a'b-Alr'c+6P, becomes . . . '7a'*-{8a^+APc-6P) ; or, again, . . . la''-8a^-{4Pc-6by These transformations consist in decomposing a polynomial into two parts, separated from each other by the sign — : they are very useful in algebra, 39. Remark. — From what has been shown in addition and sub- traction, we deduce the following principles. 1st. In algebra, the words add and sum do not always, as in arithmetic, convey the idea of augmentation ; for a—b, which results from the addition of —b to a, is properly speaking, a dif- ference between the number of units expressed by a, and the num- ber of units expressed by b. Consequently, this result is less than a. 24 ALGEBRA. To distinguish this sum from an arithmetical sum, it is called tho algebraic sum. Thus, the polynomial 2rt~— 3fl-Z»+3//c is an algebraic sum, so long as it is considered as the result of the union of the monomials 2a^, — Srt^^, -\-2,}rc, with their respective signs ; and, in its frofer ac- ceptatwn, it is the arithmetical difference between the sum of the units contained in the additive terms, and the sum of the units con- tained in the subtractive terms. It follows from this that an algebraic sum may, in the numericol applications, be reduced to a negative number, or a number affected with the sign — . 2d. The words siiUraction and difference do not always convey the idea of diminution, for the difference between +a and —b being' a + b, exceeds a. This result is an algebraic difference, and can be put under the form of a — ( — Z*). MULtlPLICATION. 40. Algebraic multiplication has the same object as arithmetical, viz. to repeat the multiplicand as many times as there are units in the multiplier. It is generally proved, in arithmetical treaties, that the product of two or moi-e numbers is the same, in whatever order the multiplii a- tion is performed ; we will, therefore, consider this principle de- monstrated. This being ndmitted, we v/ill first consider the case in which it is required to ninltiply one monomial by another. The expression for the ])roduct of . la^V^ by A,a-b may at once be written thus . . laVrx^ba^ But this may be simplified by observing that, from the preceding principles and the signification of algebraic symbols, it can be written . . . 7x^anaoahbb. Now, as t!ie co-efficients are particular numbers, nothing prrvonfs our forming a single number from them by multiplying them together, which gives 28 for thf* co-efficient of the product. As to MULTIPLICATION. 25 the letters, the product aaaaa, is equivalent to a-, and the product hbb, to IP ; therefore, the final result is . . . 2QaW. Again, let us multiply .... Vla^b'c^ by Qd^JroP. The product is V2x^aaaaabbbhbhccdd=9QaW(?cP. 41. Hence, for the multiplication of monomials we have the following RULE. I. Multiply the co-efficients together. II. Write after this product all the letters ivhich are cotnmon to the multiplicand and multiplier, affecting each letter ivith an exponent equal to the sum of the two exponents with which this letter is affected in the two factors. III. If a letter enters into but one of the factors, write it in tlw pro- duct with the exponent with which it is affected in the factor. The reason for the rule relative to the co-efRcients is evident. But in order to understand the rule for the exponents, it should be ob- served, that in general, a quantity a is found as many times a factor in the product, as it is in both the multiplicand and multiplier. Now the exponents of the letters denote the number of times they enter as factors (Art. 13.) ; hence the sum of the two exponents of the same letter denotes the number of times it is a factor in the required product. From the above rule, it follows that, 8a''bc'x'iabd?= bQaWc'cP 2laWdc X 8abc''=16&a'Pc'd ^abcx'!df= 28abcdf. Multiplv . Sa^b 12 a'x Qxy z a-xy by .' . 2b a^ 12 x^y ayH Ixf 6a!^lP 144rtV7/ Sxuy^z' 2a\v'y^ 42. We will now proceed to the multiplication of polynomials. Take the two polynomials a-{-b-}-c, and d+f com.posed entirely of additive terms ; the product may be presented under the form (a + S + c) (d-{-f). But it is often necessary to form a single 3 26 ALGEBRA. polynomial from this product, and it is in this that the multiplication of two polynomials consists. Now it is evident, that to multiply . a+b-\-c by d+f ad-{-bd-\-cd +af+bf+cf ad+bd+cd+ a/+ bf+ tf. is the same thing as taking a-\-b-\-c as many times as there are units in d, then as many times as there are units in^, and adding the two products together. But to multiply a + ^-f c by d, is to take each of the parts of the multiplicand d times and add together the partial products, which gives ad-\-bd-\-cd. In like manner, to multiply a-\-b-^c by/, is to take each of the parts of the multiplicand,/times, and add together the partial products. Hence, (a + b+c) {d-^f)T=ad-\-bd+cd+af-^bf+cf. Therefore, in order to multiply together two polynomials com- posed entirely of additive terms, multiply successively each term of the multiplicand by each term of the inultiplier, and add together all the j)roducts. If the terms are affected with co-efficients and exponents, observe the rule given for the multiplication of monomials (Art. 41). For example, multiply by . . . . . 2a+ 5b The product, after reducing, 60""+ Qa'b-^2aV' + \ba^b+2{)a¥+W becomes x+y . 6a'+23a^*+22a^'+5Z'3 a;'+ xf +lax ax +bax +^y+f x'*+xf+x'y+y' a3?+axY + 'id'3? + 5ax« +5flxy+35aV 6aa;''+6aarj/«+42aV. MULTIPLICATION. 27 43. In order to explain the most general case, we will first re- mark, that if the multiplicand contains additive and subtractive terms, it may be considered as expressing the difference between the number of units indicated by the sum of the additive terms, and the number of units indicated by the sum of the subtractive terms. The same reasoning applies to the multiplier ; whence it follows, that the general case may be reduced to the multiplication of two binomials, such as a — b and c—d', a denoting the sum of the additive terms, and b the sum of the subtractive terms of the multiplicand, c and d expressing similar values of the multiplier. We will then show how the multiplication expressed \)y{a — b)x{c — d) can be effected, a -b c -d ac — bc — ad+bd ac—bc —ad-\-bd. Now, to multiply a—b by c—d, is evidently the same thing as to take a — b as many times as there are units in c, and then diminish this product by a—b, taken as many times as there are units in d; or to multiply a—b by c, and subtract from this product that of a— J by d. But to multiply a—bhyc, is to take a—b,c times. Now if we mul- tiply a by c the product is ac, which is too large by b taken c times ; therefore cimust be taken from it : hence, the product of a— 5 by c, is ac — bc. In like manner, the product of a — bhy d, is ad — bd; and as we have just seen that this last product should be subtract- ed from the preceding ac—bc, it is necessary to change the signs of ad—bd, and write it under ac — bc, which (Art. 37), gives (a — b) (c — d)=ac—bc — ad+bd. If we suppose a and c each equal to 0, the product will reduce to +bd. 44. Hence, for the multiplication of one polynomial, by another we have the following 28 ALGEBRA. EULE. I. Multiply aU the terms of the multrpllcand, both additive and sub. tractive, by each additive term of the multiplier, and affect the partial products with the same signs as those with which the terms of the nnil- tiplicand are affected ; also midtiply all the tentis of the multiplicand by each sidHractive term of the multiplier, but affect the partial products with signs'Conirnry to those with which the terms of the multiplicand are affected. Then reduce the pohnondal result to its simplestforrn. Take, for an example, the two polynomials : 4a:' — 5a-b — 8ab''-\-2P and 2a--Sab-4I/' 8a'— lOa'b— WaW+Aa'b'' — V2a'b +lbaW+24:a~b^-Qab'' — 16a''^- + 20a'Z-H32a b' — 8l/- 8aP-22a'b—lla'l^+A.8a-b- + 2Qab' — 8lf, After having arranged the polynomials one under the other, mul- tiply each term of the first, by the tenn 2a^ of the second ; this gives the polynomial 8a^—lQa^b — \Qa¥-^AaW, the signs of which are the same as those of the multiplicand. Passing then to the term ?>ah of the multiplier, multiply each term of the multiplicand by it, and as it is affected with the sign — , affect each product with a sign contrary to that of the corresponding term in the multiplicand ; this gives — 12a*J4-15a^^+24a^5^— 6aJ* for a product, which is written under the first. The same operation is also performed with the term AJr, which is also subtractive; this gives, —\Qa''V--^r20aW + ^2ab' — 8b\ The product is then reduced, and we finally obtain, for the most simple expression of the product, 8a'-22a'b-\laW-^^8a"F + 2Qa¥-81/. The rule for the signs, which is the most important to retain, in the multiplication of two polynomials, may be expressed thus : When two terms of the multiplicand and multiplier are affected with the sa7ne sign, the corresponding j)roduct is affected with the sign +, and when 5IULTIPLICATION. 29 they are affected ivith contrary signs, the product is affected with the sign -. Again, we say in algebraic language, that + multiplied by +, or — multiplied by — , gives + ; — multiplied by +, or + multi- plied by — , gives — . But this last enunciation, which does not in itself offer any reasonable direction, should only be considered as an abbreviation of the preceding. This is not the only case in which algebraists, for the sake of brevity, employ incorrect expressions, but which have the advantage of fixing the rules in the memory. EXAMPLES. 1. Multiply V2ax by 3a. Am. dda'x. 2. Multiply U^-2y by 2y. Ans. 8x'y- -Af. 3. Multiply 2x4-4^ by 2x—4:y. Ans. AaP- ley. 4. Multiply aP J^x'y^xy'+y^ by x-y. Ans. X ^-rf. 5. Multiply x'+xy-itf by ^—xy-\-f. Ans. x'^-\-x'y =+2/^ 6. Multiply ' 2a=— 3ax+4ar^ by ba^—^ax -2x^. 7. Multiply Zx^—2xy-\-b by 3? + 2xy-[ 3. 8. Multiply 3af'+2a.-/+3/ ^y 2x'-3x' y+5f. 9. . red. 1. 2a'-5id+cf -5a2+4M-8c/. 'rod -15a^ + 37a=M-29aV- 2QU'd?-\-Ubcdf- -Srp. IC 4a^&'-5aWc+8a'bc'-: ^aV-labc^ 2a&'-Aabc-2bc^ +, 6\ r 8aW —lOaWc -^2&aWc'-S4:aWc^ Prod. red. aIf-2Qah'-lbb' 2r:>aW—20ab' — 15b* The object of this operation is, as we have already said (Art. 49), to find a third polynomial, which, multiplied by the second, shall produce the first. It follows from the definition and the rule for the multiplication of polynomials (Art. 43), that the dividend is the assemblage, after addition and reduction, of the partial products of each term of the divisor, multiplied by each term of the quotient sought. Hence, if we could discover a term in the dividend which was derived, with- out reduction, from the multiplication of one of the terms of the divisor, by a term of the quotient, then, by dividing the term of the dividend by that of the divisor, we would obtain a term of the re- quired quotient. Now, from the third remark of Art. 45, the term lOa", affected with the highest exponent of the letter a, is derived, without reduc- tion from the two terms of the divisor and quotient, affected with the highest exponent of the same letter. Hence, by dividijig the term lOo* by the term —5a\ we will have a term of the required quotient. But here another difficulty presents itself, viz. to deter- mine the sign with which the term of the quotient should be affl-cted. In order that this subject may not impede our progress hereafter, we will establish a rule for the' signs in division. DIVISION. 37 Since, in multiplication, the product of two terms having the same sign is affected with the sign +, and the product of two terms having contrary signs is affected with the sign — , we may con- clude, 1st. That when the term of the dividend has the sign +, and that of the divisor the sign -f, the term of the quotient must have the sign +. 2d. When the term of the dividend has the sign +, and that of the divisor the sign — , the term of the quotient must have the sign — , because it is only the sign — , which, combined with the sign — , can produce the sign + of the dividend. 3d. When the term of the dividend has the sign — , and that of the divisor the sign +, the quotient must have the sign — . That is, when the two terms of the dividend and divisor have the same sign, the quotient will be affected with the sign +, and when they are affected with contrary signs, the quotient will be affected with the sign — ; again, for the sake of brevity, we say that + divided by +, and — divided by — , give + ; — divided by +» and + divided by — , give — . In the proposed example, 10a* and — 5a^ being affected with contrary signs, their quotient will have the sign — ; moreover, I0a\ divided by 5a% gives 2a^ ; hence, —2a- is a term of the re- quired quotient. After having written it under the divisor, multiply each term of the divisor by it, and subtract the product, from the dividend, which is done by writing it below the dividend, conceiving the signs to be changed, and performing the reduction. Thus, the result of the first partial operation is -40a^S+57a-6--f 4fl&'- 15Z»*. This result is composed of the partial products of each term of the divisor, by all the terms of the quotient which remain to be de- termined. We may then consider it as a new dividend, and reason upon it as upon the proposed dividend. We will therefore take in 38 ALGEBRA. this result, the term — 40a'&, affected with the highest exponent of fl, and divide it by the tenn — 5a^ of the divisor. Now, from the preceding principles, —40a'*, divided by —5a^ gives -{-8ai for a new term of the quotient, which is written on the right of the first. Multiplying each term of the divisor by this term, and writing the products underneath the second dividend, and making the subtrac- tion, the result of the second operation is 25d'Ir-20aP-15¥; then dividing 25aW by —5a", we have —5b^ for the third term of the quotient. Multiplying the divisor by this term, and writing the terms of the product under the third dividend, and reducing, we ob- tain for the result. Hence, —2a^ -i-&ab—5b% or Sab— 20^—56" is the required quotient, which may be verified by multiplying the divisor by it ; the product should be equal to the dividend. By reflecting upon the preceding reasoning, it will be perceived, that, in each partial operation, we divide that term of the dividend which is affected with the highest exponent of one of the letters, by that term of the divisor affected with the highest exponent of the same letter. Now, we avoid the trouble of looking out the term, by taking care, in the first place, to write the terms of the dividend and divisor in such a manner that the exponents of the same letter shall go on diminishing from left to right. This is what is called arrung. ing the dividend and divisor with reference to a certain letter. By this preparation, the first term on the left of the dividend, and the first on the left of the divisor, are always the two which must be divided by each other in order to obtain a term of the quotient ; and it is the same in all the following operations ; because the partial quotients, and the products of the divisor by these quotients are always arranged. 55. Hence, for the division polynomials we have the following RULE. I. Arrange the dividend and divisor with reference to a certain letter, apd then divide the first term on the left of the dividend hy the first term mvisioN. 39 on the left of the divisor, the result is the first term of the quotient ,• multiply the divisor by this term, and subtract the 'product from the dividend. II. Then divide the first term of the remainder hy the first term of the divisor, which gives the second term of the quotient ; multiply the divisor by this second term, and subtract the product from the result of the first operation. Continue the same process until you obtain Ofor a resuU ; in which case the division is said to be exact. When the first term of the arranged dividend is not exactly divisi- ble by that of the arranged divisor, the complete division is impossi- ble, that is to say, there is not a polynomial which, multiplied by the divisor, will produce the dividend. And in general, we will find that a division is impossible, when the first term of one o£ the partial dividends is not divisible by the first term of the divisor. 56. Though there is some analogy between arithmetical and algebraical division, with respect to the manner in which the opera- tions are disposed and performed, yet there is this essential difference between them, that in arithmetical division the figures of the quo- tient are obtained by trial, while in algebraical division the quotient obtained by dividing the first term of the partial dividend by the first term of the divisor is always one of the terms of the quotient sought. Besides, nothing prevents our commencing the operation at the right instead of the left, since it might be performed upon the terms affected with the lowest exponent of the letter, with reference to which the arrangement has been made. In arithmetical division the quotient can only be obtained by commencing on the lefl. Lastly, so independent are the partial operations required by the process, that after having subtracted the product of the divisor by the first term found in the quotient, we could obtain another term of the quotient by dividing by each other the two terms of the new divi- dend and divisor, affected with the highest exponent of a different letter from the one first considered. If th^same letter is preserved, it is because there is no reason for changing it, and because the two 40 ALGEBRA. polynomials are already arranged with reference to it; the first terms on the left of the dividend and divisor being sufficient to obtain a term of the quotient ; whereas, if the letter is changed, it would be necessary to seek again for the highest exponent of this letter. SECOND EXAMPLE. Divide . . . 21a;y+25a;y+68ay-40/— 56x'— 18a:Vby 5y'' — 8x'' — 6xy. — ^0y'+68xy' + 25xy+21xy-18x*y-56x'\\5y''-6xy-8x"- — 40y^- + 48^^ + 64A-y —8y'+Axy?-3x-y+lx^ 1st. rem. 20 a,y— 39xy +21a;y 2 xy' — 24x-'^' — 32 xY 2d. rem. — 15a;y +53xy — ISa;^ — Wxy 4-1 8x^3;=' + 24:X*y 35xy- ^2x*y-56x' 35a,y— 42x'^y-56x' Final remainder 0. 57. Remark. — In performing the division, it is not necessary to bring down all the terms of the dividend to form the first remainder, but they may be brought down in succession, as in the example. As it is important that beginners should render themselves familiar with the algebraic operations, and acquire the habit of calculating promptly, we will treat of this last example in a different manner, at the same time indicating the simplifications which should* be" introduced. As in arithmetic, they consist in subtracting each partial product from the dividend as soon as this product is formed. — 40j/' + 68xy' + 25a''2/' +21 ^Y — 1 Sx*y — 56a;' || 5?/' — dxy — 8x^ 1st. rem. 20a;/ — 39xy+21 a^y — 8j/=+4a,y— 3ar'y+7x' 2d. rem. —iSxy-f- 53 xY — 18x*y 3d. rem. 4 ^^ a;y — 42a;'7/— 56x' Final rem. 0. DIVISION. 41 First, by dividing —40^^ by 5y^, we obtain —8/ for the quotient. Multiplying 5?/^ by — 8^, we have —40?/*, or by changing the sign, ^-402/^ which destroys the first term of the dividend. In like manner, —6xyX —^y^ gives +4:8xy* and for the subtrac- tion — 48a^*, which reduced with -\-68xy\ gives 20xy* for a remain- der. Again, —Svc'^X— 8^' gives +, and changing sign, — 64a;y, which reduced with 25x'y^, gives — 39a;y. Hence the result of the first operation is 20xy*—S9xY followed by those terms of the divi- dend which have not been reduced with the partial products already obtained. For the second part of the operation, it is only necessary to bring down the next term of the dividend, separating this new dividend from the primitive by a line, and operate upon this new dividend in the same manner as we operated upon the primitive, and so on. THIRD EXAMPLE. Divide 95a-73a'+56a^-25-59a^ by -2a^+5-Ua-\-la' 56a* — 59a' — 73a' + 95fl— 25||7a' — 3a' — lla+5 1st. rem. ^-3^5a^+lba'-\-b5a—25 8 a — 5 2d. rem. 0. EXAMPLES. 1. Divide ISa;" by. 9a;. Ans. 2x. 2. Divide lOx^ by —bx'y. Ans. —2y. 3. Divide — 9axy by Qx'^y. Ans. —ay. 4. Divide — 8x^ by —2x. Ans. -|-4r. 5. Divide 10a5+15ac by 5a. Ans. 23-j-3c. 6. Divide 30aa;— 54a; by 6a;. Ans. 5a— 9. 7. Divide \Ox^y—\by''—by by 5t/. Ans. 2x'—dy—l. 8. Divide 13a+3aa?— 17a;'' by 21a. 9. Divide Sa" — 15-t-6a+3J by 3a. 10. Divide a''+2aa;-l-a;'' by a-\-x. Ans. a-\-x. 11. Divide a^ — Za^y-\-^ay'^—y'^ by a—y. Ans. a'— 2ay+w', 4* 42 ALGEBRA. 12. Divide 1 by 1— x. Ans. l+x+x^+a;', &c. 13. Divide 6a;' — 96 by 3x— 6. Ans. 2a;'+4x''+8x+16. 14. Divide a' — 5a*x4-10aV — 10aV+5aa;*— «' by a'— 2aa;+x'. Ans. a^—Za^x+^ax^—x^. 15. Divide 48a;=— 76aa;'' — 64a'a;+105a^ by 2a;— 3a. 16. Divide/-3i/V432/V— a;^ by y'—^''x+Zyx''—x\ 58. It may happen that one, or both, of the proposed polynomials contains in two or more temis the same power of the letter with re- ference to which the arrangement is to be made. In this case, how should the arrangement be made, and the divi- sion be effected 1 Divide lla^i-19aJc + 10a'-15a^c+3aJ^ + 15Jc^-5&^c by 5a'-f3a&— 5Jc. In the first place, the two terms lla^J— ISa^'c, can be placed un- der the form (11&— 15c) a", or Hi I a", by writing the power a' -15c 1 once, and placing to the left of it, and in the same vertical column, the quantities by which this power is multiplied ; this polynomial multiplier is then called the co-efficient of a^. The second manner of connecting the terms involving the same power, is preferable to the first, for two reasons. 1st. Because where there are many terms in the dividend and divisor, it would be difficult to write all on the same horizontal line. 2d. As the co-ef- ficient of each power ought to be arranged with reference to a second letter, we are obliged, if the first term is subtractive, to sub- ject the term to a modification, which might lead to error, in employ- ing the first manner. Take, for example, —\blPa^ -\-lhca^ — Qc^a' the modification consists in putting this expression under the form — {lbb'' — 'Jbc+Sc')a' .... (Art. 38). whereas, by the second, it is written thus : —Ibh"^ a", and by this + lie — ^e manner we have the advantage of preserving to each term the sign with which it was at first affected. DIVISION. 43 In like manner, —l9abc-{-Sah^ is written : . . -\- Sb^ \ a -19bc I This being understood the operation may be performed in the following manner. -15c| -19^1 '2a+b-3c a—5¥c+l5b(? 1st. Rem. 5b — 15c -9bc 2d. Rem. 0. First divide 10 a^ by 5a-, the quotient is 2a. Subtracting the product of the divisor by '2a, we obtain the first remainder. Divi- ding the part involving a' in this remainder by 5a^, the quotient is b — 3c. Multiplying successively each term of the divisor by b—3c, and subtracting the product, we have for the result. Hence, 2a 4-^ — 3c is the required quotient. 59. Among the different examples of algebraic division, there is one remarkable for its applications. It is so often met with in the resolution of questions, that algebraists have mg.de a kind oHheorem of it. We have seen (Arj^ 46), that {a-\-by\a—b) ^a^ — J^: hence, a^-J2 --a-\-b. a —b If we divide . . a^—b^ by a—b we have a^-b-" a — b a*—¥ -.a^-\-ab-\-b'^ : also :a"+a'3+a5^+i= a —b by performing the division. These are results that may be obtained by the ordinary pro cess of division. Analogy would lead to the conclusion that what- ever may be the exponents of the letters a and b, the division could be performed exactly ; but analogy does not always lead to cer- 44 ALGEBRA. tainty. To be certain on this point, denote the exponent by m ; and proceed to divide a"'—b'" by a — b. a"'—b'^ \\a—h 1st. Rem. . . . a"^^b—b"'\a'"--^ + or .... 3(a"*-'-J'»-'). Dividing a" by a the quotient is a"'-\ by the rule for the exponents. The product of a— 3 by a™-' being subtracted from the dividend, the first remainder is a'^'i— J", which can be put under the form b (a"'-^—b'"-^). Now, if «"-' — §'»-' is divisible by a—b, then will oT'—b'" also be divisible by a— ^^ ; that is, if the difference of the similar powers of two quantities of a certain degree, is exactly divisi- ble by the difference of these quantities, the difference of the powers of a degree greater by unity, is aJso divisible by it. But it has already been shown that a*— i* is divisible hy a—b : hence, a^—b^ is also divisible by a—b. Now, if a^—¥ is divisible hy a—b, it must follow that a^ — b'' is also divisible hy a—b. In the same way ;t may be shown that the division is possible when the exponent is 7, 8, 9, &c. Hence, generally, a*"— J'" is divisible hy a—b. This proposition may be verified by actually performing the division, and then multiplying the quotient by the divisor. Thus, a -b - But . . ultiplied by . a'^-'+a^-^b+a'^-'b-' . . a - b _ a^-^h-a'"-^¥ . . . —ab^-' — b'". equal to . . . a"— *'". It will be perceived that the partial products a" and — &"• are the only ones that do not destroy each other in the reduction. For example, multiplying a""-^b by a, the product is a'"-'3 ; but by multiplying a"*^' by —b, the product is —a"'-% and this term destroys the preceding. The other terms cancel in the same way. DIVISION. 45 The beginner should reflect upon the first method of demonstrating the proposition, as it is frequently employed in algebra. 60. We have given (Art. 51. and 55.), the principal circum- stances by which it may be discovered that the division of monomial or polynomial quantities is not exact; that is, the case in which there does not exist a third entire algebraic quantity, which, multi- plied by the second, will produce the first. We will add, as to polynomials, that it may often be discovered by mere inspection that they cannot be divided by each other. When these polynomials contain two or more letters, before arrang- ing them with reference to a particular letter, observe the two terms of the dividend and divisor, which are affected with the highest exponent of each of the letters. If for either of these let- ters, one of the terms with the highest exponent is not divisible by the other, we may conclude that the total division is impossible. This remark applies to each of the operations required by the process for finding the quotient. Take, for example, 12a^—da-h-\-lab''—Ub\ to be divided by Aa' — 8ab-{-Sb\ By considering only the letter a, the division would appear pos- sible ; but regarding the letter b, the division is impossible, since — llb^ is not divisible by 35\ One polynomial A, carmot be divided by another B containing a letter which is not found in the dividend ; for it is impossible that a third quantity, multiplied by B which depends upon a certain letter, should give a product independent of this letter. A monomial is never divisible by a polynomial, because every polynomial multiplied by another, gives a product containing at least two terms which are not susceptible of reduction. 61. Remark If the letter with reference to which the dividend is arranged, is not found in the divisor, the divisor is said to be inde- pendent of that letter ; and in that case the exact division is impos- sible, unless the divisor mill divide separately the co-efficient of each term of the dividend. 46 ALGEBRA. For example, if the dividend were Sba' + 9ba^+12h, arranged with reference to the letter a, and the divisor 3b, the divisor would be independent of the letter a ; and it is evident that the exact divi- sion could not be performed unless the co-efRcient of each term of the dividend were divisible by 3Z>. The exponents of the leading letter in the quotient would be the same as in the dividend. OF ALGEBRAIC FRACTIONS. 62. Algebraic fractions should be considered in the same point of view as arithmetical fractions, such as f , |i, that is, we must con- ceive that the unit has been divided into as many equal parts as there are units in the denominator, and that one of these parts is taken as many times as there are units in the numerator. Hence, addition, subtraction, multiplication, and division, are performed ac- cording to the rules established for arithmetical fractions. It will not, therefore, be necessary to demonstrate those rules, and in their application we must follow the procedures indicated for the calculus of entire algebraic quantities. 63. Every quantity which is not expressed under a fractional form is called an entire algebraic quantity. 64. An algebraic expression, composed partly of an entire quan- tity and partly of a fraction, is called a mixed quantity. 65. When a division of monomial or poljniomial quantities cannot be performed exactly, it is indicated by means of the known sign, and in this case, the quotient is presented under the form of a frac- tion, which we have already learned how to simplify (Art. 51). With respect to polynomial fractions, the following are cases which are easily reduced. a^' — b^ Take, for example, the expression -; — — ^ — r— ' a- — 2ab-{-b' This fraction can take the form , ,~ (Art. 46). (a— by ' OF FRACTIONS. 47 Suppressing the factor a — h, which is common to the two terms, we obtain .... • , a—b 5a^ — 10a^b+5ab^ Again, take the expression r-^ — 3-77 oa — Oft This expression can be decomposed thus : — -j- rr ba(a — b)- or — — . 8a\a-b) Suppressing the common factor, a(a—b,) the result is . . 5(a-b) 8a * The particular cases examined above, are those in which the two terms of the fraction can be decomposed into the product of the sum by the difference of two quantities, and into the square of the sum or difference of two quantities. Practice teaches the manner of per- forming these decompositions, when they are possible. But the two terms of the fraction may be more complicated poly- nomials, and then, their decomposition into factors not being so easy, we have recourse to the process for finding the greatest common divisor. CASE I. Of the Greatest Common Divisor. 66. The greatest common divisor of two polynomials, is the great- est polynomial, with reference to the exponents and co-efficients, that will exactly divide the proposed polynomials. , If two polynomials be divided by their greatest common divisor, the quotients will he prime with resjject to each other ; that is, they will no longer contain a common factor. For, let A and B be the given polynomials, D their greatest com- mon divisor, A' and B' the quotients afler division*. Then * Note. — When the same letter is used to designate different quantities, as above, the quantities having a certain connexion with each other, we read A', B', A prime, B prime, and if we have A", B", we say, A second, B second, &c. 48 ALGEBRA. A B -^=A' and -^=B' Or . . . A=A'xD and B=B'xD now if A' and B' had a common factor d, it would follow that dxD would be a divisor, common to the two polynomials, and greater than D, either with respect to the exponents or the co-effi- cients, which would be contrary to the definition. Again, since D exactly divides A and B, every factor of D will have a corresponding factor in both A and B. Hence, 1st. The greater common divisor of two polynomials contains as factors, all the particular divisors common to the two polynomials, and does not contain any other factors. 67. We will now show that the greatest common divisor of two polynomials will divide their remainder after division. Let A and B be two polynomials, D their greatest common divisor, and suppose A to contain the highest exponent of the letter with re- ference to which they are arranged. Then, A B Yr=A' and — =B' or, A=A'xD and B=B'xD. Let us now represent the entire part of the quotient by Q and the remainder by R, and we shall have A A'xD R A'xD=B'xDxQ+R R hence, A'=B'xQ+^. But A' is an entire quantity, hence the quantity to which it is R equal is also entire : and since B'Q is entire, it follows, that — is entire ; that is,D will exactly divide R, We will now show that if D will exactly divide B and R that it will also divide A. For, having divided A by B we have OF FRACTIONS. 49 A=BxQ+R> and by dividing by D, we obtain A B ^ R But since we suppose B and R to be divisible by D, and know Q to be an entire quantity, the second part of the equality is entire ; hence the first part, to which it is equal, is also entire ; that is, A is exactly divisible by D. Hence, 2dly . The greatest common divisor of tivo polynomials is the same as that which exists between the least polynomial and their remainder after division. These principles being established, let us suppose that it is re- quired to find the greatest common divisor between the two poly- nomials a^ — a'h + Mh' — U^ and a' — 5rt6+4J^ First Operation. a'-a'b ^Sai'-Sb'\\a'-bab + W' ^a'b-ab^-2b' \a+ U 1st. Rem. . . 19ab'-19b' or . . . 19b%a-b) Second Operation. a'-dab+^b' \\a-b — 4:ab-\-U- 1 rt— 4i 0. Hence, a— Z» is the greatest common divisor. We begin by dividing the polynomial of the highest degree by that of the lowest degree ; the quotient is, as we see in the above table, a-\-Ab and the remainder is 19ab'^—19b^. By the second principle, the required common divisor is the same as that which exists between this remainder and the polynomial divisor. But 19ab'^—l9¥ can be put under the form I9b-(a—b). Now 5 50 ALGEBRA. the factor, 19i^ will divide this remainder without dividing a' — bah+4.h\ hence, by the first principle, this factor cannot enter into the greatest common divisor ; we may therefore suppress it, and the question is reduced to finding the greatest common divisor between' a^ — bah+lP and a—h. Dividing the first of these two polynomials by the second, there is an exact quotient, a — 4Z>; hence a—b is their greatest common divisor, and is consequently the greatest common divisor of the two proposed pohmomials. Again, take the same example, and arrange the polynomials with reference to h. — SJ'-I-SflJ'— a^^+a\ and W-bab-\-a\ First Ojoeration. lJ/-rl2ah'-4a'b+4:a' || 4 b''~5ab-\-a' 1st. Rem. — dah'—a''b +4a' -I2ah"--4:a'b+l6a' -Sb, -3a 2d. Rem. . or -19a^^ + 19a^ 19a\-b-{-a). Second Operation. 4P-^5ab+o'\\ -I +n —ab +a' I — 4tb-{-a 0. Hence, —b-^a, or n — b, is the greatest common divisor. Here we meet with a difficulty in dividing the two polymonials, because the first term of the dividend is not exactly divisible by the first term of the divisor. But if we observe that the co-efficient 4 of this last, is not a factor of all the terms of the polynomial 4^^ — 5rtZ>-|-a% and that therefore, by the first principle, 4 cannot form a part of the greatest common divisor, we can, without afiecting this common OF FRACTIONS. 51 divisor, introduce this flictor into the dividend. This gives -123' + 12aZ<=— 4a"5+4a% and then the division of the first two terms is possible. Effecting this division, the quotient is — Sb, and the remainder is — 3ab^—a'b-i-4:a\ As the exponent of 5 in this remainder is still equal to that of the divisor, the division may be continued, by multiplying this remainder by 4, in order to render the division of the first term possible. This done, the remainder becomes — 12a^^ — 4a^^+16a^ which divided by 4.b^ — dal)-\-a', gives the quotient —3a, which should be separated from the first by a comma, having no connexion with it; and the remainder is . — 19a*J+19a^. Placing this last remainder under the form 19a-( — 3+a), and sup- pressing the factor 19rt^, as forming no part of the common divisor, the question is reduced to finding the greatest common divisor between 4:P — 5ab-]-a', and •-b-\-a. Dividing the first of these polynomials by the second, we obtain an exact quotient, — 45+a; hence —b+a, or a— b, is the greatest common divisor requii'ed. 68. In the above example, as in all those in which the exponent of the prmcipal letter is greater by unity in the dividend than in the divisor, we can abridge the operation by multiplying every term of the dividend by the square of the co-efiicient of the first term of the divisor. We may easily conceive that, by this means, the first par- tial quotient obtained will contain the first power of this co-efficient. Multiplying the divisor by the quotient, and making the reductions with the dividend thus prepared, the result will still contain the co- efficient as a factor, and the division can be continued until a re- mainder is obtained of a lower degree than the divisor, with refe- rence to the principal letter. Take the same example as before, viz. —SP + Sab'^—a^b-i-a^ and 4Z>'' — 5rti+rt^ ; and multiply the dividend by the square of 4=16 : and we have 52 ALGEBRA. First Operation. -m ¥ + ^Qab- -IQa'b + 16a=||43=-5aZ>+a^ -12a// — Ad'b 4- 16a° | — 126-3a 1st. Rem. . . — IQa'^Z* + lOa^ IQa'-' (-^'+«) Second Operation. Alf — bah+a^ \\ —h+a — ab+a^ I —^b+a or 2d- Rem. ... - 0. Remark 1. When the exponent of the principal letter in the di- vidend exceeds that of the same letter in the divisor by two, three, &c. units, multiply the dividend by the third, fourth, &c. power of the co-efficient of the first term of the divisor. It is easy to see the reason of this. 2. It might be asked if the suppression of the factors, common to all the terms of one of the remainders, is absolutely necessary, or whether the object is merely to render the operations more simple. Now, it will easily be perceived that the suppression of these factors is necessary ; for, if the factor 19a^ was not suppressed in the pre- ceding example, it would be necessary to multiply the whole divi- dend by this factor, in order to render the first term of the dividend divisible by the first term of the divisor ; but then, a factor would be introduced into the dividend which was also contained in the divi- sor ; and consequently the required greatest common divisor would be combined with the factor 19a^ which should not form a part of it. 69. For another example, it is proposed to find the greatest com. mon divisbr between the two polynomials, a'+^a'b+^a'b'-Qab'+^b' and Aa''b+2ab'-^b\ or simply, 2a''+ab—b\ since the factor 2b can be suppressed, being a factor of the second polynomial and not of the first. OF FRACTIONS. 53 First Operation. 8a* + 24a'J + 32a^5' - 48a&' + 1 65* 1 1 '■Za^ +ah-lf +20a='6+36a-'^''-48a5' + 16F| 4a' + 10ai+13A=' + 26a'Z»' — 38a5' + 16^'* 1st. Rem. . . . -b\aV + -2W or, . . . —¥{bla-2U). Second Operation. Multiply by 2601, the square of 51. 5202a^+2601aZ>-2601^>'|| 51a- 2 95 5202a^-2958a5 I 102a + 109& 1st. Rem. . +5559a6- 26015^ 5559ai-316lZ'' 2d. Rem. ... + 5606^ The exponent of the letter a in the dividend, exceeding that of the same letter in the divisor by two units, we multiply the whole dividend by the cube of 2, or 8. This done, we perform three con- secutive divisions, and obtain for the first principal remainder, -51a5'+29Z>*. Suppressing b^ in this remainder, it becomes — 51a+295 for a new divisor, or, changing the signs, which is permitted, 51a— 295: the new dividend is ^a^-^-ah—h". Multiplying this dividend by the square of 51, or 2601, then effect- ing the division, we obtain for the second principal remainder, +5605^ which proves that the two proposed polynomials are prime with re- spect to each other, that is, they have not a common factor. In fact it results from the second principle (Art. 67), that the greatest com- mon divisor must be a factor of the remainder of each operation ; therefore it should divide the remainder 5605== ; but this remainder is independent of the principal letter a ; hence, if the two polyno- mials have a common divisor, it must be independent of a, and will consequently be found as a factor in the co-efficients of the different powers of this letter, in each of the proposed polynomials ; but it is 5* 54 ALGEBRA. evident that the co-efficients of these polynomials have not a com- mon factor. 70. These examples are sufficient to point out the course the be- ginner is to pursue, in finding the greatest common divisor of two polynomials, which may be expressed by the following general RULE. I. Take the first polynomial and suppress all the mo7iotniaI factors common to each of its terms. Do the same with the second polynomial, and if the factors so suppressed have a common divisor, set it aside as forming a part of the com7no7i divisor sought. II. Having done this, prepare the dividend in a such a manner that its first term shall be divisible by the first term of the divisor ; then perform the division, which gives a remainder of a degree less than that of the divisor, in which suppress all the factors that are common to the co-efiicients of the different jfowers of the j^rincipal letter. Then fake this remainder as a divisor, and the second polyno^nial as a divi- dend, and cojitinue the operation with these polynomials, in the same manner as with the p)receding. III. Continue this series of operations until a remainder is obtained which will exactly divide the preceding rcTnainder ; this last retnainder will be the greatest common divisor ; but if a remainder is obtained which is independent of the principal letter, and which will not divide the co-efficients of each of the proposed polynomials, it shoics that the jiroposed polynomials are prime with respect to each other, or that they have not a common factor. EXAMPLES, 1. Find the greatest common divisor between the two poly- nomials. ah-\-2a- — '&¥ —4bc— ac — c\ and . . 9ac+2a'— 5aJ + 4c=+8k — 12^' OF FRACTIONS. 55 First Operation. ^h a-U^ 2a^-5b a- 125" — c —Uc + 9c + 8bc — c^ + 4c^ 1st. Remainder 65 10c a +95" -125c - 5 c* (35— 5c) (2a+35+c). Second Operation. 2a^-55 +9c a- 125" +85c +4c" -85 +8c t-125= + 85c + 4c" 2a+35+c s— 45 +4c 0. Hence, 2fl+ 35 +c is the greatest common divisor. After arranging the two polynomials, the division may be perform- ed without any preparation, and the first remainder will be, 65 I a+ 95- -10c -125c ' - 5c" To continue the operation, it is necessary to take the second po- lynomial for a dividend, and this remainder for a divisor, and multi- ply the new dividend by 65— 10c, or simply 35— 5c, since 2 is a factor of the first temi of the dividend. But we are not at liberty to multiply by 35— 5c, if it is a factor of the remainder. There- fore, before effecting the multiplication, we must see if 35— 5c will exactly divide the first remainder ; we find that it does, and gives for a quotient 2a+35 + c : whence it follo\vs that the remainder can be put under the form (35-5c)(2a+35+c). 56 ALGEBRA. Now, 3b— 5c is a factor of this remainder, and is not a factor of the new dividend. For, being independent of the letter a, if it was a factor of the dividend it would necessarily divide the co- efficient of this letter in each of the terms, which it does not ; ^ve may there- fore suppress it without affecting the greatest common divisor. This suppression is indispensable, for otherwise a new factor would be introduced into the dividend, and then the two polynomials con- taining a factor they had not before, the greatest common divisor would be changed; it would be combined with the factor Sb—5c, which should not form a part of it. Suppressing this factor, and effecting the new division, M'e obtain an exact quotient ; hence 2a+3^+c is the greatest conomon divisor. Remaek. The rule for the greatest common divisor of two po- lynomials, may readily be extended to three or more polynomials. For, having the polynomials A, B, C, D, &c. if we find the greatest common divisor of A and B, and then the greatest common divisor of this result and C, the divisor so obtained will evidently be the greatest common divisor of A, B and C ; and the same process may be applied to the remaining polynomials. 2. Find the greatest common divisor of x*—l and a^^+o;'. Ans. l+x"^. 3. Find the greatest common divisor of 4a' — 2a^ — 3a + l and 3a' — 2a — 1. Ans. a — 1. 4. Find the greatest common divisor of a*—x* and a^—a'x—ax'' +x\ Ans. a''—x\ 5. Find the greatest common divisor of SGa"- 18a^— 27a^ + 9a' and 27a-h''-18a'b' — 9a'b\ Ans. 9aXa—l). 6. Find the greatest common divisor of qnp' + 2npY—^npcf—2nq'' and 2tnpY—A7np*—mp''q+2mpq\ Ans. p—q. OF FRACTIONS. 57 7. Find the greatest common divisor of the two polynomials l5a'+l0a'b-}-4:a'b^-\-6a'b' — 2ab'' 12a'b^+38a^b'+16ab*-l0b\ Ans. Sa^+2ab-b\ CASE 11. 71. To reduce a mixed quantity to the form of a fraction. RULE. Multiply ilie entire part by tlie dekominator of ihefracUon : then connect this jJrodiict with the terms of the numerator by the rules for addition, and under the result place the given denominator. EXAMPLES. 1. Reduce X- (a^-x^) X to the form of a fraction. X — a'-x' x->-{a'-x') 2x X . Ans. X X 2. Reduce X- 2a to the form of a L fraction. flX'-X^ Ans. . 2a 2x— 7 3. Reduce 5H — — to the form of a fraction ox 17x-7 Ans. — . 3x a; — a — 1 4. Reduce 1— to the form of a fraction. 2a-x+l Ans. . a X— 3 5. Reduce l+2x— r — to the form of a fraction. ox 10x=+4x4-3 ^^^- 5^ • 58 ALGEBRA. CASE III. 72. To reduce a fraction to an entire or mixed quantity. RULE. Divide the numerator by the denominator for the entire part, and place the remainder, if any, over the denominator for the fractional part. EXAMPLES. 1. Reduce ax-\-a^ X to a mixed quantity. ax+a"" ft^ =a+— Ans. 2. Reouce ax—x"^ X to an entire or mixed quantity. Ans. a—x. 3. Reduce ah-2a^ b to a mixed quantity. 2a' Ans. a——r. 4. Reduce a^-x' a—x to an entire quantity. Ans. a-\-x 5. Reduce x'-y' x—y to an entire quantity. Ans, x'+xy+y^. 6. Reduce 10a;=-5x+3 r-; to a mixed quantity. 