that of representing the operations of addition, subtraction, &c., by means of certain signs, renders both the language and algorithm of this science extremely simple and commodious. Besides the advantages which the algebraic method of notation possesses over that of numbers, it may be observed, that even in this early part of the science we are furnished with the means of obtaining several general theorems that could not be well established by the principles of Arithmetic. 102. The greater of any two numbers is equal to half their sum added to half their difference, and the less is equal to half their sum minus half their difference. Let a and b be any two numbers, of which a is the greater; let their sum be represented by s; and their difference by d: Then, a+b=s1 Cor. 1. Hence if the sum and difference of any two numbers be given, we can readily find each of the numbers; thus, if s be equal to the sum of two numbers, and d equal to the difference; then the general expression for the first, isS d second s+d 2 2 and for the Whatever may be the numeral values that we assign to s and d, or whatever valucs these letters must represent in a particular question, we have but to substitute them in the above expressions, in order to ascertain the numbers required: For example. Given the sum of two numbers equal to 36, and the difference equal to 8: Then, by substituting 36, for s, and 8 for d, in s+d S -d and 2 36+8 we have s+d 44 =22, and 2 2 2 2 Cor. 2. Also, if it were required to divide the number s into two such parts, that the first will exceed the second by d. It appears evident, that the s+d general expression for the first part is and for " 2 the second whatever. S d ;s and d representing any numbers 2 103. The general expression, s+d 2 may be found after the manner of Garnier. Thus, let represent the first part; then according to the enunciation of the question, x-d will be the second; and, as any quantity is equal to the sum of all its parts, we have therefore, xxds, or 2x-d=s. This equality will not be altered, by adding the number to each member, and then it becomes, Bx-d+d=s+d, or 2x=s+d; dividing each member by 2, we have the equality, 8+d 2 ; in which we read that the number sought is equal to half the sum of the two numbers s and d; thus the relation between the unknown and known numbers remaining the same, the question is resolved in general for all numbers s and d. 104. We have not here the numerical value of the unknown quantity; but the system of operations that is to be performed upon the given quantities; in order to deduce from them, according to the conditions of the problem, the value of the quantity sought; and the expression that indicates these operations, is called a formula. It is thus, for example, that if we denote by a the tens of a number, and the units by b, we have this constant composition of a square, or this formula, a2+2ab+b2; this algebraic expression is a brief enunciation of the rules to be pursued in order to pass from a number to its square. 105. From whence we infer that, if a number be divided into any two parts, the square of the number is equal to the square of the two parts, together with twice the product of those parts. Which may be demonstrated thus; let the number n be divided into any two parts a and b ; Then n=a+b, and n=a+b; .. by Multiplication, n2=a2+2ab+b2 (Art. 50). 106. If the sum and difference of any two numbers or quantities be multiplied together, their product gives the difference of their squares, observing to take with the sign that of the two squares whose root is subtracted. Let м and N represent any two quantities, or polynomials whatever, of which м is the greater; then (M+N)X(M-N) is equal to м2-N2; for the operation stands thus; (M+N)×(M—N)=M2+MN -MN-N2 } 107. When we put м=a3, and N=63; then, · (·a3+b3) × (a3 —b3)=a®—b® ; (See Ex. 9. page 41). Where a is the square of a3, and bo that of b3, and this last square is subtracted from the first. Reciprocally, the difference of two squares M2 —N2, can be put under the form (M+N) × (M—N). This result is a formula that should be remembered. 108. The difference of any two equal powers of differ ent quantities is always divisible by the difference of their roots, whether the exponent of the power be even or odd. For since -a5 XX &c. =x*+ax3+a2x2+a3x+x1; · = x2 + a x 1 + a 2 x3 + a3 x2 + a1x + α 3 ; &c. C We may conclude that in general, "a" is divisible by x-a, m being an entire positive number; that is, m -am -a m +am-2x+am-1... (1), 109. The difference of any two equal powers of differ ent quantities, is also divisible by the sum of their roots, when the exponent of the power is an even number. For since Hence we may conclude that, in general, x2m--a2m 2m-1. -ax2m-2+.. +a2m-2x-a2m-1 x+a 110. And the sum of any two equal powers of different • quantities, is also divisible by the sum of their roots, when the exponent of the power is an odd number. For since 3 x + a x5+as x+u 3 = x2-ax+x2; = x2-ax3+a2x2 —a3x+a1 ; Ilence, we may conclude that, in general, 111. In the formulæ (1), (2), (3), as well as in all others of a similar kind, it is to be observed, that if m be any whole number whatever, 2m will always be an even number, and 2m+1 an odd number; so that, 2m is a general formula for even numbers, and 2m+1 for odd numbers. 112. Also, if a in each of the above formula, be taken 1, and x being always considered greater than a; they will stand as follows: xm -1 -- m =xm−1+xm−2+xm−3+ X- -1 x2m 1 -=x2n-1 -x2 |
