Elements of Algebra

4 ELEMENTS SECT. I. same thing if we subtract their sum, which is $25$, at once from $50$, and the remainder will be $25$ as before. 13. It is also easy to determine the value of similar expressions, in which both the signs $+$ plus and $-$ minus are found. For example; $12 - 3 - 5 + 2 - 1$ is the same as $5$. We have only to collect separately the sum of the numbers that have the sign $+$ before them, and subtract from it the sum of those that have the sign $-$. Thus, the sum of $12$ and $2$ is $14$; and that of $3$, $5$, and $1$, is $9$; hence $9$ being taken from $14$, there remain $5$. 14. It will be perceived, from these examples, that the order in which we write the numbers is perfectly indifferent and arbitrary, provided the proper sign of each be preserved. We might with equal propriety have arranged the expression in the preceding article thus; $12 + 2 - 5 - 3 - 1$, or $2 - 1 - 3 - 5 + 12$, or $2 + 12 - 3 - 1 - 5$, or in still different orders; where it must be observed, that in the arrangement first proposed, the sign $+$ is supposed to be placed before the number $12$. 15. It will not be attended with any more difficulty if, in order to generalise these operations, we make use of letters instead of real numbers. It is evident, for example, that $a - b - c + d - e$, signifies that we have numbers expressed by $a$ and $d$, and that from these numbers, or from their sum, we must subtract the numbers expressed by the letters $b, c, e$, which have before them the sign $-$. 16. Hence it is absolutely necessary to consider what sign is prefixed to each number: for in Algebra, simple quantities are numbers considered with regard to the signs which precede, or affect them. Farther, we call those positive quantities, before which the sign $+$ is found; and those are called negative quantities, which are affected by the sign $-$. 17. The manner in which we generally calculate a person's property, is an apt illustration of what has just been said. For we denote what a man really possesses by positive numbers, using, or understanding the sign $+$; whereas his debts are represented by negative numbers, or by using the sign $-$. Thus, when it is said of any one that he has $100$ crowns, but owes $50$, this means that his real possession amounts to $100 - 50$; or, which is the same thing, $+100 - 50$, that is to say, $50$. 18. Since negative numbers may be considered as debts, because positive numbers represent real possessions, we