4.2.4 DISTRIBUTION OF FLOATING POINT NUMBERS 245 O.S,- (Web domain)
4.2.4 DISTRIBUTION OF FLOATING POINT NUMBERS 245 O.S,- 1 2 I 3 I 4 I 5 I 6 I I I 9I I 7 8 10 s Fig. 5. The probability that the leading digit is 1. The gist of Lemma Q is that we have the limiting relationship lim Pm(lOns) = Sm(s). (16) n-+oo Also, since Sm(s) is not constant as s varies, the limit lim P,(n) n–r00 (which would be our desired probability ) does not exist for any m. The situation is shown in Fig. 5, which shows the values of S*(s) when m is small and r = 2. Even though S,(s) is not a constant, so that we do not have a definite limit for P,(n), note that already for m = 3 in Fig. 5 the value of Z&(s) stays very close to log,, 2 = 0.30103.. . . Therefore we have good reason to suspect that S&s) is very close to log,, r for all large m, and, in fact, that the sequence of functions (Sm(s)) converges uniformly to the constant function log,, V-. It is interesting to prove this conjecture by explicitly calculating Q,(s) and R,,Js) for all m, as in the proof of the following theorem: Theorem F. Let Sm(s) be the limit defined in (16). For dl E > 0, there exists a number N(e) such that ISm(s) - lw&, 4 < 6, for 1 5 s 5 10, (17) whenever m > N(E). Proof In view of Lemma Q, we can prove this result if we can show that there is a number M depending on E such that, for 1 < s 5 10 and for all m > M, we have IQm(s) -loglo 4 < E and I%dS)I < E. (9