author : Ray Hemmings

Answer quickly, without thinking too long about it: what proportion of (base 10) numbers, up to 1000, and up to 10 million, contain at least one digit '9'? Many people answer "about a tenth", but that's much too small. More than a quarter of the numbers up to 1000 contain at least one '9', as do more than half the numbers up to 10 million. Images of Infinity contains a wonderful, simple, proof that most numbers, when written in base 10, contain at least one digit '9' in them. Consider the first 9ⁿ numbers written in base 9, none of which contains a '9'. Now consider the first 10ⁿ numbers written in base 10, 9ⁿ of which textually correspond to the base 9 numbers, and all the rest of which contain at least one '9'. So the fraction of base 10 numbers up to 10ⁿ digit without a '9' digit is 0.9ⁿ , which tends to zero as n tends to infinity. The same proof, with suitable choice of base, can be used to show that almost all numbers contain any given sequence of digits.

This little book contains many such gems about infinity: fractal curves; limits of strange series; "proofs" that pi equals 2, or 4; the Hilbert Hotel with its large number of rooms; a Jorge Luis Borges story about an infinite book; and more. I'm not sure who the target audience is intended to be: some of the stuff is pretty basic, some is very deep, yet there is no bibliography of further reading for the curious. But it is an interesting confection.