Elements of Algebra

42 ELEMENTS SECT. I. dition and subtraction, because we only connect the numbers by the signs $+$ and $-$: for example, $\sqrt{2}$ added to $\sqrt{3}$ is written $\sqrt{2} + \sqrt{3}$; and $\sqrt{3}$ subtracted from $\sqrt{5}$ is written $\sqrt{5} - \sqrt{3}$. 138. We may observe, lastly, that in order to distinguish the irrational numbers, we call all other numbers, both integral and fractional, rational numbers; so that, whenever we speak of rational numbers, we understand integers, or fractions. CHAP. XIII. *Of Impossible, or Imaginary Quantities, which arise from the same source.* 139. We have already seen that the squares of numbers, negative as well as positive, are always positive, or affected by the sign $+$; having shewn that $-a$ multiplied by $-a$ gives $+aa$, the same as the product of $+a$ by $+a$: wherefore, in the preceding chapter, we supposed that all the numbers, of which it was required to extract the square roots, were positive. 140. When it is required, therefore, to extract the root of a negative number, a great difficulty arises; since there is no assignable number, the square of which would be a negative quantity. Suppose, for example, that we wished to extract the root of $-4$; we here require such a number as, when multiplied by itself, would produce $-4$: now, this number is neither $+2$ nor $-2$, because the square both of $+2$ and of $-2$, is $+4$, and not $-4$. 141. We must therefore conclude, that the square root of a negative number cannot be either a positive number or a negative number, since the squares of negative numbers also take the sign *plus*: consequently, the root in question must belong to an entirely distinct species of numbers; since it cannot be ranked either among positive or negative numbers. 142. Now, we before remarked, that positive numbers are all greater than nothing, or $0$, and that negative numbers are all less than nothing, or $0$; so that whatever exceeds $0$ is expressed by positive numbers, and whatever is less than $0$ is expressed by negative numbers. The square roots of negative numbers, therefore, are neither greater nor less than nothing; yet we cannot say, that they are $0$; for $0$