= Simply normal number
{title2=$\lim_{N\to\infty}F_N(a)=1/K$}
= Simple normality
{synonym}
A <real number> is simply normal in base $K$ if each individual digit in its <positional notation> has limiting <digit block frequency> $1/K$. This is the length-one part of the definition of a <normal number>. It places no equal-frequency requirement on longer <words>.
For example, $x=0.010101\ldots_2=1/3$ is a <simply normal number> in base two: each <binary digit> has frequency $1/2$. But $00$ and $11$ never occur, and $01$ and $10$ each have frequency $1/2$. Thus $x$ is not a <normal number> in base two. Even a <sequence> that looks more complicated, such as the deterministic <Thue–Morse sequence>, must be tested against the actual <digit block frequency> definition; visual complexity is no substitute for it.
The <decimal representation> $y=0.01234567890123456789\ldots$ makes the same point in base ten. Every digit has frequency $1/10$, yet the only adjacent pairs are
$$
01,12,23,34,45,56,67,78,89,90.
$$
Each has limiting <digit block frequency> $1/10$; the other ninety pairs have frequency zero. A <normal number> in base ten requires $1/100$ for every one of the hundred pairs. This <rational number> is a <simply normal number> in base ten, but it is not a <normal number> in that base.
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