Do irrational number contain infinite/every patterns of sequences?

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exorteygrdh · Accepted Answer

No, this is not the case for every irrational number. For example, the number

1.01001000100001000001000000100...

where each run of zeroes is one longer, is clearly irrational, since the decimal expansion never repeats. But it just as clearly doesn't contain every pattern of digits, because the only digits it contains are 0 and 1.

π

in particular is suspected (but not proved) to satisfy a stronger property, namely that it is normal, which means that not only does every pattern of digits occur, but every pattern occurs infinitely many times, with the frequency one would assume in a random string of digits.

In a certain technical sense, most numbers are normal, but there are very few expressions that have been proved to produce a normal number.

Do irrational number contain infinite/every patterns of sequences?

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