Difference between revisions of "Normal number"
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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<sup>-n</sup> when looking up more and more decimal places. | 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<sup>-n</sup> when looking up more and more decimal places. | ||
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| + | 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. | Again, this concept can be taken to other bases. | ||
Revision as of 20:01, July 21, 2021
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.