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$\displaystyle{ \Large \frac{a}{b} }$
where $\displaystyle{ a }$ and $\displaystyle{ b }$ are integers and $\displaystyle{ b }$ is not zero. It can readily be shown that the irrational numbers are precisely those numbers whose expansion in any given base (decimal, binary, etc) never ends and never enters a periodic pattern, but no mathematician takes that to be a definition. Some examples of irrational numbers are $\displaystyle{ \sqrt{2} }$ or $\displaystyle{ e }$.