# 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

$$\sqrt {ab} = \sqrt {a} \cdot \sqrt {b}$$

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).