Randomness of digits of irrational numbers.

[Fallen Seraph]

How random are the digits of irrational numbers? Can it be said of them (i.e. pi=3.14159...) that given any arbitrarily long string of digits it must occur at some point in any irrational number? And would anyone know of anywhere I could find out more on this topic?

 

[HallsofIvy]

How random are the digits of irrational numbers? Can it be said of them (i.e. pi=3.14159...) that given any arbitrarily long string of digits it must occur at some point in any irrational number? And would anyone know of anywhere I could find out more on this topic?

Except for a few "made up" examples that are defined by using random numbers, no one really knows. In particular it is not known whether \pi or e or other familiar irrational numbers are "normal numbers": numbers such that every possible list of n numbers occurs, on average, 1/10ⁿ of the time: exactly what you would expect of a set of random numbers.

It can be shown, however, that, in a very specific sense, "almost all" numbers are normal! For more information, look up "normal numbers".

 

[Fallen Seraph]

Thanks a lot.