Elements of Algebra

CHAP. II. OF ALGEBRA. 5 may say that negative numbers are less than nothing. Thus, when a man has nothing of his own, and owes $50$ crowns, it is certain that he has $50$ crowns less than nothing; for if any one were to make him a present of $50$ crowns to pay his debts, he would still be only at the point nothing, though really richer than before. 19. In the same manner, therefore, as positive numbers are incontestably greater than nothing, negative numbers are less than nothing. Now, we obtain positive numbers by adding $1$ to $0$, that is to say, $1$ to nothing; and by continuing always to increase thus from unity. This is the origin of the series of numbers called *natural numbers*; the following being the leading terms of this series: $0, +1, +2, +3, +4, +5, +6, +7, +8, +9, +10$, and so on to infinity. But if, instead of continuing this series by successive additions, we continued it in the opposite direction, by perpetually subtracting unity, we should have the following series of negative numbers: $0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10$, and so on to infinity. 20. All these numbers, whether positive or negative, have the known appellation of whole numbers, or *integers*, which consequently are either greater or less than nothing. We call them *integers*, to distinguish them from fractions, and from several other kinds of numbers of which we shall hereafter speak. For instance, $50$ being greater by an entire unit than $49$, it is easy to comprehend that there may be, between $49$ and $50$, an infinity of intermediate numbers, all greater than $49$, and yet all less than $50$. We need only imagine two lines, one $50$ feet, the other $49$ feet long, and it is evident that an infinite number of lines may be drawn, all longer than $49$ feet, and yet shorter than $50$. 21. It is of the utmost importance through the whole of Algebra, that a precise idea should be formed of those negative quantities, about which we have been speaking. I shall, however, content myself with remarking here, that all such expressions as $+1-1, +2-2, +3-3, +4-4, \text{&c.}$ are equal to $0$, or nothing. And that $+2-5$ is equal to $-3$: for if a person has $2$ crowns, and owes $5$, he has not only nothing, but still owes $3$ crowns. In the same manner, $7-12$ is equal to $-5$, and $25-40$ is equal to $-15$. 22. The same observations hold true, when, to make the expression more general, letters are used instead of numbers;