Notes

Chapter 4: Systems Based on Numbers

Section 5: Mathematical Constants


Leading digits [in numbers]

Even though in individual numbers generated by simple mathematical procedures all possible digits often appear to occur with equal frequency, leading digits in sequences of numbers typically do not. Instead it is common for a leading digit s in base b to occur with frequency Log[b, (s+1)/s] (so that in base 10 1's occur 30% of the time and 9's 4.5%). This will happen whenever FractionalPart[Log[b, a[n]]] is uniformly distributed, which, as discussed on page 903, is known to be true for sequences such as r^n (with Log[b, r] irrational), n^n, n!, Fibonacci[n], but not r n, Prime[n] or Log[n]. A logarithmic law for leading digits is also found in many practical numerical tables, as noted by Simon Newcomb in 1881 and Frank Benford in 1938.


From Stephen Wolfram: A New Kind of Science [citation]