Editing Normal number (section)

===Non-normal numbers===
No [[rational number]] is normal in any base, since the digit sequences of rational numbers are [[Repeating decimal|eventually periodic]].

{{harvs|last=Martin|year=2001|txt}} gives an example of an irrational number that is absolutely abnormal.{{sfn|Bugeaud|2012|page=113}}  Let

<math display="block">f\left(n\right) = \begin{cases}
n^\frac{f\left(n-1\right)}{n-1}, & n\in\mathbb{Z}\cap\left[3,\infty\right) \\
4, & n = 2
\end{cases}
</math>

<math display="block">\begin{align} & \alpha = \prod_{m=2}^\infty \left({1 - \frac{1}{f\left(m\right)}}\right) = \left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)\left(1-\frac{1}{64}\right)\left(1-\frac{1}{152587890625}\right)\left(1-\frac 1{6^{\left(5^{15}\right)}}\right)\ldots = \\ &=0.6562499999956991\underbrace{99999\ldots99999}_{23,747,291,559}8528404201690728\ldots\end{align}</math>

Then &alpha; is a [[Liouville number]] and is absolutely abnormal.