In Article 175 it is proved that the binomial theorem is true for any exponent whatever; the transformations 1, 2, 3, 4, are therefore true for every value of n; but when n is negative or fractional, the values of A, B, are composed of an infinite number of terms. 173. When it is required to reduce the radical expression a+b=1 to the form A+B−1 the radical is replaced by the fractional power (a+b√−1)", which is developed in the manner already mentioned. Algebra supplies no other general method for this transformation; but when n is a power of 2 the transformation can be effected without the help of series. Taking, first, the two radicals a+b√−1, √/a-b√—1. in which, whatever be the sign of a, the value of x2 is positive, and that of y2 negative. Now, since x2=2a+2√/a2+b2, x=+√/2a+2√✅a2+b2 and since y=2a-2/a2+b2, y=+√−2a+2√a2 +b2 √—) } 7 Substituting for x, y in these expressions the values of x, y, equation 7, √a+b√−1={√2a+2√/a2+b2+{√−2a+2√ a2+b2 √−1 √a-b√-1/2a+2√a2+b2 — }√−2a+2√/a2+b2 √−1 from 8 9 If the radical expressions, a+b√-1, Va+b√√-1, a+b√-1, &c. are considered, it is evident that the extraction of a root whose index is a power of 2, may be replaced by successive extractions of the square root; and that, by consequence, the repeated employment of formule 8 and 9 must always reduce expressions such as the preceding to the form A+B√−1. BINOMIAL THEOREM, THE EXPONENT BEING FRACTIONAL OR NEGATIVE. 174. Let the expressions A+B+Cx2+Dx3+ A'+B'x+C+D'x3+ ... (in which A, A', B, B', &c. do not contain x) be equal to each other, whatever the value of x, then, if x=0, the terms which contain x are reduced to zero, and A=A'. Again, since A=A', Bx+Cx2+Dx3+ =B'x+C'x2+D′x3+ .. .... ... Dividing all the terms of this equality by x, B+Cx+Dx2+ . . . =B ́+C ́x+D2x2+ Therefore, if x=0, B=B′; and, similarly, C=C', D=D'. Wherefore, if two polynomial expressions, which are arranged according to the powers of x, are equal, whatever be the value of x, the coefficients of the same powers of x in the two expressions are equal. m 175. a+ being a binomial, and a positive fractional exponent, (a+x)"="√(a+x)"=√ √ n Then, since the first and second terms of the nth root of the quantity under the radical sign are a", a r, and the succeeding terms are of the -3 form Aa" x2, Ba" x3, &c. (A, B, . . denoting coefficients which contain neither a nor x), the development of (a+x) is of the form When the exponent is any negative number, integral or fractional, (a+x)== 1 1 ́a”+maTM¬1x+m(m−1)aTM−2x2, &c. 1.2 Performing the division indicated in this expression, the first and second terms of the quotient are a", -ma-x; and the quotient, which is composed of an indefinite number of terms, is of the form a ̄”—ma ̄”—1x+Aa¬~2x2+Ba ̄”¬3μ3+ The first and second terms of the development (a+x)" being aTM and ma"-1x, and the third, fourth, &c. being of the form Aa-2x2, Ba®-3r3, &c., it appears that if m denote any exponent, positive or negative, integral or fractional, the development of (a+x)" may be put under the form in which the first and second terms are determined; and it is necessary to find the coefficients A, B, . . . To render the investigation more general, let (a+x)"a"+ma"-1x+ +Ma"""+Na+1+. . « Ma"-"x", Na"-"-1"+1, being two consecutive terms of any rank. If x is changed into x+y (as the indeterminate coefficients contain neither a nor r) the result is (a+x+y)=a+ma"-1(x+y)+ Na"-"-1(x+y)+1+ And if a is changed into a+y, the result is (a+x+y)=(a+y)"+m(a+y)"1x+ N(a+y)" "+1+ +Ma* ̄*(x+y)*+}- 1 +M(a+y)=~***+ } 2 The first members of equalities 1, 2 are identical, therefore the second members must be equal, whatever the values of x, y, and also identical, if arranged according to the powers of y. The second members involve certain powers of the binomials x+y, a+y; but the first and second terms of each of these binomials are known, so that the polynomial coefficient of the first power of y can be formed in both the second members, and this is sufficient. The parts of the binomials which contain the first power of y are, Equation 1. maTM~ly from the term 'maTM~1(x+y). Mnaly MaTM~"(x+y)". * from the term (a+y)”. Equation 2. ma"-ly M(m—n)a”—n—1x′′y 'M(a+y)="x". N(m-n-1)a-"-22"+1y N(a+y)+1. Denoting the assemblage of these parts by Yy, Y'y respectively, Yy=(ma-1+. ·+Mna"-""1+N(n+1)a”—”−1x" +....)y, Y'y=(ma"-1+...+M(m—n)aTM~~~1x2+N(m—n−1)a”—n−2x”+1+........)y. These quantities, being the coefficients of the same power of y, are equal; and since they must be equal, whatever be the value of x, the coefficients of the same powers of x in each of the expressions Yy, Y'y are also equal (Art. 174); therefore N(n+1) the coefficient of a"-"-1" in the first expression is equal to M(m-n) the coefficient of a"-"-1" in the second. And since N(n+1)=M(m-n), Now M, N are the coefficients of two consecutive terms; m-n is the exponent of a, and n the exponent of x, in the term whose coefficient is M; it therefore appears that the coefficient of any term is formed from the coefficient of the preceding term by the rule (Art. 141) for the case of an integer positive exponent. Whence, since the first and second terms of the development are composed in the same manner, whatever be the exponent, it follows that the coefficients of all the terms of a binomial are formed by the rule of Article 141, whether the exponent be positive or negative, integer or fractional. That is, When m is integer and positive, the last term of the formula is xTM; in all other cases the number of terms is infinite. 