Main Concept

A numeral system is a way of representing real numbers as an ordered sequence of symbols called digits from a finite ordered set. The number, $b$, of symbols in this set is called the base. The symbols themselves represent the number zero, followed by the first $b-1$ positive integers.

For any base $b$, any real number can be written as a sum of the form $\sum _{i\&equals;-\mathrm{\&infin;}}^n{x}_i\mathbf{}{b}^i$, where $0 \le x_i<b$.

The corresponding base $b$ representation of this number is:

$x_n\mathbf{}{x}_{n-1}\cdot \cdot \cdot x_1\mathbf{}{x}_{0\mathbf{·}}{x}_{-1}{x}_{-2}\mathbf{}\cdot \cdot \cdot$

The standard numeral system around the world is the base ten decimal system, which uses the digits {0,1,2,3,4,5,6,7,8,9}. Systems using a base other than ten are used commonly in computing, including:

In number theory, a real number $\mathit{x}$ is called normal in base $\mathit{b}$ if the sequence of digits in its representation in base $b$ appears random, in the following sense: The density of any length $k$ digit subsequence $x_{i\&plus;1}\cdot \cdot \cdot x_{i\&plus;k}$  in the representation of $x$ is $b^{-k}$. The number $x$ is normal if it is normal in every base $b\&gt;1$.

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