3 Ans. 2a;- 1+—. ,^K^ CASE IV. 73. To reduce fractions having different denominators to equiva- lent fractions having a common denominator. OF FRACTIONS 59 RULE. Multiply each numenttor into all the denominators except its oton, for the new numerators, and all the denominators together for a com- mon denominator. EXAMPLES. a h 1. Reduce — and — to equivalent fractions having a com- mon denominator. ay^c^ac , , , „ . the new numerators. hxl>=h^ ) and . . hxc=bc the common denominator. a a-\-b ' ^ to tractions, having a C( ab+P 2. Reduce -r- and to fractions, having a common de- be ' » Ans. -r- and be be 3x 2b 3. Reduce — -, 7;—, and d, to fractions having a common dc- 9cx Aab Gacd nominator. Ans. — — , — — and — — . Qac Qac (\ac „ , 3 2a; 2x 4. Reduce — -, — , and a-\ , to fractions having a com- 9a Qax 12a=+24x mon denommator. Ans. , , and 12a 12a 12a r r, ^ la' a'+x' 5. Reduce -— , -— and ■ , to fractions having a com- 2 3 a+x ° mon denominator. 3a+3a: 2a^+2a^x 6a=+6a;' Ans. - — — -, — — — — , and Qa + Qx' Qa-\-Gx ' 6a+6.T CASE V. 74. To add fractional quantities together. 60 ALGEBRA. RULE. Reduce the fractions, if necessary, to a common denoyninator : then add the numerators togetJier and place their sum over the common rfe- nominator. EXAMPLES. 1. Find the sum of — , — , and -j. Here, . axdxf=adf' cxixf—cbf y the new numerators. exbxd=ebd And . . bxdxf=idf the common denominator. adf cbf eld adf+cbf-\-ebd "'"^"' -w^W^W^ ' bdf — '^^ '""^- ^ Sx^ ^ ^ ^ 2ax 2. To a 7- add b-\ . b c 2abx—'6cx'' Ans. a+6+- bc XXX X 3. Add — -, — and — together. Ans. x+—-. 4> o 4 12 ^ ^ ,^ x-2 4a; , ^ iar-14 4. Add — - — and -— together. \ns. — . o 1 21 x—2 2a,'— 3 5. Add xH — — to ^x-\ — . lOx-lT Ans. Ax-\- 12 5a;* x-^a 6. It is required to add 4a;, — - — , and — - — together. 2a 2a; ° 5a;^+aa;+a' Ans. 4a;4— 2aa; « T ■ ■ •■ ,, 2.r 7.T , 2a;+l 7. It IS required to add — , — , and — together. tj 4 u 49a;+12 Ans. 2a;+- 60 OF FRACTIONS. 61 8. It is required to add 4x, — , and 2+— together. 441;+ 90 Atis Ax-\— 45 9. It is required to add 3a: +— and x — — together 2x 8x together. 23a; Am. 2x-{ — 7=— 45 CASE VI. 75. To subtract one fractional quantity from another. RULE. I. Reduce the fractions to a common denominator. II. Subtract the numerator of the fraction to he subtracted f'om the numerator of the other fraction, and place the difference over tJie coip~ mon denominator. EXAMPLES. X *~- Qi 2rt ^— 4^ 1. Find the difference of the fractions ^, and — 2b 3c Here, (x— a)x 3c=:3cx— 3ac , , ^ ^ ^ the numerators (2a— 4a;)x2J=4a3— 85a; And, 25 x3c=65c the common denominator. ^cx — Zac 4ab—8bx 3cx—Sac — 4:ab-\-8bx "^"'^^' —Wc 6br-= eTc -• '^"^• 12.r 3.r 2. Required the difference of — — and — 3. Required the difference of 5i/ and — . 39a; Ans. -3^. Ans. —- — , 62 Sx 2x 4. Required the difference of — and — . ^"^- -63- 5. Requia-ed the difference between — ^ — and —-. a dx-\-ad—bc Alls. bd Sx+a 2x+l 6. Required the difference of — ^ — and Ans bb 8 24a;+8a— lOZ'a; — 355 40^> X X — a 7. Required the difference of 3a;+— and x . cx-\-bx—ab Ans. 2x+- bc CASE VII. 76, To multiply fractional quantities together. RULE. If the quantities to he multiplied are mixed, reduce them to a frac tionalform ; then multiply the numerators together for a numerator and the denominators together for a denominator. EXAMPLES. 1. Multiply bx -y-a- bx a^+l a X en 3e, . . . • • a^+bx X a c 1^' a^c-{-bcx ad 2. Required 3a; the product of — - and Sa ~b' Ans. 9ax OF FRACTIONS. 63 2a; Sx' 3. Required the product of — and — . Ans. -^— . 5a 2a; 2ab Sac 4. Find the continued product of — , , and ^, . * a c 2b Alls. 9ax. hx a 5. It is required to find the product of b-\- — and — . ab-{-bx Ans. X x'^ — P x'^+b' 6. Required the product of — r and — y— — . x'—b- Ans. x-}-l x—1 7. Required the product of x-i , and , Ans. b\+bc'' ax"—ax-\-x'^—\ a--\-ab d^ 0^ y^ 8. Required the product of a-\ ■ by a—x a; -\-x' Ans. ax-{-ax'—x^—x* CASE VIII. 77. To divide one fractional quantity by another. RULE. Reduce the mixed quantities, if there are any, to a fractional form : then invert the terms of the divisor and multiply the fractions together as in the last case. EXAMPLES. 1. Divide .... a——- by — . 2c g b _2ac—b . ^~2^~~2r~ 64 ALGEBRA. Hence, b . / 2ac-b g " 2c- g- 2c ^f 2acg- - 2c/ :^. Ans. 2. Let 7x 12 — be divided by — . O lo ^MS. 91a; 60 * 3. Let 4a-^ — -— be divided by 5x. 4x 4. Let — - — be aivided by -— . b 6 ^ns. a:+l 4a; * 5. Let X x be divided by -— . X — 1 2 ^?JS. 2 X-\' 6. Let — be divided by —^. Ans. bbx 2a ' 7. Let -r—-r be divided by , , . Serf •' 4(Z Ans. x-b 6c'x' 8. Let ..■-2te+*' "'"'■"■ledby- Ans. 78. We will add but a single proposition more on the subject of fractions. It is this. If the same number be added to each of the terms of a projier fraction, the new fraction resulting from this addition will be greater than the first ; but if it be added to the terms of an hwproper fraction, the re- sulting fraction will be less than the first. a Let the fraction be expressed by — , and suppose aa. Hence the second fraction is greater than the first. If the given fraction is improper, or a>5, it is plain that the nu- merator of the second fraction will be less than that of the first, once bm would be less than am. CHAPTER n. Of Equations of the First Degree. 79. An Equation is the expression of two equal quantities with the sign of equality placed between them. Thus, x^=a-{-b is an equation, in which x is equal to the sum of a and b, 80. By the definition, every equation is composed of two parts, separated from each other by the sign =. The part on the left of the sign, is called the first member, and the part on the right, is called the second member ; and each member may be composed of one or more terms. 81. Every equation may be regarded as the enunciation, in alge- braic language, of a particular problem. Thus, the equation a;+a;=: 30, is the algebraic enunciation of the following problem ; To find a number which, being added to itself, shall give a sum equal to 20. Were it required to solve this problem we should first express it in algebraic language, which would give the equation x-\- a;=:30. By adding a; to itself, we have 2a;=:30. and by dividing by 2, we obtain .... x= 15. 6* 66 ALGEBRA. Hence we see that the solution of a problem by algebra, consists of two distinct parts. 15^. To express algebraically tlie relation between the known and uiiknown quantities. 2d. To find a value for the unknown quantity, in terms of those which are known, which substituted in its place in the given equation vnll satisfy the equation ; that is, render tJie first meinber equal to the second. This latter part is called the solution of the equation. 82. An equation is said to be verified, when such a value is sub- stituted for the unknown quantity as will prove the two members of the equation to be equal to each other. 83. Equations are divided into different classes. Those which contain only the first power of the unknown quantity, are called equations of the first degree. Thus, ax -\- b =1 cx+d is an equation of the 1st. degree. 2x^—'6x =5 —2a;'' is an equation of the 2d. degree. 4a;' — 5a;'' =3x4- 11 is an equation of the 3d. degree. In general, the degree of an equation is denoted by the greatest of the exponents with which the unknown quantity is affected. 84. Equations are also distinguished as numerical equations and literal equations. The first are those which contain numbers only, with the exception of the unknown quantity, which is always de- noted by a letter. Thus, 4a;— 3=2a;-f 5, 3a;' — a;=8, are numerical equations. They are the algebraical translation of problems, in which the known quantities are particular numbers. The equations ax—b=:zcx-\-d, ax^-{-bx-=c, are literal equations, in which the given quantities of the problem are represented by letters. 85. It frequently occurs in algebra, that the algebraic sign + or — , which is written, is not the true sign of the term before which it is placed. Thus, if it were required to subtract —b from a, we should write EQUATIONS OF THE FIRST DEGREE. 67 Here the true sign of the second term of the binomial is pkis, al- though its algebraic sign, which is written in the first member of the equation, is — . This minus sign, operating upon the sign of h, which is also negative, produces a plus sign for b in the result. The sign which results, after combining the algebraic sign with the sign of the quantity, is called the essential sign of the term, and is often different from the algebraic sign. By considering the nature of an equation, we perceive that it must possess the three following properties. 1st. The two members are composed of qu antities of the same kind. 2d. The two members are equal to each other. 3d. The essential sign of the two members must be the same. Equations of the First Degree involving but one unknown quantity. 86. An axiom is a self-evident proposition. We may here state the following. 1. If equal quantities be added to both members of an equation, the equality of the members will not be destroyed. 2. If equal quantities be subtracted from both members of an equation, the equahty will not be destroyed. 3. If both members of an equation be multiphed by the same number, the equality will not be destroyed. 4. If both members of an equation be divided by the same num. bar, the equality will not be destroyed. 87. The transformation of an equation consists in changing its form without affecting the equality of its members. The following transformations are of continued use in the resolu- tion of equations. First Transformation. 88. When some of the terms of an equation are fractional, to re- duce the equation to one in which the terms shall be entire. • 68 Take the equation, 2x 3 X First, reduce all the fractions to the same denominator, by the knowTi rule ; the equation becomes 48a; 54a; 12a; "72 72""*" 72 ~^^ and since we can multiply both inembers by the same number with- out destroying the equality, we will multiply them by 72, which is the same as suppressing the denominator 72, in the fractional terms, and multiplying the entire term by 72 ; the equation then becomes 48x'— 54a;+12a;=792. or dividing by 6 8a;— 9a;+ 2a;=:132. 89. The last equation could have been found in another manner by employing the least common multiple of the denominators. The common multiple of two or more numbers is any number which they will both divide without a remainder ; and the least common multiple, is the least number which they will so divide. The least common multiple will be the pz'oduct of all the numbers, when, in comparing either with the others, we find no common fac- tors. But when there are common factors, the least common mul- tiple will be the product of all the numbers divided by the product of the common factors. The least common multiple, when the numbers are small, can generally be found by inspection. Thus, 24 is the least common multiple of 4, 6, and 8, and 12 is the least common multiple of 3, 4 and 6. 2a; 3 a; Take the last equation — — a;+— =11. We see that 12 is the least common multiple of the denomina- tors, and if we multiply all the terms of the equation by 12, and divide by the denominators, we obtain 8a;— 9.r+2x=132. the same equation as before found. EQUATIONS OF THE FIRST DEGREE. 69 90. Hence, to make the denominators disappear from an equation, we have the following RULE. I. Form the least common multiple of all the denominators. II. Multiply each of the entire terms by this multiple, and each of the fractional terms hy the quotient of this multiple divided by the de- nominator of the term thus multiplied, and omit the denominators of the fractional terms. EXAMPLES. 1. Clear the equation ~r-\—^ — 4=3 of its denominators. Ans. 7x+5x-140=105. 2. Clear the equation -7 —-\-fz=g. Alls. ad—bc-\-ldf=zbdg. 3.. In the equation ax 2c^x Abc'^x 5a^ 2c^ b ab a? W a the least common multiple of the denominators is (V'V ; hence clear- ing the fractions, we obtain a''bx-2a%c''x^\aW=M''&x-ha^-^2a'^We—2,aW. Second Ti'ansformatiun. 91. When the two members of an equation are entire polynomials, to transpose certain terms from one member to the other. Take for example the equation .... 5x — 6=84-2x. If, in the first place we subtract 2a? from both members, the equality will not be de- stroyed, aud we have 5a;— 6 — 2a;=:8. Whence we see that the term 2a;, which was additive in the second member becomes subtractive in the first. '!0 ALGEBRA. In the second place if we add 6 to both members, the equahty will still exist and we have 5x— 6 — 2a; + 6=8+6. Or, since —6 and +6 destroy each other 5a;— 2a;=8+6. Hence the term which was subtractive in the first member, passes into the second member with the sign of addition. Again, take the equation ax-\-b=d—cx. If we add ex to both members and subtract b from them, the equation becomes .... ax-\-b-\-cx—b=d—cx+cx—b. or reducing ax-\-cx=d—b. Therefore, for the transposition of the terms, we have the following RULE. Any term of an equation may be transposed from one member to the other by changing its sign. 92. We will now apply the preceding principles to the resolution of the equation, 4a;— n=2,r+5. by transposing the terms — 3 and 2a; it becomes 4a;— 2a;=54-3 Or reducing . 2x=;8 8 Dividing by 2 . a;=— =4. Now, if 4 be substituted in the place of x in the first equation, it becomes 4x4—3=2x4+5 or .... 13=13. Hence, the value of x is verified by substituting it for the unknown quantity in the given equation. For a second example, take the equation 5a; Ax 7 13a; 1 o 12 3 8 6 • EQUATIONS OF THE FIRST DEGREfc. 71 By making the denominators disappear, we have lOx — 32j;— 312= 21 — 52x or, by transposing . lOx'— 32a;+52a;= 21 +312 by reducing . . . 300;= 333 333 111 dividing by 30 . . a,'=-— -=:— — — =11,1. oO 10 a result which may be verified by substituting it for x in the given equation. For a third example let us take the equation (3a— ic) (a—b)+2ax=4:h{x-[-a). It is first necessary to perform the multiplications indicated, in or- der to reduce the two members to two polynomials, and thus be able to disengage the unknown quantity x, from the known quantities. Having done that, the equation becomes, Sa'' — ax—2ab-\-bx-\-2ax=z4:bx-\-4:ab. or by transposing . — cw;+Z'a;+2avr— 4ix =4a5+3ai— 3a' by reducing . . ax—Sbx =lab— So' Or, (Art. 48). . . {a-Sb)x=:7ab-Sa^ Dividing both members by a—Zb we find 7a5— 3a^ ^= a-3b ' 93. Hence, in order to resolve any equation of the first degree, we have the following general RULE. I. If there are any denominators, cause them to disappear, and per- form, in both members, all the algebraic operations indicated : we thus obtain an equation the tico members of which are entire polynomials. II. Tlien transpose all the terms affected with the unknown quantity into the first member, and all the known terms into the second member. III. Reduce to a single term all the terms involving x : this term will be composed of two factors, one of which will be x, and the other all the multipliers of x, connected with their respective signs. 72 ALGEBRA. IV. Divide both members by the number or polynomial by which the unknown quantity is multiplied. EXAMPLES. 1. Given 3a;— 2+24=31 to find X. Ans. x=S. 1 2. Given a;+ 18= 3a;— 5 to find a;. Ans. a;=ll— . 3. Given 6 — 2a;+10=20 — 3a;— 2 to find x. Ans. x=2. 4. Given x+— a;+— a;=ll to find a;. Ans. x=6. Z o 1 6 5. Given 2a; — — a;+l = 5a;— 2 to find x. Ans. x=—, a 6. Given Saj:+— — S=bx—a to find x. 6-Sa Atis. X-. '6a-2b 7. Given — - — 1-^=20 — to find a;. 1 Ans. a;=23— . 4 a;+3 X x—b 8. Given ~^+"§"^'*'"~4~ ^° ^' Ans. a;=3— . lo ax—b a bx bx—a 9. Given — - — +y = y 3— ^° ^"'^ ^• 3* Ans. X- '2a — 2b' 10. Find the value of x in the equation a— A a+o a* + Sa^b+'ia''b'' — 6aP + 2b* Ans. x= 2b(2a''+ab—b') EQUATIONS OF THE FIRST DEGREE. 73 Of Questions producing Equations of the- First Degree involving hut a single unknown que 94. It has already been observed (Art. 81), that the sohition of a problem by algebra, consists of two distinct parts. 1st. To express the conditions of the problem algebraically ; and 2d. To disengage the unknown from the known quantities. We have already explained the manner of finding the value of the unknown quantity, after the question has been stated ; and it only remains to point out the best methods of enunciating a problem in the language of algebra. This part of the algebraic resolution of a problem, cannot, like the second, be subjected to any well defined rule. Sometimes the enunciation of the problem furnishes the equation immediately ; and sometimes it is necessary to discover, from the enunciation, new con- ditions from which an equation may be formed. The conditions enunciated are called explicit conditions, and those which are de- duced from them, implicit conditions. In almost all cases, however, we ai-e enabled to discover the equa- tion by applying the following RULE. Consider the problem solved ; and then indicate, hj means of alge. hraic signs, upon the known and unknown quantities, the same course of reasoning and operations which it ivould he necessary to perform-, in order to verify the unknown quantity, had it been given. QUESTIONS. 1. Find a number such, that the sum of one half, one third, and one fourth of it, augmented by 45, shall be equal to 448. Let the required number be denoted by . . . x , 7 74 A.LGEBRA. X Then, one half of it will be denoted by . . . — . X one third of it . . by . . • -^. o X one fourth of it . . by . . . — . And by the conditions, —-+— +-^+45=448. ^ o 4 Or by subtracting 45 from both members, XXX _+_+_=403. B^ clearing the terms of their denominators,we obtain 6a;+4a;+3a;=4836. or . . 13a;=:4836'. 4836 Hence . 0;=— — -=372. lo Let this result be verified. ^72 S72 S72 -^-+-— +-^-l-45=-186 + 124 + 93 + 45=448, 2 o 4 2. What number is that whose third part exceeds its fourth, by 16. Let the required number be represented by x. Then, -3-^= = the third part. 1 the fourth part And by the question 1 1 -x--.= 16. or, . 4a;-3a:=192. a;=192. Verification. 192 192 -^ T-=64-48=16. 3 4 EQUATIONS OF THE FIRST DEGREE. 75 3, Divide $1000 between A, B, and C, so that A shall have i72 more than B, and C $100 more than A. Let . . x= B's share of the $1000. Then . . x-\- 12= A's share. And . . x+n2= C's share. Their sum 3x4-244=1000. Whence, 3a;= 1000-244=756 or 756 a;=— ;— -=$252= B's share. 3 X-}- 72=252+ 72=$324= A's share. And a;+172=252 + 172=$424= C's share. Verification. 252+324+424=1000. 4. Out of a cask of wine which had leaked away a third part, 21 gallons were afterwards drawn, and the cask being then gauged, appeared to be half full : how much did it hold ? Suppose the cask to have held x gallons. a; Then, — = what leaked away. And — + 21= all that was taken out of it. o x 1 Hence, — + 21=— a; by the question. o Z or 2a;+126=3x-. or — .-c = — 126. or X = 126, by changing the signs of both members, which does not destroy their equahty. Verification. 126 126 __+21=42+21=63=-^. 5. A fish was caught whose tail weighed 9lb. ; his head weighed 76 ALGEBRA. as much as his tail and half his body, and his body weighed as much as his head and tail together ; M'hat was the weight of the fish? Let . 2 a;= the weight of the body. Then . 9+a;= weight of the head. And since the body weighed as much as both head and tail 2a;=9+ 9+a; or . . 2x— a;=:18 and a;=18. Veiification. 2a;=:36Z^'= weight of the body. 9+a;=27/3= weight of the head. 9Zi= weight of the tail. Hence, . 72 lb= weight of the fish. 6. A person engaged a workman for 48 days. For each day that he laboured he received 24 cents, and for each day that he was idle, he paid 12 cents for his board. At the end of the 48 days, the account was settled, when the labourer received 504 cents. Re. quired the number of working days, and the number of days lie was idle. If these two numbers were known, by multiplying them respec- tively by 24 and 12, then subtractmg the last product from the first, the result would be 504. Let us indicate these operations by means of algebraic signs. Let . . X = the number of working days. 48— a; = the number of idle days. Then 24X'i-' = the amount earned, and 12(48— x)= the amount paid for his board. Then 24a;— 12(48— x) =504 what he received, or 24x-576 + 12x=504. or 36a;=504+576 = 1080 1080 and a;=— ^j^=30 the working days. whence, 48—30=18 the idle days. EQUATIONS OF THE FIRST DEGREE. 77 Verification. Thirty day's labor, at 24 cents a day, amounts to 30x24=720 cts. And 18 day's board, at 12 cents a day, amounts to 18x12=216 cts. And 720—216=504, the amount received. This question may be made general, by deno- ting the whole number of working and idle days. The amount received, for each day he worked, The amount paid for his board, for each idle day, And the balance due the laborer, or the result of the account, As before, let the number of working days be represented ....... The number of idle days will be expressed Hence, what he earns will be expressed and the sum to be deducted, on account of his board. The equation of the problem therefore is, ax — h(^n — x)^c whence ax—h n+lx=c {a-\-h)x=c +^n c -{-hn c -\-hn nn-\-ln—c — hn by n. by a. by h. by c. by X. by n—x. by ax. by b{n--x). and consequently, a +b a-i-b 071 — c a+b 7. A fox, pursued by a greyhound, has a start of 60 leaps. He makes 9 leaps while the greyhound makes but 6 ; but three leaps of the greyhound are equivalent to 7 of the fox. How many leaps must the greyhound make to overtake the fox ? From the enunciation, it is evident that the distance to be passed 78 ALGEBRA. over by the greyhound is composed of the 60 leaps which the fox is in advance, plus the distance that the fox passes over from the mo- ment when the greyhound starts in pursuit of him. Hence, if we can find the expression for these two distances, it will be easy to form the equation of the problem. Let «=: the number of leaps made by the greyhound before he overtakes the fox. Now, since the fox makes 9 leaps while the greyhound makes 9 3 but 6, the fox will make — or — leaps while the greyhound makes 1 ; and, therefore, while the greyhound makes x leaps, the 3 fox will make —x leaps. Hence, the distance which the greyhound must pass over, will be 3 expressed by 60-\-—x leaps of the fox. It might be supposed,that in order to obtam the equation, it would 3 be sufficient to place x equal to 60+- ^a;; but in doing so, a manifest error would be committed ; for the leaps of the greyhound are greater than those of the fox, and we would then equate hetero- geneous numbers, that is, numbers referred to different units. Hence it is necessary to express the leaps of the fox by means of those of the greyhound, or reciprocally. Now, according to the enunciation, 3 leaps of the greyhound are equivalent to 7 leaps of 7 the fox, then 1 leap of the greyhound is equivalent to — leaps of the fox, and consequently a; leaps of the greyhound are equivalent Ix to -r- of the fox. or reducing to a common denominator, a7i-\-x—a =lst part. In order to form the 2d part, this first part and 2a must be subtract- (an-\-x—a) , . ed from x : this gives x—2a -, or reducing to a com- 80 ALGEBRA. mon denominator and subtracting, nx — ^an — x-\-a 2d. remainder. N ow, the second part is composed of 2a, plus the nih. part of this nx—^an—x-\-a remainder; therefore, it is 2a-\ , or reducing to a common denominator, 2an^-\-nx—^an—x-{-a —^ =2d. part. Subtracting the two first parts pkis 3fl, from x, we have {an+x—a) (2avF-\-nx—^an—x-\-a) n ir ' Or, reducing to a common denominator, and performing the opera- tions indicated, n^x—Qarv' —^nx-^-^an-^x—a 7. 3d. remainder. „ , ^ . n^x—6a7r — 27ix-\-4.a7i-\-x—a Hence the 3d part is 3a-] Or, reducing to a common denominator, tian^-\-n'^x—6an^—2nx-\-4ian-\-x—a :3d part. But from the enunciation, the estate of the father is found to be entirely divided. Hence, the difference between x, and the sum of the three parts should be equal to zero. This gives the equation an 4-x— a 2an^+ nx—San—x-\-a , =0. 2,a')^-\-ivx —%aT?—2nx-\-^:an-\-x—a by making the denominators disappear, and performing the opera- tions indicated, we have n='x— 6a7i3— S/i^'a'+lOan^+Snx— 5an— a;+a=0. EQUATIONS OF THE FIRST DEGREE. 81 Whence, _6an'-10«n2+5an-a_a(6n='— 1071^ + 571-1) ^^ iv'—Sn^ + dn — l ~ rv'—Sn^-i-Sn—l A more simple equation and result may be obtained, by observ- ing, that the part which goes to the third child is composed of 3a, plus the 7ith part of what remains, and that the estate is then entirely exhausted ; that is, the third child has only the sum 3a, and the re- mainder just mentioned is nothing. Now the expression for this remainder has been found to be ri^x—6an^—2nx-\-4:an-\-x—a n^ • Placing this equal to zero, and making the denominator disappear, we have n^x—6an^—2nx-\-4ia7i-\-x—a^0. 6an^—4:an-{-a a(6n^—4:n-\-l) Whence x= t, — - — — — = ^ — „ , , 7r— 2/t+l 7r— 2/1+1 Ve}ificatio7i. To prove the numerical identity of this expression with the pre- ceding, it is only necessary to show that the second can be deduced from the first, by suppressing a factor common to its numerator and denominator. Now if we apply the rule for finding the greatest common divisor (Art. 70.), to the two polynomials a(67i2- 1071^+571-- 1) and v? —Zii' J^^n—X, it will be seen that ?i— 1 is a common factor, and by dividing the numerator and denominator of the first expression by this factor, the result will be the second. This problem shows the beginner how important it is to seize upon every circumstance in the enunciation of a question, which may facilitate the formation of the equation, otherwise he runs the risk of arriving at results more complicated than the nature of the the question requires. The conditions which have served to form successively the ex- 82 ALGEBRA. pressions for the three parts, are the explicit conditions of the pro- blem ; and the condition which has served to determine the most simple equation of the problem, is an implicit condition, which a little attention has sufficed to show, was comprehended in the enun- elation. To obtain the values of the three parts, it is only necessary to substitute for x its value in the three expressions obtained for these parts. a(6n2— 471+1) . , , Ai)p]y the formula x=-^ — ^ , , to a particular example. Let a=10000, n=5. We have 10000(6x25-4x5 + 1) 10000x131 1310000 25-10 + 1 16 16 To verify the enunciation in this case : = 81875. 81875-10000 The first child should have, lOOOOH r , or 24375. o There remains then 81875 — 24375, or 57500, to divide between the other two children. 57500-20000 The second should have, 20000 H z , or 27500. o Then there remains 57500 — 27500, or 30000, for the third child. Now 30000 is triple of 10000 ; hence the problem is verified. We can give a more simple and elegant solution to this problem, but it is less direct. It also depends upon the remark, that after hav- ing subtracted 3a and the two first parts from the whole estate, no- thing remains. Denote the three remainders mentioned in the enunciation by r, r', r" . The algebraic expressions for the three parts will be r r' r" fl+— , 2«+— , 3a+— . n n n Now, 1st. From the enunciation, it is evident that r"=0. Therefore the third part is 3a. EQUATIONS OF THE FIRST DEGREE. 83 r' 2d. What remains after giving to the second child 2a-] / (n—iy can be represented by r — — r or . Moreover, this remainder also forms the third part. Therefore we have (n— l)r' San -=Sa; whence /=■ n n— 1 3an 2a Then the second part is 2a-\ --^n=2a-j- -, or convert- ^ n— 1 «— 1 2«n+of ing the whole number into a fraction, and reducing, — . r 3d. The remainder, after giving to the first a-] , can be ex- r (n — l)r pressed by r or . Now this remainder should form 2an-\-a the two other parts, or 3a-) — . _ (n—l)r 2an-\-a ban— 2a Therefore, -^ '—=2a+- - 11 n — \ 11—1 ban — 2a n 5an^ — 2aii Hence, r= — X 7 ^77-= — -, r-^ — . And consequently the first part is 5an^—2an ban— 2a «H 7 T^ — -^n=a-\- {n-lf ■ "~"^ (n-iy ' ban — 2a an" -{-San — a ~""^n2-2n + l "^ n"—2n+l ' Ther the whole estate is 2an-\-a an" -{-San— a Sa+ V+- n— 1 n^— 2n+l Or, by reducing the whole number and fractions to a common denominator, 84 ALGEBRA. 3a(w^ — 2w+l) + (2an+a) (n — l)+a7v' + San—a H^— 27J + 1 * Or performing the operations indicated and reducing 6an^ — 4an+a _o(6?j='— 4n+l) 7i^-2rt+l ~ (n-lf ' which agrees with the preceding result. This solution is more complete than the preceding, since we obtain from it the estate of the father, and the expressions for the three parts. 9. A father ordered in his will, that the eldest of his children should have a sum a, out of his estate, plus the «th part of the re- mainder ; that the second should have a sum 2a, plus the nth part of what remained after having subtracted from it the first part and 2a ; that the third should have a sum Sa, plus the nth part of the new re- mainder — and so on. It is moreover supposed that the children share equally. Required, the value of the father's estate, the share of each child, and the number of children. This problem is remarkable, because the number of conditions con- tained in the enunciation is greater than the number of unknown values required to be found. Let the estate of the father be represented by x : then will x—a express what remains after having taken from it the sum a. There- fo)-e the share of the eldest is x—a an-\-x—a a-\- or =lst. part. n n ^ Subtracting the first part, and 2a, from a;, we have -x—a) nx— ^or, n nx—^an — x-\-a the nth part of which is, . tt Hence, the share of the second child is nx — San — x+a 2an^-\-nx—^an — r+a 2a-] , or ■ ^ = 2d part. (an-{-x—a) nx—San—x-{-a 2a — ^^ or, n n EQUATIONS OF THE FIRST DEGREE. 85 In like manner, the other parts might be formed, but as all the parts should be equal, it suffices to form the equation of the problem, to equate the two first parts, which gives an + a; — a 2an^-\- nx — Ban —x-\-a n ~ r? ' whence, x=an^—2an+a=a{n— If. Substituting this value of a; in the expression for the first part, we find an + an^ — 2an -{-a — a or reducing. 71 ■an—a=^a{n — \) ; and as the parts are equal, by dividing the whole estate by the first part, we will obtain a quotient that will show the number of child. an^—2an-\-a , , , ^ ren; therefore, , or n—l, denotes the number of an— a children. The father's estate, . . an" —2an-\-a=a{n — lf. The share of each child, . rt(w— 1). Whole number of children, . {i^—^)' It yet remains to be shown, that the other conditions of the pro. blem are satisfied ; that is, that by giving to the second child, 2a plus the nth part of what remains ; to the third, 2>a plus the nth part of what remains, &c., the share of each child is in fact (n— 1)«. The difference between the estate of the father and the first part being «(n— 1)^— a(n— 1), the share of the second child will be a(n-lY-a(n-\)-2a 2a(n-l)+a{n-lf-a{n-\) 2a-\-- ^ , or -^ ^ ^ -, n n and reducing a(n-\)-\-a{n-\y a{n-\) {\+n-l) or , n n 8 or fl(n — 1). In like manner, the difference between a{n—iy and the two first parts being, a{n—iy—2a{n—l), the third part will be a{n-lY-2a{n-l)-2a Sa-\ , n which being reduced, becomes a{n-l)+a{n-iy a(n-l). In the same way we would obtain for the fourth part fl(7i_l)2_3rt(,i_l)_4a a(,n—l)+ci(.7i—iy 4a -1 , or , and so on. 71 n Hence all the conditions of the enunciation are satisfied. 10. What number is that from which, if 5 be subtracted, | of the remainder will be 40 ? Am. 65. 11. A post is \ in the mud, i in the water, and ten feet above the water : what is the whole length of the post ? Ans. 24 feet. 12. After paying I and \ of my money, I had 66 guineas left in my purse : how many guineas were in it at first ? Ans. 120. 13. A person was desirous of giving 3 pence a piece to some beggars, but found he had not money enough in his pocket by 8 pence : he therefore gave them each 2 pence and had 3 pence re- maining : required the number of beggars. Ans. 11. 14. A person in play lost \ of his money, and then won 3 shil- lings ; after which he lost \ of what he then had ; and this done, found that he had but 12 shillings remaining : what had he at first ? Ans. 205. 15. Two persons, A and B, lay out equal sums of money in trade ; A gains $126, and B loses $87, and A's money is now double of B's : what did each lay out ? Ans. $300. 16. A person goes to a tavern with a certain sum of money in his pocket, where he spends 2 shillings ; he then borrows as much mo- EQUATIONS OF THE FIRST DEGREE. 87 ney as he had left, and going to another tavern, he there spends 2 shillings also ; then borrowing again as much money as was left, he went to a third tavern, where likewise he spent two shillings and borrowed as much as he had left ; and again spending 2 shillings at a fourth tavern, he then had nothing remaining. What had he at first ? -471S. 3*. 9d. Of Equations of the First Degree involving two or more unknown quantities. 95. Although several of the questions hitherto resolved, contain- ed in their enunciation more than one unknown quantity, we have resolved them by employing but one symbol. The reason of this is, that we have been able, from the conditions of the enunciation, to express easily the other unknown quantities by means of this syra- bo] ; but this is not the case in all problems containing more than one unknown quantity. To ascertam how problems of this kind are resolved : first, take some of those which have been resolved by means of one unknown quantity. 1. Given the sum a, of two numbers, and their difference h, it is required to find these numbers. Let X— the greater, and y the less number. | Then by the conditions .... x+y= a. and .... x—y=h. By adding (Art. 86. Ax. 1.) . . . 2x=a+i. By subtracting (Art. 86. Ax. 2.) . . 2y—a—b. Each of these equations contains but one unknown quantity. a+b From the first we obtain . . • x= . a — b And from the second .... u=— -— . 88 ALGEBRA. Verification, a+i a — b 2a , o + i a — b 2b -2-+^-=¥ = '^' ^"^ "2 ^=Y=^- For a second example, let us also take a problem that has been already solved. 2. A person engaged a workman for 48 days. For each day that he labored he was to receive 24 cents, and for each day that he was idle he was to pay 12 cents for his board. At the end of the 48 days, the account was settled, when the laborer received 504 cents. Required the number of workmg days and the number of days he was idle. Let X — the number of working days. y = the number of idle days. 71 — the whole number of days = 48. a = what he received per day for work = 24 cts. b = what he paid per day for board = 12 cts. c = what he received at the end of the time = 504. Then, ax = what he earned, And by = what he paid for his board. x-\- y=n. We have by the question . . . . , •' ^ ( ax—by:= c. It has already been shown that the two members of an equation can be multiplied by the same number, without destroying the equal- ity ; therefore the two members of the first equation may be multi- plied by b, the co-efRcient of y in the second, and we have The equation ..... bx-{-by=bji. Which, added to the second . . ax— by= c. Gives ...... ax-\-bx^bn-\-c. bn-\-c Whence ...... a;= — — ;. a+b In like manner, multiplying the two members of the first equa- tion by a, tlie co-efficient of x in the second, it becomes EQUATIONS OF THE FIRST DEGREEo 89 From which, subtract the second equation, And we obtain Whence By introducing a symbol to represent each of the unknown quan- titles in the preceding problem, the solution which has just been given has the advantage of making known the two required num. bers, independently of each other. Elimination. 96. The method which has just been explained of combining two equations, involving two unknown quantities, and deducing there- from a single, equation involving but one, may be extended to three, four, or any number of equations, and is called elimination. There are three principal methods of elimination : 1st. By addition and subtraction. 2d. By substitution. 3d. By comparison. We will consider these methods separately. Eliminalion hy Addition and Subtraction. 97. Take the two equations . . , ,, which may be regarded as the algebraic enunciation of a problem containing two unknown quantities. If, in these equations, one of the unknown quantities was affected with the same co-efRcient, we might, by a simple subtraction, form a new equation which would contain but one unknown quantity, and from which the value of this unknown quantity could be deduced. Now, if both members of the first equation be multiplied by 9, the co-efficient of y m the second, and the two members of the second by 7, the co-efficient of y in the first, we will obtain 8* 90 ALGEBRA. 45a,'+63^=:387, equations which may be substituted for the two first, and in which y is affected with the same co-efficient. Subti'acting, then, the first of these equations from the second, there results 32a;=96, whence x=3. Again, if we muhiply both members of the first equation by 11, the co-efficient of x m the second, and both members of the second by 5, the co-efficient of x in the first, we will form the two equations 55a;+77?/=473, o/ic; \ which may be substituted for the two ODX-\-HtOy — o45, ; proposed equations, and in which the co-efficients of « are the same. Subtracting, then, the second of these two equations from the first, there results 32^=128, whence 2/=4. Therefore a;=3 and y=,4:, are the values of x and y, which should verify the enunciation of the question. Indeed we have, 1st. 5x3+7x4=15+28=43; 2d. 11x3 + 9x4=33+36=69. The method of elimination, just explained is called the method by addition and subtraction, because the unknown quantities disappear by additions and subtractions, after having prepared the equations in such a manner that one unknown quantity shall have the same co-efficient in two of them. Elimination by Substitution. 5a;+7t/=43. 98. Take the same equations . . # , , . „ ^ ( lla;+9j/=69. Find the value of x in the first equation, which gives 43-7y Substitute this value of x m the second equation, and we have 43 -7y nX—~+9y=Q9. EQUATIONS OF THE FIRST DEGREE. 91 or or Hence And 473-77?/+45j/=345. — d2y = — l28. y=4. 43-28 This method, called the method by substitution, consists in finding the value of one of the unknown quantities m one of the equations, as if the other unknown quantities were already determined, and in substituting this value in the other equations; in this way new equa- tions are formed, which contain one unknown quantity less than the given equations, and upon which we operate as upon the proposed equations. Elimination by Comparison. 5x+72/=43 llx+9y=e9. Finding the value of a; in the first equation, we have 43 -7z/ 99. Take the same equations have And finding the value of a; in the second, we obtain 69 — 9y Let these two values of x be placed equal to each other, and we 43— 7y_ 69- 9y ~ IT Or, . Or, . Hence, And, 473-771/ = 345-45?/ -d2y = -128. y= 4 69-36 11 This method of elimination is called the method by comparison, and consists in finding the value of the same unknown quantity in all the equations, placing them equal to each other, two and two, which 92 ALGEBRA. necessarily gives a new set of equations, containing one unknown quantity less than the other, upon which we operate as upon the proposed equations. But there is an inconvenience in the two last methods, which the method by addition and subtraction is not subject to, viz. : they pro- duce new equations, containing denominators, which it is afterwards necessary to make disappear. The metliod by substitution is, how- ever, advantageously employed whenever the co-efficient of one of the unknown quantities is equal to unity in one of the equations, be- cause then the inconvenience of which we have just spoken does not occur. We shall sometimes have occasion to employ it, but gene- rally, the method by addition and subtraction is preferable. It more- over presents this advantage, viz. : when the co-efficients are not too great, we can perform the addition or subtraction at the same time with the multiplication which is necessary to render the co-ef- ficients equal to each other. 100. Let us now consider the case of three equations involving three unknown quantities. {5a;-6j/+42=15. 7x-{-4^y—Sz=19. 2x+ ?/+6s=46. To eliminate 2 b)' means of the first two equations, multiply the first by 3 and the second by 4, then since the co-efficients of z have contrary signs, add the two results together : this gives a new equation ...... 43a;— 2]/= 121 " Multiplying the second equation by 2, a fac- tor of the co-efficient of s; in the third equation, and adding them together, we have . . 16x-\-9y= 84 The question is then reduced to finding the values of x and y, which will satisfy these new equations. Now, if the first be multiplied by 9, the second by 2, and the re- sults be added together, we find 419a;=rl257, whence x=3. EQUATIONS OF THE FIRST DEGREE. 93 We might, by means of the two equations involving x and y, de- termiiie y in the same way we have determined x ; but the value of y may be determined more simply, by observing that the last of these two equations becomes, by substituting for x its value found above, 84-48 48+9?/=84 whence y= — — =4. In the same manner the first of the three proposed equations, be- comes, by substituting the values of x and y, 24 15—24+4^=15, whence z—~=6. 101. Hence, if there are m equations involving a like number of unknown quantities, the unknown quantities may be eliminated by the following RULE. I. To eliminate one of the unknown quantities, comhine any one of the equations with each of the m— 1 others ; there will thu^ he obtain- ed m— 1 new equations containing m— 1 unknown quantities. II. Eliminate another unknown quantity hy combining one of these new equations with the m— 2 others ; this will give m— 2 equations containing m— 2 unknoion quantities. III. Continue this series of operations until a single equation con. taining but one unknown quantify is obtained, from which the value of this unknoion quantity is easily found. Then hy going back through the series of equations which have been obtained, the values of the other unknoion quantities may be successively determined. 102. It often happens that each of the proposed equatiotis does not contain all the unknown quantities. In this case, with a little address, the elimination is very quickly performed. Take the four equations involving four unknown quantities : 2x-Zy+2z=\Z\ . . (1) 4i/+2sr=14 . . (3). ^u-2x=Zo\ . . (2) 5i/ + 3M=32 . . (4). By inspecting these equations, we see that the ehmination of % in the two equations, (1) and (3), will give an equation involving .r and y ; and if we eliminate u in the equations (2) and (4), we will ob- tain a second equation, involving x and ?/. These two last unknown quantities may therefore be easily determined. In the first place, the elimination of z in (1) and (3) gives . . 7^— 2x=l That of M in (2) and (4), gives . . . 20j/+6a;=38 Multiplying the first of these equations by 3, and adding Whence Substituting this value in 7?/— 2x=l, we find Substituting for x its value in equation (2), it becomes 4m— 6=30, whence And substituting for y its value in equation (3), there results ...... 41t/=41 y= 1 x= 3 EXAMPLES. yy 1. Given 2a;+3?/=16, and 2>x—2y=\\ to find the values of a; and y. Ans. a;=5, 3/=2. 2. Given '2x 3i/ 9 3x 2j/ 61 5 + 4 ~20 ^"^ 4 + 5 = 120 ^° ^^ ^^'^ ''^^"'' of X and y. 1 1 Ans. x^—, y=Y- 3. Given X y y+7?/=99, and Y+7a;=51, to find the values of X and y. Ans. xz=zl, y=\A. 4. Given ^-..=-^-,8, .a t''+:-B=v+-. to find the values of x and y. Ans, x—%^, ?/=40. ' ^+ y+ z=29^ 5. Given x-\- 2*/+ 32=62 1 , ^ , , < > to find X, y and z. Am. x=Q, y=9, z=12. EQUATIONS OF THE FIRST DEGREE. 95 6. Gi 7. Given <' 2x+ Ay— 3z=22 4a;— 2y-\- 5z = 18 6a;+ 7y— 2=63 "+Y^+T^=3"^ T^'+T^+-5-^=^''^ to find a.', y and t. iln5. a;=3, y='!, 2=4. > to find X, y and z. T^+T^+^ 12 2z+ 3w=17^ x=12, y=20, 5=30. > to find X, y, z, u, and I. Ix- Ay— 2z+ <=11 8. Given r+— a; — — =4. Therefore . . 15a;+10.T;-24.i-=120. Or, . . x= 120 the number of eggs of each sort. 3. A person possessed a capital of 30,000 dollars for which he drew a certain interest ; but he owed the sum of 20,000 dollars, for which he paid a certain interest. The intei-est that he received ex- ceeded that which he paid by 800 dollars. Another person pos. 9 Let 2x-- Then X-. Then will 1 And 1 3^^ But 5 : 2 : : sold. 98 ALGEBRA. sessed 35,000 dollars, for which he received interest at the second of the above rates, but he owed 24,000 dollars, for which he paid interest at the first of the above rates. The interest that he re- ceived exceeded that which he paid by 310 dollars. Required, the two rates of interest. Let X and y denote the two rates of interest : that is, the interest of Si 00 for the given time. To obtain the interest of $30,000 at the first rate denoted by x, we form the proportion 30,000a; 100 : a; : : 30,000 : : —vw^ or 300a;. And for the interest $20,000, the rate being y. 20,000y 100 : 3/ : : 20,000 : : -JT^ or 200y. But from the enunciation, the difference between these two in- terests is equal to 800 dollars. We have, then, for the first equation of the problem, 300x-200(/=800. By writing algebraically the second condition of the problem, we obtain the other equation, 350?/-240x=310. Both members of the first equation being divisible by 100, and those of the second by 10, we may put the following, in place of them : Sx—2y=8, S5y—24x=3l. To eliminate x, multiply the first equation by 8, and then add it to the second ; there results l9y—95, whence i/=5. Substituting for y, in the first equation, its value, this equation becomes 3a;— 10=8, whence a;=6. Therefore, the first rate is 6 per cent., and the second 5. EQUATIONS OF THE FIRST DEGREE. 