176. Writing the development of (x+a)" as in Article 144, ma m(m-1) a2 m(m−1)(m—2) a3 (x+a)=(1+ it 1 and replacing m by, (x+a)" becomes (x+a)" or √x+a. 1 a 1-1 a2, 1—1—2 a2 + Nx n 2 x2 n 2 3x2+, &c. Reducing the coefficients, √x+a=x*(1+ 1 a 12-1 &c. n 2 1n-1 a2 1 n-1 2n-1 a3 nx n 2n x2+ n 2n 3nx3+, &c.) The fifth term of this expression is evidently obtained by multiplying the As an application of the binomial formula thus reduced, let it be required to find the cube root of 31: 2560 11 4 112640 and the sixth term= 430467211527-17433922005 Reducing the terms to decimals, and forming their algebraic sum, +3·00000 = +0.14815 -0.00731 2187 To determine the degree of approximation obtained by taking any number of terms of a series like the preceding, in which the terms are alternately positive and negative, and the absolute value of each term is less than that of the preceding term, Let the series be expressed by a−b+c―d+e—f+g-h..... and let ≈ denote the value of the series. Taking the two consecutive sums, a-b+c-d, the terms which come after a-b+c−d are +(e−ƒ)+(g—h)+(k−1)+, &c. and, since the series is decreasing, the differences e-f, g-h, k-l, &c. are positive numbers; whence, to obtain the complete value of x, it is necessary to add an absolute number to the sum a-b+c-d; therefore a-b+c-d<x. Again, the terms which follow +e are (−ƒ+g), (−l+m), &c.; and the differences g-f, m-l are negative. Consequently, to obtain the true value of r it is necessary to add a negative quantity to the sum, a-b+c-d+e, or to diminish this sum by an absolute number; therefore a-b+c=d+e>x. Consequently the numerical value of x is comprehended between any two consecutive sums of the terms of the series. Since the numerical value of the difference of the sums a-b+c-d and a-b+c-d+e is e, it follows that the error committed by taking a certain number of terms, a-b+c-d, for the value of x is numerically less than e, the term which follows the last term taken. Hence the error of the value 31-314138, to form which five terms have been taken, is less than the sixth term 112640 17433922005 =0·0000064+. SECTION VIII. OF ARITHMETICAL PROPORTION, RATIO, GEOMETRICAL PROPORTION, VARIATION, ARITHMETICAL AND GEOMETRICAL PROGRESSION, AND PILES OF BALLS. OF ARITHMETICAL PROPORTION. 177. If the difference between two quantities, a, b, is equal to the difference between two other quantities, c, d, the four quantities a, b, c, d, form an arithmetical proportion or equi-difference. This relation of the quantities a, b, c, d, is expressed thus, a.b.c.d; or by the equation a-b-c-d. By transposing the terms b, d, this equation becomes a+d=b+c. Consequently the sum of the extreme terms of an arithmetical proportion is equal to the sum of the mean terms. a. Again, from a+d=b+c or b+c=a+d is obtained, by transposing the terms a, c, b-a=d-c. Therefore, when four quantities are such that the sum of any two is equal to the sum of the other two, the first two are the extremes, and the second the means; or the first two are the means, and the second the extremes of an equi-difference or arithmetical proportion. Also, from b-a-d-c, is obtained, by transposing b, c, c-a-d-b. Whence the extreme and mean terms may be interchanged without destroying the equi-difference. b. When b=c, the proportion is termed continued. In this case the equi-difference a-b-c-d becomes a-b-b-d. .'.a+d=b+b=2b. Whence, in a continued arithmetical proportion, the sum of the extreme terms is equal to twice the mean term. OF RATIOS. 178. The term Ratio is employed to express the relation which exists between two quantities of the same kind with respect to magnitude. The first quantity is termed the antecedent, and the second the consequent, of the ratio. As in arithmetic (Part I. Art. 329) a ratio, that of a to b, for example, is expressed by the symbols a: b, or by writing the antecedent and consequent as the numerator and denominator of a fraction. The theory of ratios becomes thus reduced to the theory of fractions. 179. Since all fractions, whose terms are like multiples or like parts of the terms of a proposed fraction, are equal to that fraction, it follows that if one antecedent is the same multiple part or parts of its consequent that another antecedent is of its consequent, the ratios are equal. Consequently, if the terms of a ratio are both multiplied or both divided by the same quantity, the ratio is not altered. Ratios are compared with each other by comparing the fractions by which they are denoted. The ratios of 3 to 5 and of 7 to 8 are expressed by the These fractions are compared, in respect of their magni3 24 7 35 5-40 and 8 40 3 fractions and 7 tude, by reducing them to the common denominator 40,= The second being the greater, the ratio of 7 to 8 is greater than the ratio of 3 to 5. |