Normal number

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A number is called a simply normal number to base 10, if each of of the digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} appears with the same frequency in its decimal representation. Thus, the number 123,456,789/9,999,999,999 = 0.01234567890123456789... is simply normal.

This concept can be taken to other bases: 1/3 is not simply normal to base 10, but simply normal to base 2, as its dual expansion is 0.01010101...2.

A simply normal number to base 10 is a normal number to base 10, if not only single digits appear with the frequency of 1/10 in the long run, but each sequence of two digits {00,01,02,... ,98,99} can be found with the frequency of 1/100, each sequence of three digits with a frequency of 1/1000, and in general, each sequence of n digits will appear with a frequency of approximately 10-n when looking up more and more decimal places.

A simply normal number to base 10 has to be irrational - otherwise, a sequence of digits of some given length has to repeat itself forever - while some other sequences of this length will not appear at all.

Again, this concept can be taken to other bases.

A number is called normal, if it is normal for every base b. It is unproven if π is a normal number, but an analysis of the first trillion digits of π suggests that it is normal.[1]

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