Elements of Algebra

CHAP. XII. OF ALGEBRA. 39 fraction is still greater than the root required, though but very little so, as the difference of the two squares is only $\frac{4}{225}$. 126. We may suppose that as $3\frac{1}{2}$ and $3\frac{2}{3}$ are numbers greater than the root of $12$, it might be possible to add to $3$ a fraction a little less than $\frac{2}{3}$, and precisely such, that the square of the sum would be equal to $12$. Let us therefore try with $3\frac{2}{5}$, since $\frac{2}{5}$ is a little less than $\frac{1}{2}$. Now $3\frac{2}{5}$ is equal to $\frac{17}{5}$, the square of which is $\frac{289}{25}$, and consequently less by $\frac{11}{25}$ than $12$, which may be expressed by $\frac{300}{25}$. It is, therefore, proved that $3\frac{2}{5}$ is less, and that $3\frac{1}{2}$ is greater than the root required. Let us then try a number a little greater than $3\frac{2}{5}$, but yet less than $3\frac{1}{2}$; for example, $3\frac{5}{11}$; this number, which is equal to $\frac{38}{11}$, has for its square $\frac{1444}{121}$; and by reducing $12$ to this denominator, we obtain $\frac{1452}{121}$ which shews that $3\frac{5}{11}$ is still less than the root of $12$, viz. by $\frac{8}{121}$; let us therefore substitute for $\frac{1}{2}$ the fraction $\frac{6}{13}$, which is a little greater, and see what will be the result of the comparison of the square of $3\frac{6}{13}$, with the proposed number $12$. Here the square of $3\frac{6}{13}$ is $\frac{2025}{169}$; and $12$ reduced to the same denominator is $\frac{2028}{169}$; so that $3\frac{6}{13}$ is still too small, though only by $\frac{3}{169}$, whilst $3\frac{1}{2}$ has been found too great. 127. It is evident, therefore, that whatever fraction is joined to $3$, the square of that sum must always contain a fraction, and can never be exactly equal to the integer $12$. Thus, although we know that the square root of $12$ is greater than $3\frac{6}{13}$, and less than $3\frac{1}{2}$, yet we are unable to assign an intermediate fraction between these two, which, at the same time, if added to $3$, would express exactly the square root of $12$; but notwithstanding this, we are not to assert that the square root of $12$ is absolutely and in itself indeterminate: it only follows from what has been said, that this root, though it necessarily has a determinate magnitude, cannot be expressed by fractions. 128. There is therefore a sort of numbers, which cannot be assigned by fractions, but which are nevertheless determinate quantities; as, for instance, the square root of $12$: and we call this new species of numbers, *irrational numbers*. They occur whenever we endeavour to find the square root of a number which is not a square; thus, $2$ not being a perfect square, the square root of $2$, or the number which multiplied by itself would produce $2$, is an irrational quantity. These numbers are also called *surd quantities*, or *incommensurables*. 129. These irrational quantities, though they cannot be expressed by fractions, are nevertheless magnitudes of which we may form an accurate idea; since, however concealed