99 Verification. $30,000, placed at 6 per cent., gives 300x6, = 81800. $20,000, do. 5 do. 200x5, = 81000. And we have 1800 — 1000=800. The second condition can be verified in the same manner. 4. There are three ingots composed of different metals mixed together. A pound of the first contains 7 ounces of silver, 3 ounces of copper, and 6 of pewter. A pound of the second contains 12 ounces of silver, 3 ounces of copper, and 1 of pewter. A pound of the third contains 4 ounces of silver, 7 ounces of copper, and 5 of pewter. It is required to find how much it will take of each of the three ingots to form a fourth, which shall contain in a pound, 8 ounces of silver, 3| of copper, and 4| of pewter. Let X, y and z represent the number of ounces which it is neces- sary to take from the three ingots respectively, in order to form a pound of the required ingot. Since there are 7 ounces of silver in a pound, or 16 ounces, of the first ingot, it follows that one ounce of it contains -^^ of an ounce of silver, and consequently in a num. 7x ber of ounces denoted by x, there is — ounces of silver. In the 12^ Az same manner we would find that — — - and — -, express the num. lb 16 '^ ber of- ounces of silver taken from the second and third, to form the fourth ; but from the enunciation, one pound of this fourth ingot contains 8 ounces of silver. We have, then, for the first equation 7a; \2y Ax 16 16 16^ or, making the denominators disappear. . 7x-f 12?/+4z=128 As respects the copper, we should find . . 3a'+ 3^+7?= 60 and with reference to the pewter . . . Qx-\- ?/4-52= 68 As the co-efficients of y in these three equations, are the most simple, it is most convenient to eliminate this unknown quantity first. 100 ALGEBRA. JMultiplyiiig the second equation by 4, and subtracting the first equation from the product, we have . . . 5ic+24z=112 Multiplying the third equation by 3, aiid subtracting the second from the product . . 15a;+ 82=144 Multiplying this last equation by 3, and subtracting the preced- ing one from the product, we obtain 40a;=320, whence x=8. Substitute this value for x m the equation 15x+82=rl44 ; it be- comes 120 + 82=144, whence z=3. Lastly, the two values x=8, «=3, being substituted in the equa- tion 6a;+2/+52=68, give 48+?/+15=68, whence 2/=5. Therefore in order to form a pound of the fourth ingot, we must take 8 ounces of the first, 5 ounces of the second, and 3 of the third. Verification. If there be 7 ounces of silver in 16 ounces of the first ingot, in 8 ounces of it, there should be a number of ounces of silver ex- pressed by —r^' 12x5 , 4X3 In like manner — — — and — r^- will express the quantity lb lb of silver contained in 5 ounces of the second ingot, and 3 ounces of the third. 7x8 12x5 4x3 128 ^ ^ ^ Now, we have -r^-i r^ 1 — r^=-rz-=8; therefore, a 16 16 16 lb pound of the fourth ingot contains 8 ounces of silver, as required by the enunciation. The same conditions may be verified relative to the copper and pewter. 5. What two numbers are those, whose difference is 7, and sum 33? Ans. 13 and 20. 6. To divide the number 75 into two such parts, that three times the greater may exceed seven times the less by 15. Ans, 54 and 21. EQUATIONS OF THE FIRST DEGREE. 101 7. In a mixture of wine and cider, i of the whole plus 25 gal- lons was wine, and i part minus 5 gallons was cider ; how many gallons were there of each ? Ans. 85 of wine, and 35 of cider. 8. A bill of £120 was paid in guineas and moidores, and the number of pieces of both sorts that were used was just 100 ; if the guinea be estimated at 21*. and the moidore at 27 s. how many were there of each ? Ans. 50 of each. ► 9. Two travellers set out at the same time from London and York, whose distance apart is 150 miles ; one of them goes 8 miles a^day, and the other 7 ; in what time will they meet ? Ans. In 10 days. 10. At a certain election, 375 persons voted for two candidates, and the candidate chosen had a majority of 91 ; how many voted for each ? Ans. 283 for one, and 142 for the other. 11. A's age is double of B's, and B's is triple of C's, and the sum of all their ages is 140 ; what is the age of each 1 Ans. A's=84, B's=42, andC's=zl4. 12. A person bought a chaise, horse, and harness, for £60 ; the horse came to twice the price of the harness, and the chaise to twice the price of the horse and harness ; what did he give for each ? j'£13. 6s. 8d. for the horse. Ans. J £ 6. 135. M. for the harness. (^ £40. for the chaise. 13. Two persons, A and B, have both the same income : A saves i of his yearly, but B, by spending £50 per annum more than A, at the end of 4 years finds himself £100 in debt ; what is their income? Ans. £125. 14. A person has two horses, and a saddle worth £50 ; nolv if the saddle be put on- the back of the first horse, it will make his value double that of the second ; but if it be put on the back of the second, it will make his value triple that of the first ; what is the value of each horse ? Ans. One £30, and the other £40. 102 ^■■' ALGEBRA. 15. To divide the number 36 into three such parts that \ of the first, i of the second, and J of the third, may be all equal to each otlier. -' '- ■' : Ans. 8, 12, and 16. 16. A footman agreed to serve his master for £8 a year and a livery, but was turned away at the end of 7 months, and received only £2. 135. 4d. and his livery; what was its value? ^ '^ " " Ans. £4. 165. 17. To divide the number 90 into four such parts, that if the first be increased by 2, the second diminished by 2, the third multiplied^ by 2, and the fourth divided by 2, the sum, difference, product, and quotient so obtained, will be all equal to each other. Ans. The parts are 18, 22, 10, and 40. 18. The hour and minute hands of a clock are exactly together at 12 o'clock ; when are they next together? Ans. 111. b-^jViin. 19. A man and his wife usually drank out a cask of beer in 12 days ; but when the man was from home, it lasted the woman 30 da3-s ; how many days would the man alone be in drinking it ? Ans. 20 days. 20. If A and B together can perform a piece of work in 8 days, A and C together in 9 days, and B and C in 10 days : how many days would it take each person to perform the same work alone ? Ans. A 14f|- days, B 17|f, and C 23/,-. 21. A laborer can do a certain work expressed by a, in a time expressed by J ; a second laborei', the work c in a time d ; a third, the work e, in a time/. It is required to find the time it would take the three laborers, working together, to perform the work g. • An5. <^- ^^jj^j^^fj^i^^' Application. a=21 ; J=4 ] c=35 ; d=6 | e=40 ; /=12 ] ^=191 ; r,^' X will be found equal to 12. 22^ If 32 pounds of sea water contain 1 pound of salt, how no EQUATIONS OF THE FIRST DEGREE. 103 much fresh water must be added to these 32 pounds, in order that the quantity of salt contained in 32 pounds of the new mixtui-e shall be reduced to 2 ounces, or | of a pound ? Ans. 22A lb. 23. A number is expressed by three figures ; the sum of these figures is 11 ; the figure in the place of units is double that in the place of hundreds ; and when 297 is added to this number, the sum obtamed is expressed by the figures of this number reversed. What is the number ? Ans. 326 24. A person who possessed 100,000 dollars, placed the greater part of it out at 5 per cent, interest, and the other part at 4 per cent. The interest which he received for the whole amounted to 4640 dollars. Required, the two parts. ^ .^4:^/;,^^^^^'^ ^^ . , ^^^^ 64,000 and 36,000. 25. A person possessed a certain capital, which he placed out at a certain interest. Another person who possessed 10,000 dollars more than the first, and who put out his capital 1 per cent, more advantageously than he did, had an income greater by 800 dollars. A third person who possessed 15,000 dollars more than the first, and who put out his capital 2 per cent, more advantageously than he did, had an income greater by 1500 dollars. Required, the capi- tals of the three persons, and the three rates of interest. Sums at interest, ^30,000, $40,000, $45,000. Rates of interest, 4 5 6 per cent. 26. A banker has two kinds of money ; it takes a pieces of the first to make a crown, and b of the second to make the same sum. Some one offers him a crown for c pieces. How many of each kind must the banker give him ? , , . , a(c—h) , , . , i(a—c) Ans. Istkmd, -^ — r^ ; 2d kind, -^ =-^. a—b a—b 27. Find what each of three persons A, B, C, are worth, know- ing; 1st, that what A is worth added to Z times what B and C are worth is equal top ; 2d, that what B is worth added to m times what 104 ALGEBRA. A and C are worth is equal to q ; 3d, that what C is worth added to n times what A and B are worth is equal to r. This question can be resolved in a very simple manner, by intro- ducing an auxiliary unknown quantity into the calculus. This un- known quantity is equal to what A, B and C are worth. 28. Find the values of the estates of six persons, A, B, C, D, E, F, from the following conditions : 1st. The sum of the estates of A and B is equal to a ; that of C and D is equal to I ; and that of E and F is equal to c. 2d. The estate of A is worth m times that of C ; the estate of D is worth n times that of E, and the estate of F is worth p times that of B. This problem may be resolved by means of a single equation, involving but one unknown quantity. Theory of Negative Quantities. Explanation of the terms, Nothing and Infinity. 104. The algebraic signs are an abbreviated language. They point out in the shortest and clearest manner the operations to be performed on the quantities with which they are connected. Having once fixed the particular operation indicated by a parti- cular sign, it is obvious that that operation should always be perform- ed on every quantity before which the sign is placed. Indeed, the principles of algebra are all established upon the supposition, that each particular sign which is employed always means the same thing ; and that whatever it requires is strictly performed. Thus, if the sign of a quantity is +, we understand that the quantity is to be added ; if it is —, we understand that it is to be subtracted. For example, if we have —4, we understand that this 4 is to be subtracted from some other number, or that it is the result of a sub- traction in which the number to be subtracted was the greatest. If it were required to subtract 20 from 16, the subtraction could not be made by the rules of arithmetic, since 16 does not contain EQUATIONS OF THE FIRST DEGREE. 105 20 ; nor indeed can it be entirely performed by Algebra. We write the numbers for subtraction thus, 16 — 20=16-16— 4= -4. By decomposing —20 into —16 and —4, the —16 will cancel the +16, and leave —4 for a remainder. We thus indicate that the quantity to be subtracted exceeds the quantity from which it is to be taken, by 4. To show the necessity of giving to this remainder its proper sign, let us suppose that the difference of 16 — 20 is to be added to 10. The numbers would then be wi-itten 16-20=- 4 + 10 = + 10 26-20=+ 6 105. If the sum of the negative quantities in the first member of the equation, exceeds the sum of the positive quantities, the second member of the equation will be negative, and the verification of the equation will show it to be so. For example, if a—b=c, and we make a=15 and i=18, c will be =—3. Now the essen- tial sign of c is different from its algebraic sign in the equation. This arises from the circumstance, that the equation a—b=c ex- presses generally, the difference between a and h, without indicating which of them is the greater. When, therefore, we attribute par- ticular values to a and h, the sign of c, as well as its value, becomes known. We will illustrate these remarks t^ a few examples. 1. To find a number which, added to the number I, gives for a sum the number a. Let a;= the required number. Then, by the condition a?+J=a, whence x=a — b. This expression, or formula, will give the algebraic value of x in all the particular cases of this problem. For example, let rt=47, i=29, then x= 47— 29=18. 106 ALGEBRA. - Agaiii, let fl=:24, l=Sl ; then will x=24 — 31 = — 7. This value obtained for x, is called a negative solution. How is it to be interpreted ? Considered arithmetically, the problem with these values of a and b, is impossible, since the number b is already greater than 24. Con- sidered algebraically, however, it is not so ; for we have found the value of a; to be —7, and this number added, in the algebraic sense, to 31, gives 24 for the algebraic sum, and therefore satisfies both the equation and enunciation. 2. A father has lived a number a of years, his son a number of years expressed by h. Find in how many years the age of the son will be one fourth the age of the father. Let x=i the required number of years. Then a-\-x= the age of the father ^ , . at the end of the requir- and b-\-x= the age of the son ' ed time. a-\-x a—^b Hence, by the question — — =Z»+a^ ; whence x= — - — . 54-36 18 Suppose a=54, and b—9 : then x= — — — =——G. The father having lived 54 years, and the son 9, in 6 years the father will have lived 60 years, and his son 15 ; now 15 is the fourth of 60 ; hence, a;=6 satisfies the enunciation. 45-60 Let us now suppose a=45, and h=zl5 : then x=^ — - — =-5. If we substitute this value of x in the equation of condition, we obtain 45-5 =15-5 4 or 10=10. Hence, — 5 substituted for x verifies the equation, and therefore is the true answer. Now, the positive result which has been obtained, shows that the EQUATIONS OF THE FIRST DEGREE. 107 age of the father will be four times that of the son at the expiration of 6 years from the time when their ages were considered ; while the negative result indicates that the age of the father was four times that of his son, 5 years previous to the time when their ages were compared. The question, taken in its most general or algebraic sense, de- mands the time, at which the age of the father was four times that of the son. In stating it, we supposed that the age of the father was to be augmented ; and so it was, by the first supposition. But the conditions imposed by the second supposition, required the age of the father to be diminished, and the algebraic result confoi-med to this condition by appearing with a negative sign. If we wished the result, under the second supposition, to have a positive sign, we might alter the enunciation by demanding hoio many years since the age of the father ivasfour times that of his son. If x= the number of years, we shall have a—x 4& — a —— — b—x: hence x= — r . If a=45 and 5=15, x will be equal to 5. Reasoning from analogy, we establish the following general principles. 1st. Every negative value found for the unknown quantity in a problem of the first degree, will, when taken vnth its proper sign, verify the equation from which it was derived. 2d. That this negative value, taken with its proper sign, will also satisfy the enunciation of the problem, understood in its algebraic serise. 3d. If the emmciation is to be understood in its arithmetical sense, in which the quantities referred to are always supposed to be positive, then this value, considered without reference to its sign, may be con. spidered as the answer to a problem, of which ike enunciation only dif. fersfrom that of the proposed problem in this, that certain quantities which were additive, have become subtractive, and reciprocally. 108 ALGEBRA. 106. Take for example the problem of the labourer (Page. 88). Supposing that the labourer receives a sum c, we have the equations. '' ^-'' x+ y=n ) In^c om—c whence x- — r-, 3/=- ax—by=c ) a-{-b a+b But if we suppose that the labourer, instead of receiving, owes a an c, the equations will then bo a;4- y=n ^ C "x+ y=n, iy—ax=c ) ' ( ax—by= — c. By changing the signs of the second equation. Now it is visible that we can obtain immediately the values of x and y, which correspond to the preceding values, by merely chang- ing the sign of c in each of those values ; this gives bn—c a7i-{-c '^~ a + b ' ^~ a + f' To prove this rigorously, let us denote —chyd; \ x-{- y=n The equations then become j _ and they only differ from those of the first enunciation by having d in the place of c. We would, therefore, necessarily find bn-\-d an—d '^~ a+b ' ^~ a-\-b * And by substituting — c for d, we have bn-\-( — c) an—( — c) ""^ a+b ' ^=' a+b ' or by applying the rules of Art. 85, bn — c an+c ''- a+b ' ^^ a+b ' The results, which agree to both enunciations, may be compre- bended in the same formula, by writing bndzc anrpc y=- a+b ' ■" a+b EQUATIONS OF THE FIRST DEGREE. 100 The double sign ± is read phis or minus, the superior signs cor- respond to the case in which the labourer received, and the inferior signs to the case in which he owed a sum c. These formulas com- prehend the case in which, in a settlement between the labourer and his employer, their accounts balance. This supposes c=0, which gives in an 107. When a problem has been resolved generally, that is, by representing the given quantities by letters, it may be required to determine what the values of the unknown quantities become, when particular suppositions are made upon the given quantities. The determination of these values, and the interpretation of the peculiar results obtained, form what is called the discussion of the problem. The discussion of the following question presents nearly all the circumstances which are met with in problems of the first degree. 108. Two couriers are travelling along the same right line and in the same direction from R' towards R. The number of miles travelled by one of them per hour is expressed by m, and the number of miles travelled by the other per hour, is expressed by n. Now, at a given time, say 12 o'clock, the distance between them is equal to a number of miles expressed by a : required the time when they will be together. R' A B R. At 12 o'clock suppose the forward courier to be at B, the other at A, and R to be the point at which they will be together. Then, AB=rt, their distance apart at 12 o'clock. Let . . t= the number of hours which must elapse, before they come together. And . a;= the distance BR, which is to be passed over by the forward courier. Then, since the rate per hour, multiplied by the number of hours, will give the distance passed over by each, we have, 10 110 ALGEBRA. iX»i ■= a+x=AR. iXn — X =BR. Hence by subtractmg, t{in—n) — a Thereforfe, . . / = . m—n Now so long as m>n, i will be positive, and the problem will be solved ill the arithmetical sense of the enunciation. For, if niyn the" courier from A will travel faster than the courier from B, and will therefore be continually gaining on him : the interval which separates them will diminish more and more, until it becomes 0, and then the couriers will be found upon the same point of the line. In this case,the time i, which elapses, must be added to 12 o'clock, to obtain the time when they are together. But, if we suppose ni, or m and evidently EQUATIONS OF THE FIRST DEGREE. 115 cannot be satisfied by any finite value of x. We will however remark b that, as the equation can be put under the form — =0, if we sub- b stitute for x, numbers greater and greater, — will ditTer less and less from 0, and the equation will become more and more exact ; so b that, we may take a value for x so great that — will be less than b any assignable quantity, or — =0. It is in consequence of this that algebraists say, that infinity satis- fies the equation in this case ; and there are some questions for which this kind of result forms a true solution ; at least, it is certain that the equation does not admit of a solution m finite numbers, and this is all that we wish to prove. 2d. If we have a—Q, b=0, at the same time, the value of x takes the fonn x=-7r- In this case, the equation becomes Oxa;=0, and every finite num- ber, positive or negative, substituted for x, will satisfy the equation. Therefore iJie equation, or the probkm of which it is the algebraic translation, is indeterminate. 113. It should be observed, that the expression — , does not al- ways indicate an indetcrmination, it frequently indicates only the exist, ence of a common factor to the two terms of the fraction, which fac- tor becomes nothing, m consequence of a particular hypothesis. For example, suppose that we find for the solution of a problem, x=-^ — :r7r. If, in this formula, a is made equal to b, there results ci?—b^ x=-. But it will be observed, that a^—¥ can be put under the form {a-b) {a^'+ab+b^), (Art. 59), and that a^-J^ is equal to {a-b) 116 ALGEBRA. (a+b), therefore the vakie of ;^ becomes _{a-b) (a^+aZ.-l-i2) '"-'- {a-b) (a + b) • Now, if we suppress the common factor (a — b), before making the supposition a=J, the value of x becomes a;= —. , '^ a-\-b 3rt2 3a which reduces to a;=— - — , or x=--, when a^Z*. 2a 2 For another example, take the expression a2_J2 _ (a + b) (a-h) {a-by~ (a-b) {a-b}' Making a=b, we find ^^='7r> because the factor (a — b) is com- mon to the two terms ; but if we first suppress this factor, there re- a+b 1 . , 1 , 2« : 5-5 which reduces to x=-—, a—b From this we conclude, that the symbol — someUmes indicates the existence of a common factor to the two terms of the fraction ■which reduces to this form. Therefore, before pronouncing upon the true value of the fraction, it is necessary to ascertain whether the two terms do not contain a common factor. If they do not, we conclude that the equation is really indeterminate. If they do con- tain one, suppress it, and then make the particular hypothesis ; this will give the true value of the fraction, which will assume one of A A the three forms -^, — -, --, in which case, the equation is determi- JD note, impossible in finite numbers, or indeterminate. This observation is very useful in the discussion of problems. Of Inequalities. 114. In the discussion of problems, we have often occasion to suppose several inequalities, and to perform transformations upon them, analogous to those executed upon equalities. We are often INEQUALITIES. 117 obliged to do this, when, in discussing a problem, we wish to esta- blish the necessary relations between the given quantities, in order that the problem may be susceptible of a direct, or at least a real solution, and to fix, with the aid of these relations, the limits between which the particular values of certain given quantities must be found, in order that the enunciation may fulfil a particular condition. Now, although the principles established for equations are in general applicable to inequalities, there are nevertheless some exceptions, of which it is necessary to speak, in order to put the beginner upon his guard against some errors that he might commit, in making use of the sign of inequality. These exceptions arise from the introduction of negative expressions into the calculus, as quantities. In order that we may be clearly understood, we will take exar pies of the different transformations that inequalities may be subject- ed to, taking care to point out the exceptions to which these trans- formations are liable. 115. Two inequalities are said to subsist in the same sense, when the greater quantity stands at the left in both, or at the right in both ; and m a contrary sense, when the greater quantity stands at the right in one, and at the left in the other. Thus, 25>20 and 18>10, or 6<8 and 7<9, ai'e inequalities which subsist in the same sense ; and the inequalities 15>13 and 12<14, subsist in a contrary sense. 1. If we add the same quantity to loth members of an inequality, or subtract the same quantity from both members, the resulting in- equality will subsist in the same sense. Thus, take 8>6 ; by adding 5, we still have 8 + 5>6 + 5 and 8 — 5>6 — 5. When the two members of an equality are both negative, that one is the least, algebraically considered, which contains the great- est number of units. Thus, — 25< — 20 ; and if 30 be added to both members,- we have 5<10. This must be understood entirely in an algebraic sense, and arises from the convention before esta- 118 ALGEBRA. blished, to consider all quantities preceded by the minus sign, as subtractive. The principle first enunciated, serves to transpose certain terms from one member of the inequality to the other. Take, for ex- ample, the inequality a^ + h''y2b- — 2a^ ; there will result from it a^+2a-y2h--P, or 3a2>2^. 2. If two inequalities subsist in the same sense, and tee add them member to member, the resulting inequality will also subsist in the same sense. Thus, from ayb,cyd,eyf, there results a-\-c-{-eyb-\-d-\-f But this is not always the case, when ice subtract, member from mem- , her, two inequalities established in the same sense. Let there be the two inequalities 4<7 and 2<3, we have 4-2 or 2<7-3 or 4. But if we have the inequalities 9<10 and 6<8, by subtracting we have 9—6 or 3>10 — 8 or 2. We should then avoid this transformation as much as possible, or if we employ it, determine in what sense the resulting inequality exists. 3. If the two members of an inequality be multiplied by a positive number, the resulting inequality will exist in the same sense. Thus, from a — , we deduce, by multiply. ing by 6ad, M{a^-V)y2d{c''-d-). The same principle is true for division. But when the two members of an inequality are multiplied or di- vided hy a negative number, the inequality subsists in a contrary seme. Take, for example, 8>7; multiplying by -3, we have -24<-21. INEQUALITIES. 119 8 8 7 In like manner, 8>7 gives — — , or — — < — ^- — 3 3 3 Therefore, when the two members of an inequality are multipli- ed or divided by a number expressed algebraically, it is necessary to ascertain whether the multiplier or divisor is negative ; for, in that case, the inequality would exist in a contrary sense. 4. It is not permitted to change the signs of the two members of an inequality unless we eslahlish the resulting inequality in a contrary sense ; for this transformation is evidently the same as multiplying the two members by —1. 5. Both memlers of an inequality between positive numbers can be squared, and the inequality loill exist in the same sense. Thus from 5>3, we deduce 25>9; from a+3>c, we find 6. Whe7i both members of the inequality are not positive, we cannot tell before the operation is performed, in which sense the resulting in- equality will exist. For example, — 2<3 gives ( — 2)- or 4<9; but — 3>— 5 gives, on the contrary, ( — 3)^ or 9<( — 5)- or 25. We must then, before squaring, ascertam whether the two mem- bers can be considered as positive numbers. EXAMPLES. 1. Find the limit of the value of x in the expression 5a;— 6>1.9. Ans. x>5. 2. Find the limit of the value of x in the expression 14 3x+— a;-30>10 Ans. x>4. 3. Find the limit of the value of x in the expression 1 1 a; 13 17 T'^-Y'^+"2+'2">Y- ^'^' ^>^- 4. Find the limit of the value of a; in the inequalities 120 ALGEBRA. ax a? — ^bx—ab^—. Ix y -^-axArab25. 3x— 7<2x + 13. Resolving these inequalities, we have a;>15 and a; < 20. Any number, therefore, either entire or fractional, comprised between 15 and 20, will satisfy the conditions. 6. A boy being asked how many apples he had in his basket, re- plied, that the sum of 3 times the number plus half the number, di- minished by 5 is greater than 16 ; and twice the number diminished by one third of the number, plus 2 is less than 22. Required the number which he had. Ans. 7, 8, 9, 10, or 11. CHAPTER III. Extraction of the Square Root of Numbers. Forma- tion of the Square and Extraction of the Square Root of Algebraic Quantities. Calculus of Radi- cals of the Second Degree. Equations of the Se- cond Degree. 116. The square or second power of a number, is the product which arises from multiplying that number by itself once : for ex- ample, 49 is the square of 7, and 144 is the square of 12. EXTRACTION OF THE SQUARE ROOT OF NUMBERS. 121 The square root of a number is a second number of such a value, that, when multiplied by itself once the product is equal to the given number. Thus, 7 is the square root of 49, and 12 the square root of 144: for 7x^ = 49, and 12x12 = 144. The square of a number, either entire or fractional, is easily found, being always obtained by multiplying this number by itself once. The extraction of the square root of a number, is however, attended with some difficulty, and requires particular explanation. The first ten numbers are, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and their squares, . 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. and reciprocally, the numbers of the first line are tlie square roots of the corresponding numbers of the second. We may also remark that, the square of a number expressed by a sihgle figure, will contain no figure of a higher denomination than tens. The numbers of the last line 1, 4, 9, 16, &c., and all other num- bers which can be produced by the multiplication of a number by itself, are called perfect squares. It is obvious, that there are but nine perfect squares among all the numbers which can be expressed by one or two figures : the square roots of all other numbers expressed by one or two figures will be found between two whole numbers differing from each other by unity. Thus, 55 which is comprised between 49 and 64, has for its square root a number between 7 and 8. Also, 91 which is comprised between 81 and 100, has for its square root a number between 9 and 10. Every number may be regarded as made up of a certain number of tens and a certain number of units. Thus 64 is made up of 6 tens and 4 units, and may be expressed under the form 60+4=64. Now, if we represent the tens by a and the units by b, we shall have a-\-b = 64 and (a+&)2=(64f or . . . a- + 2ai+Z>2 =4096. 11 122 ALGEBRA. Which proves that the square of a number composed of tens and units contains, the square of the tens plus twice the product of the tens by the units, plus the square of the units. 117. If now, we make the units 1, 2, 3, 4, &c., tens, by annex- ing to each figure a cipher, we shall have, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 and for their squares, 100, 400, 900, 1600, 2500, 3600, 4900, 6400, 8100, 10000. from which we see that the square of one ten is 100, the square of two tens 400 ; and generally, that the square of tens will contain no figure of a less denoininaiion than hundreds, nor of a higher name than thousands. Example I — To extract the square root of 6084. Since this number is composed of more than two places of figures its roots will contain more than one. 60.84 But since it is less than 10000, which is the square of 100, the root will contain but two figures : that is, units and tens. Now, the square of the tens must be found in the two left hand figures which we will separate from the other two by a point. These parts, of two figures each, are called periods. The part 60 is comprised between the two squares 49 and 64, of which the roots are 7 and 8 : hence, 7 is the figure of the tens sought ; and the re- quired root is composed of 7 tens and a certain number of units. The figure 7 being found, we write it on the right of the given 60.84 | 78 number, from which we separate it 49 I by a vertical line : then we subtract 7x2 = 14.8 I 118.4 its square 49 from 60, which leaves I 118 4 a remainder of 11, to which we bi'ing down the two next figures 84. The result of this operation 1184, contains twice the product of the tens by the units plus the square of (he units. But since tens multi- plied by units cannot give a product of a less name than tens, it fol- EXTRACTION OF THE SQUARE ROOT OF NUMBERS. 123 lows that the last figure 4 can form no part of the double product of the tens by the units : this double product is therefore found in the part 118, which we separate from the units' place 4 by a point. Now if we double the tens, which gives 14, and then divide 118 by 14, the quotient 8 is the figure of the tmits, or a figure greater than the units. This quotient figure can never be too small, since the part 118 will be at least equal to twice the product of the tens by the units : but it may be too large ; for the 118 besides the dou- ble product of the tens by the units, may likewise contain tens aris- ing from the square of the units. To ascertain if the quotient 8 expresses the units, we write the 8 to the right of the 14, which gives 148, and then we multiply 148 by 8. Thus, we evidently form, 1st, the square of the units : and 2d, the double product of the tens by the units. This multiplication being effected, gives for a product 1184, a number equal to the result of the first operation. Having subtracted the product, we find the remainder equal to : hence 78 is the root required. Indeed, in the operations, we have merely subtracted from the given number 6084, 1st, the square of 7 tens or 70 ; 2d, twice the product of 70 by 8 ; and Sd, the square of 8 : that is, the three parts which enter into the composition of the square of 70 + 8 or 78 ; and since the result of the subtraction is 0, it follows that 78 is the square root of 6084. Ex. 2. To extract the square root of 841. We first separate the number into periods, as in the last example. In the 8.41 I 29 second period, which contains the square 4 | of the tens, there is but one figuz-e. The 2x2=4.9 144.1 greatest square contained in 8 is 4, the I 44 1 root of which is 2 : hence 2 is the fi- gure of the tens in the required root. Subtracting its square 4 from 8, and bringing down 41, we obtain for a result 441. 124 ALGEBRA. If now, as in the last example, we separate the last figure 1 from the others by a pohit, and divide 44 by 4, which is double the tens, the quotient figure will be the units, or a figure greater than the units. Here the quotient is 11 ; but it is plain that it ought not to exceed 9, for if it could, the figure of the tens already found would be too small. Let us then try 9. Placing 9 in the root, and also on the right of the 4, and multiplying 49 by 9, we obtain for a pro- duct 441 : hence, 29 is the square root of 841. Remark. The quotient figure 11, first found, was too large be- cause the dividend 44 contained, besides the double product of the tens by the units, 8 tens arising from the square of the units. When the dividend is considerably augmented, by tens arising from the square of the units, the quotient figure will be too large. Ex. 3. To extract the square root of 431649. Since the given number exceeds 10,000 its root will be greater than 100 ; that is, it will contain more than two places of figures. But we may still regard the root as composed of tens and units, for every number may be expressed in tens and units. For example, the number 6758 is equal to 675 tens and 8 units, equal to 6750 + 8. Now, we know that the square of the tens of the required root can make no part of the two right hand figures 49, which therefore, we 43.16.49 | 657 separate from the others by a point, and the remaining figures 4316 con- tain the square of the tens of the re- quired root. But since 4316 exceeds 100 the tens of the required root will contain more than one figure : hence 4316 must be separated into two parts, of which the right hand period 16 will contain no part of the square of that figure of the root, which is of the highest name, and for a similar reason we should separate again if the part to the left contained more than two figures. 43.16.49 12.5 36 71.6 5 62 5 130.7 9 14.9 9 14 9 EXTRACTION OF THE SQUARE ROOT OF NUMBERS. 125 Since 36 is the greatest square contained in 43, the first figure of the root is 6. We then subtract its square 36 from 43, and to the remainder 7 bring down the next period 16. Now, since the last figure 6 of the result 716, contains no part of the double product of the first figure of the tens by the second, it follows, that the second figure of the root will be obtained by dividing 71 by 12,double the first figure of the tens. This gives 5 for a quotient, which we place in the root, and at the right of the divisor 12. Then subtract the product of 125 by 5 from 716, and to the remainder bruig down the next period, and the result 9149 will contam twice the product of the tens of the root multiplied by the units, plus the square of the units. If this result be then divided by twice 65, that is, by double the tens of the root, (which may always be found by adding the last figure of the divisor to itself), the quotient will be the units of the root. Hence, for the extraction of the square root of numbers, we have the following RULE. I. Separate the given number into periods of two figures each be- ginning at the right hand, — the period on the left will often contain but one figure. II. Find the greatest square in the first period on the left, and place its root on the right after the manner of a quotient in division. Sub- tract the square of the root from the first period, and to the remainder bring down the second period for a dividend. III. Double the root already found and place it on the left for a di- visor. Seek how many times the divisor is contained in the dividend, exclusive of the right hand figure, and place the figure hi the root and also at the right of the divisor. IV. Multiply the divisor, thus augmented, by the last figure of the root, and subtract the product from the dividend, and to the remainder bring doion the next period for a new dividend. V. Double the whole root already found, for a new divisor, and continue the operation as before, until all the periods are brought doicn. 11* 126 ALGEBRA. 1st. Remark. If, after all the periods are brought down, there is no remainder, the proposed number is a perfect square. But if there is a remainder, you have only found the root of the greatest perfect square coutauied m the given number, or the entire part of the root sought. For example, if it were required to extract the square root of 665, we should find 25 for the entire part of the root and a remainder of 40, which shows that 665 is not a perfect square. But is the square of 25 the greatest perfect square contained in 665 ? that is, is 25 the entire part of the root ? To prove this, we will first show that, the difference between the squares of two conseciUive numlers, is equal to twice the less numher augmented hy unity. Let . . . a = the less number, and . . . a + 1 = the greater. Then . . (a+l)2=a''+2a+l and . . . {af=a- Their difference is . ■ = 2a+l as enunciated. Hence, the entire part of the root cannot be augmented, unless the remainder exceed twice the root found, plus unity. But 25 X 2+1 =51 > 40 the remauider : therefore, 25 is the en- tire part of the root. 2d. Remark. The number of figures m the root will always be equal to the number of periods into which the given number is separated. EXAMPLES. 1. To find the square root of 7225. 2. To find the square root of 17689. 3. To find the square root of 994009. 4. To find the square root of 85678973. 5. To find the square root of 67812675. 118. The square root of a number which is not a perfect square, is called incommensuraljle or irrational, because its exact root can- EXTRACTION OF THE SQUARE ROOT OF NUMBERS. 127 not be found in terms of the numerical unit. Thus, V^ Vs, V77 are incommensurable numbers. They are also sometimes called surds. In order to prove that the root of an imperfect power cannot be expressed by exact parts of unity, we must first show that. Every numher P, lohich loill exactly divide the product A xB of two numbers, and which is prime with one of them, will divide the other. Let us suppose that P will not divide A, and that A is greater than P. Apply to A and P the process for finding the greatest com- mon divisor, and designate the quotients which arise by Q, Q', Q" . . . and the remainders R, R', R" . . . respectively. If the division be continued sufficiently far, we shall obtain a remainder equal to unity, for the remainder cannot be 0, since by hypothesis A and P are prime with each other. Hence we shall have the following equations. A=P Q +R P =R Q' +R' R=iR'Q"+R" R'=R"Q"'+R"' Multiplying the first of these equations by B, and dividing by P, we have AB ^ BR — =BQ+—. AB . , . r. But, by hypothesis, „ is an entire number, and since B and Q are entire numbers, the product BQ is an entire number. Hence BR it follows that „ is an entire number. If we multiply the second of the above equations by B, and divide by P, we have BRQ' BR' 128 ALGEBRA. RR But we have already shown that is an entire number ; B^Q' • • . mu- . • , BR' hence — 5 — is an entire number. 1 his being the case, -r^— must also be an entire number. If the operation be continued until Bxl the number which multiplies B becomes 1, we shall have — - — equal to an entire number, which proves that P will divide B. In the operations above we have supposed A>P, but if P> A we should first divide P by A. Hence, if a number P xvill exactly divide the product of iico num.- hers, and is prime with one of them, it loill divide the other. We will now show that the root of an imperfect power cannot be expressed by a fractional number. Let c be an imperfect square. Then if its exact root can be ex- pressed by a fractional number, we shall have / — « or . • . . ^^="17 by squaring both members. But if c is not a perfect power, its root will not be a whole num- ber, hence -7- will at least be an irreducible fraction, or a and b will be prime to each other. But if a is not divisible hj b, axo- ox c? will not be divisible by b, from what has been shown above ; neither then can c^ be divisible by ¥. Smce to divide by ¥ is but a^ to divide cP twice by b. Hence, -j^ is an irreducible fraction, and therefore cannot be equal to the entire number c : therefore, we . — a cannot assume v c—-j, or the root of an miperfect power can- not be expressed by a fractional number that is rational. EXTRACTION OF THE SQUARE ROOT OF FRACTIONS. 129 Extraction of the square root of Fractions. 119. Since the square or second power of a fraction is obtained by squaring the numerator and denominator separately, it follows that the square root of a fraction will be equal to the square root of the numerator divided by the square root of the denominator. a^ a For exainple, the square root of 7^ is equal to -r- '• ^or a a b'^b b' But if neither the numerator nor the denominator is a perfect square, the root of the fraction cannot be exactly found. We can however, easily find the exact root to within less than one of the equal parts of the fraction. To effect this, multiply both terms of the fraction by the denomina- tor, which makes the denominator a perfect square without altering the value of the fraction. Then extract the square root of the perfect square nearest the value of the numerator, and place the root of the denominator under it; this fraction will be the approximate root. 3 Thus, if it be required to extract the square root of — , wemul- 15 tiply both terms by 5, which gives — : the square nearest 15 is 4 16 : hence — is the required root, and is exact to within less o 1 than — . 5 120. We may, by a similar method, determine approximatively the roots of whole numbers which are not perfect squares. Let it be required, for example, to determine the square root of an entire number a, nearer than the fraction — : that is to say, to find a 130 ALGEBRA. number which shall differ from the exact root of a, by a quantity less than — . n It may be observed that a=:— j-. If we designate by r the entire part of the root of air, the number an^ will then be compris- ed between r^ and. {r+Vf\ and — — will be comprised between ' -"^ — j-^ ; and consequently the true root of a is com- and prised between the root of -^ and 5- — ; that is, between ' ir n^ — and . Hence — will represent the square root of a n n n within less than the fraction — . Hence to obtain the root : n Multiply the given number hij the square of the denominator of the fraction which determines the degree of approximation : then extract the square root of the product to the nearest unit, and divide this root by the denominator of the fraction. Suppose, for example, it were required to extract the square root 1 of 59, to within less than — . Let us repeat on this example, the demonstration which has just been made. 59x(12f The number 59 can be put under the form — — -^ — , or by 8496 multipliying by (12)2, —-. But the root of 8496 to the near- 8496 est unit, is 92 : hence it follows that — -r^ or 59, is comprised be- tween 7— ^ and )— ^. Then, the square root of 59 is itself ( 12 ) ( Iz) EXTRACTION OF THE SQUARE ROOT OF FRACTIONS. 131 92 93 comprised between — and — : that is to say, the true root 92 , 1 differs from — by a fraction less than — . 12 I'* 92 93 8464 8649 Indeed the squares of — and — are ^ and , num. 8496 bers which comprise or 59. {i-'i) 2. To find the VTT to within less than t-t. 15 , 1 3. To find the V223 to within less than ' — . 4 Ans. 3—. 15 37 Ans. 14-—. 40 121. The manner of determining the approximate root in deci- mals, is a consequence of the preceding rule. . . 