Let’s begin with a nice example, the proof that there are infinitely manyprime numbers. If asked for a typical bit of real mathematics, yourfriendly neighbourhood mathematician is as likely to give this example as any. First, we need to know that some numbers, called ‘composite’, can be divided without remainder or broken into factors (e.g. 6 2 3, 561 3 11 17), while other numbers, called ‘prime’, cannot (e.g. 2, 3, 5, 7, 11, 13, 17, . . .). Now we can ask: How many primes are there? The answer is at least as old as Euclid and is contained in the following.