chapter  18
8 Pages

Most Numbers are Normal, but it’s Tough to Find One

WithMalcolm E Lines

If you look at a long run of digits in a number like pi or the square root of two, it seems as though each numeral occurs about as frequently as any other. Lacking any specific reason for supposing otherwise, one would not therefore be unduly surprised if a careful statistical analysis of, say, the first one million digits of pi showed that there were about 100,000 ones, 100,000 twos and so on. This does, in fact, appear to be the case. Thus, although the numeral 1 (for example) appears in pi in a most unorderly fashion (showing up in the 1st, 3rd, 37th, 40th, 49th, 68th, 94th, 95th, …. decimal places) it does appear approximately a fixed fraction of the time on average, and this fraction is, of course, one tenth. For the other numerals this same overall frequency of appearance also seems to occur, although there are some unusual patterns of digits in the first one million decimal places of pi, such as the seven consecutive threes which follow the 710,099th digit.