This factor can evidently be removed and its root placed
before the radical as a coefficient. It is called the rational
factor, and the factor whose root cannot be extracted is
called the irrational factor.
232. When the coefficient of the surd is unity, it is said to be entire.
233. When the irrational factor is integral, and all rational factors have been removed, the surd is in its simplest form.
234. When surds of the same order contain the same irrational factor, they are said to be similar or like.
Thus 5 3, 2 3, 3 are like surds. But 3 2 and 2 3 are unlike surds.
235. In the case of numerical surds such as 2, 3 5, ... , although the exact value can never be found, it can be determined to any degree of accuracy by carrying the process of evolution far enough.
Thus 5 = 2.236068... ; that is 5 lies between 2.23606 and 2.23607; and therefore the error in using either of these quantities instead of 5 is less than .00001. By taking the root to a greater number of decimal places we can approximate still nearer to the true value.
It thus appears that it will never be absolutely necessary to introduce surds into numerical work, which can always be carried on to a certain degree of accuracy; but we shall in the present chapter prove laws for combination of surd quantities which will enable us to work with symbols such as 2, 3 5, 4 a, ... with absolute accuracy so long as the symbols are kept in their surd form. Moreover it will be found that even where approximate numerical results are required, the work is considerably simplified and shortened by operating with surd symbols, and afterwards substituting numerical values, if necessary.
