Difference between revisions of "Normal number"
(just a start) |
(pi -> π) |
||
| (5 intermediate revisions by 2 users not shown) | |||
| Line 3: | Line 3: | ||
This concept can be taken to other [[base]]s: 1/3 is not simply normal to base 10, but simply normal to base 2, as its [[dual expansion]] is 0.01010101...<sub>2</sub>. | This concept can be taken to other [[base]]s: 1/3 is not simply normal to base 10, but simply normal to base 2, as its [[dual expansion]] is 0.01010101...<sub>2</sub>. | ||
| − | 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. |
| + | |||
| + | 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. | ||
| − | A number is called '''normal''', if it is normal for every base b. | + | A number is called '''normal''', if it is normal for every base b. It is unproven if [[pi|π]] is a normal number, but an analysis of the first trillion digits of π suggests that it is normal.<ref>https://phys.org/news/2016-03-pi-random-full-hidden-patterns.html</ref> |
==See further== | ==See further== | ||
*[https://mathworld.wolfram.com/NormalNumber.html Normal Number] at [[Wolfram Math World]] | *[https://mathworld.wolfram.com/NormalNumber.html Normal Number] at [[Wolfram Math World]] | ||
| + | |||
| + | [[Category:Number Theory]] | ||
Latest revision as of 09:12, August 8, 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. 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]