Base and Normality - Maple Help

Base and Normality

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 ${\sum }_{i=-\mathrm{∞}}^{n}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{x}_{i}\mathbf{}{b}^{i}$, where .

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:

 • binary (base two), binary digits (bits) = {0,1}
 • octal (base eight), octal digits = {0,1,2,3,4,5,6,7}

In number theory, a real number $\mathbit{x}$ is called normal in base $\mathbit{b}$ 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 ${x}_{i+1}\cdot \cdot \cdot {x}_{i+k}$  in the representation of $x$ is ${b}^{-k}$. The number $x$ is normal if it is normal in every base $b>1$.

Input a Maple expression in the box below (or choose one from the drop-down box) that evaluates to a real number. Choose a base b > 1, and Maple will find the base b representation for your number. Use the slider to adjust the number of significant figures, and look at the graph to see if your number is normal in that base.

 Pi22/7sqrt(2)exp(1)ln(3)Other base = # of significant figures =





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