With regard to
real numbers:
A real number x is normal in base b if in its representation in base b all digits occur, in an
asymptotic sense, equally often. In addition, for each m, the bm different m-strings must occur
equally often. In other words, lim n->infinity N(s,n)/n = b-m for each m-string s, where N(s,n) is
the number of occurrences of s in the first n base-b digits of x. A number that is normal in all bases
is called normal.
Often a person will ask whether the decimal expansion of some real number is truly random. This is a misguided question when applied to such mathematical constants as pi; since they are by definition not random. The question is rather whether or not they are distinguishable by inspection from a truly random number sequence, and whether or not it is normal is one part of answering that question. This concept was formalized by E. Borel in 1909, who also proved that there are lots and lots of normal numbers. The conjecture that pi is normal remains unproven, though it is widely believed and has held true for all digits yet computed.