Elements of Algebra

76 ELEMENTS SECT. II. SECTION II. Of the different Methods of calculating Compound Quantities. CHAP. I. Of the Addition of Compound Quantities. 256. When two or more expressions, consisting of several terms, are to be added together, the operation is frequently represented merely by signs, placing each expression between two parentheses, and connecting it with the rest by means of the sign $+$. Thus, for example, if it be required to add the expressions $a+b+c$ and $d+e+f$, we represent the sum by $(a+b+c) + (d+e+f)$. 257. It is evident that this is not to perform addition, but only to represent it. We see, however, at the same time, that in order to perform it actually, we have only to leave out the parentheses; for as the number $d+e+f$ is to be added to $a+b+c$, we know that this is done by joining to it first $+d$, then $+e$, and then $+f$; which therefore gives the sum $a+b+c+d+e+f$; and the same method is to be observed, if any of the terms are affected by the sign $-$; as they must be connected in the same way, by means of their proper sign. 258. To make this more evident, we shall consider an example in pure numbers, proposing to add the expression $15-6$ to $12-8$. Here, if we begin by adding $15$, we shall have $12-8+15$; but this is adding too much, since we had only to add $15-6$, and it is evident that $6$ is the number which we have added too much; let us therefore take this $6$ away by writing it with the negative sign, and we shall have the true sum, $12-8+15-6$; which shews that the sums are found by writing all the terms, each with its proper sign.