1 1 To obtain the square root of an entire number within — , r-T-r-r, «Sjc. — it is neccssary according to the preceding rule to mul- tiply the proposed number by (10)^ (100)^ (1000)^ . . . or, which is the same thing, add to the riglit of the number, two, four, six, ^c. ciphers : then extract the root of the product to the nearest unit, and divide this root hy 10, 100, 1000, &c., which is effected hy pointing off one, two, three, ^ is a perfect square, but 98a&* is not a perfect square, because 98 is not a perfect square, and a is affected with an uneven exponent. In the latter case, the quantity is introduced into the calculus by affecting it with the sign V , and it is written thus, VdSah*. Quantities of this^ kind are called radical quantities, or irrational qvMntities, or simply radicals of the second degree. 125. These expressions may sometimes be simplified, upon the principle that, the square root of the product of two or more factors is equal to the product of the square roots of these factors ; or, in alge- braic language, Vabcd . . . = y/a. y/h. \/c. ^/d. . . . To demonstrate this principle, we will observe, that from the de- finition of the square root, we have ( Valcd . . . .f=abcd .... Again, (v/aX v/^X v/cX W . . f=(%/«)'x(%/Jfx(v/c;)'x( W)' • • • =zal)cd .... Hence, since the squares of Vabcd . . . ., and, -v/a. ■s/b. -v/c. \/d. . . ., are equal, the quantities themselves are equal. This being the case, the above expression, ■V98ab*, can be put under the form VA9b'x2a= V^9^X V2a. Now Vio^may be reduced to IP ; hence VoSab^z^lb^ V2a. In like manner, V^So^ZiVd^ V9a-lr'c''x5bd = Sabc V5bd, Vse4:aWc''= VTUa^b^>^ is either +baV^ or —ha¥. Whence we may conclude, that if a monomial is positive, its square root may be affected either with the sign + or — ; thus, VdcF^±2a", for +3a2 or —Za^, squared, gives 9a*. The double sign ± with which the root is affected is read plus or minus. If the proposed monomial were negative, it would be impossible to extract its root, since it has just been shown that the square of every quantity, whether positive or negative, is essentially positive. Therefore, V— 9, V— 4a^ V — Sa% are algebraic symbols which indicate operations that cannot be performed. They are called imaginary quantities, or rather imaginary expressions, and are frequently met with in the resolution of equations of the second 136 ALGEBRA. degree. These symbols can, however, by extending the rules, be simplified in the same manner as those irrational expressions which indicate operations that cannot be performed. Thus, V — 9 may be reduced to (Art. 125.) Vox V^7or,3 V-l; V-Aa^^ VlaFx V-l = 2a \/^ 127. Let us now examine the law of formation for the square of any polynomial whatever ; for, from this law, a rule is to be de- duced for extracting the square root. It has already been shown that the square of a binomial (a + b) is equal to a^-\-2ah+P (Art. 46.). Now to form the square of a trinomial a-{-b-{-c, denote a +3 by the single letter s, and we have (a+J+c)^=(5+c)-=:^+25c+c2. Bui s^^(a+bY^a^ + 2ab+Ir' ; and 2sc=2{a+b)c=2ac-]-2bc. Hence (a+b+cy=a"+2ab + I^+2ac+2bc+c^ ; that is, tJie square of a trinomial is composed of the sum of the squares of its three terms, and twice the products of these terms multiplied together two and two. If we take a polynomial of four or more terms, and square it, we shall find the same law of formation. We may, therefore, suppose the law to be proved for the square of a polynomial of m terms ; and it then only remains to be shown that it will be true for a poly- nomial of m+1 terms. Take the polynomial {a-\-b+c . . . +?), having m terms, and denote their sum by s\ then the polynomial (a-\-b-{-c . . . +i+k) having ??i + l terms, will be denoted by (*+/t). Now, {s-{-kf=^+2sk-]-k~, or by substituting for s. {s-\-kf=(a+b-{-c . . . +iy+2{a+b+c . . .-?-i)k+k''. But by hypothesis, the first part of this expression is composed of the squares of all the terms of the first polynomial and the double SQUARE ROOT OF ALGEBRAIC QUANTITIES. 137 products of these terms taken two and two ; the second part contains the double products of all the terms of the first polynomial by the additional term Tc ; and the third part is the square of this term. Therefore, the law of composition, announced above, is true for the new polynomial. But it has been proved to be true for a trinomial ; hence it is true for a polynomial containing four terms ; being true iovfour, it is necessarily true for^t-e, and so on. Therefore it is general. This law can be enunciated in another manner : viz. The square of any polynomial contains the square of tJiefrtt term, plus twice the product of the first hy the second, plus the square of the second ; plus twice the product of tfie first two terms ly the third, plus the square of the third ; plus twice the product of the first three terms by the fourth, plus the sqiuire of the fourth ; and so on. This enunciation which is evidently comprehended in the first, shows more clearly the process for extracting the square root of a polynomial. From this law, (^a+i+cf=a'^+2ab-\-lr+2{a-^h)c+c^ {a+b+c+df=d'+2ab-{-lr'+2{a + b)c + c''+2{a + b+c)d+d^. 128. We will now proceed to extract the square root of a poly- nomial. Let the proposed polynomial be designated by N, and its root, which we wUl suppose is determined, by R ; conceive, also, that these two polynomials are arranged with reference to one of the letters which they contain, a, for example. Now it is plain that the first term of the root R may be found by extracting the root of the first term of the polynomial N ; and that the second term of the root may be found by dividing the second term of the polynomial N, by twice the first term of the root R. If now we form the square of the binomial thus found, and sub- tract it from N, the first term of the remainder will be twice the product of the first term of R by the third term : hence, if this first 12* 138 ALGEBRA. term be divided by double the first term of R, the quotient will be the third term of R. In order to obtain the fourth term of R, form the double products of the first and second terms, by the third, plus the square of the third ; then subtract all these products from the remainder before found, and the first term of the result will be twice the product of the first term of the root by the 4th : hence, if it be divided by double the first term, the quotient will be the fourth term. In the same manner the next and subsequent terms may be found. Hence, for the extraction of the square root of a polynomial we have the following RULE. I. Arrange the polynomial with reference to one of its letters and extract the square root of the first term : this will give the first term of the root. II. Divide the second term of the polynomial by double the first term, of ilie root, and tlie quotient will be the second term of the root. III. Then form the square of the two terms of the root found, and subtract it from the first polynomial, and then divide the first term of the remainder by double the first term of the root, and the quotient vnll he the third term. IV. Form the double products of the first and second terms, by the third, plus the square of the third ; then subtract all these products from the last remaiJider, and divide the first term of the result by dou- ble tJie first term of the root, and the quotient toill be the fourth term. Then proceed in the same manner to find the other terms. EXAMPLES. 1. Extract the square root of the polynomial 49a^Ir'-24:aP+25a''-S0a''b + 16b*. First arrange it with reference to the letter a. SQUARE ROOT OF ALGEBRAIC QUANTITIES. 139 25a*-S0a'b-{-A9a^--24aP+16b* 5a' — Sab+4,P 25a*— 30a^5+ Qa^^ lOa^ 4:0a-b'-24.aP+16M 1st. Rem. ^0aW-24aP + 16¥ . . . 2cl. Rem. After having arranged the polynomial with reference to a, extract the square root of SSa", this gives 5a^, which is placed to the right of the polynomial; then divide the second term, —SOa% by the double of 5a^, or lOa^; the quotient is —3ab, and is placed to the right of 5a^. Hence, the first two terms of the root are 5a^—3ab. Squaring this binomial, it becomes 25a*~30a^i+9a^Z^, which, sub- tracted from the proposed polynomial, gives a remainder, of which the first term is AOaW. Dividing this first term by lOa^ (the double of 5a^), the quotient is +4^- ; this is the third term of the root, and is written on the right of the first two terms. Forming the double product of 5a^—3ab by 4J^, and the square of 45^, we find the poly- nomial 40a^^— 24aZ'^+16Z>*, which, subtracted from the first re- mainder, gives 0. Therefore 5a^—Sab-i-4P is the required root. 2. Find the square root of a* + 'ia\v + ea^r" + 4fl.r^ + x*. 3. Find the square root of a*-2a='a,'+3aV-2aar'+«*. 4. Find the square root of Ax^ + l2x^ + 5x*-2x^+7x^-2x+l. 5. Find the square root of 9a*-12a'b+28a'P-16aP+16b*. 6. Find the square root of 25a* J2_ 40a^52c + 760=^^0^- 48aJ2c='+ 36 J^c* - 30a*3c + 24a33c^ -36a25c='+9aV. 189. We will conclude this subject with the following remarks. 1st. A binomial can never be a perfect square, smce we know that the square of the most simple polynomial, viz. a binomial, con- 140 ALGEBRA. tains three distinct parts, Mhich cannot experience any reduction amongst themselves. Thus, the expression ar-\-¥ is not a perfect square ; it wants the term ±2a5 in order that it should be the square of aztil). 2d. In order that a trinomial, when arranged, may be a perfect square, its two extreme terms must be squares, and the middle term must be the double product of the square roots of the two others. Therefore, to obtain the square root of a trinomial when it is a per- fect square ; Extract the roots of the two exireme terms, and give these roots the same or contrary signs, according as the middle term is positive or negative. To verify it, see if the double product of the two roots gives the middle term of the trinomial. Thus, 9a^— 48a*Z»2+64a^^* is a perfect square, suice •v/9a^=3aS and V6^a^^=^—8aP, and also 2 x 3a= X — Sai^= — 48a'*^'^ the middle term. But 4a- + 14aZ' + 9Z»^ is not a perfect square : for although 4a^ and +9^"^ are the squares of 2a and 3 J, yet 2x2«X 3<5> is not equal to l^ab. 3d. In the series of operations required in a general problem, when the first term of one of the remainders is not exactly divisi- ble by twice the first term of the root, we may conclude that the proposed polynomial is not a perfect square. This is an evident consequence of the course of reasoning, by which we have arrived at the general rule for extracting the square root. 4th. When the polynomial is not a perfect square, it may be sim- phfied (See Art. ^25.). Take, for example, the expression V a^^+4rt^i^+4ai^. The quantity under the radical is not a perfect square ; but it can be put under the form al{a^-\-4tab+4:P). Now, the factor between the parenthesis is evidently the square of a-\-2b, whence we may conclude that, Va='&+4a^Z^+4ai*= {a+2b) Vab. RADICALS OF THE SECOND DEGREE. 141 Of the Calculus of Radicals of the Second Degree. 130. A radical quantity is the indicated root of an imperfect power. The extraction of the square root gives rise to such expressions as va , 3 V 6 , 7 V 2 , which are called irrational quantities, or radicals of the second degree. We will now establish rules for per- forming the four fundamental operations on these expressions. 131. Two radicals of the second degree are similar, when the quantities under the radical sign are the same in both. Thus, zVh and 5c V~T are similar radicals ; and so also are 9 V 2 and 7 VT. Addition and Subtraction. 132. In order to add or subtract similar radicals, add or subtract their co-efficients, then prefix the siwi or difference to the common radical. Thus, . . . 3a VT+5c A/T=(3a+5c) VT. And . . . 3a VT-5c VT=(3a-5c)'/T. In like manner, 7 V2a+S V2a=(l-{-S) V2a—10 V2a. And ... 7'/2^-3-v/2^=(7-3) V^= 4^2^. Two radicals, which do not appear to be similar at first sight, may become so by simplification (Art. 125). For example, V^8aP-{-l? VT5^=U V2a + 5b V3a=9b Vs^, and 2 -/is — 3 a/5^6 -/s"— 3 a/5^3 VsT When the radicals are not similar, the addition or subtraction can only be indicated. Thus, in order to add 3 Vb to 5 Va7 we write 5Va+2Vb7 142 Multiplication. 133. To multiply one radical by another, multiply the two quan- tities under the radical sign together, and place the common radical over the product. Thus, Vax Vh= "v/o^; this is the principle of Art. 125, taken in an inverse order. When there are co-efficients, we Jirst multiply them together, and write the product before the radical. Thus, 3 Vbab'x^ V20a =12 Vl00a''b'=120a Vb7 2a V^ x3a VTc=6a^ y/Vc" =6a^c. 2a VaF+¥x—^a Va-+^= — 6a=(a^+^). Division. 134. To divide one radical by another, divide one of the quantu ties under the radical sign hy the other and place the common radical over the quotient. V a vf Thus, - ^ — N/ -j- ; for the squares of these two expres- a sions are equal to the same quantity — ; hence the expressions themselves must be equal. When there are co. efficients, write their quotient as a co-ejicient of the radical. For example, — 5a / h baVh^2hVc=~s/- 2b \2ac Vohc^^c a/2^=3« V-— -=3a V^c. 2o 135. There arc two transformations of frequent use in finding the numerical values of radicals. RADICALS OF THE SECOND DEGREE. 143 The first consists in passing the co-efficient of a radical under the sign. Take, for example, the expression 3a V 56 ; it is equiva- lent to V 9a- X V^ib, or V Qa?.bb — V 4:5a% by applymg the rule for the multiplication of two radicals ; therefore, to pass the co-efficient of a radical under the sign, it is only necessary to square it. The principal use of this transformation, is to find a number which shall differ from the proposed radical, by a quantity less than unity. Take, for example, the expression 6 VlS; as 13 is not a perfect square, we can only obtain an approximate value for its root. This root is equal to 3, plus a certain fraction ; this being multiplied by 6, gives 18, plus the product of the fraction by 6 ; and the en- tire part of this result, obtained in this way, cannot be greater than 18. The only method of obtaining the entire part exactly, is to put 6 Vl3 under the form Vg-xIS = VS6xl2= V 468. Now 468 has 21 for the entire part of its square root ; hence, 6 VTs is equal to 21, plus a fraction. In the same way, we find that 12 ^7=31, plus a fraction. 136. The object of the second transformation is to convert the « « . . denominators of such expressions as — ; — — , — , into rational P+ Vq V— Vq quantities, a and p bemg any numbers whatever, and q not a per. feet square. Expressions of this kind are often met with in the resolution of equations of the second degree. Now this object is accomplished by multiplying the two terms of the fraction by p— y/q, when the denominator is ^+ y/q, and by i>+ \/?j when the denominator is _p— ^q. For multiplying in this manner, and recollecting that the sum of two quantities, multiphed by their difference, is equal to the difference of their squares, we have a o-iV— \fq) (i{p— Vq) _ ap—a Vq P+Vq~{p+Vq){p-Vq)'~ f-q ~ f-q ' .» «»— ^»r. *J^-«^ ^- -- — If — I. frbcsiHL ^ If*- aoa tae -rstOK ic ,'5,11-,'^, T^'S-T^'IS -.':— ^'? -i Ifev 4i= '• ^3lS^r=^l^'L »xiai Mi, 7^*5 sue— rrjLl i4^ ^4l RADICALS OF THE SECOND DEGREE. 145 3+2 v/7 : 2,123, exact, to within 0,001. 5x/12-6V5 Remark. Expressions of this kind might be calculated by ap- proximating to the value of each of the radicals which enter the numerator and denominator. But as the value of the denominator would not be exact, we could not form a precise idea of the degree of approximation which would be obtained, whereas by the method just indicated, the denominator becomes rational, and we always know to what degree the approximation is made. The principles for the extraction of the square root of particular numbers and of algebraic quantities, being established, we will pro- ceed to the resolution of problems of the second degree. Examples in the Calculus of Radicals. 1. Reduce V 125 to its most simple terms. Ans. 5 V~b. v/^ 2. Reduce v — rr- to its most simple terms. 147 ^ 5 /— Ans. ^ v 6 . 3. Reduce V QSa^x to its most simple terms. Ans, la V2Jc. 4. Reduce V{x^—a-s(?) to its most simple terms. 5. Required the sum of Vl^ and V 128 . Ans. 14 V~2. 6. Required the sum of V^ and V 147 . Ans. 10 VY . /T . /27 7. Required the sum of \/ — and \^ — . 50 19 /_ Ans. -VT. 13 8. Required the sum of 2 "v/ arb and 3 V^^bx'^. 9. Required the sum of 9 V 243 and 10 V 363. 10. Required the difference of \/ — and S/ ^. 4 ,_ A71S. T^ V 15. 45 11. Required the product of 5 V 8 and 3 V 5 . Arts. 30 VlO. 2 /~1 3 /^ 12. Required the product of -^r-W -^ and —r\/ 777. 00 4 10 Ans. — -/35. 40 13. Divide 6 Vlo by 3 V~b. Ans. 2 VT. Of Equations of the Second Degree. 137. When the enunciation of a problem leads to an equation of the form ax^=b, in which the unknown quantity is multiplied by itself, the equation is said to be of the second degree, and the princi- ples established in the two preceding chapters are not sufficient for the resolution of it ; but since by dividing the two members by a, it b becomes x^= — , we see that the question is reduced to finding the b square root of — . a 138. Equations of the second degree are of two kinds, viz. equa- tions involving two terms, or incomplete equations, and equations in- volving three terms, or co?iipIete equations. The first are those which contain only terms involving the square of the unknown quantity, and known terms ; such are the equa- tions, 1 5 7 299 3,^=5; -^-3 + --.---.-+—. EQUATIONS OF THE SECOND DEGREE. 147 These are called equations involving two terms because they may- be reduced to the form aaP—b, by means of the two general trans- formations (Art. 90 & 91). For, let us consider the second equa- tion, which is the most complicated ; by clearing the fractions it be- comes 8x2_72 + 10a^=7-24r'+299, or transposing and reducing 42x^=378. Equations involving three terms, or complete equations, are those which contain the square, and also the first power of the unknown quantity, together with a known term ; such are the equations 5 13 2 , 273 5x2-7x=34 ; y^- Y''»''+^=Q-y^'-^ +-12~' They can always be reduced to the form a3r+hx=c, by the two transformations already cited. Of Equations involving two terms. 139. There is no difficuhy in the resolution of the equation b ax?=h. We deduce from it ar= — , whence x-. ' ■ V^- When — is a particular number, either entire or fractional, we can obtain the square root of it exactly, or by approxmiation. If — is algebraic, we apply the rules established for algebraic quantities. But as the square of +m or —m, is +m^, it follows that |dtv/ — j is equal to — . Therefore, x is susceptible of two values, viz. a;=+Y/ — , and x= — \/ — . For, substituting either of these values in the equation aa^= J, it becomes 148 ALGEBRA. ax( + \/-)=h,ovax~=b, I /* \^ * and . . . flXf — \/ — ) —h or ax — =b. For another example take the equation 4a;^— 7=3a;^+9 ; by- transposing, it becomes, a;^=rl6, whence a;=± vl6=±4. Again, take the equation 1 5 7 299 We have already seen (Art. 138.), that this equation reduces to 378 42a;^=378, and dividing by 42, x^r=— — =9; hence a;=±3. Lastly, from the equation Z3?=h ; we find .= ±V/-|-=±|\/15. As 15 is not a perfect square, the values of x can only be deter- mined by approximation OJ complete Equations of the Second Degree. 140. In order to resolve the general equation we begin by dividing both numbers by the co-efficient of i^, which gives, x^4- — X— — , or o(f+x>x=q a a b ^ c by makmg — =p and — =?• Now, if we could make the first member aP+px the square of a binomial, the equation might be reduced to one of the first degree, by simply extracting the square root. By comparing this member with the square of the binomial (x+a), that is, with x^+Sax+a-, It is plam that x^+px is composed of the square of a first term x, EQUATIONS OF THE SECOND DEGREE. 149 plus the double product of this first term x by a second, which must P P r P P^ 1. be ^, since px=2^x', therefore, if the square of — or — , be added to x'^+px, the first member of the equation will become the square of a;+^ ; but in order that the equality may not be destroy. ed — must be added to the second member. 4 By this transformation, the equation x^+px=q becomes p^ p^ Whence by extracting the square root The double sign ± is placed here, because either + V ?+-7-> or —V $+"r' squared gives 5+—. p Transposing — , we obtain ■I-±n/«+^' « + '4 From this we derive, for the resolution of complete equations of the second degree, the following general RULE. After reducing the equation to the form x^-{-^x= ^+22==^'^-22- + y' Therefore, ^=-22^^-22- + (22)' which agrees ^vith the enunciation given above for the double value of a;. It remains to perform the numerical operations. In the first 360 / 1 \^ place, 00 "^(oo) "''ust be reduced to a single number, having 2* \Zii (22)^ for its denominator. 360 /I v2_360x22 + l_792I ^°^^' ^2~ ^ (22/ ~ (22)2 - (2^ 5 extracting the square root of 7921, we find it to be 89 ; therefore, ^ 22 ^V22/ 22 EQUATIONS OF THE SECOND DEGREE. 151 1 . 89 Consequently, X- ~~22 22' Separating the t\v ■0 values, we have 1 '"=-22- 89 88 "^22~22~' 1 "=-22 89 45 22~-ll' Therefore, one of the two values which will satisfy the proposed equaticJn, is a positive whole number, and the other a negative frac- tion. For another example, take the equation which reduces to ar. 37 57 ~~6' 37 /37s If we add the square of — , or ( — j to both members, it be- comes 37 /37\2 57 /37\2 whence, by extracting the square root 37 ^ / 57 737^2 Consequently, 37^ . / 57 /37x2 In order to reduce l—j — — to a single number, wo will ob- serve, that (12)2=12x12=6x24; therefore, it is only necessary to multiply 57 by 24, then 37 by itself, and divide the difference of the two products by (12)*. Now, 37x37=1369; 57x24=1368; 152 ALGEBRA. therefore, /37\2 57_ 1 \12/ '~"6"~(12)''* 1 the square root of which i 12 37 1 Hence, ^=j;^±j^, or 37 1 _38_19 'l2"^12~12~'6"' 37 1 _36_ 'l2'~12~"l2~ This example is remarkable, as both of the values are positive, and answer directly to the enunciation of the question, of which the proposed equation is the algebraic translation. Let us now take the literal equation 4a=^— 2a;2+2aa;=:18aJ— 18&2. By transposing, changing the signs, and dividuig by 2, it becomes whence, completing the square, ar'-ax+—=— Oah+dl/'. extracting the square root, 2 4 9a2 3a Now, the square root of — 9ab+9P, is evidently, -^ — 3o. Therefore, a , /3a \ ( 0.'= 2a — db, x=—±{—-Sb), or , „, 2 \ 2 / ( x=— a+2b. These two values will be positive at the same time, if 2a>3i, and dbya, that is if the numerical value of b is greater than a 2a -77 and less than — . Jo EQUATIONS OF THE SECOND DEGREE. 153 EXAMPLES, x=2 a:^— 7x+10=0 .... values . 1 ^ 4 ^ i x= 7,12 } to within 3 '5 ^ ( a;=-5,73 ) 0,01.- 3. Given a,-^— 8a;+10=19, to find a-. Ans. x=9. ' 4. Given a;^—a;— 40=170, to find a;. Ans. a;r=15. 5. Given Sa'^+Sa-— 9=76, to find «. Ans. x=5. 6. Given ix^— i.r+7f=8, to find a\ Ans. a?=li. 7. Given a^-{-P—2bx+x^=—^ to find x. n Ans. x=-^—(b7iziz Va'nv'+b^m^-a'lA. n-'—iir \ I QUESTIONS. 1. Find a number such, that twice its square, increased by three times this number, shall give 65. Let a: be the unknown number, the equation of the problem will be 2a^+3a^=65, whence. Therefore, 3 23 3 23 13 a;= — —+—=5, and x=— — = — — . 4 4 4 4 2 Both these values satisfy the question in its algebraic For, 2x(5)='+3x5=2x25 + 15=:65. / 13x2 13 109 39 130 and 2(--) +3x-y=— -y-^~=65. But, if we wish to restrict the enunciation to its arithmetical sense, we will first observe, that when k is replaced by —a?, in the 154 ALGEBRA. equation 2x^+3a-'=65, the sign of the second term 3x only, is chang- ed, because {—xy=x^. 3 23 Therefore, instead of obtaining x= — T^X' ^^ would find 3 2S 13 x=—±—, or x=—- and a;=— 5, values which only differ from the preceding by their signs. Hence, we may say that the nega- 13 tive solution — — , considered independently of its sign, satisfies this new enunciation, viz. : To find a number such, that twice its square, diminished hy three times this numler, shall give 65. In fact, we have /13v2 13 169 39 2. A certain person purchased a number of yards of cloth for 240 cents. If he had received 3 yards less of the same cloth, for the same sum, it would have cost him 4 cents more per yard. How many yards did he purchase ? Let x= the number of yards purchased. 240 Then will express the price per yard. If, for 240 cents, he had received 3 yards less, that is x— 3 yards, the price per yard, in this hypothesis, would have been repre- 240 sented by -. But, by the enunciation this last cost would ex- X — o ceed the first, by 4 cents. Therefore, we have the equation 240 240 _ x-Z ^"^^^ whence, by reducing a,^— 3a;=180, 3 . ./''d _ 3±27 x=— ±V -r+i80=- 2 ' 4 ' 2 therefore a'=15, and x= — 12. EQUATIONS OP THE SECOND DEGREE. 155 The value a;=15 satisfies the enunciation ; for, 15 yards for 240 240 cents, gives , , or 16 cents for the price of one yard, and 12 yards for 240 cents, gives 20 cents for the price of one yard, which exceeds 16 by 4. As to the second solution, we can form a new enunciation, with which it will agree. For, go back to the equation, and change x into —X, it becomes, 240 240 240 240 _ an equation which may be considered the algebraic translation of this problem, viz. : A certain person 'purchased a numler of yards of cloth for 240 cents : if he had paid the same sum for 3 yards more, it would have cost Mm 4 cents less per yard. How many yards did he purchase ? Ans. «=12, and a:= — 15. Remark. Hence the principles of (Arts. 104 and 105.) are confirmed for two problems of the second degree, as they were for all problems of the first degree. 3. A merchant discounted two notes, one of $8776, payable in nine months, the other of $7488, payable in eight months. He paid $1200 more for the first than the second. At what rate of interest did he discount them ? To simplify the operation, denote the interest of $100 for one month by x, or the annual interest by 12x ; 9x and 8a; are the in- terests for 9 and 8 months. Hence 100+9a;, and 100+8a;, repre- sent what the capital of $100 will be at the end of 9 and 8 months. Therefore, to determine the present values of the notes for $8776, and $7488, make the two proportions, 100 + 9x : 100 : : 8776 100 + 8x : 100 : : 7488 877600 100 + 9X 748800 100 + 8X ' and the fourth terms of these proportions will express what the mer- 1 56 ALGEBRA. chant paid for each note. Hence, we have the equation 877600 748800 1200; 100 + 9a; lOO+So; or, observing that the two members are divisible by 400, 2194 1872 100 + 9a; 100 + 8a; Clearing the fraction, and reducing, it becomes, 216a'2+4396x==2200; whence 2198/2200 (2198)2 ^ OTA ' 216 ^ 216 (216)2 Reducmg the two terms under the radical to the same denomi- nator. — 2198± V 5306404 ar— — 216 or multiplying by 12, -2198d=V 5306404 12.= . To obtain the value of 12a; to within 0,01, we have only to ex- tract the square root of 5306404 to within 0,1, since it is afterwards to be divided by 18. This root is 2303,5 ; hence -2198±2303,5 12a;= 18 and consequently. 12a;= 105,5 18 .-:.5,86, and 12x— - -4501,5 = -250,08 18 The positive value, 12a;=5,86, therefore represents the rate of interest sought. EQUATIONS OF THE SECOND DEGREE. 157 As to the negative solution, it can only be regarded as connected with the first by an equation of the second degree. By going back to the equation, and changing x into —x, we could with some trou-^ ble, translate the new equation into an enunciation analogous to that of the proposed problem. 4. A man bought a horse, which he sold after some time for 24 dollars. At this sale, he loses as much per cent, upon the price of his purchase, as the horse cost him. What did he pay for the horse ? Let X denote the number of dollars that he paid for the horse, a:— 24 will express the loss he sustained. But as he X lost X per cent, by the sale, he must have lost -^^ upon each ar dollar, and upon x dollars he loses a sum denoted by -y-^- ; we have then the equation J?^=a;— 24, whence 0,-2—1000;= -2400. and a;=50± V2500-2400=50±10. Therefore, = 60 and a;=40. Both of these values satisfy the question. For, in the first place, suppose the man gave $60 for the horse and sold him for 24, he loses 36. Again, from the. enunciation, he 60 60x60 should lose 60 per cent, of 60, that is, -— - of 60, or ■ , which reduces to 36 ; therefore 60 satisfies the enunciation. If he paid $40 for the horse, he loses 16 by the sale ; for, he 40 should lose 40 per cent, of 40, or 40 X ,^.. > which reduces to 16 ; therefore 40 verifies the enunciation. 5. A grazier bought as many sheep as cost him £60, and after 14 158 ALGEBRA. reserving fifteen out of the number, he sold the remamder for £54, and gained 2s a head on those he sold : how many did he buy ? Alls. 75. 6. A merchant bought cloth for which he paid £33 155, which he sold again at £2 8s per piece, and gained by the bargain as much as one piece cost him : how many pieces did he buy ? Ans. 15. 7. What number is that, which, being divided by the product of its digits, the quotient is 3 ; and if 18 be added to it. the digits will be iirverted ? Ans. 24. 8. To find a number such that if you subtract it from 10, and multiply the remainder by the number itself, the product shall be 21. Ans. 7 or 3. 9. Two persons, A and B, departed from different places at the same time, and travelled towards each other. On meeting, it ap- peared that A had travelled 18 miles more than B ; and that A could have gone B's journey in 15f days, but B would have been 28 days in performing A's journey. How far did each travel ? > - jr ^^^ , ( A 72 miles. ^ ( B 54 miles. Discussion of the General Equation of the Second Degree. 141. As yet Ave have only resolved problems of the second de- gree, in which the known quantities were expressed by particular numbers. To be able to resolve general problems, and interpret all of the results obtained, by attributing particular values to the given quantities, it is necessary to resume the general equation of the second degree, and to examine the circumstances which result from every possible hypothesis made upon its co-efficients. This is the object of the discussion of the equation of the second degree. 142. A root of an equation of the second degree, is such a num. ber as being substituted for the unknown quantity, will satisfy the equation. EQUATIONS OF THE SECOND DEGIIEE. 159 It has been shown (Art. 138), that every equation of the second degree can be reduced to the form x'+px^q .... (1), p and q being numerical or algebraic quantities, whole numbers or fractions, and their signs plus or minus. - If, in order to render the first member a perfect square, we add — to both members, the equation becomes ir jr (.+1)^=,+^ 4* Whatever may be the value of the number expressed by q+~ri its root can be denoted by m, and the equation becomes (x+^) =m\ or {x+^) -«r=0. But as the first member of this equation is the difference between two squares, it can be put under the form {x+^-m).(x+^+ii^=Q ; . . . (2). in which the first member is the product of two factors, and the second is 0. Now we can render the product equal to 0, and con- sequently satisfy the equation (2), in two different ways : viz. P 'P By supposing x-{-— — m=0, whence x= — ^+'>n. % P , P or supposing x+— 4-m=0, whence x=—— — m. At * Or substituting for m its value. 160 ALGEBRA. Now, either of these values, being substituted for x in its cor- responding factor of equation (2) will satisfy that equation ; and as equation (1) will always be satisfied when the derived equation (2) is satisfied, it follows, that either value will satisfy equation (1). Hence we conclude, 1st. That every equation of the second degree has two roots, and only two. .2d. That every equation of the second degree may he decomposed into two binomial factors of the first degree with respect to x, having X for a common term, and the two roots, taken with their signs changed, for the second terms. For example, the equation o?+^x—2S = being resolved gives .T=4 and a;= — 7 ; either of which values will satisfy the equation. We also have (a;_4) (x + 7) =.x^ + 3.r-28. 143. If we designate the two roots by x' and x", we have .■=_f+^;7? and .■=-lL^V^, by adding the roots we obtain ^''+""= 2 — - 4 and by multiplying them together, we have 4-('4)=-*- Hence, 1st. The algebraic sum of the two roots is equal to the co. efficient of the second term of the equation, taken ivith a contrary sign. 2d. The product of the two roots is equal to the second mem- ber of the equation, taken also with a contrary sign. EQUATIONS OF THE SECOND DEGREE. 161 Remake. The preceding properties suppose that the equatiou has been reduced to the form c(r-{-px=q ; that is, 1st. That every term of the equation has been divided by the co-efficient of x^. 2d. That all the terms involving x have been transposed and ar- ranged in the first member, and x^ made positive. 144. There are four forms, under which the equation of the se- cond degree may be written. x^-\-px= q (1) a^—px = q (2) s^+px = —q (3) 3?—px= — q (4). In which we suppose p and q to be positive. These equations being resolved, give, =-f-v/ .4 (1) =+f-v ,4 (2) =-f-^-4 (3) -+^±V- •?+T (*)• In order that the value of x, in these equations, may be found, cither exactly or approximatively, it is necessary that the quantity under the radical sign be positive (Art. 126). f . Now, — being necessarily positive, whatever may be the sign of ^, it follows, that in \he first and second forms all the values of x will be real. They will be determined exactly, when the quan- tity ?+-T- is a perfect square, and approximatively when it is not so. In the first form, ihe first value of x, that is, the one arising from 14* 162 ALGEBRA. taking the plus value of the radical, is always positive ; for the radical S/ q+—, being numerically greater than — , the ex- pression — — iir\/ q+— is necessarily of the same sign tis that of the radical. For the same reason, the second value is es- sentially negative, since it must have the same sign as that with which the radical is affected : but each root, taken with its proper sign,will satisfy the equation. The positive value will, in general, alone satisfy the problem understood in its arithmetical sense ; the negative value, answering to a similar problem, differing from the first only in this ; that a certain quantity which is regarded as ad- ditive in the one, is subtractive in the other, and the reverse. In the second form, the first value of x is also positive, and the second negative, the positive value being the greater. In the third and fourth forms, the values of x will be imaginary when 5'>— , and reaZ when 2'<--t-- . / ^ p And since v — ?+-t- is less than — , it follows that the real values of x will both be negative in the third form, and both positive in the fourth. 145. The same general consequences which have just been re- marked, would follow from the two properties of an equation of the second degree demonstrated in (Art. 143). The properties are : TJie algehraic sum of the roots is equal to the co-efficient of the se- cond term, taken with a contrary sign, and their product is equal to the second member, taken also with a contrary sign. For, in the first two forms, q being positive in the second mem- ber, it follows that the product of the two roots is negative : hence, they have contrary signs. But in the third and fourth forms q being EQUATIONS OF THE SECOND DEGREE. 163 negative in the second member, it follows that the product of the two roots will be positive : hence, they will have like signs, viz. both negative in the third form, where p is positive, and both positive in the fourth form where p is negative. Moreover, since the sum of the roots is affected with a sign con- trary to that of the co-efficient p ; it follows, that, the negative root will be the greatest in the first form, and the least in the second. 146. We will now show that, when in the third and fourth forms, p^ we have ?>"^» the conditions of the question will be incompa- tible with each other, and therefore, the values of x ought to be imaginary. Before showing this it will be necessary to establish a proposition on which it depends : viz. If a given number be decomposed into two parts and those parts multiplied together, the product will be the greatest possible when the parts are equal. Let p be the number to be decomposed, and d the difference of the parts. Then p d -^+— = the greater part (Art, 32). p d and — — — = the less part. their product (Art. 46). rf d^ and ^_ =P, 4 4 Now it is plain that P will increase as d diminishes, and that it will be the greatest possible when d=0 : that is, p p p^ -^X— =— is the greatest product. 147. Now, since in the equation a^ — px= — q p is the sum of the roots, and q their product, it follows that q can 164 ALGEBRA. never be greater than — . The conditions of the equation there- fore fix a limit to the vakie of q, and if we make ?>^5 we express by the equation a condition which cannot be fulfilled, and, this con- tradiction is made apparent by the values of x becoming imaginary. Hence we may conclude that, The value of the unknown quantity loill always he imaginary when the conditions of tlie question are incompatible with each other. Remark. Since the roots of the equation, in the first and second forms, have contrary signs, the condition that their sum shall be equal to a given number p, does not fix a limit to their product : hence, in those two forms the roots are never imaginary. 148. We will conclude this discussion by the following remarks. 1st. If in the third and fourth forms, we suppose q=-ri the ra- dical part of the two values of x becomes 0, and both of these p values reduce to x= ——•.the two roots are then said to le equal. p^ In fact, by substituting -— for q in the equation, it becomes P' x^-\-px— — —, whence P^ I P\^ a^+pa;+— =0, or \x+-^) =0. In this case, the first member is i\ie product of two equal factors. Hence we may also say, that the roots of the equation are equal, since in this case the two factors being placed equal to zero, give the same value for x. 2d. If, in the general equation, x^-^px=q, we suppose q^O, P P the two values of x reduce to x= — ^+~, or x=0, and to 2 2 P P x=-—-Y, or x=-p. EQUATIONS OF THE SECOND DEGREE. 165 In fact, the equation is then of the form !x^-{-px=0, or x{x+p) = 0, which can be satisfied either by supposing x=0, or x-\-p—0, whence x= —p : that is, one of the roots is 0, and the other the co-efficient of x taken with a contrary sign. 3d. If in the general equation oc^+px=q, we suppose p=0, there will result x"=q, whence x=± y/q ; that is, in this case tlie Lwo values of x are equal, and have contrary signs, real in the first and second forms, and imaginary in the third and fourth. The equation then belongs to the class of equations involving two terms, treated of in (Art. 139). 4th. Suppose we have at the same time p=0, ^'=0 ; the equa- tion reduces to x? — Q, and gives two values of x, equal to 0. 149. There remains a singular case to be examined, which is often met with in the resolution of problems of the second degree. To discuss it, take the equation ax^-\-hx=c. This equation gives '= 2-a • Suppose now, that from a particular hypothesis made upon the given quantities of the question, we have a=0 ; the expression for X becomes -i±:b I """¥' -Q-' ^^^'"^" 1 __2b " 0' The second value is presented under the form of infinity, and may be considered as an answer when the proposed questions will admit of answers in infinite numbers. As to the first — , we must endeavour to interpret it. By multiplying the numerator and denominator of the 2d mem- ber of the equation -b+ Vlr' + iac -b- V¥T^c '= 2-a ^^ 2a 166 we obtain h2—(^p^Aac) — 4ac 2a{-i— Vl^'+^ac ■2a{ — b— V¥+lac -2c Vb"-{-^ac c by dividing by 2a, — by making a=0. Hence we see that the indetermination arises from a common fac- tor in the numerator and denominator. If we had at the same time a=0, b—0, c=0, the proposed equation would be altogether indeterminate. This is the only case of indetermination that the equation of the second degree presents. We are now going to apply the principles of this general discus- sion to a problem which will give rise to most of the circumstances which are commonly met with in problems of the second degree. Problem of the Lights. C" A C B a 150. Find upon the line which joins two lights, A and B, of dif- ferent intensities, the point which is equally illuminated ; admitting the following principle of physics, viz. : The intensity of the same light at two different distances, is in the inverse ratio of the squares of these distances. Let the distance AB between the two lights be expressed by a ; the intensity of the light A, at the units distance, by b ; that of the light B, at the same distance, by c. Let C be the required point, and make AC=ix, whence BC:=a—x. From the principle of physics, the intensity of A, at the zmity of distance, being b, its intensity at the distances 2, 3, 4, &:c., is b b b —5 — > — , &c., hence at the distance x it will be expressed by PROBLEM OF THE LIGHTS. 167 h -5-. In like manner, the intensity of B at the distance a—x, is c 7 — ; but, by the enunciation, these two intensities are equal (a— a;)- ' ' -^ ^1 to each other, therefore we have the equation I c x^ (a—xy Whence, by developing and reducing, (h — c)xr — 2aix= —a^h. This equation gives ab ^ / d'W ceh h-c ^ {i-cf b-c or reducing, a(h± Vhc) b—c This expression may be simplified by observing, 1st. that ~b± Vbc can be put under the form y/b. y/b±z ^b. v/c, or ■//>( v/^± ^c) ; 2d. that b—c={^bf—{'^cy={y/b+^c).{^b—^c.) There- fore, by first considering the superior sign of the above expression, we have a^b{s/b+Vc) cisfb ""^ ( Vb+ v/c).( Vb- v/c) " Vb- Vc ' In like manner we obtain for the second value, a Vb( Vb— v/f ) _ a^b '"'"'( Vb+ n/c).( Vb- Vc) ~ Vb+ Vc ' Hence, we have aVb^ f a Vc 1st . . . X-- 2d . . . X-. Vb-\- Vc^ I from which j ' Vb+ Vc' aVb I we obtain | —aVc Vb-Vcj [ Vb-Vc 1st. Suppose that Z>>c. 168 ALGEBRA. The first value of x, is then positive and less than a, because — -, is a proper fraction ; thus this value gives for the required point, a point C, situated between the points A and B. We see moreover, that the point is nearer to B than A ; for since J>c, we have s/h+ ^h or 2 v/3>( VJ+ v/c) ; whence ->— and consequently, ^"o"' ought to be the case, since the intensity of A was supposed to be greater than that of B. The corresponding value of a—x, , is also positive, a and less than — , as may easily be shown. a\/h . , The second value of x, ——, -, is also positive, but greater ■\/o — vc than a : because — -, r>l- Hence this second value gives a y/O— \/C second point C, situated upon the prolongation of AB, and to the right of the two lights. We may in fact conceive that the I'wo lights, exerting their influence in every direction, should have upon the prolongation of AB, another point equally illuminated ; but this point must be nearest that light whose intensity is the least. We can easily explain, why these two values are connected by the same equation. If, instead of taking AC for the unknown quan- tity X, we had taken AC, there would have resulted BC'=x—a ; b c and the equation -^^^-r-^:^- Now, as {x — ay is identical with {a—xy, the new equation is the same as that already established, which consequently should have given AC as well as AC. And since every equation is but the algebraic enunciation of a problem, it follows that, when Ihe same equation enunciates several problems, it ought by its different roots to solve them all. EQUATIONS OF THE SECOND DEGREE. 169 When the unknown quantity x represents the liiie AC, the second value of a — x, — — , is negative, as it should be, since y/l) — y/C we have a;>a ; but by changing the signs in the equation — ay/c as/c It becomes a;— a= — ~ — ; and this value of x—a represents the positive value of BC . 2d. Let 3( v/Z»+ ^l) or than 2 ^l. The corresponding value of a—x, or — — is positive, and y/0-\- y/c greater than — . Therefore in this hypothesis, the point C, situated between A and B, must be nearer A than B. rri 1 1 r "-^^ —as/l) . i he second value of x, — — or — —, is essentially ne- gative. To interpret it, let us take for the unknown quantity the distance AC", and let us represent this distance by a;, and at the same time consider, as we have a right to do, x as essentially ne- gative. Then the general expression for BC" being a — x, if we regard x as essenticdly negative, the true numerical value of a—x is expressed by a+x. Hence as before, the equation or algebraic expression will be be b or x^ {a-xf a-2 (a + xf in the first of which equations x is essentially negative. This equation ought to give a negative value for x, and a posi- tive value for BC"=a-\-x. Indeeed, since the intensity of the light B is greater than that of A, the second required point ought to be 15 170 ALGEBRA. nearer A than B. The algebraic value for BC", which is — a Vc a Vc ■\/h — Vc~ Vc — Vb~ 3d. Let h=c. positive. a The first two values of a; and a—x reduce to — , which gives the middle of AB for the first required point. This result agrees with the hypothesis. The two other values reduce to — - — , or infinity ; that is, the second required point is situated at a distance from the two points A and B, greater than any assignable quantity. This result agrees perfectly with the present hypothesis, because, by supposing the difference b—c to be extremely small, without being absolutely nothing, the second point must be at a very great distance from the a^/b lights ; this is indicated by the expression — —, the denomi- nator of which is extremely small with respect to the numerator. And if we finally suppose Z»=c, or y/b— ^/c=0, the required point cannot exist for a finite distance, or is situated at an infinite distance. We will observe, that in the case of b=c, if we should consider the values before they were simplified, viz. a(b+-ybc) a(b— y/bc) x=- - , and X— J b~c b—c aVb the first, which corresponds to a;— — -, — , would become 2ab a s/b — r-, and the second, which corresponds to — ; — , would be- come — . But — would be obtained in consequence of the exist- ence of a common factor, yjb— -/c, between the two terms of the value of X (see Art. 113). EQUATIONS OF THE SECOND DEGREE. 