### Theory:

The decimal expansion of an irrational number is non-terminating and non-recurring. Conversely, a decimal expansion of a number is non-terminating and non-recurring is an irrational number.
Example:
Many square roots and cube roots are irrational numbers.

$\begin{array}{l}\sqrt{3}\phantom{\rule{0.147em}{0ex}}=1.7320508075...\\ \\ \sqrt{5}\phantom{\rule{0.147em}{0ex}}=2.2360679774...\phantom{\rule{0.147em}{0ex}}\\ \\ \sqrt[3]{3}=1.4422495707...\end{array}$
Some of the famous irrational numbers:

 Irrational Number Its non-terminating and non-recurring decimal value $\mathrm{\pi }\phantom{\rule{0.147em}{0ex}}\left(\mathit{pi}\right)$ $$3.141592653589793$$... $\mathrm{e}\phantom{\rule{0.147em}{0ex}}\left(\mathit{Eulers}\phantom{\rule{0.147em}{0ex}}\mathit{number}\right)$ $$2.718281828459045$$... $\mathrm{\varphi }\phantom{\rule{0.147em}{0ex}}\left(\mathit{Golden}\phantom{\rule{0.147em}{0ex}}\mathit{ratio}\right)$ $$1.618033988749894$$...