Elements of Algebra

38 ELEMENTS SECT. I. evident that if the root have the sign $+$, that is to say, if it be a positive number, its square must necessarily be a positive number also, because $+$ multiplied by $+$ makes $+$: hence the square of $+a$ will be $+aa$: but if the root be a negative number, as $-a$, the square is still positive, for it is $+aa$. We may therefore conclude, that $+aa$ is the square both of $+a$ and of $-a$, and that consequently every square has two roots, one positive, and the other negative. The square root of $25$, for example, is both $+5$ and $-5$, because $-5$ multiplied by $-5$ gives $25$, as well as $+5$ by $+5$. CHAP. XII. Of Square Roots, and of Irrational Numbers resulting from them. 123. What we have said in the preceding chapter amounts to this; that the square root of a given number is that number whose square is equal to the given number; and that we may put before those roots either the positive, or the negative sign. 124. So that when a square number is given, provided we retain in our memory a sufficient number of square numbers, it is easy to find its root. If $196$, for example, be the given number, we know that its square root is $14$. Fractions, likewise, are easily managed in the same way. It is evident, for example, that $\frac{4}{7}$ is the square root of $\frac{16}{49}$; to be convinced of which, we have only to take the square root of the numerator and that of the denominator. If the number proposed be a mixed number, as $12\frac{1}{4}$, we reduce it to a single fraction, which, in this case, will be $\frac{49}{4}$; and from this we immediately perceive that $\frac{7}{2}$, or $3\frac{1}{2}$, must be the square root of $12\frac{1}{4}$. 125. But when the given number is not a square, as $12$, for example, it is not possible to extract its square root; or to find a number, which, multiplied by itself, will give the product $12$. We know, however, that the square root of $12$ must be greater than $3$, because $3 \times 3$ produces only $9$; and less than $4$, because $4 \times 4$ produces $16$, which is more than $12$; we know also, that this root is less than $3\frac{1}{2}$, for we have seen that the square of $3\frac{1}{2}$, or $\frac{7}{2}$, is $12\frac{1}{4}$; and we may approach still nearer to this root, by comparing it with $3\frac{4}{7}$; for the square of $3\frac{4}{7}$, or of $\frac{25}{7}$, is $\frac{625}{49}$, or $12\frac{37}{49}$; so that this