171 Let l=:c, and a = 0. The first system of values for x and a—x, reduces to 0, and the second to — . This last symbol is that of indeiermination ; for, resuming the equation of the problem, {b—c)xP—'iabx=—a?l, it reduces, in the present hypothesis to O.or — 0.a;=0, which maybe satisfied by giving x any value whatever. In fact, since the two lights have the same intensity, and are placed at the same point, they ought to illuminate equally each point of the line A B. The solution 0, given by the first system, is one of those solutions in infinite numbers, of which we have spoken. Finally, suppose a=0, and h and c, unequal Each of the two systems reduces to 0, which proves that there is but one point in this case equally illuminated, and that is the point in which the two lights are placed. In this case, the equation reduces to {h—c)s^—0, and gives the two equal values, a;=0, a;=0. The preceding discussion presents another example of the pre- cision with which algebra responds to all the circumstances of the enunciation of a problem. Of Equations of the Second Degree, involving two or more unknown quantities, 151. A complete theory of this subject cannot be given here, be- cause the resolution of two equations of the second degree involv- ing two unknown quantities, in general depends upon the solution of an equation of the fourth degree involving one unknown quantity ; but we will propose some questions, which depend only upon the solution of an equation of the second degi-ee involving one unknown quantity. 1. Find two numbers such that the sum of their products by the respective numbers a and l, may be equal to 2^, ansl that their product may be equal to p. 172 ALGEBRA. Let X and y be the required numbers, we have the equations, ax-\-hy^2s. xy=p. 2s— ax h rom the first y— — -- — ; whence, by substituting in the se- cond, and reducing, aa^—2sx=z—hp. Therefore, - a'= and consequently. 1 , — V s^ — abp, a ^ This problem is susceptible of two direct solutions, because S IS evidently > Vs^~abj), but in order that they may be real, it is necessary that ^> or ^abp. Let a=5=l ; the values of x, and y, reduce to x=s±: Vs^—p and y—szp Vs^—p Whence we see, that the two values of x are equal to those of y, taken in an inverse order ; which shows, that if «+ Vs^—p repre- sents the value of a;, s— Vs^—p will represent the corresponding value of y, and reciprocally. This circumstance is accounted for, by observing, that in this par- ticular case the equations reduce to < ' and then the question is reduced to, finding two numbers of which the sum is 2s, and their product p, or in other words, to divide a number 2s, into two such parts, that their product may be equal to a given number p. 2. Find four numbers in proportion, knowing the sum 2s of their extremes, the sum 2s' of the means, and the sum 4c^ of their squares. Let u, X, y, z, denote the four terms of the proportion ; the cqua- tions of the problem will be EQUATIONS OF THE SECOND DEGREE. 173 u+z=2s x+y=2s' uz=xy At first sight, it may appear difficult to find the values of the un- known quantities, but with the aid of an unknown auxiliary they are easily determined. Let p be the unknown product of the extremes or means, we have 1st. The equations c u+z=2s, , . , . ( «=*+ V s^-p, \ which give , \ , I uz=p, ° i z=s-V s'-p. 2d. The equations ix+y=2s', . tx=s'+Vs'^-p, \ which give \ , \ ^=P, iy=s'-Vs'^-p. Hence, we see that the determination of the four unknown quan- tities depends only upon that of the product p. Now, by substituting these values of u, x, y, z in the last of the equations of the problem, it becomes + (s'- V7^^py=^c' ; or, developing and reducing, 452^45'2_4^_4c . hence p=s^+s"^—c''. Substituting this value for p, in the expressions for u, x, y, z, we find , u=s-\- V c'-s"', ( x=s'-\- V c'-s', \ z=s- V c^-T^, \ y=s'— V c^-s^- These four numbers evidently form a proportion ; for we have 15* 174 ALGEBRA. Tliis problem sliows how much the mtroduction of an unknown auxiliary facilitates the determination of the principal unknown quan- tities. There are other problems of the same kind, which lead to equations of a degree superior to the second, and yet they may be resolved by the aid of equations of the first and second degrees, by introducing unknown auxiliaries. 152. We will now consider the case in which a problem leads to two equations of the second degree, involving two unknown quan- tities. An equation involving two unknown quantities is said to be of the second degree, when it contains a term in which the sum of the expo- nents of the two unknown quantities is equal to 2. Thus, S3r'—Ax+f—xy—5y+Q=0, 7xt/— 4x+3/=0, are equations of the second degree. Hence, every general equation of the second degree, involving two unknown quantities, is of the form ay" + bxy + cx^ + dy +fx +g=0, a, h, c, . . . representing known quantities, either numerical or al- gebraic. Take the two equations af+bxy + cx''-\-dy+fx+g=0, aY+i'xy-{-c'a^-\-d'y-\-f'x-\-g'=0. Arranging them with reference to x, they become c x/' + {by+f )x+af+dy+g =0, c'oir' + {h'y+f')x-{-ay-{-d'y+g'z=0. Now, if the co-cfficients of x^ in the two equations were the same, wc could, by subtracting one equation from the other, obtain an equation of the first degree in x, which could be substituted for one of the proposed equations ; from this equation, the value of x could be found in terms of y, and by substituting this value in one of the proposed equations, we would obtain an equation involving only the unknown quantity y. EQUATIONS OF THE SECOND DEGREE. 175 By multiplying the first equation by c', and the second by c, they become ccV+(5?/+/)c'a;+(a/+cZy+g)c'=0, cc'a^ + {h'y-{-f')c x+(ay +d't/+^')c =0, and these equations, in which the co-efficients of x^ are the same, may take the place of the preceding. Subtracting one from the othei', we have \{hc' — cb')y-\-fc' — cf''\x-\-{ac' — ca')'f -\-{dc' — cd')y-\-gc' — eg' =0, which gives {ca' — ac')y^-\-{cd'—dc')y-\-cg'—gc' ~ {hc'—cb')y+fc'-cf' This expression for x, substituted in one of the proposed equa- tions, will give a final equation, involving y. But without effecting this substitution, which would lead to a very complicated result, it is easy to perceive that the equation involving y will be of the fourth degree ; for the numerator of the expres- sion for X being of the form my^-\-ny-\-p, its square, or the expres- sion for x^, is of the fourth degree. Now this square forms one of the parts of the result of the substitution. Therefore, in general, the resolution of two equations of the se- cond degree, involving two unknown quantities, depends upon that of an equation of the fourth degree, involving one unknown quantity. 153. There is a class of equations of the fourth degree, that can be resolved in the same way as equations of the second degree ; these are equations of the form x'^-\-poc^-\-q=.0. They are called trinomial equations, because they contain but three kinds of terms ; viz. terms involving a;*, those involving x^, and terms entirely known. In order to resolve the equation x'^-\-px^-{-q=Q, suppose a^=y» we have Pa./ V^ f^Vy^i=^^ whence 3/=-y±V —^.^-^^ But the equation ar^=y, gives a;=± s/y. 176 Hence, x=±V -^±\/_^+^. We perceive that the unknown quantity has four values, since each of the signs + and — , which affect the first radical, can be combined successively with each of the signs which affect the se- cond ; but these values taken two and two are equal, and have contra, ry signs. Take for example the equation a;* — 25a;^=: — 144 ; by supposing s^—y, it becomes ?/^— 25^/= — 144 ; whence ^=16, y^Q. Substituting these values in the equation x^=y there will result 1st. 3?=IQ, whence a;=:±4; 2d. a^—g^ whence a;=±3. Therefore the four values are +4, —4, +3 and —3. Again, take the equation a;^— 7x^=8. Supposing s?=y, the equation becomes /— 7^=8; whence y=Q, y= — l. Therefore, 1st. !c^=8, whence a;=±2^/2; 2d. ar^=-l; whence x= ± v/— 1 ; the two last values of x are imaginary. Let there be the algebraic equation x'^—(2ic-\-4a'')x^=—lr'c'; taking x^=y, the equation becomes f—{2ie-\-4:a^)y=—i^c''; from which we deduce y= U + 2a2±2a VlT^^T^, And consequently x=±\y be ^2a'^±2a^/lc + a^. 154. Every equation of the form y^''+py''+q=0, in which the exponent of the unknown quantity in one term is double that of the other, may be solved by the rules for equations of the second degree. For, put y"=x, then f"=3?, and y^" -{-fy^ ■\-q=x'^ ^'px-{-q=^. Hence Or And a; = - -^4. r = - --!• = V - ■i-^- ^4- EQUATIONS OF THE SECOND DEGREE. 177 Extraction of the Square Root of Binomials of the form a ± VT^ 155. The resolution of trinomial equations of the fourth degree, gives rise to a new species of algebraic operation : viz. the extrac- tion of the square root of a quantity of the form ad= Vb, a and h being numerical or algebraic quantities. By squaring the expression 3± Vs, we have (3zfc Vy)2=9±6 VT+5=14±6 Vb~: hence, reciprocally \/ 14±6 V 5 =3± V 5. In like manner, ( v/7± ^/ll)2=7±2^/7x \/ll + ll = 18±2^/77. Hence reciprocally V18±2 V77= n/7± v/11. Whence we see that an expression of the form v a=fc \/h, may sometimes be reduced to the form a'± y/h' or -/a'db V^' ; and when this transformation is possible, it is advantageous to effect it, since in this case we have only to extract two simple square roots, whereas the expression Vai \/b requires the extraction of the square root of the square root. 156. If we let p and q denote two indeterminate quantities, we can always attribute to them such values as to satisfy the equations Va+Vh:=p+q (1). Va— y/b=p—q (2). These equations, being multiplied together, give Va'-b^p^-q' (3). Now, if p and q are irrational monomials involving only single ra- dicals of thesecond degree, orif one is rational and the other irration- al, it follows that p'^ and q^ will be rational ; in which case, p^ — q", or its value, Va'—b, is necessarily a rational quantity, or a?—h is a perfect square. 178 ALGEBRA. When this is the case, the transformation can always be effected. For, take o?—b, a perfect square, and suppose Va^—l=c', the equation (3) becomes Moreover, the equations (1) and (2) being squared, give f+q^+2fq=a+ ^/&, p^J^q^ — 2pq=a—^/h^, whence, by adding member to member, f+f=^ (4) ; but y^—f=c (5). Hence, by adding these last equations, and subtracting the se- cond from the first, we obtain 25'2=a-c; and consequently p=±' q^±.' 2 '■ 2 ' ~2 f + C or Therefore, Va-\-Vb, or p+g-^riV — ^db' V a— \/b, or p—q—±:\/ — ^"+v*=±(V-i-+V— ) / /* /o+c . /a — c\ a — c ~2~' 'a—c 2 '■ . . (6\ . . (7). These two formulas can be verified ; for by squaring both mem- bers of the first, it becomes a-\-c a — c . / a^—c^ .— — - a+ v/i--^+-^- + 2V — ^— =:a+ Va'-c' ; but the relation Va^—h — c, gives c^z=a^—b. EQUATIONS OF THE SECOND DEGREE. 179 Hence, a+ Vi=a+ Va''—a^+b=a-{- -Jh. The second formula can be verified in the same manner. 157. Remark. As the accuracy of the formulas (6) and (7) is proved, whatever may be the quantity c, or Va^—h, it follows, that when this quantity is not a perfect square, we may still replace the expressions Va+ y/b and Va— \/h, by the second inembers of the equalities (6) and (7) ; but then we would not simplify the expression, since the quantities p and q would be of the same form as the proposed expression. We would not, therefore, in general, use this transformation, unless c?—!) is a perfect square. EXAMPLES 158. Take the numerical expression 94+42 v'5, which reduces to 94+ V8820. We have a=94, 5=8820, whence c= ■\/'cf—b= V'8836 — 8820=4, a rational quantity ; therefore the formula (6) is applicable to this It becomes / /. /94+4 •+' '/94+42 v/5= ± ( V ^^^ 01-, reducing, =^( V^+ ^45) ; therefore, ■v/94T42V5= ±(7+3 v/5). In fact, (7 + 3 v/5)-=49 + 45 + 42 v/5 = 94+42 v/5. Again, take the expression S/ np+2m^—2m Vnp+m^; we have a=np-\-2)H^, b—'lm^{np-\-m^), whence a^—i=7i^p^, and c or Va' — 3=7ip; 180 ALGEBRA. therefore the formula (7) is applicable. It gives for the required root or, reducing, ±( V np-\-7n^—m). In fact, ( Vnp + 7n^—my=np+2m^—2}n V np+n?. For another example, take the expression V 16 + 30 V^T+Vie-so V~^, and reduce it to its simplest terms. By applying the preceding formulas, we find V 16 + 30 a/-1 = 5 + 3V-1, 1/16-30^-1 = 5-3^31^ Hence, V 16 + 30 '/^+ \/ 16-30 -v/^ = 10. This last example shows, better than any of the others, the utili- ty of the general problem ; because it proves that imaginary ex- pressions combined together, may produce real, and even rational results. \/28 + 10 VT=5+ VT; V 1+4 V-3=:2+ V -S, \/ bc + 2b V bc-W + \/ hc-2b V hc-lF^^2b; \/ ab + ^c^-d?+2 Viabc^-ab(P= Vab+ VAit-'^. Examples of Equations of the Second Degree, which either involve Radicals, or tivo unknown quantities. 2a^ 1. Given x-\- V a~+3r — — == to find x. Va^-\-y? X Vc^x^+a'+x'=2d' X 'y/a^-\-a?=a^—3? by transposing. EQUATIONS OF THE SECOND" DEGREE. 181 henco or a!'a^+x*=a* — 2a^x^+x\ by squaring, 2a\^=xK 2. Given x=dz' \/ ^^j^b'^-S/ ^-¥=h to find X. J+^=v/^_^+,, by transposing. ^+^=^-^+2* .h' + h\ hence hence V=2b\/ -^-y. ar'=- 4a^ hence a;=±- 2a J VT 3. Given 1 =-r to find a?. a,' X Ans. a;=3± V2ab — b^. 4. Given ««/ VT and :48 :24 > to find X and ?/. Dividing the first equation by the second, we have IG 182 V- V y =2, or t/=4. 4a; Whence from the second equation =4 V~x=24, to fino X. G. Given x + -X^-^-y = 19 ) and x^+ xy+f ^13S\ to find a: and y. Dividbg the second equation by the first, we have X— ■Vxy-\-y= 7 but ... x-\- Vxy+y=l9 hence .... 2a:+2!/=26 by addition, or .... a;+ ?/=13 and .... Vxy-\-13 = 19 by substituting in the 1st eq. or .... ■Vxy= 6 and .... xy=:26 From 2d equation, x^+xy-{-y'^=123 and from the last Sxy =108 Subtracting . . oi^ — 2xy+y^— 25 Hence x—y=zt: 5 But x-\-y= 13 Hence . . x=9 or 4; and «/=4 or 9. 7. Gi a— V cr 1+ -/a^-x^ :i, to find X. Ans. x=±- 2a VT EQUATIONS OF THE SECOND DEGREE. 183 8. Given =- to find x. V X — V x—a *'~^ a(l±nY Ans. — l±2w ^. V a-\-x V a — x . /~x 9. Given — ^H ;z^ = V ^ to find x. V X V X ^ Ans. x=±2Vab-¥. 10. Given j ^,3^^^^^i2^ | to find x and y. ( a;z=2 or 1 Ans. { ' y=l or 2. 11. Given j , „ 2, , - _„_ to find a; and y. ( x=ll or 5 Ans. \ ^ ,, ( y=b or 11. a+a;+ V2aa;+x2 12. Given =h, to find x. a-\-x ±«(lq= V2b-b'') Ans. x= y — \ V25-^'2 13. Given ^ ^ to find a; and y i xy=z 6 > ^ ( x=3 or 2 or — 3± VY Ans. { , — { y=2 or 3 or -3ip VT. 14. Given the sum of two numbers equal to a, and the sum of their cubes equal to c, to find the numbers „ , ,. . ( X +7/ =a By the conditions { „ ( ar + 2/-'=c. Putting x=s-\-z, and y=s—z, we have a=25, C r'=«='+3^«+352='+r» Hence, by addition, x^-]-y^=2s^ -\-6sz^=c 184 Whence 2^= — r and z=±\/ — , ^ . / c-2s' , . /T^T" or a;=:5±V — g^— 5 »nd ?/=5;pV — g^^ — , Or by putting for s its value, c -|-v/(-3r^)=|-N/- 4c — G^ a // 4 \ « /4c— «^ and ^=_:p^(____j=._:p^___. QUESTIONS. 1. There are two numbers whose difference is 15, and half their product is equal to the cube of the lesser number. What are those numbers? Ans. 3 and 18. 2. What two numbers are those whose sum, multiplied by the greater, is equal to 77 ; and whose difference, multiplied by the lesser, is equal to 12 ? Ans. 4 and 7, or | s/2 and y ^/2. 3. To divide 100 mto two such parts, that the sum of their square roots may be 14. A7is. 64 and 36. 4. It is required to divide the number 24 into two such parts, that their product may be equal to 35 times their difference. A71S. 10 and 14. 5. The sum of two numbers is 8, and the sum of their cubes is 152. What are the numbers ? Ans. 3 and 5. 6. The sum of two numbers is 7, and the sum of their 4th powers is 641. What are the numbers? A71S. 2 and 5. 7. The sum of two numbers is 6, and the sum of their 5th pow- ers is 1056. What are the numbers? A71S. 2 and 4. 8. Two merchants each sold the same kind of stuff; the second sold 3 yards more of it than the first, and together, they receive 35 FORMATION OF FOWERsO- •••••«- 185 crowns. The first said to the second, I would have received 24 crowns for your stuff; the other replied, and I would have received 121 crowns for yours. How many yards did each of them sell ? ( 1st merchant a;=15 x=5 } ^"^- i 2d . . . y=18 °^- y=s\' 9. A widow possessed 13,000 dollars, which she divided into two parts, and placed them at interest, in such a manner, that the incomes from them were equal. If she had put out the first portion at the same rate as the second, she would have drawn for this part 360 dollars interest, and if she had placed the second out at the same rate as the first, she would have drawn for it 490 dollars interest. What were the two rates of interest ? Ans. 7 and 6 per cent. CHAPTER IV. Formation of Powers, and Extraction of Roots of any degree whatever. 159. The resolution of equations of the second degree supposes the process for extracting the square root to be known ; in like man- ner the resolution of equations of the third, fourth, &c. degree, re- quires that we should know how to extract the third, fourth, d;c. root of any numerical or algebraic quantity. It will be the principal object of this chapter to explain the rais- ing of powers, the extraction of roots, and the calculus of radicals. Although any power of a number can be obtained from the rules of multiplication, yet this power is subjected to a certain law of com- position which it is absolutely necessary to know, in order to dedtice the root from the poioer. Now, the law of composition of the square of a numerical or algebraic quantity, is deduced from the expression for the square of a binomial (Art. 117) ; so likewise, the law 16* ♦ ........ r -Tt ^ 186 ALGEBRA. . • • • t)f 'a power 'df any degree, is deduced from the same power of a binomial. We will therefore determine the development of any power of a binomial. 160. By multiplying the binomial x-\-a into itself several times the following results are obtauied ; (x+a)=x+a, {x+ay=x'^+2ax+a^, {x+af=x^+3a3r'+Sa''x+a^ (a; + a)* = a* + 4aa;' + 6a V + 4a='a; + «*, (a; +a)5 =0,-5 + 5ax' + 1 OaV + lOaV + 5a*a; + a' By inspecting these developments it is easy to discover a law ac- cording to which the exponents of x and a decrease and increase in the successive terms; it is not, however, so easy to discover a law for the co-efficients. Newton discovered one, by means of which, any power of a binomial can be formed, without first obtain- ing all of the inferior powers. He did not however explain the course of reasoning which led him to the discovery of it ; but the existence of this law has since been demonstrated in a rigorous manner. Of all the known demonstrations of it, the most elemen- tary is that which is founded upon the theory of comhinations. How- ever, as it is rather complicated, we will, in order to simplify the ex- position of it, begin by resolving some problems relative to combi- nations, from which it will be easy to deduce the formula for the hi- nomial, or the development of any power of a binomial. Theory of Permutations and Comhinations. 161. Let It be proposed to determine the whole number of ways in which several letters, a, b, c, d, &c. can be written one after the other. The results corresponding to each change in the position of any one of these letters, are called permutations. Thus, the two letters a and b furnish the two permutations ab and ba. PERMUTATIONS AND COMBINATIONS. 187 In like manner, the three letters a, b, c, furnish six permutations. ' abc acb cab bac bca ^ cba Permutations, are the results obtained by writing a certain number of letters one after the other, in every possible order, in such a man. ner that all the letters shall enter into each result, and each letter enter but once. Problem 1. To determine the number of -permutations of which n letters are susceptible. In the first place, two letters a and b evidently ( ab give two permutations. ( ba Therefore, the number of permutations of two letters is 1 X 2« Take the three letters a, h, and c. Reserve r c either of the letters, as c, and permute the other two, < ab giving o , \ ba Now, the third letter c may be placed before ab, between a and b, and at the right of ab ; and the same for ba : that is, in one of the first permutations the reserved letter c may have three different places, giving three 'permutations. Now, as the same may be shown for each of the first permutations, it fol- lows that the whole number of permutations of three letters will be expressed by 1x2x3. If now, a fourth letter d be introduced, it can have four places in each of the six permutations of three letters : hence all the per- mutations of four letters will be expressed by 1x2x3x4. In general, let there be n letters a, b, c, &c. and suppose the total number of permutations of n— 1 letters to be known; and let Q denote that number. Now, in each of the n— 1 permutations the reserved letter may have n places, giving n permutations : hence, cab acb abc cba bca bac xOO ALGEBRA. when it is so placed in all of them, the number of permutations will be expressed by Qxn. Let n=2. Q will then denote the number of permutations that can be made with a single letter; hence Q=l, and in this particu- lar case we have Q X n~ 1x2. Let 7i='6. Q will then express the number of permutations of 3 — 1 or 2 letters, and is equal to 1x2. Therefore Qx^ is equal to 1x2x3. Let 71=4:. Q hi this case denotes the number of permutations of 3 letters, and is equal to 1x2x3. Hence, Qxn becomes 1X2x3x4, and similarly when there are more letters. 162. Suppose we have a number m, of letters a, I, c, d, &c., if they are written one after the other, 2 and 2, 3 and 3, 4 and 4 . . . in every possible order, in such a manner, however, that the num. bar of letters in each result may be less than the number of given letters, we may demand the whole number of results thus obtamed. These results are called arrange7nenls. Thus ah, ac, ad, . . . ba, he, hd, . . . ca, ch, cd, . . . are arrange- t)ie7its of 7)1 letters taken 2 and 2, or in sets 6f 2 each. In like manner, ahc, ahd, . , . hac, bad, . . . ach, acd, . . . are ar- rangemeTits taken in sets of 3. Arrangements, are the results obtained by writing a number m of letters one after the other in every possible order, in sets of 2 and 2, 3 and 3, 4 and 4 . . . n and n ; m being >n : that is, the num- ber of letters in each set being less than the whole number of letters considered. However, if we suppose n=?n, the arrangeme7its taken n and n, will become simple per7nutations. Problem 2. Having give 71 a number m of letters a, b,'C, d . . ., to determine the total 7iumber of arra7igements that may be for7ned of them by taki7ig them n at a time ; m being supposed greater than n. Let it be proposed, in the first place, to arrange the three letters a, h, and c in sets of two each. PERMUTATIONS AND COMBINATIONS. 189 First, arrange the letters in sets of one each, in which r a case we say there are two letters reserved : the reserved } b letters for either arrangement, being those which do not ( c enter it. ah Now, to any one of the letters, as a, annex, in suc- cession, the reserved letters b and c : to the second ar- rangement b, annex the reserved letters a and c ; and to the third arrangement c, annex the reserved letters a and b : this gives Hence, we see, that the arrangements of three letters taken two in a set, loill be equal to the arrangements of the same number of letters taken one in a set, multiplied by the number of reserved letters. Let it be required to form the arrangement of four letters, a, b, c, and d, taken 3 in a set. First, arrange the four letters two in a set : there will r ab then be two reserved letters. Take one of the sets and write after it, in succession, each of the reserved letters : we shall thus form as many sets of three letters each as there are reserved letters ; these sets differing from each other by at least the last letter. Take another of the first arrangements, and annex in succession the reserved letters ; we shall again form as many different arrange- ments, as there are reserved letters. Do the same for all of the first arrangements, and it is plain, that the whole number of arrangements which will be formed, of four letters, taken 3 and 3, will be equal to the arrangements of the same letters, taken two in a set, multiplied by the num- ber of reserved letters. In order to resolve this question in a general manner, suppose the total number of arrangements of the m letters taken n— 1 in a set to be known, and denote this number by P. Take any one of these arrangements, and annex to it each of the reserved letters, of which the rumber is m—{n—\), or ba ac ca ad da be cb hd dh cd L dc 190 ALGEBRA. ?M — n + l ; it is evident, that we shall thus form a number m— n+1 of arrangements of n letters, differing from each other by the last let- ter. Now take another of the arrangements of w— 1 letters, and an- nex to it each of the m—n+l letters which do not make a part of it; we again obtain a number m—n-\-\ of arrangements of n let- ters, differing from each other, and from those obtained as above, at least in the disposition of one of the n— 1 first letters. Now, as we naay in the same manner take all the P arrangements of the m letters, taken n—\ in a set, and annex to them successively the m— n + 1 other letters, it follows that the total number of ar- rangements of m letters taken n in a set, is expressed by P{in-n-\-\). To apply this to the particular cases of the number of arrange- ments of m letters taken 2 and 2, 3 and 3, 4 and 4, make n=2, whence w— n+l=wi— 1; P will in this case express the total num- ber of arrangements, taken 2—1 and 2—1, or 1 and 1, and is con- sequently equal to m ; therefore the formula becomes m{m—\). Let n= 3, whence m— n + l=m— 2; P will then express the number of arrangements taken 2 and 2, and is equal to m{m—\) ; therefore the formula becomes m{m—\) (m— 2). Again, take 7i=4, whence m— n+l=m— 3 ; P will express the number of arrangements taken, 3 and 3, or is equal to 7«(?H— 1) (m — 2) : therefore the formula becomes m{jn — \) (/n — 2) (?h — 3). Remark. From the manner in which the particular cases have been djeduced from the general formula, we may conclude that it reduces to m{ni—\) (ot— 2) (wi— 3) .... (m — « + l) ; that is, it is composed of the product of the n consecutive numbers comprised between m and rn— n + 1, inclusively. PERMUTATIONS AND COMBINATIONS. 191 From this formula, that of the preceding Art. can easily be de- duced, viz. the development of the value of Qx^- For, we see that the arrangements become permutations when the number of letters composing each arrangement is supposed equal to the total number of letters considered. Therefore, to pass from the total number of arrangements of m letters, taken n and n, to the number of permutations of n letters, it is only necessary to make m=?i in the above development, which gives n(n-l) (n-2) («-3) 1. By reversing the order of the factors, observing that the last is 1, the next to the last 2, which is preceded by 3 . . ., it becomes 1, 2, 3, 4 (^-2) {n-l)n, for the development of Q X ^^' This is nothing more than the series of natural numbers compris- ed between 1 and n, inclusively. 163. When the letters are disposed, as in the arrangements, 2 and 2, 3 and 3, 4 and 4, &c., it may be required that no two of the results, thus formed, shall be composed of the same letters, in which case the products of the letters will be different ; and we may then demand the whole number of results thus obtained. In this case, the results are called combinations. Thus, ai, ac, be, . . . ad, bd, . . . are combinations of the letters taken 2 and 2. In like manner, abc, abd, . . . acd, bed . . . are combinations of the letters taken 3 and 3. Combinations, are arrangements in which any two will differ from each other by at least one of the letters lohich enter them. Hence, there is an essential difference in the signification of the words, permutations, arrangements, and combinations. Problem 3. To determine the total number of different combina- tions that can be formed of m letters, taken n in a set. Let X denote the total number of arrangements that can bo formed of m letters, taken n and n : F the number of permutations 192 ALGEBRA. of n letters ; and Z the total number of different coinUnations taken « and 11. It is evident, that all the possible arrangements of m letters, taken w at a time, can be obtained, by subjecting the n letters of each of the Z combinations, to all the permutations of which these letters are susceptible. Now a single combination of n letters gives, by hypothesis Y permutations ; therefore Z combinations will give Yx Z . . . arrangements, taken n and n ; and as X denotes the total number of arrangements, it follows that the three quantities X X, Y, and Z, give the relations X= YxZ; whence Z=^. But we have (Art. 162), X=P(m-n+l) and (Art. 161), Y=Qxn. P(m—n + l) P m—n+\ Therefore, Z= ^—-: ' = — X . QXn Q n Since P expresses the total number of arrangements, taken n— 1 and^i— 1, and Q the number of permutations of w— 1 letters, it P follows that — expresses the number of different combinations of m letters taken n— 1 and n— 1. To apply this to the particular case of combinations of m letters taken 2 and 2, 3 and 3, 4 and 4 . . . P Make n=2, in which case — expresses the number of com- binations of the letters taken 2—1 and 2—1 or 1 and 1, and is equal to m ; the above formula becomes m — \ m{m—\) mx- 2 1.2 P Let n=S, — will express thcvnumber of combinations taken 7w(m— 1) 2 and 2, and is equal to — — — ; and the formula becomes m{m—l) (m—2) 1.2.3 • BliVOMIAL THEOREM. 193 In like manner, we would find the number of combinations of letters taken 4 and 4, to ??i(m— 1) (m—2) (m — 3) 1.2.3.4 ; and in ge- neral, the number of combinations of m letters taken n and n, is ex- pressed by m(m—l) {m — 2) (m—S) . . . (m— n + 1) 1.2.3.4 . . . {n-l)7n ' which is the development of the expression P{m—7i+l) Qxn~' Demonstration of the Binomial Theorem. 164. In order to discover more easily the law for the develop- ment of the mth power of the binomial x-}-a, we will observe the law of the product of several binomial factors x+a, x-\-bf x+c, x-\-d . . . oi^ which the first term is the same in each, and the se- cond terms different. X + a X + b 1st. product 2d. 3d. x^ + a + i X -\- c x' + a + h + c X -{■ d X + ab X? + ab + ac + be X + abc a + b + c + d x^ + (lb + ac 4- ad + be + bd + cd 17 x^ + abc + ahd + acd + bed X + c-bcd 194 ALGEBRA. From these products, obtained by the common rule for algebraic multiplication, we discover tlie following laws : 1st. With respect to the exponents ; the exponent of x, in the first term, is equal to the number of binomial factors employed. In the following terms, this exponent diminishes by unity to the last term, where it is 0. 2d. With respect to the co-efRcients of the different powers of a;; that of the first term is unity ; the co-efficient of the second term is equal to the sum of the second terms of the binomials ; the co-effi- cient of the third term is equal to the sum of the products of the different second terms taken two and two ; the co-efficient of the fourth term is equal to the sum of their different products taken three and three. Reasoning from analogy, we may conclude that the co-efficient of the term which has n terms before it, is equal to the sum of the different products of the m second terms of the bi- nomials taken 71 and n. The last term is equal to the continued pro- duct of the second terms of the binomials. In order to be certain that this law of composition is general, sup- pose that it has been proved to be true for a number m of binomials ; let us see if it be true when a new factor is introduced into the pro- duct. For this purpose, suppose a?--hAa;-"-*+Bx'"-2+ar"'-^ . . . +Ma;— "+i+N-r"'-'^-f . . . +U, to be the product of ?« binomial fectors, Nx"—" representing the term 'which has n terms before it, and Ma;"'-"+^ that which immedi- diately precedes. Let x+K be the new factor, the product when arranged according to the powers of x, will be +kI +ak1 +bkI +mkI +UK. From which we perceive that the law of the exponents is evident- ly the same. With respect to the co-efficients, 1st. That of the first term is BINOMIAL THEOREM. 195 ttnity. 2d. A+K, or the co-efficient of x'", is also the sum of the second terms of 'the m+1 binomials. 3d. B is by hypothesis the sum of the difTerent products of the second terms of the m binomials, and A.K expresses the sum of the products of each of the second terms of the' m first binomials, by the new second term K ; therefore B-^-AK is tJie sum of the dif- f event products of the second terms of the m+1 binomials, taJcen two and two. In general, since N expresses the sum of the products of the se- cond terms of the m first binomials, taken n and n ; and as 3IK re- presents the sum of the products of these second terms, taken n— 1 and n— 1, multiplied by the new second term K, it follows that N-\-]MK, or the co-efficient of the term which has n terms before it, is equal to the sum of the difierenl products of the second terms of the m+1 binomials, taken n and n. The last term is equal to the contmued product of the m+1 second terms. Therefore, the law of composition, supposed true for a number m of binomial factors, is also true for a number denoted by m + 1. It is therefore general. Let us suppose, that in the product resulting from the multiplica- tion of the m binomial factors, x-\-a, x-\-b, x-\-c, x-\-d ... we make a=b=c=d . . ., the indicated expression of this product, {x-\-a) (.T+&) (x+c), will be changed into (x+a)"'. With respect to its de- velopment, the co-efficients being a+5+c + cPc-2hciy={2(v'cy+A{Za^cf{-2bd)-\-Q{3cv'cf{-2hdf + 4(.3a2c) {-2hdf + {-2bd)\ or, by performing the operations indicated {3c^c-'2My^Q\a^c*-2lQa<'6'bd + '2lQa*c'lrd"-QQarchH\ + lQbhl\ The terms of the development are alternately plus and minus, as they should be, since the second term is — . 17* 198 ALGEBRA. 167. The powers of any polynomial may easily be found by the binomial theorem. For example, raise a+Zi+c to the third power. First, put .... Z' + c=d. Then (a+b+cf={a+df=a'+da^d+Sa(P+d^. Or, by substituting for the value of d, {a+b+cy=a'+Sa^+Zab^+P Sa^'c + Si'^c +eahc + 2ac^+Sbc'' + c\ This expression is composed of the cubes of the three terms, plus three times the square of each term by the first powers of tlie two others, plus six times the product of all three terms. It is easily proved that this law is true for any polynomial. To apply the preceding formula to the development of the cube of a trinomial, in which the terms are affected with co-efficients and exponents, designate each term by a single letter, then replace the let. ters introduced, by their values, and perform the operations indicated. From this rule, we will find that (2a2-4a5+3Z^f=8a«-48a'J+132a^J^-208a='53 + 1 QSa'b" _ 1 OSaJs + 21b\ The fourth, fifth, &c. powers of any polynomial can be develop, ed in a similar manner. Consequences of the Binomial Formula. 168. First. The expression (x+a)'" being such, that x may be substituted for a, and a for x, without altermg its value, it fol- lows that the same thing can be done in the development of it ; therefore, if this development contains a term of the form Ka"a;'"^'', it must have another equal to Kx^a"^'" or Ka"'"""a;''. These two terms of the development are evidently at equal distances from the two extremes ; for the number of terms which precede any term, being indicated by the exponent of a in that term, it follows that BINOMIfVL THEOREM. 199 the term K^''^;'""" has n terms before it ; and that the term Ka"'~"oif' has m—n terms before it, and consequently n terms after it, since the whole number of terms is denoted by m + 1. Therefore, in the development of any poiver of a hinomiul, the co- efficients at equal distances from the two extremes are equal to each other. Remark. In the terms Ka^T""-", Krt™-"af , the first co-efficient ex- presses the number of different combinations that can be formed with m letters taken n and n ; and the second, the number which can be formed when taken m—n and ?n—n; we may therefore conclude that, the number of different combinations of m letters taken n and n, is equal to the number of combinations of m letters taken m—n and m—n. For example, twelve letters combined 5 and 5, give the same number of combinations as these twelve letters taken 12 — 5 and 12 — 5, or 7 and 7. Five letters combined 2 and 2, give the same number of combinations as five letters combined 5—2 and 5—2, or 3 and 3. 169- Second. If in the general formula, m — 1 {x-\-aY^x''' -{-max'"~^ -\-m — -— a^a;'"""^+, &c. we suppose x=l, 0=1, it becomes m — \ 711—1 m — 2 (1 + 1)- or 2'"=l+m+m— ^— +7n-— — . ~^— +, &c. That is, the sum of the co-efficients of the different terms of the formula for the binomial, is equal to the mth power of 2. Thus, in the particular case (x + a)5 = 0,-5 + ^ax" + 1 Oa V + 1 Oa^'ar + ba'x + a\ the sum of the co-efficients 1+5+10 + 104-5+1 is equal to 2^ or 32. In the 10th power developed, the sum of the co-efficients is equal to 2" or 1024. 170. Third. In a series of numbers decreasing by unity, of which 200 ALGEBRA. the first term is m and the last m—p, m and p being entire numbers, the continued product of all these numbers is divisible by the con- tinued product of all the natural numbers from 1 to ^+1 inclu- sively. Thnt i« »<^^-l) {m-2) (m-3) . . . {m-p) . That IS, 1.2 ^ 3 ^ ^ __ (^^^^) IS a whole num. ber. For, from what has been said in (Art. 16-3), this expression represents the number of different combinations that can be formed of m letters taken p+1 and p+1. Now this number of combina- tions is, from its nature, an entire number ; therefore the above ex- pression is necessarily a whole number. Of the Extraction of the Roots of jJ articular numbers. 171. The third power or cube of a number, is the product arising from multiplying this number by itself twice ; and the thirds or cube root, is a number which, being raised to the third power, will produce the proposed number. The ten first numbers being 1, 2, 3, 4, &, 6, 7, 8, 9, 10. their cubes are 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. Reciprocally, the numbers of the first line are the cube roots of the numbers of the second. By mspecting these lines, we perceive that there are but nine perfect cubes among numbers expressed by one, two, or three figures ; each of the other numbers has for its cube root a whole number, plus a fraction which cannot be expressed exactly by means of unity, as may be shown, by a course of reasoning entirely similar to that pursued in the latter part of (Art. 118). 172. The difference between the cubes of two consecutive num- bers increases, when the numbers are increased. Let a and a+1, be two consecutive whole numbers ; we have (rt + lf=a^ + 3a' + 3a + l; whence {a-\-lf-a''^'^a'+^a+\. EXTRACTION OF ROOTS. 201 That is, the difference between the cubes of tioo consecutive whole numbers, is equal to three times the square of the least number, plus three times this number, plus 1. Thus, the difference between the cube of 90 and the cube of 89, is equal to 3(89)2 + 3x89 + 1 = 24031. 173. In order to extract the cube root of an entire number, we will observe, that when the figures expressing the number do not exceed three, its root is obtained by merely inspecting the cubes of the first nine numbers. Thus, the cube root of 125 is 5 ; the cube root of 72 is 4 plus a fraction, or is within one of 4 ; the cube root of 841 is within one of 9, since 841 falls between 729, or the cube of 9, and 1000, or the cube of 10. When the number is expressed by more than three figures, the process will be as follows. Let the proposed number be 103823. 103.823 47 64 8 1 398.23 . 48 47 48 47 384 329 192 188 2304 2209 48 47 18432 15463 9216 8836 110592 103823 This number being comprised between 1,000, which is the cube of 10, and 1,000,000, which is the cube of 100, its root will be ex- pressed by two figures, or by tens and units. Denoting the tens by a, and the units by b, we have (Art. 160), {a-^bf=:a? + 2,a?b + 2a¥-\-V\ Whence it follows, that the cube of a number composed of tens and units, is equal to the cube of the tens, plus three times the product 202 ALGEBRA. of the square of the tens ly the units, plus three times the product of the tens by the square of the units, plus the cube of the units. This being the case, the cube of the tens, giving at least, thou- sands, the last three figures to the right cannot form a part of it : the cube of the tens must therefore be found in the part 103 which is separated from the last three figures by a point. Now the root of the greatest cube contained in 103 being 4, this is the number of tens in the required root ; for 103823 is evidently comprised be- tween (40)=* or 64,000, and (50)=* or 125,000 ; hence the required root is composed of 4 tens, plus a certain number of units less than ten. Having found the number of tens, subtract its cube 64 from 103 ; there remains 39, and bringing down the part 823, we have 39823, which contains three times the square of the tens by the units, plus the two parts before mentioned. Now, as the square of a number of tens gives at least hundreds, it follows that three times the square of the tens by the units, must be found in the part 398, to the left of 23, which is separated from it by a point. Therefore, dividing 398 by three times the square of the tens, which is 48, the quotient 8 will be the unit of the root, or something greater, since 398 hun- dreds is composed of three times the square of the tens by the units, together with the two other parts. We may ascertain whether the figure 8 is too great, by forming the three parts which enter into 39823, by means of the figure 8 and the number of tens 4 ; but it is much easier to cube 48, as has been done in the above table. Now the cube of 48 is 110592, which is greater than 103823 ; therefore 8 is too great. By cubing 47 we obtain 103823 ; hence the pro- posed number is a perfect cube, and 47 is the cube root of it. Remark. The units figures could not be first obtained ; because the cube of the units might give tens, and even hundreds, and the tens and hundreds would be confounded with those which arise from other parts of the cube. EXTRACTION OF ROOTS. >03 Affain, extract the cube root of 47954 47.954 36 27 36 3^X3==27 1 209 37 36 37 47954 216 259 46656 108 111 1298 1296 1369 36 37 7776 9583 3888 4107 46656 50653 The number 47954 being below 1,000,000, its root contains only two figures, viz. tens and units. The cube of the tens is found m 47 thousands, and we can prove, as in the preceding example, that 3, the root of the greatest cube contained in 47, expresses the tens. Subtracting the cube of 3 or 27, from 47, there remains 20 ; bring- ing down to the right of this remainder the figure 9 from the part 954, the number 209 hundreds, is composed of three times the square of the tens by the units, plus the number arising from the other two parts. Therefore, by forming three times the square of the tens, 3, which is 27, and dividing 209 by it, the quotient 7 will be the units of the root, or something greater. Cubing 37, we have 50653, which is greater than 47954 ; then cubing 36, we obtain 46656, which subtracted from 47954, gives 1298 for a remainder. Hence the proposed number is not a perfect cube ; but 36 is its root to within unity. In fact, the difference between the proposed number and the cube of 36, is, as we have just seen, 1298, which is less than 3(36)^+3x36 + 1, for in verifying the result we have obtained 3888 for three times the square of 36. 174. Again, take for another example, the number, 43725658 containing more than 6 figures. 204 3^X3= 35^X3 = 3675 Rem. ALGEBRA. 