A calculus of the absurd

4.5 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 integers1313 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 5.4.1).

4.5.1 Rationalising the denominator

Sometimes surds can be the same, even if it’s not immediately apparent. For example

\begin{equation} \frac {1}{\sqrt {2}} = \frac {\sqrt {2}}{2} \end{equation}

Why? We can just multiply by \(1\),

\begin{align} \frac {1}{\sqrt {2}} & = \frac {1}{\sqrt {2}} \cdot \frac {\sqrt {2}}{\sqrt {2}} \\ & = \frac {\sqrt {2}}{\sqrt {2}\sqrt {2}} \\ & = \frac {\sqrt {2}}{(\sqrt {2})^2} \\ & = \frac {\sqrt {2}}{2} \end{align}

Note that through this process we have removed the irrational part from the denominator or, alternatively, we have rationalised the denominator.

We can also do this in the case where we have slightly more complex expressions in the form

\begin{equation} \frac {a \pm \sqrt {b}}{c\pm \sqrt {d}} \end{equation}

For example, we might want to make the denominator of one of these expressions rational

\begin{align} & \frac {3}{7-\sqrt {11}} \\ & \frac {4 - \sqrt {2}}{\sqrt {3} + 44} \\ & \frac {1 + \sqrt {5}}{1 + \sqrt {2}} \end{align}

The solution here is to use the difference of two squares (see Section 4.3.3 if unsure on what this is).

\begin{align} \frac {1 + \sqrt {5}}{1 + \sqrt {2}} & = \frac {1 + \sqrt {5}}{1 + \sqrt {2}} \frac {1 - \sqrt {2}}{1 - \sqrt {2}} \\ & = \frac {(1+\sqrt {5})(1-\sqrt {2})}{(1+\sqrt {2})(1-\sqrt {2})} \\ & = \frac {1 - \sqrt {2} + \sqrt {5} - \sqrt {10}}{1^2 - (\sqrt {2})^2} \\ & = \frac {1 - \sqrt {2} + \sqrt {5} - \sqrt {10}}{-1} \end{align}