This tag is used for questions about normal, simply normal, and absolutely normal numbers. Discussions regarding normality, simply normality, or absolutely normality of numbers are included (e.g., "Prove that a particular number is not absolutely normal").
A real number $x$ is said to be simply normal in a given base $b\in\mathbb{Z}_{>1}$ if, for each integer $t$ such that $0\leq t<b$, $t$ occurs in the sequence of the digits of $x$ in the base $b$ with the natural density $\dfrac{1}{b}$. For example, in the base $b=10$, $$x=0.\dot{0}12345678\dot{9}$$ is simply normal in the base $10$. On the other hand, $$x=0.\dot{1}$$ is clearly not simply normal in the base $10$.
We say that $x$ is normal in the base $b$, all possible string of a given length $n\in\mathbb{Z}_{>0}$ occurs in the base-$b$ representation of $x$ with the natural density $\dfrac{1}{b^n}$. The example $$x=0.\dot{0}12345678\dot{9}$$ is simply normal, but not normal, in the base $10$. On the other hand, Champernowne's constant $$C_{10}=0.1234567891011121314151617181920212223242526272829\ldots$$ (obtained by concatenating the representations of the natural numbers in order) is normal in the base $10$. It is known that almost all real numbers are normal in a given base $b$ (i.e., the set of all non-normal real numbers in the base $b$ has Lebesgue measure zero).
We say that $x$ is absolutely normal if it is normal in all bases $b\in\mathbb{Z}_{>1}$. Sometimes, an absolutely normal number is also called a normal number. Becher and Figueira proved that there exists a computable absolutely normal number.
A number is absolutely non-normal or absolutely abnormal if it is not simply normal in any base. If $f(2):=4$ and $f(n):=n^{\frac{f(n-1)}{n-1}}$ for every integer $n\geq 3$, then the number $$\alpha:=\prod_{m=2}^\infty\,\left(1-\frac{1}{f(m)}\right)$$ is known to be absolutely abnormal.