43.725.658 352 27 1 167 35 352 35 352 43 725 175 704 42 875 105 1225 1760 8506 1056 35 123904 43725658 6125 352 43614208 3675 247808 111450 42875 619520 371712 43614208 Now the required root contains more than one figure, and may be considered as composed of units and tens only, the tens being expressed by one or more figures. Since the cube of the tens gives at least thousands, it must be found in the part which is to the lefl; of the last three figures 658. I say now that if we extract the root of the greatest cube contain- ed in the part 43725, considered with reference to its absolute value, we shall obtain the whole number of tens of the root ; for let a be the root of 43725, to within unity, that is, such that 43725 shall be comprised between a^ and (a + 1)^ ; then will 43725000 be compre- hended between a?X 1000 and (a+l)^x 1000 ; and as these two last numbers differ from each other by more than 1000, it follows that the proposed number itself, 43725658, is comprised between a^x 1000 and (a 4-1)^X1 000 ; therefore the required root is comprised be- tween that of a^x 1000, and (a + l)^X 1000, that is, between ax 10 and (a+l)xlO- It is therefore composed of a tens, plus a certain number of units less than ten. The question is then reduced to extracting the cube root of 43725 ; but this number having more than three figures, its root will con- EXTRACTION OF ROOTS. 205 tain more than one, that is, it will contain tens and units. To ob- tain the tens, point off the last three figures, 725, and extract the root of the greatest cube contained in 43. The greatest cube contained in 43 is 27, the root of which is 3 ; this figure will then express the tens of the root of 43725, or the figure in the place of hundreds in the total root. Subtracting the cube of 3, or 27, from 43, we obtain 16 for a remainder, to the right of which bring down the first figure 7, of the second period 725, which gives 167. Taking three times the square of the tens, 3, which is 27, and dividing 167 by it, the quotient 6 is the unit figure of the root of 43725, or something greater. It is easily seen that this number is in fact too great ; we must therefore try 5. The cube of 35 is 42875, which, subtracted from 43725, gives 850 for a remainder, which IS evidently less than 3 x (35)^+3x35 + 1. Therefore, 35 is the root of the greatest cube contained in 43725 ; hence it is the number of tens in the required root. To obtain the units, bring down to the right of the remainder 850, the first figure, 6, of the last period, 658, which gives 8506 ; then take 3 times the square of the tens, 35, which is 3675, and divide 8506 by it ; the quotient is 2, which we try by cubing 352 : tiiis gives 43614208, which is less than the proposed number, and sub- tracting it from this number, we obtain 111450 for a remainder. Therefore 352 is the cube root of 43725658, to within unity. Hence, for the extraction of the cube root we have the following RULE. I. Separate the given number into periods of three figures each, he. ginning at the right hand : the left hand period will often contain less than three places of figures. II. Seek the greatest cube in the first period, at the left, and set its root on the right, after the manner of a quotient in division. Subtract the cube of this figure of the root from the first period, and to the re- 18 206 ALGEBRA. mainder bring down the first figure of the next period, and call this number the dividend. III. Take three times the square of the root just found for a divi. sor, and see haw often it is contained in the dividend, and place the quotient for a second figure of the root. Then cube the figures of the root thus found, and if their cube be greater thajUhe first two periods of the given number, diminish the last figure ; bulif it be less, subtract it from the first tioo periods, and to the reinainder bring down the first figure of the next period, for a new dividend. ■ IV. Take three times the square of the whole root for a new divi- sor, and seek how often it is contained in the new dividend : the quo- tient will be the third figure of the root. Cube the whole root and subtract the result from the three first periods of the given number, and proceed in a similar way for all the periods. Remark. If any of the remainders are equal to, or exceed, three times the square of the root obtained plus three times this root, plus one, the last figure of the root is too small and must be aug- mented by at least unity (Art. 172). EXAMPLES, 1. V48228544=364. 2. V27054036008 = 3002. 3. V483249=78, with a remainder 8697; 4. '/91632508641 = 4508, with a remainder 20644129 • 5. V32977340218432=:: 32068. To extract the n"" root of a ivhole number. 175. In order to generalize the process for the extraction of roots, we will denote the proposed number by N, and the degree of the root to be extracted by n. If the number of figures in iV, does not exceed n, the root will be expressed by a single figure, and is ob- tained immediately by forming the n"' power of each of the whole EXTRACTION OF ROOTS. 207 numbers comprised between 1 and 10 ; for the n"' power of 9 is the largest perfect power which can be expressed by n figures. When N contains more than n figures, there will be more than one figure in the root, which may then be considered as composed of tens and units. Designating the tens by a, and the units by b, we have (Art. 166), n—\ iV=(a + Z»)''=a'' + na''-V>'+n— ~a"-2^2+, &,c. ; that is, the proposed number contains the n*'' power of the tens, plus n times the product of the n — 1 power of the tens by the units, plus a series of other parts which it is not necessary to consider. Now, as the »"' power of the tens cannot give units of an order inferior to unity followed by n ciphers, the last n figures on the right, cannot make a part of it. They must then be pointed off, and the root of the greatest n"" power contained in the figures on the left should be extracted ; this root will be the tens of the required root. If this part on the left should contain more than n figures, the n figures on the right of it, must be separated from the rest, and the root of the greatest n"" power contained in the part on the left ex- tracted, and so on. Hence the following RULE. I. Divide the number N into periods of x\ figures each, beginning at the right hand ; extract the root of the greatest n"' power contained ill the left hand period, and subtract tJie n"" poioer of this figure from the left hand period. [I. Bring down to the right of the remainder corresponding to the first period, the first figure of the second period, and call this number the dividend. III. For?n the n— 1 power of the first figure of the root, multiply it by n, and see how often the product is contained in the dividend : the quotient will be the second figure of the root, or something greater. IV. liaise the number thus formed to the n'** power, then subtract this result from the two first periods, and to the new remainder bring down the first figwe of the third period : then divide the number thus 20S ALGEBRA formed by n times the n— 1 power of the two figures of the root al- ready found, and continue this operation until all the periods are brought down. EXAMPLES. Extract the 4th root of 531441. 53.1441 I 27 2^= 16 4X2^=32 I 371 (27)"= 531441. We first divide off, from the right hand, the period of four figures, and then find the greatest fourth root contained in 53, the first period to the left, which is 2. We next subtract the 4th power of 2, which is 16, from 53, and to the remainder 37 we bring down the first figure of the next period. We then divide 371 by 4 times the cube of 2, which would give 8 for a quotient ; but by raising 28 to the 4th power, we discover that 8 is too large, then trying 7 we find the exact root to be 27. 176. Remark. When the degree of the root to be extracted is a multiple of two or more numbers, as 4, 6, . . . ., the root can he oh- tuined by extracting the roots of more simple degrees, successively. To explain this, we will remark that, and that in general (a'")''=a'"xa'"Xa'"Xa"' • • • =«"""■ (Art. 13). Hence, the n'*" power of the m"" power of a number, is equal to the mn"" power of this number. Reciprocally, the mn"' root of a number is equal to the n"" root of the m"' root of this number, or algebraically "'V~a= V V a = V Va^ For, let . . . Vv«=«'> raising both membei-s to the n'* power there will result . . . V^=«'" ; for from the definition of a root, we have ( Vir)''=K. EXTRACTION OF ROOTS. 209 Again, by raising both members to the m"' power, we obtain a= (a'")'" =:«""". Extracting the /»«'" root of both members, "'!ya=a' ; but we already have V'^a=a' ; hence "■;/«= V'^a. In a similar manner we might find v a= \/ Va. By this method we find that V256 =\/ V 256 =r 'V/16 = 4 ; V 2985984 = \/ ^ 2985984 = Vl728= V 1771561 =V V 1771561 =11 ; V 1679616 = Vl296 = ' Remark. Although the successive roots may be extracted in any order whatever, it is better to extract the roots of the lowest degree first, for then the extraction of the roots of the higher de- grees, which is a more complicated operation, is effected upon num. bers containing fewer figures than the proposed number. Extraction of Roots by approximation. 177. When it is required to -extract the n"" root of a number which is not a perfect power, the method of (Art. 175), will give only the entire part of the root, or the root to within unity. As to the fraction which is to be added, in order to complete the root, it cannot be obtained exactly, but we can approximate as near as we please to the required root. Let it be required to extract the n"' root of the whole number a, to within a fraction — ; that is, so near it, that the error shall be P 1 less than — . P (ly^pT We M'ill observe that a can be put under the form . If p" 18* 210 ALGEBRA. axp" we denote the root of ap" to within unity, by r, the number P" or a, will be comprehended between — and — ; there- fore the Va will be comprised between the two numbers, r r+1 r — and . Hence — is the required root, to within the p p P fraction — . P Hence, to extract the root of a whole number to within a fraction — , multiply the number by p" ; extract the n"" root of the product to within unity, and divide the result by p. 178. Again, suppose it is required to extract the n"" root of the fraction -r-. Multiply each term of the fraction by a ab"-^ b"-^ ; It becomes -7-= , . b ft" Let r denote the n"" root of aft"-', to within unity; or — , will be comprised between -j- and — — — Therefore, after having made the denominator of the fraction a per- fect power of the n'*" degree, extract the n^^ root of the numerator, to within unity, and divide ilie result by the root of the new denominator. When a greater degree of exactness is required than that indi- cated by -^, extract the root of aS"-' to within any fraction — ; •' b p and designate this root by — . Now, since — is the root of the & P P 1 / numerator to within — , it follows, that -7- is the true root of p' bp 1 the fraction to within -r-. bp EXTRACTION OF ROOTS. 211 179. Suppose it is required to extract the cube root of 15, to within — . We have 15xl2'=15x 1728=25920. Now the cube root of 25920, to within unity, is 29 ; hence the required root 29 5 '^ 1^ '' 'l2- I • Again, extract the cube root of 47, to within — . We have 47x20^=47x8000=376000. Now the cube root , — 72 12 of 376000, to within unity, is 72; hence 3/47=— =3—, to 1 withm -. Fmd the value of V25 to within 0,001. To do this, multiply 25 by the cube of 1000, or by 1000000000, which gives 25000000000. Now, the cube root of this number, is 2920 ; hence V25= 2,920 to within 0,001. In general, in order to extract the cube root of a whole number to within a given decimal fraction, annex three times as many ciphers to the number, as there are decimal places in the required root ; extract the cube root of the number thus formed to within unity, and point off from the right of this root the required number of decimals. 180. We will now explain the method of extracting the cube root of a decimal fraction. Suppose it is required to extract the cube root of 3,1415. As the denominator 10000, of this fraction, is not a perfect cube, it is necessary to make it one, by multiplying it by 100, which amounts to annexing two ciphers to the proposed decimal, and we have 3,141500. Extract the cube root of 3141500, that is, of the num- ber considered independent of the comma, to within unity ; this gives 146. Then divide by 100, or VlOOOOOO, and we find V3,1415=l,46 to within 0,01. 212 ALGEBRA. Hence, to extract the cube root of a decimal number, we have the following RULE Annex ciphers to the decimal part, if necessary, until it can be divided into exact periods of three figures each, ohserving that the number of periods must be made equal to the number of decimal places required in the root. Then, extract the root as in entire num. bers, and pioint off as many places for decimals as there are periods in the decimal part of the number. To extract the cube root of a vulgar fraction to within a given decimal fraction, the most simple method is to reduce the proposed fraction to a decimal fraction, continuing the operation until the num. ber of decimal places is equal to three times tlie number required in the root. The question is then reduced to extracting the cube root of a decimal fraction. 181. Suppose it is required to find the sixth root of 23, to within 0,01. Applying the rule of Art. 177 to this example, we multiply 23 by 100% or annex twelve ciphers to 23, extract the sixth root of the number thus formed to within unity, and divide this root by 100, or point off two decimals on the right. In this way we will find that V23r=l,68, to within 0,01. EXAMPLES. 1. Find the V473 to within J^. Ans. 7^. 2. Find the V79 to within ,0001. Ans. 4,2908. 3. Find the Vl3 to within ,01. Ans. 1,53. 4. Find the V3,00415 to within ,0001. Ans. 1,4429. 5. Find the VOjOOlOl to within ,01. Ans. 0,10. to within ,001. Ans. 0,824. EXTRACTION OF ROOTS. 213 Formation of Powers and Extraction of Roots of A/gebraic Quantities. Calculus of Radicals. We will first consider monomials. 182. Let it be required to form the fifth power of 2a^^. We have i^o'l/f == 2a?¥ X 'ia?!)" X 2a^Z^ x 2a^¥ X 2a^^^ from which it follows, 1st. That the co-efficient 2 must be multi- plied by itself four times, or raised to the fifth power. 2d. That each of the exponents of the letters must be added to itself four times, or multiplied by 5. Hence, (2a='Zr')5=2^a^ ^ ^P>^^='^2a^^b^''. In like manner, {Sa''U'cf=&'.a'>^W>^''c''=bl2a^h^c'. Therefore, in order to raise a monomial to a given power, raise the co-efficient to this poioer, and multiply the eoqionent of each of the letters by the exponent of the power. Hence, reciprocally, to extract any root of a monomial, 1st. Extract the root of the co-efficient. 2d. Divide the exponent of each letter hy the index of the root. V64a^^= 4a='<5c2 ; Vl6a«^''V= ^a'^Pc. From this rule, we perceive, that in order that a monomial may be a perfect power of the degree of the root to be extracted, 1st. its co-efficient must be a perfect power ; and 2d. the exponent of each letter must be divisible by the index of the root to be extracted. It will be shown hereafter, how the expression for the root of a quantity which is not a perfect power is reduced to its simplest terms. 183. Hitherto, we have paid no attention to the sign with which the monomial may be affected ; but if we observe, that what- ever may be the sign of a monomial, its square is always positive, and that every power of an even degree, 2n, can be considered as the n** power of the square, that is, a^'=(o^)", it will follow that, 214 ALGEBRA. every power of a quantity, of an even degree, whether positive or negative, is essentially positive. Thus, {±2aWcy= + lGaWc\ Again, as a power of an uneven degree, 2nH-l, is the product of a power of an even degree, 2», by the first power, it follows that, every power of an uneven degree cf a monomial, is affected with the same sign as the monomial. Hence, (+4a2^)='=+64rtW ; {^-^a^bf^-QW^P. From this it is evident, 1st. That when the degree of the root of a monomial is uneven, the root .will be affected with the same sign as the quantity. Therefore, V+8a^ = + 2« ; V-Sa^^ - 2a ; V-32a"6== - 2a^^. 2d. When the degree of the root is even, and the monomial a positive quantity, the root is affected either with + or — . Thus, V81a^=±3aZ'='; V64^^=±2a^ 3d. When the degree of the root is even, and the monomial nega- tive, the root is impossible ; for, there is no quantity which, raised to a power of an isven degree, can give a negative result. Therefore, V— a, V — b, V — c, are symbols of operations which it is impossible to execute. They are, like V —a, V —h, imagina- ry expressions (Art. 126). 184. In order to develop {a+y-{-z\^, we will place y-\-z=u, and we have (a +«)='= a^* + Sa^M + 3a?i^ + M^ or by replacing u by its value, y+z {a+y+zf=.a^ + Za\y+z) + 2a{y+zf+{yJrzf, or performing the operations indicated {a+y+zf=a'+2,a^y-\-Za\ + ^af+Qayz-\-Zaz'^+f-\-^fz-{- Syz'^+z^. When the polynomial is composed of more than three terms, as EXTRACTION OF ROOTS. 215 a-\-y-\-z-\-x . . . . p, let, as before, u= the sum of all the terms after the first. Then, a+w will be equal to the given polynomial, and From which we see, that by cubing a polynomial, we obtain the cube of the first term, plus three times ike square of the first term multiplied hy each of the remaining terms, plus other terms. It often happens that u contains a, as in the polynomial a^-\-ax-\-b, where u=zax+i. But since we suppose the polynomial arranged with reference to a, it follows that a will have a less exponent in u than in the first term. In this case also, the co-efficient of u, multiplied by the first term of u, will be irreducible with the remaining terms of the develop, ment, because that product will mvolve a to a higher power than the other terms : and when a does not enter u, the product of that co-efficient by all the terms of w, will be irreducible with all the other terms of the development. 185. As to the extraction of roots of polynomials, we will first explain the method for the cube root ; it will afterwards be easy to generalize. Let N be the polynomial, and R its cube root. Conceive the two polynomials to be arranged with reference to some letter, a, for ex- ample. It results from the law of composition of the cube of a po- lynomial (Art. 184), that the cube of R contains two parts, which cannot be reduced with the others ; these are, the cube of the first term, and three times the square of the first term by the second. Hence, the cube root of that term of N which contains a, affect- ed with the highest exponent, will be the first term of R : and the second term of jR will be found by dividing the second term of N by three times the square of the first term of R. If then, we form the cube of the two terms of the root already found, and subtract it from N, and divide the first term of the re- mainder by 3 times the square of the first term of R, the quotient will be the third term of the root. Therefore, having arranged the terms of N, we have the following 216 ALGEBRA. RULE. I. Extract the cube root of the first term. II. Divide the second term of N by three times the sqiuire of the first term of R : the quotient will be the second term of R. III. Having found the two first terms of R,form the cube of the binomial and subtract it from N ; after which, divide the first term of the remainder by three times the square of the first terin of R : the quotient will be the third term of R. IV. Cube the three terms of the root found, and subtract the cube from N ; then divide the first term of the remainder by the divisor already used : the quotient will be the fourth term of the root, and the remaining terms, if there are any, may be found in a similar manner. EXAMPLES. 1. Extract the cube root of a;«— Gx^+lSx* — 20xHl5a:* — Gx + 1. (sr'-2xf=x^-ex^ + 12x*- Sa;^ 3^^ 1st Rem. . ~. '. '. 3.r*— 12x3+, &c. (r'-2x+lf=x''-6x' + 15a'-20x^-{-15x^—6x+l. In this example, we first extract the cube root of x', which gives x^, for the first term of the root. Squaring a^, and multiplying by 3, we obtain the divisor 3a;'* : this is contained in the second term — 6x^, —2x times. Then cubing the root, and subtracting, we find that the first term of the remainder 3x*, contains the divisor once. Cubing the whole root, we find the cube equal to the given polyno- mial. Remark. The rule for the extraction of the cube root is easily extended to a root with a higher index. For, Let a+i-j-c-j- . . f, be any polynomial. Let s= the sum of all the terms after the first. Then a + s= the given polynomial : and (a+5)" = a" + na'' *5+ other terms- CALCULUS OF RADICALS. 217 That is, the n"" poiver of a polynomial, is equal to the n"' pouter of the first ter7n, plus n times the first term raised to the power n— 1, multiplied hy each of the remaining terms ; + other terms of the de- velopment. Hence, we see, that the rule for the cube root will become the rule for the n"" root, by first extracting the n'^ root of the first term, taking for a divisor n times this root raised to the n^l power, and raising the partial roots to the ?i"' power, instead of to the cube. 2. Extract the 4th root of lQa*-QQa''x^2\Qa'x'-2\Gax' + Qla\ (2a-3a;)'' = 16a''-96a^x'+216aV-216ca;='+81a''l32a3=4x(2a)3 We first extract the 4th root of 16a*, which is 2a. We then raise 2a to the third power, and multiply by 4, the index of the root : this gives the divisor 32a^. This divisor is contained in the second term — 96a^a;, — 3a; times, which is the second term of the root. Raising the whole root to the 4th power, we find the power equal to the given polynomial. 3. Find the cube root of a;6 + 6a;5— 40a^ + 96a; - 64. 4. Find the cube root of Ibx^-Qx+x^-Qx" -203^ + 10x^ + 1. 5. Find the 5th root of 320^* - SOx* + 80.r^ - 40a;2 + 1 Ox - 1 . Calculus of Radicals. 186. When it is required to extract a certain root of a monomial or polynomial which is not a perfect power, it can only be indicated by writing the proposed quantity after the sign V, and placing over this sign the number which denotes the degree of the root to be ex- tracted. This number is called the index of the root, or of the radical. A radical expression may be reduced to its simplest terms, by 19 218 ALGEBRA. observing that, the n"" root of a product is equal to the product of the n^^ roots of its different factors. Or, in algebraic terms : Vabcd=:i/aXV^X V^X V^. For, raising both members to the n"" power, we have for the first, ( V abed) ^ahcd . . ., and for the second, (V«X7&X Vcx7^---)"=(^«)'"-(V^)"-(Vc)''.(yrf)"...=a^c/¥= ''-v/2^='^/l; and V3=V3^=:V'9: Whereas, they should have, by the foregoing rule, ''^r2=W~{2f='^r^, and V^ = '\^J^=V^. Reduce V2, %/4, W\, to the same index. Addition and Subtraction of Radicals. 189. Two radicals are similar, when they have the same index, and the same quantity, under the sign. Thus, 3 -^/ab and 7 Vah, are similar radicals, as also 3a- V^^j and 9c^ %/P. Therefore, to add or subtract similar radicals, add or subtract their co-efficients, and prefix the sum or difference to the common radical. Thus, 2,Wb + )lWb=bWb, ^Wb-2Wb=Wb, 3aVbdz2cVb=(3adz2c)Vb. CALCULUS OF RADICALS. 221 Sometimes when two radicals are dissimilar, they can be reduced to similar radicals by Arts. 186 and 187. For example, W8a'b+l6a''-Vb' + 2aP=2aVb + 2^-b VM^«~ =.{2a-b)Vb+2^; 3 V4^+2 V2^=3 V2^+2 V2a=5 V2^. When the radicals are dissimilar, and irreducible, they can only be added or subtracted by means of the signs + or — . Multiplication and Division. 190. We will first suppose that the radicals have a common index. Let it be required to multiply or divide Va by V^. We have Vax Vb= Vab, and Va -^ Vb=\/-r. For by raising Va . "Vb and 'Vab to the n"" power, we obtain the same result ab ; hence the two expressions are equal. r/ffl 1 " /^ . , , . . « In Uke manner, — r and \/ - raised to the n''' power give -^: iy ^ . hence these two expressions are equal. Therefore we have the following RULE. Multiply or divide the quantities tinder the sign by each other, and give to the product, or quotient, the common radical sign. If they have co-efficients, first multiply or divide them separately. Thus, 2a V _Z_x-3aV ^ T ^ =-6a^V ^ j^-. c a ca or, reducing to its simplest terms, Qa^a^ + V) 19* 222 ALGEBRA. 3fl VSafx^i V4aF^=6ai VS2a*c=l2a''bV2'c V \/^ aW+b' _ ' /Qb {aW + h^) _^^l /g!±jl Sb When the radicals have not a common index, they should be re- duced to one. For example, 3aVbx5b''V^c=ldabx'Vsb^' EXAJtfPLES, 1. Multiply \/2X -V3 by V yX' ^-w Alls. 'ViT 2. Multiply 2 Vis by sVTo An*. 6V337500. 5 /T . /y 3. Multiply 4V Y by 2 V -j- ': / 27 ^-- ^v ,5,. 2 'v/ 3 V -v/ 4 4. Reduce = to its lowest terms. x*-v/TxV3 ^715. 4'V288. /V J- X2V 3 5. Reduce \/ ^ ^z to its lowest terms. ^ 4V2 X -/s 6. Multiply VT, VT, and 'V^ to ether. An*. 'V648000. CALCULUS OP RADICALS. 223 7. Multiply V -r-, 'v 17 and 'V6 together. 3' " 2 Ans. V ^. 8. Multiply (4V-^+5\Ai) by (V Y+2V y) 9. Divide yV Y by ( -/T+sVy) 43 13 ^- y +Y v42. ^n.. 1 10. Divide 1 by *VT+V3 . *V^- *V~^+ W^- V"^ Ans. a—b 11. Divide *V~a" + *-v/X by W a —*V1)' . a-\-h-\-2 V^+2Va^+2*Va^ -4ns. a-b Formation of Powers, and Extraction of Roots. 191. By raising V« to the n"" power, we have (Va)''=V«XV«XV« • • • ='Va", by the rule just given for the multiplication of radicals. Hence, for raising a radical to any power, we have the following RULE. Raise the quantity under the sign to the given power, and affect the result with the radical sign, having the primitive index. If it has a co.efficient, first raise it to the given pozver. Thus, ( *V^f = V{^ay = Vl6a«= 2a V^; (3V2^)5=35.V(2^'=243V32^5=:486aV4^ 224 ALGEBRA. When the index of the radical is a multiple of the power, the re- sult can be reduced. For, *-\/2a='\/ V2a (Art. 176) : hence, to square *V2a, we have only to omit the first radical, which gives ( V2a) = V2a. Again, to square Wsb, we have V 3b=\/ VsZ* • hence Consequently, when the itidex of the radical is divisible by the ex- ponent of the power, perform this division, leaving the quantity under the radical unchanged. To extract the root of a radical, multiply the index of the radical by the index of the root to be extracted, leaving the quantity under the sign unchanged. Thus, V *V3^="V^; S/^VTc^WVc. This rule is nothing more than the principle of Art. 176, enun- dated in an inverse order. When the quantity under the radical is a perfect power, of the degree of either of the roots to be extracted, the result can be re- duced. Thus, \/ VSa^ bemg equal to \/ VSo^ it reduces to V2a. In like manner, \/ W^^s/ Vdc^^^VSa. It is evident that y'l^a— V ya ; because both expressions are equal to "'ya~(Art. 176). 192. The rules just demonstrated for the calculus of radicals, principally depend upon the fact that the ji"* root of the product of several factors is equal to the product of the n"" roots of these fac- tors ; and the demonstration of this principle depends upon this : When the powers, of the same degree, of two expressions are equal, CALCULUS OF RADICALS. 225 the expressions are also equal. Now this last proposition, which is true for absolute numbers, is not always true for algebraic expres- sions. To prove this, we will show that the same number can have more titan one square root, cuie root, fourth root, ^c. For, denote the general expression of the square root of a by x, and the arithmetical value of it by ^ ; we have the equation x^=a, or xi^z=p^, whence x=zkp. Hence we see that the square of p, which is the root of a, will give a, whether its sign be + or — . In the second place, let x be the general expression of the cube root of a, and p the numerical value of this root ; we have the equation x^=a, or x^^p^. This equation is satisfied by making x=p. Observing that the equation cc^=^p^ can be put under the form sP—p^=0, and that the expression x^—p^ is divisible by x—p, (Art. 59), which gives the exact quotient, x^-^-px+p^, the above equation can be transformed into (x—p) (x^ -{-px-{-p^)^0. Now, every value of x which will satisfy this equation will satis- fy the first equation. But this equation can be verified by suppos- ing x—p^O, whence x=p ; or by supposuig x^+px+p'^—O, from which last we have x=--±-V-3, or x:=p[ j. Hence, the cube foot of a, admits of three different algebraic va- lues, viz. P> P{ 2 J' ^""^ P\ 2 /• Again, resolve the equation x'^^p*, in which p denotes the arith- metical value of \/a. This equation can be put under the form x*— p''=0. Now this expression reduces to (a^— p^) (a^+P^)* 226 ALGEBRA. Hence the equation reduces to (a^— p^) {xr'-\-p^)=0, and can be satisfied by supposing x^—p^—0, whence x=±p ; or by suppos- ing a^-{-p-=0, whence x=± V —p'^—dzp V—1. We therefore obtain four different algebraic expressions for the fourth root of a. For another example, resolve the equation .... x^z=p^, which can be put under the form :x^—p^=0. Nowa;®—2>^ reduces to {s(P—p^) (aP-\-p^), therefore the equation becomes .... (x^—p^) {xP+p^)=0. But x^—p^—0, gives /-1± V~^3. x=p, and x=pl ; I. And if in the equation x^+p^ = 0, we make p~—p', it becomes aP—p'^—0 from which we deduce a;=|)', and -=P( ^ ); or, subslituthig for p' its value, —p, /-1± a/^ x=—p and x——pl J. Therefore the value of x, in the equation a;"— ^"=0, and conse- quently the 6th root of a, admits of six values, p, ap, a'p, —p, — a]), —a'p, by making -= 2- , a'= . We may then conclude from analogy, that x in every equation of the form x'^—a—O, or a^'"— ^"=0, is susceptible of mdifferent va- lues, that is, the m"" root of a number admits of m different alge. braic values. 193. If in the preceding equations and the results corresponding to them, we suppose as a particular case a=l, whence p=l, we shall obtain the second, third, fourth, &c. roots of unity. Thus + 1 and —1 are the two square roots of unity, because the equation ar*— 1 = 0, gives a:=zhl. CALCrLTJS OF RADICALS. 227 ^ ,., -l+V-3 -1— VT In like manner +1, , , are the three cube roots of unity, or the roots of a;^— 1=0. And + 1,-1, + V — 1, — V — 1, are the four fourth roots of unity, or the roots of x*— 1 = 0, 194. It results from the preceding analysis, that the rules for the calculus of radicals, which are exact when applied to absolute num- bers, are susceptible of some modifications, when applied to expres- sions or symbols which are purely algeiraic ; these modifications are more particularly necessary when applied to imaginary expressions, and are a consequence of what has been said in (Art. 192). For example, the product of V —ahy V —a, by the rule of (Art. 190), would be Now, Va" is equal to ±a (Art. 192) ; there is, then, apparent- ly, an uncertainty as to the sign with which a should be affected. Nevertheless, the true answer is —a ; for, in order to square -y/m, it is only necessary to suppress the radical ; but the V — « X V' — a reduces to ( V — «} , and is therefore equal to —a. Again, let it be required to form the product V —a x v/ —h, by the rule of (Art. 190), we shall have V —a X V—b— V+ab. Now, Vab=±p (Art. 192), p being the arithmetical value of the square root of ah ; but I say that the true result should be —p or — ■yob, so long as both the radicals V —a and V—b are con. sidered to be affected with the sign +. For, V—a— y/a. V^l and V^F^ y/b. V^ ; hence -/^x V^= va. V-ix V^-bx V^r= V~^h{ V^lf = Vabx—l=— Vab. 228 Upon this principle we find the different powers of v— 1 to be, as follows : V-i= V^-i^ {V^^Y=-i, and (a/3I)4^( V^)^(-v/^)2=-lX-l= + l• Again, let it be proposed to determine the product of V — a by the V —h which, from the rule, will be V +ah, and consequently will give the four values (Art. 192). + Vo^, . - Wab, + Wab. V -I, — Wab. V^T. To determine the true product, observe that But V^^Tx *V^i^{V^y= (v V~^J = V^T^ hence V — a .*V^^=iWab. V—1. We will apply the preceding calculus to the verification of the expression , considered as a root of the equation a;3_l = 0, that is, as the cube root of 1 (Art, 192). From the formula {a+by=a'' + ^a''b + Sab''+P, we have (_i)3+3(-i)^ V-3+3(-i).( V^y+i V-ny 8 — I+SV^— 3x— 3 — 3 V^-3 = 1. i_ V-s The second value, manner. may be verified in the same Theory of Exponents. 195. In extracting the n"' root of a quantity a", we have seen that when m is a multiple of n, we should divide the exponent m by THEORY OF exponents; . 229 11 the index uf the ruuL j but whon tti Is not divisible by n, in which case the root cannot be extracted algebraically, it has been agreed to indicate this operation by indicating the division of the two ex. ponents. Hence, "V a"'=a~, from a convention founded upon the rule for the exponents, in tlie extraction of the roots of monomials. In such expressions, the numerator indicates the power to which the quantity is to he raised, and the denominator, the root to be extracted. 2 7 Therefore, ya^'z^a^ ; V«'— «*• In like manner, suppose it is required to divide a"" by a". We know that the exponent of the divisor should be subtracted from the a'" exponent of the dividend, when m>w, which gives —z=a"' ". But when ?ra^c2=6a"'^*5«c^ Division. 197. To divide one monomial by another when both are affected with any exponent whatever, follow the rule given in Art. 50 for quantities affected with entire and positive exponents ; that is, sub. tract the exponents of the letters in the divisor from the exponents oj the same letters in the dividend. For, the exponent of each letter in the quotient must be such, that added to that of the same letter in the divisor, the sum shall be equal to the exponent of the letter the dividend ; hence the ex- ponent in the quotient is equal to the difference between the expo- nent in the dividend and that in the divisor 232 ALGEBRA. EXAMPLES. 3 i. 3_4 __1_ /7*_l./75— «* 5—^ 20 . 2 3 _1 7 _9_ _ Formation of Powers. 198. To form the n"' power of a monomial, affected with any- exponent whatever, observe the rule given in Art. 182, viz. multi. ply the exponent of each letter by the exponent m of the power ; for, to raise a quantity to the m"' power, is the same thing as to multi- ply it by itself m — 1 times; therefore, by the rule for multiplica- tion, the exponent of each letter must be added to itself m—1 times, or multiplied by ?n< ^ / sXs 15/2X3 6 Thus, \a*} =a* ; {a"") =a^ = a^ ; (-XsXe -0-9 /_s\i2 _10 2a ^b*) =64a '^^ . [^^ <^) =a . Extraction of Roots. 189. To extract the n"" root of a monomial, follow the rule given in Art. 182, viz. divide the exponent of each letter by the index of the root. For, the exponent of each letter in the result should be such, that multiplied by n, the index of the root to be extracted, there will be produced the exponent with which the letter is affected in the proposed monomial ; therefore, the exponents in the result must be respectively equal to the quotients arising from the division of the exponents in the proposed monomial, by n, the index of the root. Thus, V. 3 _a 1-3 a'b =a'b ^ THEORY OF EXPONENTS. 233 The last three rules have been easily deduced from the rule for multiplication ; but we might give a direct demonstration for them, by going back to the origin of quantities affected with fractional and negative exponents. We will terminate this subject by an operation which contains implicitly the demonstration of the two preceding rules. Let it be required to raise a" to the — — power; We say ihat, For, by going back to the origin of these notations, we find that The advantage derived from the use of exponents consists prin- cipally in this : The operations performed upon expressions of this kind require no other rules than those established for the calculus of quantities affected with entire exponents. Besides, this calculus is reduced to simple operations upon fractions, with which we are already familiar. 200. Remark. In the resolution of certain questions, we shall be led to consider quantities affected with incommensurable expo, nents. Now, it would seem that the rules just established for com- mensurable exponents, ought to be demonstrated for the case in which the exponents are incommensurable ; but we will observe, that an incommensurable, such as V 3 , Vll, is by its nature com- posed of an entire part, and a fraction which cannot be expressed exactly, but to which it is possible to approximate as near as we please, so that we may always conceive the incommensurable to be replaced by an exact fraction, which only differs from it by a quan- 20* 234 ALGEBRA. tity less than any given quantity ; and in applying the rules to the symbol which designates the incommensurable, it is necessary to un- derstand that we apply it to the exact fraction which represents it approximatively. EXAMPLES. iveduce — — to its simplest terms. 2 V * \ 2V2(3)2 J Ans. 4 V 3 . Reduce ^ ^^ ' y to its sii-nplest terms :V2(3)' 1 Am 384 ^A ar+ V3i \ ^ ( 2v/2.(f)^ ) Reduce \/ .< ^-' — V to its simplest terms. Ans. \/y(-^VT+V'2l). Demonsti'ation of the Binomial Theorem in the case of any Exponent whatever. 201. Since the rules for the calculus of entire and positive expo- nents may be extended to the case of any exponent whatever, it is natural to suppose that the binomial formula, which serves to deve- lop the m"' power of a binomial when m is entire and positive, will also effect this when m is any exponent whatever. In fact, analysts have discovered that this is the case, and they have deduced im- portant consequences from it, both for the extraction of roots by dp- j)roximation, and the development of algebraic expressions into series. The following is a modification of Euler's demonstration. We will remark, in the first place, that the bmomial x+a can be put under the form x(l-\ — ) ; whence there results BINOMIAL THEOREM. 235 {x+ay"=x-^(l+-^^ =a;'"(l+2)% by making —=z. Therefore, if the formula m — 1 „ 711— I m — 2 {l+2)"'=l+m2+m — - — s?-\-m. — ~ — . — g— s^+,&c. (A) is proved to be correct for any value of ?/?, we may consider the formula. 711—1 (x-\-a)'"=x'"+7nax^~^+vi . — — — a^x'^~^ 7)1 — 1 7)1 — 2 +m . — -— . — ^—a'x^-'+, &c. (B) exact for any value of m. For, by substituting — for z in the formula (A)^ and multiplying by a;", we obtain (x+ay=x'"{l+m—+7)i .—J— .—+, &c.], from which, by performing the operations mdicated, we obtain the formula (B). Now, when m is a whole number, we have m— 1 „ 7)1—1 7)1—2 (l+zf^l+mz + m. Z^ + 7)l. — -—. — ^—2' + , &c. P But, if m is a fraction — , we do not know from what algebraic expression the development 7)1-1 7)1—1 7)1 — 2 l+7)iz+7)i — — — z-+7)i. — - — . — - — 2^.^, &c. ... is derived. Denoting this unknown expression by y, we have the equation 7)1-1 „ TO— 1 7)1 — 2 y==l+niz+7)i.—^—z-+))i.—^—.—^—r^+, &c (1). and it is now required to prove that y=(l+z)"'. If m' is another fractional exponent, we shall have in like manner, m' — 1 „ 7)i' — l 7)i'—2 3/'=l+m'2+OT' . -^— z^+m' .-—— . — — — r^+,&c. . . (2). 236 ALGEBRA. Multiplying the equations (1) and (2), member by member, we shall have for the first member of the result yy'. As to the second, it would be very difficult to obtain its true form, by the common rule for the multiplication of polynomials ; but by observing that the form of a product does not depend upon the particular values of the letters which enter into its two factors (Art. 47), we see that the above product will be of the same form as in the case where m and m' are positive whole numbers. Now in this case we have \^mz+m .— ^— z2+ . . . ={l+zY, Ij^rn'z+m' . ^— z'+ • • • ={l+zY', m—\ m! — \ whence l+mz+7n. — - — 2^+ . . . j{\+m'z+m' . — - — z'+ j m+m' — \ = {l+zY+^'=lJf.{m+m')z + {ra+m') z^+ ; Therefore this form is true in the case in which m and m' are any quantities whatever, and we have m+m' — 1 yy'=l + {m+m')z+{m-^m') ^ . 5r+ (3); Let m" be a third positive fractional exponent, we shall have m" — \ y' =l+ni"z+m" ^— 5^^+ • • • Multiplying the two last equations member by member, we have , „ , m-\-7n' +m" —1 yyy =l-^{m+7n' +m")z-\-{m+m' +m") z-+ V Suppose the fractional exponent m= — . Take as many exponents m, m', m", m'", &c. as there are units in q ; we shall have, by mak- ing r equal to the sum of the exponents wi+m'+?«"+OT"'+ . • • yy 3/ y"=i +r2+/- . -^^H^ • —^ • ~3~^+ • • • (4)- BIN03IIAL THEOREMS. 237 And by supposing m—m'=m"—7n"' ... in which case r=m+m+?n+??i+ . . . =^m and y=l—pxr; whence, by addition, x-\-y=a-{-l. which demonstrates the proposition. This being the case, write the progression below itself, but in an inverse order, viz. a.h .c.d.e .f . . . . i.k.l. I .k .i c.b .a. Calling S the sum of the terms of the first progression, 2S will be the sum of the terms in both progressions, and we shall have 2S={a + l) + {b+k) + {c+i) . . . +{i+c) + (k + b) + (l+a); or, since the number of the parts a-\~l, b-\-k, c-{-i is equal to n, 2S=ia+l)n, or sJ-^^-^^ That is, the sum of a progression by differences, is equal to half the sum of the two extremes, multiplied by the number of terms. If in this formula we substitute for I its value, a + (n— l)r, we obtain [2a + (n-l)r]n . S= , but the first expression is the most useful. {a+l)n 218. The formulas Z=a4-(n— l)r, S= — - — , contain five 252 ALGEBRA. quantities, a, r, n, I and S, and consequently give rise to the follow- ing general problem, viz. : Any three of these jive quantities being given, to determine the other two. There will, therefore, be as many different cases as there can be formed combinations of five letters taken three in a set : that is, 5_1 5-2 5.— —.-^-=10. (Art. 163). 2 o Of these cases we shall consider only the most important. We already know the value of S in terms of a, n and r. From the formula l=a+{n—l)r, we find a=l—{n—l)r. That is, the first term of an increasing arithmetical progression is equal to the last term, minus the product of the common difference by the number of terms less one. From the same formula, we also find, l-a That is, in any arithmetical progression, the common difference is equal to the difference between the two extremes divided by the number of terms less one. 219. The last principle affords a solution to the following ques- tion. To find a number m of arithmetical means bettveen tioo given mun- hers a and b. To resolve this question, it is first necessary to find the common difference. Now we may regard a as the first term of an arith. metical progression, b as the last tei-m, and the required means as intermediate terms. The number of terms of this progression will be expressed by ??i+2. Now, by substituting in the above formula, b for /, and wi+2 h—a b—a . for n, it becomes r= — r> or ^= TT" 5 that is, the com. wt+2 — 1 wi+1 mon difference of the required progression is obtained by dividing PROGRESSIONS BY DIFFERENCES. 253 the difference between the given numbers a and b, by one more than the required number of means. Having obtained the common difference, form the second term of the progression, or the Jirst arithmetical mean, by adding r, or :r-> to the first term a. The second mean is obtained by aug. m + 1 menting the first by r, &c. For example, let it be required to find 12 arithmetical means be- 77-12 65 tween 12 and 77. We have r= — — — =—=5, which gives the lo xO progression 12 . 17 . 22 . 27 ... 72 . 77. 220. Remark. If the same number of arithmetical means are inserted" between all of the terms, taken two and two, these terms, and the arithmetical means united, will form but one and the same progression. For, let a . b . c . d . e ./ . . . . he the proposed progression, and m the number of means to be inserted between a and b, b and c, c and d . . . From what has -just -been said, the common difference of each b—a c—b d—c partial progression will be expressed by — , -, . . . ., *^ ^ ° r ^ m + 1' m+1 7n + l which are equal to each other, since a, b, c . . . are in progression : therefore, the common difference is the same in each of the partial progressions ; and since the last term of the first, forms the^r*^ term of the second, 1 when the progression is increasing, and $'<1 when it is decreasing : we deduce from the definition, the following equalities. b=aq, c=bq=aq^, d=cq—aq^, e=dq=aq* . , . and in general, any term 7i, that is, one which has n — 1 terms be- fore it, is expressed by aq"''^. Let I be this term ; we have the formula l=aq''~^, by means of which we can obtain the value of any term without being obliged to find the values of all those which precede it. That is, the last term of a geometrical progression is equal to the first term mvitiplied by the ratio raised to a "power whose exponent is one less than the number of terms. For example, the 8* term of the progression 2": 6^ 18 : 5C . ., is equal to 2x3^=2x2187=4374. In like manner, the 12* term of the progression . ^ . . ., 64 : 16 : 4 : 1 : — ... is equal to 64(-i)"=!l=l=_^. V4/ 4" 4» 65536 223. We will now proceed to determine the sum of n terms of the progression a : b : c : d : e :f: . . ..:i:k:ly I denoting the n"" term. We have the equations (Art. 222), b=aq, c=hq, d=cq, e=dq, . . . k=iq, l=kq; and by adding them all together, member to member, we deduce b+c+d+e+ . . . +k + I={a + h + c+d+ . . . +i-{-k)q; or, representing the required sum by S, S.-ra = {8-l)q=Sq-]q, or Sq-S=lq-a', Iq — a whence S = —] q-l That is, to obtain the sum of a certain number of terms of a pro- gression by quotients, inultijyly the last term by the ratio, subtract the first term from this product, and divide the remainder by the ratio di. vtinisJied by unity. When the progression is decreasing, we have 5<1 and Z9 . Finally, when n is taken greater than any given number, a or n = 00, then - — ^X which is nothing more than the proposed series, having b, c, d . . . replaced by their values in functions of a. PROGRESSIONS B-Y QUOTIENTS. f^V a 227. When the series is increasing, the expression cannot be considered as Or limit of the partial sums ; because, the sum of a a aq" given number of terms being S=t— — — j-^, (Art. 225), the second part augments numerically, in proportion to the in- crease of n ; hence the greater the number of terms teken, the a more the expression of their sum will differ numerically from ■— — . a The formula S=:j is, in this case, merely the algebraic ex- pression which, by its development, gives the series- a+aq+aq'^+aq^ . . . There is another circumstance presents itself here, which appears very singular at first sight. Since is the fraction which generates the above series, we should have j—=a + aq+aq''-{-aq^+aq^-{- . . . Now, by making a=l, q — 2, this equality becomes y— g. or -l^l+2.