Irrational Numbers and Surds - SS3 Mathematics Lesson Note

Recall, that rational numbers are numbers that can be expressed in the form \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b \neq 0\). Numbers of form \(17\frac{1}{3} = 17.33333\ldots = 17.\dot{3}\) are called recurring decimals and are also considered to be rational. It is often possible to express recurring decimals in the form \(\frac{a}{b}\).

Example 1 Express \(4.\dot{3}\) as a rational number (in the form \(\frac{a}{b}\))

Solution

\[4.\dot{3} = 4.33333\ldots\]

Let this equal \(x\)

\(x = 4.\dot{3} = 4.33333\ldots\) [1]

Multiply both sides by 10

\(10x = 43.33333\ldots\) [2]

Subtract [1] from [2]

\[9x = 39\]

\[x = \frac{39}{9} = \ \frac{13}{3}\]

However, when we have to deal with non-terminating and non-recurring numbers, we are dealing with irrational numbers. For instance, \(\frac{1}{17} = 0.058823529411764705\ldots\) is a non-rational/irrational number. These numbers cannot be expressed in the form \(\frac{a}{b}\).

Square roots of all non perfect squares are irrational: \(\sqrt{2},\ \sqrt{3},\ \sqrt{5},\ \sqrt{6},\ \sqrt{7},\ \sqrt{8},\ \sqrt{10},\ \sqrt{11},\ \ldots\). Also, multiples of irrational numbers are irrational: \(3\sqrt{11}\), as well as fractions of irrational numbers \(\frac{5\sqrt{4}}{3}\).

So, \(\sqrt{9} = 3 = \frac{3}{1}\) which is a rational number but \(\sqrt{5} = \ 2.2360679774997\ldots\) is not! The best we can get when dealing with irrational numbers are approximations like \(\pi = \frac{22}{7} = 3.142\) or leaving them in the form \(\sqrt{n}\) called a surd. Surds can be added, subtracted, divided and multiplied just like normal numbers. Surds can be simplified as well by looking for factors that are perfect squares. Example, \(\sqrt{72} = \ \sqrt{2 \times 36} = \ \sqrt{2 \times 6^{2}} = \ \sqrt{2} \times \sqrt{6^{2}} = \ \sqrt{2} \times 6 = 6\sqrt{2}\)