Elements of Algebra

CHAP. I. OF ALGEBRA. 77 259. If it were required therefore to add the expression $d-e-f$ to $a-b+c$, we should express the sum thus; $a-b+c+d-e-f$; remarking, however, that it is of no consequence in what order we write these terms; for their places may be changed at pleasure, provided their signs be preserved; so that this sum might have been written thus; $c-e+a-f+d-b$. 260. It is evident, therefore, that addition is attended with no difficulty, whatever be the form of the terms to be added. Thus, if it were necessary to add together the expressions $2a^3+6\sqrt{b}-4\log. c$ and $5\sqrt[5]{a}-7c$, we should write them $2a^3+6\sqrt{b}-4\log. c+5\sqrt[5]{a}-7c$, either in this or in any other order of the terms; for if the signs are not changed, the sum will always be the same. 261. But it frequently happens that the sums represented in this manner may be considerably abridged, as is the case when two or more terms destroy each other; for example, if we find in the same sum the terms $+a-a$, or $3a-4a+a$; or when two or more terms may be reduced to one, &c. Thus, in the following examples: $3a+2a=5a$, $7b-3b=+4b$ $-6c+10c=+4c$, $4d-2d=2d$ $5a-8a=-3a$, $-7b+b=-6b$ $-3c-4c=-7c$, $-3d-5d=-8d$ $2a-5a+a=-2a$, $-3b-5b+2b=-6b$. Whenever two or more terms, therefore, are entirely the same with regard to letters, their sum may be abridged; but those cases must not be confounded with such as these, $2a^2+3a$, or $2b^3-b^4$, which admit of no abridgment. 262. Let us consider now some other examples of reduction, as the following, which will lead us immediately to an important truth. Suppose it were required to add together the expressions $a+b$ and $a-b$; our rule gives $a+b+a-b$; now $a+a=2a$, and $b-b=0$; the sum therefore is $2a$: consequently, if we add together the sum of two numbers $(a+b)$ and their difference $(a-b)$, we obtain the double of the greater of those two numbers. This will be better understood perhaps from the following examples: $3a-2b-c$ $a^3-2a^2b+2ab^2$ $5b-6c+a$ $-a^2b+2ab^2-b^3$ ---------- ----------------- $4a+3b-7c$ $a^3-3a^2b+4ab^2-b^3$ ---------- -----------------