+ 4+8 + 16+32+ . . . an equation of which the first member is negative whilst the second is positive, and greater in proportion to the value of q. To interpret this result, we will observe that, when in the equa- tion =:a+aq-{-aq''-\-aq^-{- . . ., we stop at a certain term of the series, it is necessary to complete the quotient in order that the equality may subsist. Thus, in stopping, for example, at the fourth term, aq^. ALGEBRA. a + aq + 0^^ 1-q Ist remainder 2d. aq* a+aq+aq^+aq^+j— 3d. + aq' 4th. + aq' aq^ It is necessary to add the fractional expression to the quo- lient, which gives rigorously, a aq* ——=a + aq+aq^+aq=+YZ:^' If in this exact equation we make a=l, q=2, it becomes 16 -1 = 1+2 + 4+8+— Y=l+2 + 4 + 8-16, which verifies itself. In general, when an expression involving x, designated by /(x), which is called a function of x, is developed into a series of the foiTn a-]-ix-\-ca^-\-dx^-{- , we have not rigorously f(x)=a-\-bx-{-cx^-{-dx^-}- . . ., unless we conceive that, in stopping at a certain term in the second member, the series is completed by a certain expression involving x. When, in particular applications, the series is decrea,s«?^ (Art. 203 j, the expression which serves to complete it may be obtamed as near as we please, by prolongmg the series ; but the contrary is the case when the series is increasing, for then it must not be neglected. This is thereason why increasmg series cannot be used for approxi- mating to the value of numbers. It is for this reason, also, that algebraists have called those series which go on diminishing from term to term, converging series, and those in which the terms go on increasing, diverging series. In the first, the greater the number of terms taken, the nearer the sum approximates numerically to the expression of which this series is the development ; whilst in the others, the more terms we take, the more their sum differs from the numerical value of this expression. FROORESSIONS BY QUOTIENTS. 263 228. The consideration of the five quantities a, q, n, I and S, Iq — a which enter in the two formulas l=aq''-^, S= — (Arts. 222 & 223), gives rise to ten problems, as in the progression by differences (Art. 218). Of these cases, we shall consider here, as we did there, only the most important. We will first find the values of S and q in terms of a, I and n. I "-' / Z The first formula gives q"^^= — , whence q= \/ — . Substi- tuting this value in the second formula, the value of S will be ob- tained. "^^ / I The expression q= \/ — furnishes the means for resolving the following question, viz. To find m mean proportionals between two given numbers a and b ; that is, to find a number m of means, which will form with a and b, considered as extremes, a progression by quotients. For this purpose, it is only necessary to know the ratio ; now the required number of means being m, the total number of terms is equal to m + 2. Moreover, we have Z=&, therefore the value of q "+' /Z» becomes 9= \/ — ; that is, we must divide one of the given numbers (b) by the other (a), then extract that root of the quotient wJwse index is one more than the required number of means. Hence, the progression is "'+1 / ]) "•+! /Jr ">+! /P a -.a \/---a V ^ : « s/ ^ : . . . b. Thus, to insert six mean proportionals between the numbers 3 ' / 384 7 / and 384, we make ?tt=6, whence q=K/ — — — = y 128 = 2 ; whence we deduce the progression 3 : 6 : 12 : 24 : 48 : 90 : 192 : 384. Remark. When the same number of mean proportionals are in- 264 ALOEBR^. serted between all the terms of a progression by quotients, taken two and two, all the progressions thus formed will constitute a single pro- gression. 229. Of the ten principal problems that may be proposed in progressions, /our are susceptible of being easily resolved. The following are the enunciations, with the formulas relating to them. 1st. a, q, n, being given, to find I and S. l^a "-»• S- ^^""^ ^ "^^"^-^^ 2d. a, n, I, being given, to find q and S. 3d. q, n, I, being given, to find a and -S*. 4th. q, n, S, being given to find a and I. S{q-1) Sq'^-\q-l) , I- r-i ' 9"-i Two other problems depend upon the resolution of equations of a degree superior to the second ; they are those in which the un- known quantities are supposed to be a and q, or I and q. For, from the second formula we deduce a—lq — Sq+S; Whence, by substituting this value of a in the first l=aq"'^, l^{lq-Sq + S)q'-\ or, (S-l)q-'-Sq''~'+l=^0. an equation of the 7i"' degree. In like manner, in determining I and q,we should obtain the equa- tion aq"—Sq + S — a=0. 230. Finally, the other four problems lead to the resolution of CONTINUED FRACTIONS 265 equations of a peculiar nature ; they are those in which n and one of the other four quantities are unknown. From the second formula it is easy to obtain the value of one of the quantities a, q, I, and S, in functions of the other three ; hence the problem is reduced to finding n by means of the formula Iq Now this equality becomes 5-"= — , an equation of the form a'^=^b, a and b being known quantities. Equations of this kind are called exponential equations, to distinguish them from those previous, ly considered, in which the unknown quantity is raised to a power denoted by a known number. Before, however, we can solve the exponential equation a*=i, we must understand the elementary properties of Continued Frac- tions, which are now to be explained. Of Continued Fractions. 65 231. Having given a fraction of the form , in which the terms are large, and prime with respect to each other, we are una- ble to discover its precise value, either by inspection or by any mode of reduction yet explained. The manner of approximating to the value of such a fraction gives rise to a series of numbers, which taken together, form what is called a contimied fraction. 65 232. If we take, for example, the fraction , and divide both its terms by the numerator 65, the value of the fraction will not be changed, and we shall have 65 _ 1 149 ~ 149' 65 65 1 or effecting the division, T4q"~"2~Xiq 23 19 Now, if we neglect the fractional part — of the denominator, bo we shall obtain — for the approximate value of the given fraction. But this value would be too large, since the denominator used was too small. 19 If, on the contrary, mstead of neglecting the part — , we were bo -to replace it by 1, the approximate value would be — , which must o be too small, since the denominator 3 is too large. Hence 65 1 65 1 U9-<-2- ""^ lA9>r therefore the value of the fraction is comprised between — and — . If we wish a nearer approximation, it is only necessary to ope- 19 J. , . . . 65 rr as we did on the given fraction -—tt: 65 ° 149 we obtain 19 1 65 3 + 8 19' hence 65 1 149 2 + 1 3+8 19' 8 If now, we neglect the part — , the denominator 3 will be less than the true denominator, and — will be larger than the number 3 which ought to be added to 2 ; hence, 1 divided by 2+— will be o less than the value of the fraction : that is, if we reject the frac CONTINUED FRACTIONS. 267 tional part after the second reduction, we shall have 65 3 149 "^T* If we wish to approximate still nearer to the value of the given fraction, we find 8 1 19 2+£^ T» and by substituting this value, we have 65 _ 1 149 ~ 2+1 3 + 1 2+3 T 3 Now, if we neglect the fractional part — , after the third reduc- o tion, we see that 2 will be less than the real denominator ; hence — will be larger than the number to be added to 3 : that is, 1 7 3+—=— is too large ; hence 1 2 is too small, and T 2 16 2+y=y is too small ; therefore 1 7 16~16 is too large, and hence T 65 7 149 ^16" Now, as the same train of reasoning may be pursued for the re- ductions which follow, and as all the results are independent of par- 268 ALGEBRA. ticular numbers, it follows that, if we stop at an odd reduction, and neglect the fractional part, the result will he too great ; but if we stop at an even reduction, and neglect the fractional part, the result xcill be too small. Making two more reductions, in the last example, we have, 65 _ 1 149 ""2 +1 3 + 1 2 + 1 2 + 1 1+2. 2. 233. Let us take, as a general case, the continued fraction 1 h+i c + l f d+l /+r&c. Hence we see, that a continued fraction has for its mimerator the unit 1, and for its denominator a whole number, plus a fraction which has I for its numerator and for its denominator a ivhole number plus a fraction, and so on. 234. The fractions 1 1 1 a' a + 1 a + 1 T' b+1 c, &c. are called approximating fractions, because each affords, in succes- sion, a nearer value of the given fraction. 1 1 1 The fractions — , -7-, — , &c. are called integral fractions, a c <=. . When the continued fraction can be exactly expressed by a vulgar CONTINUED FRACTIONS. 269 fraction, as in the numerical examples already given, the integral frac- 1 1 1 tions — , — , — , &c. will ternninate, and we shall obtain an expres- sion for the exact value of the given fraction by taking them all. 235. We will now explain the manner in which any approximat- ing fraction may be found from those which precede it. 1 1 a 1 1 a 1st. app. fraction. h 2d. app. fraction ic+1 (nhA.'WrJ-, - 3d. app. fraction. b + l_ c By examining the third approximating fraction, we see, that its numerator is formed by multiplying the numerator of the preceding fraction by the denominator of the third integral fraction, and add- ing to the product the numerator of the first approximating frac- tion : and that the denominator is formed by multiplying the deno- minator of the last fraction by the denominator of the third integral fraction, and adding to the product the denominator of the first ap- proximating fraction. We should infer, from analogy, that this law of formation is ge- P Q R neral. But to prove it rigorously, let — , — , — , be the three 1^ ii K approximating fractions for which the law is already established. Since c is the denominator of the last integral fraction, we have from what has already been proved R Qc+P R'~ Q'c+P' ' 1 action • 23* Let us now add a new integral fraction — to those already re- 370 ALGEBRA. o duced, and suppose — to express the 4th approximating fraction. S' -,., ivill . . ,, R' ^' R S It is plain that jp will becoi:'r -7 by simply substituting for c. c^ — —: hence, a Q{c+^)+P S _ ^V^ d/^^ _{Q c+P)d+Q _ Rd-\-Q ^'~Q'(c+-)+P~ (^'^+^')rf+Q' ~ K'd+Q' Hence we see that the fourth approximating fraction is deduced from the two immediately preceding it, in the same way that the third was deduced from the first and second ; and as any fraction may be deduced from the two immediately preceded in a similar manner, we conclude, that, the numerator of the n"" approximating fraction is formed by multiplying the numerator of the preceding frac- tion by the denominator of the n"" integral fraction, and adding to the product the numerator of the n — 2 fraction ; and the denominator is formed according to the same law, from the two preceding denomina- tors. 236. If we take the difference between any two of the consecu. tive approximating fractions, we shall find, after reducing them to a common denominator, that the difference of their numerators will be equal to ±1 ; and the denominator of this difference will be the product of the denominators of the fractions. 1 h Taking, for example, the consecutive fractions — , and — 5 , we have, 1 h ab-\-\—db +1 And a ab-\-\ a{ab+l) a(a*+l)' h hc+\ -1 06 + 1 (a*+l)c-fa (ai-|-l)((ai-f l)c+a)' CONTINUED FRACTIONS. 271 m P Q Ji To prove this property in a general manner, let pT' Tv' "p^' ^^ three consecutive approximating fractions. Then P Q PQ'-P'Q But P' Q' P'Q' Q R R'Q-RQ' Q' R' Q'R' But R=Qc-irP and R'=Q'c+P' (Art. 235). Substituting these values in the last equation, we have Q R' R'Q or reducing Q 22 P'Q-PQ! p Q P' Q' Q R' Q' R' Q' R' R'Q' From which we see that the numerator of the difference is equal, with a contrary sign, to that of the difference That is, the difference hettoeen the nmnerators of any tivo consecutive approximating fractions, when reduced to a common denominator, is the same with a contrary sign, as that which exists between the last numerator and the numerator of the fraction immediately following. But we have already seen that the difference of the numerators of the 1st and 2d fractions is equal to +1 ; also that the difference between the numerators of the 2d and 3d fractions is equal to — 1 ; hence the difference between the numerators of the 3d and 4th is equal to +1 ; and so on for the following fractions. Since the odd approximating fractions are all greater than the true value of the continued fraction, and the even ones all less (Art. 232), it follows, that when a fraction of an even order is subtracted from one of an odd order, the difference should have a plus sign ; and on the contrary, it ought to have a minus sign, when one of an odd order is subtracted from one of an even. 272 ALGEBRA. 237. It has already been shown (Art. 232), that each of the ap^ proximating fractions corresponding to the odd numbers, exceeds the true value of the continued fraction ; while each of those corres- ponding to the even numbers is less than it. Hence, the difference between any two consecutive fractions is greater than the difference between either of them and the true value of the continued frac- tion. Therefore, stopping at the n"' fraction, the result will be true to within 1 divided by the denominator of the n"" fraction, multipli- ed by the denominator of the fraction which follows. Thus, if Q and R are the denominators of consecutive fractions, and we stop at the fraction whose denominator is Q', the result will be true to within ^QTgT* But since a, b, c, d, &:c. are entire numbers, the de- nominator R' will be greater than Q', and we shall have 1 1 hence, if the result be true to within yp^ it will certainly be true to within less than the larger quantity 1 that is, the approximate result which is ohtained, is true to within unity divided by the square of the denoniinator of the last approxi- mating fraction that is employed. 829 If we take the fraction ^,„ we have 347 829 _ 1 "347""^"*" 2 + 1 T+i 1+2 3+J_ 19* Here we have in the quotient the whole number 2, which may EXPONENTIAL QUANTITIES. 273 either be set aside and added to the fractional part after its value shall have been found, or we may place 1 under it for a denomina- tor and treat it as an approximating fraction. Of Exponential Quantities. Resolution of the Equation a'^=b 238. The object of this question is, to find the exponent of the power to which it is necessary to raise a given number a, in order to produce another given number b. Suppose it is required to resolve the equation 2^=64. By rais- ing 2 to its different powers, we find that 2^=64 ; hence x=6 will satisfy the conditions of the equation. Again, let there be the equation 3-^=: 243. The solution is x=5. In fact, so long as the second member Z* is a perfect power of the given number a, x will be an entire number which may be obtained by raising a to its successive powers, commencing at the first. Suppose it were required to resolve the equation 2^=6. By making x—'2, and .'c=3, we find 2^=4 and 2^=8 : from which we perceive that x has a value comprised between 2 and 3. Suppose then, that a;=2-| — y, in which case x'>l. Substituting this value in the proposed equation, it becomes, _i_ _i_ J. 3 2^"*'^'=6 or 2^x2 ^'=6; hence 2^'=—, or (— ) =2, by changing the members, and raising both to the x' power. To determine a;', make successively x'=^l and 2; we find / 3 \' 3 /3 v^ 9 I — j =— less than 2, and (— j =— ,. which is greater than 2 ; therefore x' is comprised between 1 and 2. 1 Suppose x'=l-\-j;, inwhicha:>l. 274 ALGEBRA. By substituting this value in the equation (—1 =2 /3\,4.-L 3 /3\-l / 4 \ --" 3 ^-g-j =-2 by reducing. The two hypotheses x"=l and x"=2, give — which is less than o 3 /4\^ 16 7 3 — , and ( — 1 =— =1+— which is greater than — ; therefore x" is comprised between 1 and 2. Let x"=l-\ ;;— , there will result X /4\,+-L 3 4 /4\-i- 3 /9\^"' 4 , . whence I— j =— by reducmg. Making successively x"'=l, 2, 3, we find for the two last hypo. /9\2 81 17 1 , theses ^—j =—=1+—, which is 1+— : therefore x'" is com- 8 / 512 512 o prised between 2 and 3. Let x"'=2-\-—, the equation involving x'" becomes 2+- / y \ :r'V 4 81 / 9 \ xiv 4 Kq) ^IT'^^eiU) ="3 / 256 \ x'v 9 and consequently ( g.o ) ~~q' Operating upon this exponential equation in the same manner as upon the preceding equations, we shall find two entire num. EXPONENTIAL QUANTITIES. 276 bera k and &+1, between which a;'' will be comprised. Making 0^=44 — -, xv can be determined in the same manner as x^^, and x^ so on. Making the necessary substitutions in the equations x=2 + i a^'=l+^, a."=l+4r' ^'"=2-f ^ we obtain the value of x under the ibrm of a continued fraction 1 x=2+ J- Hence we find the first three approximating fractions to be. L L 1. 1 ' 2 ' 5"' and the fourth is equal to 3x24-1 7 ^, -5^m:2-=12 (^^^- 235), which is the value of the fractional part to within (12f ""' "lii" (^'^- 2^^)- Therefore ^=2+72=^7^ to within -rrr, and if a greater de- gree of exactness is required, we must take a greater number of integral fractions. EXAMPLES. 3» = 15 X = 2,46 to within 0,01. 10* = 3 X = 0,477 0,001. 2 5' = — x=- 0,25 0,01. 276 ALGEBRA. Theory of Logarithms. 239. If we suppose a to preserve the same value in the equation and y to be replaced by all possible positive numbers, it is plain that X will undergo changes corresponding to those made in y. Now, by the method explained in the last Article, we can determine for each value of y, the corresponding value of x, either exactly or ap- proximatively. First suppose ay.t Making in succession x =0, 1, 2, 3, 4, 5 , . . . &c. there will result y=a'=l, a, a', a', a*, a*, . . . &c. hence, every value of y greater than unity, is produced by the pow' ers of a, the exponents of which are positive nuvibers, entire or frac- tional ; and the values of y increase with x. Make now x =0, —1, —2, —3, —4, —5, . . . &;c. 11111 there will result y=a'=l, — , — ;, — ^, —,, — r, . . . &c. a a' a^ a* a* hence, every value of y less than unity, is produced by the powers of a, of which the exponents are negative ; and the value of y dimin' ishes as the value of x increases negatively. Suppose al. Hence, every possible positive number can be formed with any con- stant positive number whatever, by raising it to suitable powers. Remark. The number a must always be different from unity, because all the powers of 1 are equal to 1. 240. By conceiving that a table has been formed, containing in one column, every entire number, and in another, the exponents of the powers to which it is necessary to raise an invariable number, to form all these numbers, an idea will be had of a table of logarithms. Hence, The logarithm of a number, is the exponent of the potver to which it is necessary to raise a certain invariable number, in order to pro- duce the first number. Any number, except 1, may be taken for the invariable number ; but when once chosen, it must remain the same for the formation of all numbers, and it is called the base of the system of logarithms. Whatever the base of the system may be, its logarithm is unity, and the logarithm of 1 is 0. For, let a be the base : then 1st, we have a^=a, whence log a=l. ' 2d, «"=!, whence log 1=0. The word logarithm is commonly denoted by the first three letters log, or simply by the first letter Z. We will now show some of the advantages of tables of logarithms in making numerical calculations. Multiplication and Division. 241. Let a be the base of a system of logarithms, and suppose the table to be calculated. Let it be required to multiply together a series of numbers by means of their logarithms. Denote the numw bers by y, y', y", y'" . . . &;c., and their corresponding logarithms 24 278 ALGEBRA. by X, sf, x", x'", &c. Then by definition (Art. 240), we have a'=y, a"=y', a"'=3/", a""=y[" . . . &c. Multiplying these equations together, member by member, and applying the rule for the exponents, we have ^r+x'+^//+x'// , , , =y y'y'y'" or x-\-x' +x" -\-x"' . . . =log?/+ log 3/'+ logi/"+ log 3/'" . . . = log. yy'y"y"', that is, the sum of the logarithms of any number of factors is equal to the logarithm of the product of those factors. 242. Suppose it were required to divide one number by another. Let y and y' denote the numbers, and x and x' their logarithms. We have the equations a"'— 2/ and a'''=y' ; y hence by division a'-^'=-— , y y or x—x'=:i log y— log y'= log -;-, that is, the difference between the logarithm of the dividend and the logarithm of the divisor, is equal to the logarithm of the quotient. Consequences of these properties. A multiplication can be per- formed by taking the logarithms of the two factors from the tables, and adding them together ; this will give the logarithm of the pro- duct. Then finding this new logarithm in the tables, and taking the number which cortesponds to it, we shall obtain the required pro- duct. Therefore, by a simple addition, we find the result of a 7nid- tiplication. In like manner, when one number is to be divided by another, subtract the logarithm of the divisor from that of the dividend, then find the number corresponding to this difterence ; this will be the required quotient. Therefore, by a simple subtraction, we obtain the quotient of a division. THEORY OF LOGARITHMS. 279 Formation of Powers and Extraction of Roots. 243. Let it be required to raise a number y to any power de- Ttl noted by — . If a denotes the base of the system, and x the loga- rithm of y, we shall have a'=y or y=a', m whence, raising both members to the power — , y'^=a^\ — m m Therefore, logy"— — . x= — .log?/, that is, if the logarithm of any number he multiplied by the exponent, of the power to which the number is to be raised, the product will be equal to the logarithm of that power. As a particular case, take n=l ; there will result »z. log y=: log 3/"' ; an equation which is susceptible of the above enunciation. 244. Suppose, in the first equation, m=l ; there will result 1 log y= log y"z= log V that is, the logarithm of any root of a number is obtained by divid- ing the logarithm of the number by the index of the root. Consequence. To form any power of a number, take the loga- rithm of this number from the tables, multiply it by the exponent of the power ; then the number corresponding to this product will be the required power. In like manner, to extract the root of a number, divide the loga- rithm of the proposed number by the index of the root, then the number corresponding to the quotient will be the required root. Therefore, by a simple multiplication, we can raise a quantity to a power, and extract its root by a simple division. 280 ALGEBRA. 245. The properties just demonstrated are independent of any system of logarithms ; but the consequences which have been de- duced from them, that is, the use that may be made of them in nu- merical calculations, supposes the construction of a table, contain- ing all the numbers in one column, and the logarithms of these num. bers in another, calculated from a given base. Now, in calculating this table, it is necessary, in considering the equation a'=y, to make y pass through all possible states of magnitude, and determine the value of a; corresponding to each of the values of ?/, by the method of Art. 238. The tables in common use, are those of which the base is 10 and their construction is reduced to the resolution of the equation 10*=?/. Making in this equation, y successively equal to the series of natural numbers, 1, 2, 3, 4, 5, 6, 7 . . ., we have to resolve the equations 10^=1, 10^=2, 10^=3, 10^=4 . . . We will moreover observe, that it is only necessary to calculate directly, by the method of Art. 238, the logarithms of the pi'ime numbers 1, 2, 3, 5, 7, 11, 13, 17 ... ; for as all the other entire numbers result from the multiplication of these factors, their loga- rithms may be obtained by the addition of the logarithms of the prime numbers (Art. 241). Thus, since 6 can be decomposed into 2x3, we have log 6= log 2+ log 3 ; in like manner, 24=2='x3 ; hence log 24=3 log 2+ log 3. Again, 360=23 xS^'X 5 ; hence log 360=3 log 2+2 log 3+ log 5. It is only necessary to place the logarithms of the entire num. bers in the tables ; for, by the property of division (Art. 242), we obtain the logarithm of a fraction by subtracting the logarithm of the divisor from that of the dividend. THEORY OF LOGARITHMS. 281 246. Resuming the equation 10''=y, if we make x=0, 1, 2, 3, 4, 5, .. . n-1, n. we have y=l, 10, 100, 1000, 10000, 100000, . . 10-', 10". And making x=0, -1, -2, -3, -4, -5, . . . -(n-1), — n. we have V 1 ^ 1 1 100 ' 1000' 1 1 1 10"-" 1 y-^^ 10' 10000' 100000' • ' 10" • From which we see that, the logarithm of a whole numher will 3e« come the logarithm of a corresponding decimal by changing its sign from plus to minus. 247. Resume the equation a''=y, in which we will first suppose a>l. Then, if we make y= 1 we shall have a''=l. If we make 2/+f=^) = -^H-.(f^)Vc.(f=i)V... Substituting this development for log (1+7/)— log(l+2) in the equation (3), and dividing both members by y—z, it becomes =A+B{y+z) + C{y'+yz-\-z')+ . . . Since this equation, like the preceding, must be verified by all 284 ALGEBRA. values of y and *, make y=s, and there will result ^-p^=^+2%+3C/+4Z>y'+5£2/*+ . . . Whence, clearing the fraction, and transposing 0=^+25 I 2/+3C I f+AD I f+^E I i/*+ . . . Putting the co-efficients of the different powers of y equal to zero, we obtain the series of equations A-A=Q, 2B+A=0, 3C+25=0, 4Z>+3C=0 . . . ; whence A 25 A SC A B=-^, C=- 2 3 ~ 3' ~ 4 ~ 4 A=A, The law of the series is evident ; the co-efficient of the n"^ term A IS equal to qz — , according as n is even or odd ; hence we shall ob- tain for the development of log (\-\-y), A A . A 2^ log {\^y)=Ay-—f^—f-—t =^(^-Y+-3-T+-5----) (^)- If we substitute —y for y, we shall have log(l-2/)=A(-r/-^-^-?^+&c.) (5). Hence, although the logarithm of a number cannot be developed in the powers of that number, yet it may be developed in the powers of a number greater or less by unity. By the above method of development, the co-efficients B, C, D, E, &c. have all been determined in functions of A ; but the rela- tion between A and the base of the system is yet undetermined. The number A is called the modulus of the system of logarithms in which the log (1+y), or log (1— y), is taken. Hence, THEORY OF LOGARITHMS. '4St5^ The modulus of a system of logarithns depends for its value on the base, and if a certain function of any number be multiplied by it, the product will be the logarithm, of that number augmented by unity. 250. If we take the logarithm of 1 +^ in a new system, and de- note it by l'{\-^y), we shall have '•(i+J')='i'(i'-y+T-T+T- *=•) <«> in which A' is the modulus of the new system. If we suppose y to have the same value as in equation (4), we shall have \\\+y):\{\+y)::A' :A, for, since the series in the second members are the same they may be omitted. Therefore, The logarithms of the same number, taken in two different systems, are to each other as the moduli of those systems. 1f)\. If we make the modulus A'—\, the system of logarithms which results is called the Naperian System. This was the first system known, and was invented by Baron Napier, a Scotch Ma- thematician. With this modification the proportion above becomes V{l+y):\{l+y):'. 1 : A or A.V{l+y)=\{l+y). Hence we see that, the Naperian logarithm of any number, muL tiplied by the modulus of another system, will give for a product the logarithm of the same number in that system. 252, Again, A .\'{\-^ij)=\{\-\-y) gives That is, tlie logarithm of any number divided by the modulus of the system, is equal to the Naperian logarithm of the same number. 253. If we take the Naperian logarithm and make y=l, equa- tion (6) becomes 1111 a series which does not converge rapidly, and in which it would be necessary to take a great number of terms for a near approxima- tion. In general, this series will not serve for determining the loga- rithms of entire numbers, since for every number greater than 2 we should obtain a series in which the terms would go on increasing continually. The following are the principal transformations for converting the above series into converging series, for the purpose of obtaining the logarithms of entire numbers, which are the only logarithms placed in the tables. First Transformation. Taking the Naperian logarithm in equation (6), making y= — , and observing that r(l4-— )=!'(! +^)—l'2^» it becomes l'(l+.)-l'z=i-^+^-^+ &c. (7). This series becomes more converging as z increases ; besides thfef first member of this equation expresses the difference between two consecutive logarithms. Making z=l, 2, 3, 4, 5, &c, we have 1111 1111 THEORY OF LOGARITHMS. 367 1111 M-r3=---+^j_— + . . . ,.6-l-4=l-i-+-! L.. 4 32^ 192 1024 The first series will give the logarithm of 2 ; the second series will give the logarithm of 3 by means of the logarithm of 2 ; the third, the logarithm of 4, in functions of the logarithm of 3 . . . &c. The degree of approximation can be estimated, since the series are composed of terms alternately positive and negative (Art. 203). Second Transformation. A much more converging series is obtained in the following man- ner. In the series x" aP X* r(i+.)=.--+---+... substitute — x for a; ; and it becomes ar" 3P X* !'(!-)= ~ 2 3 4 '•• Subtracting the second series from the first, observing that 1+x r(l4-x)— r(l— a;)=r- , we obtain This series will not converge very rapidly unless a; is a very small fraction, in which case, will be greater than unity, but will differ very little from it. l-\-x 1 Take =1H , 2 being an entire number : 1 — X z * we have (14-x)z=(l— a;)(j;+l): whence x-. 22+1 299 ALGfBBRA. Hence the preceding series becomes l'(l+' — ) or Vi,+i^-n=i{^+^- + g(^+ . . .) This series also gives the difference between two consecutive logarithms, but it converges much more rapidly than the series (7). Making successively 2=1, 2, 3, 4, 5 . . ., we find /I 1 1 1 \ r3--r2=2(i-+3i^3+^+y^+ • • •)' Let 2=100 ; there will result noi=noo+2(-^+^-+^+ . . .) ; whence we see, that knowing the logarithm of 100, the first term of the series is sufficient for obtaining that of 101 to seven places of decimals. The Naperian logarithm of 10 may be deduced from the third and fourth of the above equations, by simply adding the logarithm of 2 to that of 5 (Art. 241). This number has been calculated with great exactness, and is 2,302585093. There are formulas more converging than the above, which serve to obtain logarithms in functions of others already known, but the preceding are sufficient to give an idea of the facility with which tables may be constructed. We may now suppose the Naperian logarithms of all numbers to be known. THEORY OP LOGARITHMS. »»» 254. We have already observed that the base of the common system of logarithms is 10 (Art. 245). We will now find ita modulus. \'{l+y) : \{l+y) : : 1 : A (Art. 250). If we make y=9, we shall have no : no : : 1 : ^. But the 1'10 = 2,302585093 . . . and 110 = 1 (Art. 245); hence A= — ——=0,434294482 the modulus of the common sys- 2,302585093 ' '' tern. If now, we multiply the Naperian logarithms before found, by this modulus, we shall obtain a table of common logarithms (Art. 251). 255. All that now remains to be done is to find the base of the Napeiian system. If we designate that base by e, we shall have (Art. 250), I'e : le : : 1 : 0,434294482. But 1V = 1 (Art. 240): hence 1 : 1 e : : 1 : 0,434294482, or 16=0,434294482. But as we have already explained the method of calculating the common tables, we may use them to find the number whose loga- rithm is 0,434294482, which we shall find to be 2,718281828 : hence 6=2,718281828. We see from the last equation but one that, the modulas of the common system is equal to the common logarithm of the Naperian base. 25 290 CHAPTER VI. General Theory of Equations. 256. The most celebrated analysts have tried to resolve equa- tions of any degree whatever, but hitherto their efforts have been unsuccessful with respect to equations of a higher degree than the fourth. However, their investigations on this subject have conduct- ed them to some properties common to equations of every degree, which they have since used, either to resolve certain classes of equations, or to reduce the resolution of a given equation to that of one more simple. In this chapter it is proposed to make known these properties, and their use in facilitating the resolution of equa- tions. 257. The development of the properties relating to equations of every degree, leads to the consideration of polynomials of a parti- cular nature, and entirely different from those considered in the first chapter. These are, expressions of the form Ax"'-fBx''""'+ac'"-=+ . . . +Tt + U, in which m is a positive whole number ; but the co-efficients A, B, C, . . . T, U, denote any quantities whatever, that is, entire bv fractional quantities, commensurable or incommensurable. Now, in algebraic division, as explained in Chapter I, the object was this, viz. : £iven two polynomials entire, with reference to all the letters and particular numbers involved in them, tojind a third polynomial of the same kind, ichich, mulliplied by the second, would produce t'lc first. But when we have two polynomials, Ax'" + B.r"-'+Cx'"-='+ . . . +Tx + U, AV+B'o.-'-'+C'a;'' =+ . . . +T'a'+U , which are necessarily entire only with respect to x, and in which the co-efficicnts A, B, C . . ., A', R', C . . ., may be any quantities GKNKIIAL PliOPilUTIKS OF KiiLATIO.\S. 291 whatever, it may be proposed to find a third polynomial, of the samo tbrm and nature as the two preceding, W«cA mtdtiplied hythcsecondy will re-produce the first. The process for effecting this division is analogous to that for common division ; but there is this difference, viz. : In this last, the first term of each partial dividend must be exactly divisible by the first term of the divisor ; whereas, in the new kind of division, we divide the first term of each partial dividend, that is, the part affect- ed with the highest power of the principal letter, by the first term of the divisor, whether the co-efficient of the corresponding partial quotient is entire or fractional ; and the operation is continued until a quotient is obtained, which, multiplied by the divisor, will cancel the last partial dividend, in which case the division is said to be exact ; or, until a remainder is obtained, of a degree less than that of the divisor, with reference to the principal letter, in which case the di- vision is considered impossible, since by continuing the operation, quotients would be obtained containing the principal letter affected with negative exponents, or this same letter in the denominator of them, which would be contrary to the nature of the question, which requires that the quotient should be of the same form as the pro- posed polynomials. 258. To distinguish polynomials which are entire with reference to a letter, x for example, but the co-efficients of which are any quantities whatever, from ordinary polynomials, that is, from poly, nomials which are entire with reference to all the letters and parti, cular numbers involved in them, it has been agreed to call the first entire functions of x, and the second, rational and entire polynO' mials. General Properties of Equations. 259. Every complete equation of the ?«"' degree, m being a po- sitive whole number, may, by the transposition of terms, and by the division of both members by the co-efficient of af", be put under the form 292 ALGEBRA. 3r-\-?x^-'+Qx'^-^+ . . . +Ta;+U=0; P, Q, R . . . T, U, being co-efficients taken in the most general al- gebraic sense. Any expression, whatever the nature of it may be, that is, numeri- cal or algebraic, real or imaginary, which, substituted in place of x in the equation, renders Us first member equal to 0, is called a root of this equation. 260. As every equation may be considered as the algebraic trans- lation of the relations which exist between the given and unknown quantities of a problem, we are naturally led to this principle, viz. EVERY EauATiON hos at least one root. Indeed, the conditions of the enunciation may be incompatible, but then we must suppose that we shall be warned of it by some symbol of absurdity, such as a formula, containing as a necessary operation, the extraction of an even root of a negative quantity ; yet there will still exist an ex- pression which, substituted for x in the equation, will satisfy it. We will admit this principle, which we shall have occasion to verify here- after for most equations. The following proposition may be regarded as the fundamental property of the theory of equations. First Property. 261. If a is a root of the equation x'"+Pa;"-'+Qx"'--+ . . . Tx + U = 0, the first member of it is divisible by x—a ; and reciprocally, if a factor of the form x—a, will divide the first member of the proposed equation, a is a root of it. For, perform the division, and see what takes place when the ope- ration is continued until the exponent of x, in the first term of the dividend, becomes 0. GENERAL PROPERTIES OF EQUATIONS. 293 This operation is of thu nature of that spoken of in Art. 257, since a, P, Q, . . . are any quantities whatever. + p| +pl +Pg +Va^~' + Pa 4-T By reflecting a little upon the manner in which the partial quo- tients are obtained, we shall first discover from analogy, and after- wards by a method employed several times (Arts. 59 & 127), a law of formation for the co-efficients of these quotients ; and we may conclude, 1st. that there will be m partial quotients, 2d. that the co- efficient of the m"' quotient, that is of x°, must be a-> + Pa'"-=^+Qa'"-='+ . . . +T, T being the co-efficient of the last term but one of the proposed equation. Hence, by multiplying the divisor by this quotient, and reducing it with the dividend, we obtain for a remainder a-+Pa--«+Qa"-2+ • • • +Ta + U. Now, by hypothesis a is a root of the equation ; hence, this re- mainder is nothing, since it is nothing more than the result of the sub- stitution of a for X in the equation ; therefore the divisio7i is exact. Reciprocally, if x— a is an exact divisor of a;"" + Pa;'^-~'+ . . ., the remainder a"'+Pa'"^*+ . . . will be nothing ; therefore (Art. 259), a is a root of the equation. 262. From this it results that, in order to discover whether a bi- nomial of the form x—a is an exact divisor of a polynomial involv- 25* 294 ALGEBRA. ing X, it will be sufficient to see it' the result of the substitution of a for X, is equal to 0. To ascertain whether a is a root of a polynomial involving x, which is placed equal to 0, it wii! be sufficient to try the division of it by x—a. If it is exact, we may be certain that a is a root of the equation. 263. Remark. By inspecting the quotient of the division in Art. 261, we perceive the following law for the co-efficients : Each co. efficient is obtained hy multiplying that which precedes it by the root a, and adding to the product that co. efficient of the proposed equation which occupies the same rank as that which we wish to obtain in the quotient. Thus, the co-efficient of the 3d term, a^+Pa+Q, is equal to (a-f P)a+Q, or to the product of the preceding co-efficient a+P, by the root a, augmented by the co-efficient Q. of the 3d term of the proposed equation. The co-efficient of the 4th term is (a2 + Pa+Q)a-f-R, or a'+Pa^+Qa + R. This law should be remembered. Second Propei'ty. 264. Every equation involving hut one unknown quantity, has as many roots as there are units in the exponent of its degree, and no more. Let the proposed equation be a;"' + Px'''-''+Qa;'-2-|- . . . -fTa;+U = 0. Since every equation has at least one root (Art. 260), if we de note that root by a, the first member will be divisible by x—a, and we shall have the identical equation a;'"+Px" '+ . . . ={:r-a) (-r" ^+?'x^--+ ...)... (1). But by supposing a- »4-PV 2-f ... =0, we olitaiti :i!i tiiiation whiih has at least oiic root. GENERAL PROPERTIES OF EQUATIONS. 295 Denote this root by b, we have (Art. 261), a,.— i + I^'x^-^^- . . . =(x-*) (x"'-- + P"a;'"-^+ . . .). Substituting the 2d member for its vtUue, in equation (1), and we have, x"'+Pa;--«+ . . . =(x-a) {x-l) {x^ s+P'V^^^ ...)... (2). Reasoning upon the polynomial sf^ -\-V"'ic'^~^ -\- ... as upon the preceding polynomial, we have a;"'-HP"x'"^'+ . . • ={x-c) (a;--- + P"'a;'"-''+ . . .), and by substitution a;"+Px— '+ . . . ={x-a) (x-b) (x-c) (x-=*+ ...)... (3). Observe that for each indicated factor of the first degree with reference to x, the degree of x in the polynomial is diminished by unity ; therefore, after m—2 factors of the first degree have been divided out, the exponent of x will be reduced to m— (wi— 2), or 2 ; that is, we shall obtain a polynomial of the second degree with refe- rence to X, which can be decomposed into the product of two factors of the first degree, (x—k) (x—l) (Art. 142). Now, as the m—'2 factors of the first degree have already been indicated, it follows that we have the identical equation, x-^+Px— >+ . • . =(x-a) (x-b) (x-c) . . . (x-k) (x-l). From which we see, that the Jirst member of the proposed equU' Hon is decomposed into m factors of the first degree. As there is a root corresponding to each divisor of the first de. gree (Art. 261), it follows that the m factors of the first degree X— a, x—b, X— c . . ,, give the m roots a, b, c . . . for the proposed equation. Hence, the equation can have no other roots than a, b, c . . . k, I, since if it had a root «, different from a, b, c . . . 1, it would follow that it would have a divisor x— a, different from x—a, x—b, x—c . . . X— /, which is impossible. Finally, every equation of the m"" degree has m roots, and can have no more. 296 ALGEBRA. 265. There are some equations in which the number of roots la apparently less than the number of units in the exponent of their degree. They are those in which the first member is the product of equal factors, such as the equation {x-ay{x-bf{x-cy{x-dy^Q, which has hut/our different roots, although it is of the 10th degree. It is evident that no quantity a, different from a, b, c, d, can veri- fy it ; for if it had this root a, the first member would be divisible by x—cc, which is impossible. But this is no reason why the proposed equation should not have ten roots, /our of which are equal to a, three equal to b, two equal to c, and one equal to d. 266. Consequence of the second property. The first member of every equation of the m"" degree, havmg m divisors of tjie first degree, of the form x—a, x—b, x—c, . . ". x—k, x—l, if we multiply these divisors together, two and two, three and three . . ., we shall obtain as many divisors of the second, third, &c., degree with reference to x, as we can form different combinations of m quantities, taken two and two, three and three, &c. Now the number of these combinations is expressed by m—\ 7n — 2 m.-^-,m.-^... (Art. 163). Thus, the proposed equation has m . — - — divisors of the se- 7,n — 1 7)1 — 2 ,. . ^ , , . , 1 1 cond degree, m . — - — . — ^ — divisors of the third degree, and 80 on. Composition of Equations. 267. If in the identical equation x"+Pi'"-»+ . . . =(x— a) (x—b) (x—c) . . . (x—l), we perform the multiplication of four factors, we have COMPOSITION OF EQUATIONS. 297 x'-a aP+ab x'-abc x-\-abcd \ -b +ac -abd — c ■i-ad — acd -d + bc -bed + bd i +cd ) =0. If we perform the multiplication of the m factors of the second member, and compare the terms of tlie two members, we shall find the following relations between the co-efficients P, Q, R, . . . T, U, and the roots a, b, c, . . . k, /, of the proposed equation, viz. -a-b-c . . . -k-l=^P, or a + b + c+ . . . +A-+/=-P; ab-\-ac-\- . . . +^'/=Q — abc—abd . . . — zX-Z=R, or abc-\-abd -{-ikl^—K ; ±abcd . . . kl^\], or abed . . . /:/=±U. The double sign has been placed in the last relation, because the product —ax—bx—c ... X—l will he plus or minus according as the degree of the equation is even or odd. Hence, 1st. The algebraic sum of the roots, taken with contrary signs, is equal to the co-efficient of the second term ; or, the alge- braic sum of the roots themselves, is equal to the co-efficiciit of the second term taken with a contrary sign. 2d7 The sum of the products of the roots taken two and two, with their respective signs, is equal to tlie co-efficient of the third term. The sum of the products of the roots taken three and three with their signs changed, is equal to the co-efficient of the fourth term ; or the co-efficient of the fourth term, taken with a contrary sign, is equal to the sum of the products of the roots taken tliree and three; and so on. Finally, the product of all the roots, is equal to the last term ; that is., the product of all the roots, taken with their respective signs, 298 ALGKBRA. is equal to the last term of the equation, taken with its sign, ichen the equation is of an even degree, and with a contrary sign, when the equation is of an odd degree. If one of the roots is equal to 0, the absolute term will be 0. The properties demonstrated (Art. 142), with respect to equations of the second degree, are only particular cases of the above. The last term, taken with its sign, is equal to the product of the roots themselves, because the equation is of an even degree. Remarks on the Greatest Common Divisor. 