A calculus of the absurd

4.2 The root function

One useful property of the root function is that any root of \(a\) multiplied by \(b\) is the same as the root of \(a\) multiplied by the square root of \(b\).

In the case where we are interested in the square root, we have

\begin{equation} \sqrt {ab} = \sqrt {a} \cdot \sqrt {b} \end{equation}

This makes it possible to simplify expressions inside square root, for example

\begin{align} \sqrt {8} & = \sqrt {4}\sqrt {2} \\ & = 2\sqrt {2} \end{align}

4.2.1 Surds

A "surd" is an irrational square root (i.e. a number which cannot be expressed in the form \(\frac {p}{q}\) where \(p\) and \(q\) are positive integers1212 The set of positive integers contains \(1, 2, 3, 4\), and so on).

For example, \(\sqrt {2}\), \(\sqrt {3}\), \(\sqrt {5}\) are all irrational numbers. We can prove this (and there is in fact a proof of this in 5.3.1).