268. The properties of the greatest common divisor of two poly- nomials, were explained in Arts. 66 & 67. We shall here offer a kw remarks to serve as a guide in determining it. Let A be a rational and entire polynomial, supposed to be arranged with reference to one of the letters involved in it, a, for example. If this polynomial is not absolutely prime, that is, if it can be de- composed into rational and entire factors, it may be regarded as the product of three principal factors, viz. 1st. Of a monomial factor A,, common to all the terms of A. This factor is composed of the greatest common divisor of all the numerical co-efficients, multiplied by the product of the literal fac tors which are common to all the terms. 2d. Of a polynomial factor A^, independent of o, which is com- mon to all the co-efficients of the different powers of a, in the ar- ranged polynomial. 3d. Of a polynomial factor A3, depending upon a, and in which the co-efficients of the different powers of a are prime with each other ; so that we shall have Azz^AjXA^xAg. Sometimes one or both of the factors A,, A^ reduce to unity, but this is the general form of rational and entire polynomials. It GREATEST COMMON DIVISOR. 299 follows from this, that when there is a greatest common divisor of two polynomials A and B, we shall have D=D,.D,.D3; D, denoting the greatest monomial common factor, D^ the greatest polynomial factor independent of a, and D3 the greatest polynomial factor depending upon this letter. In order to obtain D ^, find the monomial factor Aj common to all the terms of A. This factor is in general composed of literal fac- tors, which are found by inspecting the terms, and of a numerical co-efficient, obtained by finding the greatest common divisor of the numerical co-efficients in A. In the same way, find the monomial B, common to all the terms of B; then determine the greatest factor Dj common to Aj a?uZB,. This factor Dj, is set aside, as forming the first part of the re- quired common divisor. The factors A, and Bj are also suppressed in the proposed polynomials, and the question is reduced to finding the greatest common divisor of two new polynomials A' ani B' which do not contain a common monomial factor. It is then to be understood that the process developed below, is to be applied to these two polynomials. 269. Several circumstances may occur as regards the number of letters that may be contained in A' and B'. \st. When A' and B' contain but one letter a. When A' and B' are arranged with reference to a, the coetli- cients will necessarily be frime with each other; therefore in this case, we shall only have to seek for the greatest common factor de- pending upon a, viz. D3. In order to obtain it, we must first prepare the polynomial of the highest degree, so that its first term may be exactly divisible by the first term of the divisor. This preparation consists in midtiply. ing the whole dividend by the co-efficient of the first term of the divi- sor, or by a factor of this co-efficienl, or by a certain pmver of it, in 300 ALGEBRA. order that we may be able to execute several operations, without any new preparations (Art. 68). The division is then performed, continuing the operation until a remainder is obtained of a lower degree than the divisor. If there is a factor common to all the co-efficients of the remainder, it must be suppressed, as it cannot form a part of the required divi- sor ; after which, we operate with the second polynomial, and this remainder, in the same way we did with the polynomials A' and B' . Continue this series of operations until a remainder is obtained which will exactly divide the preceding remainder, this remainder will be the greatest common divisor D^ of A' and B' ; and D, XD3 will express the greatest common divisor of A and B ; or, continue the operation until a remainder is obtained independent of a, that is, a numerical remainder, in which case, the two polynomials, A' and B' will be prime with each other. 2d. When A' and B' contain two letters a andh. After having arranged the polynomials with reference to a, we first find the polynomial factor \\\\\c\\ is independent of a, if there is one. To do this, we determine the greatest common divisor A, of all the co-efficients of the different powers of a in the polynomial A'. This common divisor is obtained by applying the rule for finding the greatest common divisor of several polynomials, as well as the rule for the last case, since these co-efficients contain only one let- ter b. In the same way we determine tlie greatest common divisor B, of all the co-efficients ofB'. Then comparing A^ and B^,wcset aside their greatest common divisor D2, as forming a part of the re- quired greatest common divisor ; and we also suppress the factors A 2 and B2, in A' and B'; which produces two new polynomials A" (md B", the co-efficients of which are prime ivith each other, and to which we may consequently apply the rule for the first case. Care must ahoays he talen to ascertain, in each remainder, whether GREATEST COMMON DIVISOR. 301 the co-efficients of the different powers of the letter a, do not contain a common factor, which jiiust be suppressed, as not forming a part of the common divisor. We have already seen that the suppression of these factors is absolutely necessary (Art. 68). We shall in this way obtain the common divisor D2, of A" and B", and D, xDg XD3J for the greatest common divisor of the polyno- mials A and B. Remark. In applying the rule for the first case to A'' and B", we could ascertain when these two polynomials were prime with each other, from this circumstance, viz : a remainder would he oh. tained which would he either numerical, or a function ofh, hut inde- pendent of a. The greatest common divisor of A and B would then beD^XD^. 3d. When A' and B' contain three letters, a, b, c. After arranging the two polynomials with reference to a, we de- termine the greatest common divisor independent of a, which is done by applying to the co-efficients of the different powers of a, in both polynomials, the process for the second case, since these polyno- mial co-efficients contain but two letters, i and c. The independent polynomial D^ being thus obtained, and the fac- tor Ag and B^, which have given it, being suppressed in A' and B', there will result two polynomials A" and B", having their co-effi- cients j?n me with each other, and to which the rules for the preced- ing cases may be applied, and so on. EXAMPLES. 1. Let there be the two polynomials aW—c''d--a-c' + c\ and 4a'd—2ac' + 2c^—4acd. The second contains a monomial factor 2. Suppressing it, and arranging the polynomials with reference to d, we have {a''-c^)d''-a-c'' + c*, and {2a^-2ac)d-ac''+c^. 302 ALGEBRA. It is first necesssLTy to ascertain whether there is a common divi- sor independent of d. By considering the co-efficients a^—c^, and —a'c'+c*, of the first polynomial, it will be seen that —a-c'^+c* can be put under the form —c^{a- — c^) ; hence a^—c^ is a common factor of the co-effi- cients of the first polynomial. In like manner, the co-efficients of the second, 2a= — 2ac, and —ac^+c^, can be reduced to 2a(a—c), and —(^{a—c); therefore a—c is a common factor of these co- efficients. Comparing the two factors a'^—c'^ and a—c, as this last will di- vide the first, it follows that a—c is a common factor of the propos- ed polynomials, and it is that part of their greatest common divisor which is independent of d. Suppressing a^ — c^ in the first polynomial, and a-c in the second, we obtain the two polynomials d'^—c^ and 2ad — c^, to which the or- dinary process must be applied. d=-c^ \2ad-c^ ia"d- — 4:a-c- 2ad + c^ + 2ac-d—4.a-c^ — 4a-c^ +c*. Expkination. After having multiplied the dividend by 4a^, and performed two consecutive divisions, we obtain a remainder — 4aV+cS independent of the letter d ; hence the two polynomials d= — c*, and 2ad—c^, are prime with each other. Therefore the greatest common divisor of the proposed polynomials is a — c. Again, taking the same example, and arranging with reference to a, it becomes, after suppressing the factor 2 in the second poly- nomial, {d''-c')a'-c=d--\-c\ and 2(/a='-(2aZ + c=)a + c'. It is easily perceived, that the co-efficient of the different powers of a in the second polynomial are prime with each other. In the first polynomial, the co-efficient — c'd^+c*, of the second term, or GREATEST COMMON DIVISOR. 303 B. Now it results from the nature of the quantities A and B, that they both increase as x in- creases, since they contain only positive numbers, and positive and entire powers of x ; therefore, by making x augment by insensible degrees, from p to q, the quantities A and B will also increase by in- sensible degrees. Now smce A, by hypothesis, from being less than B, afterwards becomes greater than it, A must necessarily have a more rapid increment than B, which insensibly destroys the excess that li liad over A, and finally produces an excess of A over B. From this, we conceive that in the passage from AB, RESOLUTION OF NUMERICAL EQUATIONS. 327 there must be an intermediate value for which A becomes equal to B, and the value which produces this result is a root of the equa- tion, since it verifies A— B=0, or the proposed equation. Hence, the proposition is proved. In the preceding demonstration, p and q have been supposed to be positive numbers ; but the proposition is not less true, whatever may be the signs with which p and q are affected. For we will re- mark, in the first place, that the above reasoning applies equally to the case in which one of the numbers p and q, p for example, is ; that is, it could be proved, in this case, that there was at least one real root between and q. Let both p and q be negative, and represent them by — p' and -q'. If, in the equation x^'+Pa;'"-'+Qa;'"-»+ . . . Ta;+U=0, we change x into —y, which gives the transformation (_3/)".+p(_y)-»+Q(_y)-2+ . . . T(-t/)+U = 0, it is evident that substituting —p' and —q' in the proposed equation, amounts to the same thing as substituting p' and q' in the transfor- mation, for the results of these substitutions are in both cases (-p')'"+P(-p')'"-^+Q(-p')"-^+ . . . T(-p')+U, and (_5')'"+P(-j'r-^+Q(-?'r ■'+ • • • T(-?')+U ; Now, since |> and q, or —p' and —q', substituted in the proposed equa- tion, give results with contrary signs, it follows that the numbers p' and q, substituted in the transformation, also give results with con- trary signs ; therefore, by the first part of the proposition, there is at least one real root of the transformation contained between p' and q' ; and in consequence of the relation x= —y, there is at least one value of x comprehended between —p' and —q', or p and q. This demonstration applies to cases in which ^=0 or ^=0. Lastly, suppose jj po«. , b, c, d . . •> but < , b, e, d.. • 5 whence we deduce P- -a, p- ■b, p-c, . . .>o, and 1- -a, q- -b, q — c, . . ■> hence, since p—a and q — a are affected with contrary signs, as well Q.sp—b and q—b,p — c and q—c . . ., the partial quotients p-a p-b p-c ^ q — a'' q — b' q—c' Y' are all negative ; moreover - is essentially positive, since Y' and Y" are affected with the same sign ; therefore the product p—a p—b p — c Y' X^—tX X • . . -^r^, q—a q—b q—c l" will be negative, when the number of roots, a, b, c . . ., compre- bended between p and q, is uneven, and positive when the number is even. 28* 330 ALG£BRA. Consequently, the two results (p—a) (p—b) (p— c) . . . xY', and (q—a) (q—b)(Px'"-»+Q'"-='+ . . . +Ta;+U. Let k denote the greatest co-efficient, and substitute it in place of the co-efficients ; the inequality will become It is evident that every number substituted for x which will satisfy this condition, will for a stronger reason, satisfy the preceding. Now, dividing this inequality by x"", it becomes k k k k k k Makmg x=k, the second member becomes—, or 1 plus a series of positive fractions ; then the number k will not satisfy the ine- quality ; but by supposing x=^k+\, we obtain for the second mem- ber the series effractions k k k k k ■i+l+(A:+l)2+ {k+lf + • • • +(i-4-ir-^+ {k+ir ' which, considered in an inverse order, is an increasing geometrical ]^ progression, the first term of which is . t\„. > the ratio ^+1, and k the last term , , ; hence the expression for the sum of all the terms is, (Art. 223), ^-^■'^""-Wir „,, k+i-i {k+iy rhich Ls evidently less than unity. 332 ALOCBRA. Any number ^k-\~l, put in place of x, will render the sum of the k k fractions f— t+ . • • still less. Therefore, X ar The greatest co-efficient of the equation plus unity, or any greater number, being substituted for x, will render the first term x"* greater than the arithmetical sum of all the others. Ordinary limit of the Positive Roots. 293. The number obtained above may be considered a prime limit, since this number, or any greater number, rendering the first term superior to the sum of all the others, the results of the sub- stitution of these numbers for x must be constantly positive ; but this limit is commonly much too great, because, in general, the equation contains several positive terms. We will, therefore, seek for a limit suitable for all equations. Let a;"*"" denote the power of x, corresponding to the first nega- tive term which follows x", and we will consider the most uniavour- able case, viz. that in which all of the succeeding terms are nega- tive, and affected with the greatest of the negative co-efficients in the equation. Let S be this co-efficient, and try to satisfy the condition a;">Sa-"'-''+Sx'"-"-i+ . . . Sx+S ; or, dividing both members of this inequality by nf, S_ S _S_ _S_ S_ Now by supposing x''=S or x= "V^, the second member be- S comes -^, or 1, plus a series of positive fractions ; but by making x= VS + 1, or (supposing, for simplicity, VS = S', whence 8 = 8'"), a:=S' + l, the second member becomes -+ ... , ..„., + (S' + l)-. ' (S' + l)"-^*^ • • • ' (S' + ir-^ (S' + l)' #-* U> *y>SIJ[NARY LIMIT OF THE POSITIVE ROOTS. 333 . . y s" which is a ^ogression by quotients, —^, — -— being the first term, S'" S' + l the ratio, and -7^7 — r-— the last term. Hence the expres- (b +1)" sion for the sum of all these fractions is S'- ^ S'" (S' + l)"*^^' + ^^~ (S' + l)"* S'--* S'"-» S' + l-l (S' + l)"-' (S' + lf which is evidently less than 1. Moreover, every number >S' + 1 or \^S-\-l, will, when substi- S S tuted for x, render the sum of the fractions 1 Tr4- • • • • still smaller, since the numerators remaining the same, the denominator will increase. Hence VS + 1, and any greater number, will ren- der the first term x" greater than the arithmetical sum of all the negative terms of the equation, and will consequently give a posi- tive result for the first member. Therefore Vs + l, or wiity increased by that root of the greatest negative co-ejicient icJiose index is the number of terms which precede the first negative term, is a superior limit of the positive roots of the equation. Make 7i=l, in which case the first negative term is the second term of the equation ; the limit becomes VS + 1, or S + 1 ; that is, the greatest negative co-efficient plus unity. Let 71 = 2, then the two first terms are positive, or the term x"~' is wanting in the equation ; the limit is then V^S + l. When 71 = 3 the limit is ^VS + l . . . Find the superior limits for the positive roots in the following ex- amples : c4—5x' + 37.r'- 3^+39=0; VS + l='V~S'-\-l = 6 ; a^+Tx*- 12x2- 49ar'+52x- 13=0 . Vs4.1= V49+l = 8 ; 334 ALGEBRA. Vs+i=)^- ft?^^ a;^ + llx=_25x-67=0; VS + 1 = ^;)^ + 1 or „ A— 11 3x3— 2ar'-lla: + 4=0; VS + 1=— +1 or 5. o Newton's method for determining the smallest limit in entire numbers. 294. Let X=0, be the proposed equation ; if in this equation we make x=x'+u, x' being indeterminate, we shall obtain (Art. 274), X'+Y'u+yu-> . . . +w"=0. (1) Conceive, that after successive trials we have determined a number Z' for X, which, substituted in X', Y', — . . ., renders all these co-effi- cients positive at the same time ; this number will be greater than the greatest positive root of the equation X=0. For, the co-efficients of the equation (1) being all positive, no positive number can verify it ; therefore all of the real values of u must be negative; but from the equation a'=x'+if, we have xi^=x—x' ; and in order that the values of it corresponding to each of the values of X and x' (already determined) may be negative, it is absolutely necessary that the greatest positive value of x should be less than the value of x'. EXAMPLE. x^-Sx'-ex^— 19x4-7=0. As x' is indeterminate, the letter x may be retained in the forma- tion of the derived polynomials, and we have X ^x^-Sr'— 6x2-19x-|r7, Y =:4r'-15r'-12x-19, Z _ =6x2-15x-6, V =4x— 6. 2.3 SMALLEST LIMIT OP THE ROOTS. 335 The question is, as stated above, reduced to finding the smallest number which, substituted in place of a>, will render all of these po- lynomials positive^ It is plain that 2 and every number >2, will render the polyno- mial of the first degree positive. But 2, substituted in the polynomial of the second degree, gives a negative result; and 3, or any number >3, gives a positive result. Now 3 and 4, substituted in the polynomial of the third degree, give a negative result ; but 5 and any greater number, give a posi- tive result. Lastly, 5 substituted in X, gives a negative result, and so does 6 ; for the three first terms x* — 5a;^— 6x^ are equivalent to the expres- sion 3^(x—5) — 6aP, which is reduced to when x=6 ; but x=7 evi- dently gives a positive result. Hence 7 is a superior limit of the positive roots of tJie proposed equation ; and since it has been shown that 6 gives a negative result, it follows that there is at least one real root between 6 and 7. Applying this method to the equation a;5_3x''-8x''— 25x2 + 43;- 39=0, the superior limit will be found to be 6. We should find 7, for the superior limit of the positive roots of the equation xs_5x*- 1.3x^ + 17x2-69=0. This method is scarcely ever used, except in finding incommen- surable roots. 295. It remains to determine the superior limit of the negative roots, and the inferior limits of the positive and negative roots. Hereafter we shall designate the superior limit of the positive roots of an equation by the letter L. 1st. If in the equation X=0, we make x=— y, which gives the transformed equation Y=0, it is clear that the positive roots of this new equation, taken with the sign — , will give the negative roots of 336 ALGEBRA. the proposed equation ; therefore, determining, by the known me- thods, the superior limit L' of the positive roots of the equation Y=0, we shall have — L' for the superior limit (numerically) of the nega- tive roots of the proposed equation. 2d. If in the equation X = 0, we make j;= — , which gives the equation Y=0, it follows from the relation x= — that the greatest positive values of y correspond to the smallest of x ; hence, desig- nating the superior limit of the positive roots of the equation Y=0 by L", we shall have ^ „ for the inferior limit of the positive rods of the proposed equation, 1 3d. Finally, if we replace x, in the proposed equation, by , and find the superior limit L"' of the transformed equation Y = 0, — :jr-;77- will be the inferior limit (numerically) of the negative roots of the proposed equation. 296. Every equation in which there are no variations in the . that is, in which all the terms are positive, must have all of its real roots negative ; for every positive number substituted for x will ren- der the first member essentially positive. Every complete equation, having its terms alternately positive and negative, must have its real roots all positive ; for every negative number substituted for x in the proposed equation, would render all the terms positive, if the equation was of an even degree, and all of them negative if it was of an odd degree. Hence the sum would not be equal to zero in either case. This is also true for every incomplete equation, in which there results, by sulstituting — y for x, an equation having all of its terms affected with the same sign. SMALLEST LIMIT OP THE ROOTS. 337 Consequences deduced from the preceding Principles. First. 297. Every equation of an odd degree, the co-efficients of which are real, has at least one real root affected with a sign contrary to thai of its last term. For, let a;'"+Pa;'"-'+ . . . Ta;±U=0, be the proposed equation; and first consider the case in which the last term is negative. By making .t=0 the first member becomes — U. But by giving a value to x equal to the greatest negative co-efficient plus unity, or (K-f-1), the first term .r™ will become greater than the arithmetical sum of all the others (Art. 292), the result of this substitution will therefore be positive ; hence, there is at least one real root compre- hended between and K + 1, which root is positive, and consequently affected with a sign contrary to that of the last term. Suppose now that the last term is positive. Making .r=0, we obtain +U for the result; but by putting — (K + 1) in place of a;, we shall obtain a negative result, since the first term becomes negative by this substitution ; hence the equa- tion has at least one real root comprehended between and — (K+1), which is negative, or affected toith a sign contrary to that of the last term. Second. 298. Every equation of an even degree, involving only real co. efficients of which the last term is negative, has at least two real roots, one positive and the other negative. For, let — U be the last term ; making x'=0, there results — U. Now substitute either K + 1, or — (K + 1), K being the greatest negative co-efficient of the equa- tion : as ?n is an even number, the first term x'^ will remain positive ; besides, by these substitutions, it becomes greater than the sum of all the others ; therefore the results obtained by these substitutions are both positive, or affected with a sign contrary to that given by the hypothesis x=0 ; hence the equation has at least tioo real ro-^'v 20 33o ALGEBKA. one comprehended between and K-j-l, or positive, and the other between and — (K + 1), or negative. Third. 299. If an equation, involving only real co-efficients, contains imaginary roots, the number of these roots must he even. For, conceive that the first member has been divided by all the simple factors corresponding to the real roots ; the co-efficients of the quotient will be real (261) ; and the equation must also he of an even degree; for if it was uneven, by placing it equal to zero, we should obtain an equation that would contain at least one real root, which, from the nature of the equation, it cannot have. Remark. 300. There is a property of the above polynomial quo- tient which belongs exclusively to equations containing only imagi- nary roots ; viz. every such equation ahvays remains positive for any real value substituted for x. For, if it could become negative, since we could also obtabi a posi- tive result, by substituting K + 1 or the greatest negative co-efficient plus unity for x, it would follow that this polynomial placed equal to zero, would have at least one real root comprehended between K-fl and the number which would give a negative result. It also follows, that the last term of this polynomial must be posi. live, otherwise a;=0 would give a negative result. Fout^th. 301. When the last term of an equation is positive, the number of its real positive roots is even ; and when it is negative this number is uneven. For, first suppose that the last term is +U, or positive. Since by making a;=0, there will result +11, and by making a;=K + l, the result will also be positive, it follows that and K + 1 give two results affected with the same sign, and consequently (289), the numbor of real roots, (if any), comprehended between them, is even. Descartes' rule. 339 When the last term is — U, then and K+l give two results affected with contrary signs, and consequently comprehend either a single real root, or an odd number of them. The reciprocal of this proposition is evident. Descartes^ Rule. 302. An equation of any degree whatever cannot have a greater number of positive roots than there are variations in the signs of Us terms, nor a greater number of negative roots than there are perma- nences of these signs. In the equation x—a=0, there is one variation, that is a change of sign in passing along the terms, and one positive root, x=a. And in the equation x+b=0, there is one permanence, and one negative root, x=—b. If these equations be multiplied together, there will result an equa- tion of the second degree, x^ — a I x — ab If fl is less than b, the equation will be of the first form (Art. 144) ; and if a>J the equation will be of the second form : that is ai x^—px—q = In either case, there is one variation, and one permanence, and since in either form, one root is positive and one negative, it follows that there are as many positive roots as there are variations, and as many negative roots as there are permanences. The proposition would evidently be demonstrated in a general manner, if it were shown that the multiplication of the first member by a factor x— fl,corresponding to a positive root, would introduce at least one variation, and that the multiplication by a factor x-\-a, would introduce at least one permanence. :0. 340 ALGEBRA. Let there be the equation af'±Ax'^»±Bx'"-2±Ca;""-3± . . . ±Ta:±U=0, in which the signs succeed each other in any manner whatever ; by niiiltiplying it by x—a, we have -a I ipAa I zpBa | z^Ta \ q=Ua The co-efficients which form the first horizontal line of this pro- duct, are those of the proposed equation, taken with the same sign ; and the co-efficients of the second line are formed from those of the first, multiplied by a, taken with contrary signs, and advanced one rank towards the right. Now, so long as each co-efficient of the upper line is greater than the corresponding one in the lower, it will determine the sign of the total co-efficient ; hence, in this case there will be, from the first term to that preceding the last, inclusively, the same variations and the same perm.anences as in the proposed equation ; but the last term zpUa having a sign contrary to that which immediately pre- cedes it, there must be one or more variations than in the proposed equation. When a co-efficient in the lower line is affected with a sign con- trary to the one corresponding to it in the upper, and is also greater than this last, there is a change from a permanence of sign to a variation; for the sign of the term in which this happens, being the same as that of the inferior co-efficient, must be contrary to that of the preceding term, which has been supposed to be the same as that of its superior co-efficicnt. Hence, each time we descend from the upper to the lower line, in order to determine the sign, there is a variation which is not found in the proposed equation ; and if, after passing into the lower line, we continue in it throughout, we shall find for the remaining terms the samevariatipns and the same perma- nences as in the proposed equation, sil^ce tKe co-efficients of this line are all affected with signs contrary to those of the primitive co-effi- cients. This supposition would therefore give us one variation for DESCARTES UULE. 341 each positive root. But if we ascend from the lower to the upper line, there may be either a variation or a permanence. But even by supposing that this passage produces permanences in all cases, since the last term zpUa forms a part of the lower line, it will be necessary to go once more from the upper line to the lower, than from the lower to the upper. Hence the new equation must have at least one more variation than the proposed ; and it will be the same for each positive root introduced into it. It may be demonstrated, in an analogous manner, that the multi- plication by a factor x-\-a, corresponding to a negative root, would introduce one permanence more. Hence, in any equation the num- ber of positive roots cannot be greater than the number of varia- tions of sign, nor the number of negative roots greater than the number of permanences. 303. Consequence. When the roots of an equation are all real, the number of positive roots is equal to the number of variations, and the number of negative roots is equal to the number of permanences. For, let m denote the degree of the equation, n the number of variations of the signs, p the number of permanences ; we shall have m=n-[-p. Moreover, let n' denote the number of positive roots, and p' the number of negative roots, we shall have m=n'+p'; whence n-{-p=.n' -\-p or, n—n'=p'—p. Now, we have just seen that n' cannot be >n, and p' cannot be >_p ; therefore we must have n'=n, and p'=p. Remark. 304. When an equation wants some of its terms, we can often discover the presence of imaginary roots, by means of the above rule. For example, take the equation oi?-\-px-{-q=0, p and q being essentially positive ; introducing the term which is 29* 842 ALGEBRA. wanting, by affecting it with the co-efficient ±0, it becomes By considering only the superior signs we should only obtain per- manences, whereas the inferior sign would give two variations. This proves that the equation has some imaginary roots ; for if they were all three real, it would be necessary by virtue of the superior sign, that they should be all negative, and, by virtue of the inferior sign, that two of them should be positive and one negative, which are contradictory results. We can conclude nothing from an equation of the form x^—px-\-q=zO ; for introducing the term ±0.3^, it becomes x^=tO . x^—px+q=0, which contains one permanence and two variations, whether we take the superior or inferior sign. Therefore this equation may have its three roots real, viz. two positive and one negative ; or, two of its roots may be imaginary and one negative, since its last term is ne- gative (Art. 301). Of the Commensurable Roots of Numerical Equations. 305. Every equation in which the co-efficients are whole num- bers, that of the first term being unity, can only have whole num- bers for its commensurable roots. For, let there be the equation a;-+Pa;"-i+Qa;"— ^+ . . . +T.r + U = ; in which P, Q . . . T, U, are whole numbers, and suppose that it a could have a commensurable fraction -j- for a root. Substituting this fraction for x, the equation becomes COMMENSURABLE ROOTS OP EQUATIONS. 343 whence, multiplying the whole equation by Z*""', and transposing, ^=_Pa--i_Qa— ^'5- . . . -Tai'^-'-Vb'^-' ; o but the second member of this equation is composed of a series of entire numbers, whilst the first is essentially fractional, for a and b being prime with each other, a" and b will also be prime with each other (Art. 118), hence this equality cannot exist ; for, an irreduci- ble fraction cannot be equal to a whole number. Therefore it is impossible for any commensurable fraction to sa- tisfy the equation. Now it has been shown (Art. 277), that an equation containing rational, but fractional co-efficients, can be transformed into another in which the co-efficients are whole num- bers, that of the first term being unity. Hence the research of the commensurable roots, entire or fractional, can alicays be reduced to that of the entire roots. 306. This being the case, take the general equation ^m_^Pa;m-i+Qa;— 2+ . . . +R,^''+Sr'+Ta;+U=0, and let a denote any entire number, positive or negative, which will verify it. Since a is a root, we shall have the equation a'"+Pa"-»+ . . . +Ra='+Sa^+Ta+U=0 . . • (1) ; replacing a by all the entire positive and negative numbers between 1 and the limit +L, and between —1 and — L', those which verify the above equality will be the roots of the equation. But these trials being long and troublesome, we will deduce from equation (1), other conditions equivalent to this, and easier verified. Transposing all the terms except the last, and dividing by a, the equation (1) becomes — =-a"-^-Pa"-2_ . . . _Ra2_Srt-T . . . (2) ; a now, the second member of this equation is an entire number, hence 344 ALGEBRA. — must be an entire number ; therefore the entire roots of the equa- tion are comprised among the divisors of the last term. Transposing — T in the equation (2) and dividing by a, and ma- U king |-T=T'; it becomes T' — =-a"'-2-Pa"'-3 . . . -Ra-S . . . (3) ; T' the second member of this equation being an entire number, — U' or, the quotient of the division of \-T by a, is an entire numler. Transposing the term — S and dividing by a, it becomes, by sup- T' posing f-S=S', ?-=_a-3_Pam-4_ , . . _R . . . (4), a S' the second member of this equation being an entire number, — T' or, the quotient of the division of \-^ by a, is an entire number. By continuing to transpose the terms of the second member into the first, we shall, after m— 1 transformations, obtain an equation of Q' the form — =— a— P, a Then, transposing the term — P, dividing by a, and making Q' F P' [-P=P'» we shall find —=-1, or [-1=0. a a a This equation, which is only a transformation of the equation (1), is the last condition which it is requisite and necessary that the en- tire number a should satisfy, in order that it may be known to be a root. From the preceding conditions we may conclude that, in order COMMENSURABLE ROOTS OP EQUATIONS. 345 that an entire number a, positive or negative, may be a root of the proposed equation, it is necessary That the quotient of the last term, divided by a, should be an en- tire number ; Adding to this quotient the co-efficient of .r', taken with its sign, the quotient of this sum divided by a, must be entire ; Adding the co-efficient of x^ to this quotient, the quotient of this new sum by a, must be entire; and so on. Finally, adding the co-efficient of the second term, or of a;"""', to the preceding quotient, the quotient of this sum divided by a, 7nust be entire and eqiud to —1 • or, the result of the addition of unity, or the co-efficient of x"\ to the preceding quotient, must be equal to 0. Every number which will satisfy these conditions will be a root, and those which do not satisfy them should be rejected. All the entire roots may be determined at the same time, as fol- lows. After having determined all the divisors of the last term, tcrite those which are comprehended between the limits +L and — L' upon the same horizontal line ; then underneath these divisors write the quo- tients of the last term by each of them. Add the co-efficient of x' to each of these quotients, and write the sums underneath the quotients ichich correspond to them ; then divide these siuns by each of the divisors, and write the quotients underneath the corresponding sums ; taking care to reject the fractional quo- tients and the divisors which produce them ; and so on. When there are terms wanting in the proposed equation, their co-efficients, which are to be regarded as equal to 0, must be taken into consideration. x*—a.'^—13x= + 16a:— 48=0. The superior limit of the positive roots of this equation is 13-f 1 or 14 (Art. 293). The co-efficient 48 is not considered, since the 346 ALGEBRA. two last terms can be put under the form 16(x— 3) ; hence when a;>3 this part is essentially positive. The superior limit of the negative roots is —(1+ V48), or —8 (Art. 295). Therefore, the divisors are 1, 2, 3, 4, 6, 8, 12 ; moreover, neither -f 1, nor —1, will satisfy the equation, because the co-efficient —48 is itself greater than the sum of all the others ; we should therefore try only the positive divisors from 2 to 12, and the negative divisors from —2 to —6 inclubi -ely. By observing the rule given above, we have 12, 8, 6, 4, 3, 2, — 2, - 3, - 4, - 6 - 4, - 6, -8, -12, -16, -24, +24, + 16, + 12, + 8 + 12, + 10, +8, + 4, 0, - 8, + 40, + 32, + 28, + 24 + 1, .., + 1, 0, - 4, -20, .., - 7, - 4 -12, ••J -12, -13, -17, -33, .., -20, — 17 - 1, .., - 3, .., .., .., ••> + 5, .. - 2, - 4, — 1, -, • •> ••' ••' + 4, — 1, •• The Jirst line contains the divisors, the second contains the quo- tients of the division of the last term —48, by each of the divisors. The third line contains these quotients augmented by the co-efficient + 16, and the fourth the quotients of these sums by each of the di- visors ; this second condition excludes the divisors +8, +6, and —3. The fifth is the preceding line of quotients, augmented by the co- efficient — 13, and the sixth is the quotients of these sums by each of the divisors ; this third condition excludes the divisors 3, 2, —2 and —6. Finally, the seventh is the third line of quotients, augmented by the co-efficient —1, and the eighth is the quotients of these sums by each of the divisors. The divisors +4 and — 4 are the only ones which give —1 ; hence +4 and —4 are the only entire roots of the equation. In fact, if we divide a;*— r"- 13ar'+16x— 48, by the product INCOMMENSURABLE ROOTS. 347 («— 4) (x+4), or x'—ie, the quotient will be x^—x+S, which placed equal to zero, gives therefore, the four roots are EXAMPLES. 1st. X'* — 5a;='+25a;— 21 = 0. 2d. 15a:^-19a;''+6af'+15ic2— 19a;+6=0. 3d. 9x<'+30x5+22x*+10af'+17ar'— 20a;+4=0. Of Real and Incommensurable Roots. 307. When an equation has been freed from all the divisors of the first degree which correspond to its commensurable roots, the resulting equation contains the incommensurable roots of the pro- posed equation, either real or imaginary. The true form of the real incommensurable roots of an equation will remain unknown, so long as there is not a general method for resolving equations of the higher degrees. AUhough this problem has not been resolved, yet there are methods for approximating as near as we please to the numerical values of these roots. We shall here consider only the case in which tlie difference be- tween any two roots of the proposed equation is greater than unity, omitting as too difficult for an elementary treatise, the cases in which this difference is less than unity. We will also suppose, in what follows, that we have obtained the narrowest limits 4-L and — L', by Newton's method (Art. 294). 308. Each of the incommensurable roots being necessarily com- posed of an entire part and a part less than unity, we shall first deter- mme the entire part of each root. 348 ALGEBRA. For this purpose, it is necessary to substitute, in the equation, for X, the series of natural numbers 0, 1, 2, 3 . . . and — 1, — 2, — 3 . . ., comprised between 4-L and — L'. Since there must be a real root between two numbers, which, by their substitution produce results affected with different signs (Art. 288), it follows that each pair of consecutive numbers giving results affected with contrary signs, toill comprehend a real root, and but one, since by hypothesis the difference between any two of the roots is greater than unity. The entire part of the root will be the smallest of the two numbers substituted. There are two cases which may occur; viz. by these diffei'ent substitutions there may be as many changes of sign as there are units in the degree of the equation ; in which case we may con- clude that all the roots are real. Or, the number of changes of the sign will be less than the degree of the equation, and, in this case, it will have as many real roots as there are changes of sign ; the other roots will be imaginary. In both cases, this method makes known the entire part of each of the real roots. It now remains to determine the part which is less than unity. Newton's Method of Approximation. 809. In order that this method may be more easily comprehend- ed, we shall take the equation a;3_5a;-3=0 . . . (1). The superior limits of the positive and negative roots being -f-3 and —2, we make x= — 2, —1, 0, 1, 2, 3; whence a;= — 2 the result is —1, x=-\ ... +1, x= . . . -3, a:= 1 . . — "7, x= 1 . . . -5, x= 3 . . . +9. Newton's method of approximation. 349 As tliere are three changes of sign, it follows that the three roots of the equation are real ; viz. ojie positive contained between 2 and 3, two negative, one of which is contained between and —1, the other between —1 and —2. We shall first consider the positive value between 2 and 3. The required root being between 2 and 3, we will try to contract these limits, by taking the mean 2i, or 2,5, and substituting it in the equation a;^— 5x— 3=0 ; the result of which is +0,125. Now 2 has already given —5 for a result, therefore the root is between 2 and 2,5. We will now consider another number, between 2 and 2,5 ; but as, from the results given from 2 and 2,5, it is to be presumed that the root is nearer 2,5 than 2, suppose x=2,4 ; we shall obtain — 1,176; whereas 2,5 has given +0,125. Therefore the root is between 2,4 and 2,5. By contmuing to take the means, we should be able to contract the two limits of the roots more and more. But when we have once obtained, as in the above case, the value of x to at least 0,1, we may approximate nearer in another way, and it is in this that Newtoti's method principally consists. In the equation a;^ — 5a'— 3 = 0, make a;=2,4+M. There will result (Art. 274), the transformation Z' X'+Y'it + — w2+u^=0; in which X' =(2,4)^— 5(2,4)-3= — 1,176, Y' =3(2,4)2-5=12,28, Z' — - = 3(2,4) = 7,2. The equation involving u, being of the third degree, cannot be resolved directly, but by transposing all the terms except Y'm, and dividing both members by Y', it can be put under the form X' Z' 1 "=-y^-2:y"'-y^"^- 30 350 ALGEBRA. This being the case, since one of the three roots of this equation must be less than — , from the relation ic=2,4+M, the correspond- ing values of u^ and v? are less than and . Moreover, the inspection of the numerical values of Y' and Z', proves that Z' is <1 ; therefore the value of u only differing numerically X' Z' 1 from —^, by the quantity v?+~u^, (which most frequently 1 \ X' . . is less than J, is expressed by ——to withm 1,01. As, in this example, X' +1,176 1176 12,28 12280 :0,09 there will result «=0,09, to within , and consequently 1 a:= 2,4 4- 0,0 9= 2,49, to within — — . In fact, 2,49 substituted in the first member of the proposed equa- tion, gives —0,011751 ; whilst 2,250 gives +0,125. To obtain a new approximation, make a;=2,49+w' in the pro- posed equation, and we have X"+Y'V+— «'H«'^=0; in which X"= (2,49f-5(2,49)-3=-0,011751, ¥"=3(249)2-5=13,6003, ?- = 3(2,49) = 7,47. Newton's method of approximation. 351 But the equation involving u' may be written thus : , X." Z" ,^ 1 ,3 And since one of the values of u' must be less than^^, the 1 1 corresponding values of u'^, M'^ are less than j^^qq-' ioqqooo ' X" 1 hence — ^^7, will represent the value of m' to within———. Since we have X" 0,011751 11751 ^^^^^ = — =z = 0,0008 . . ., Y" 13,6003 13600300 ' it follows that m'= 0,0008, to within , and consequently a;=2,49+0,0008=2,4908, to within 10000 Again, by supposing a;= 2,4908 +m", we could obtain a value of 1 X to within 100000000 Each operation commonly gives the root to twice as many places of decimals as the previous operation. 310. Generally, let;? andjp+1 be two numbers between which one of the roots of the equation X=0 is comprised. First determine the value of this root to loithin — , by substituting a series of numbers comprised between p and p+1, until two numbers are obtained which do not differ from each other by more than -. 352 Then, calling x' the value of x obtained to within — suppose x=ix'-\-u in the equation X = ; which gives Z' X'+Y'W + — M^ . . . +M'»=:0: whicJi can he put under the form X', Y' Z' . . . being easily calculated. (Art. 309). Since the sum of the terms, which follow —:^, in the second mem- ber of this equation is, commonly, less than , they can be neglected, and calculating — ^7 to within , we add the result to x', which gives a new value x" approximating to within of the exact value. To obtain a 3d approximation, we suppose x=x"-\-u' in the pro- posed equation, which gives Z" X'+Y'V + ym'^+ . . .m''"=0; X" Z" ,, 1 whence rt = — :r— , — ——-u'^— . . . _-—«''*. Z" 1 Neglecting the terms — --tttjm'-— . . . ——,%''" which are suppo- sed to be less than 0,0001, we calculate the value of — t?7,j continuing Newton's jiethod of approximation. 353 the operation to the place of decimals, and add the result to 1 x" ; this gives a third approximation x'", exact to within — • Repeat this series of operations for each of the positive roots. As for the negative roots, they are found m the same way as the positive roots, by changing x into —x in the proposed equation, which then becomes, _a;3_|_5_^,_3_0^ or x^ — 5x+S=0 in which the positive roots taken with a negative sign, are the nega- tive roots of the proposed equation. These roots are . a;= — 1,8342 and ic— = 0,6566 to within 0,0001. .\^ «v > ii^^a>' \ « I