How Dirac Made Every Number

A couple weeks back I offered a challenge taken from Graham Farmelo’s biography (The Strangest Man) of the physicist Paul Dirac. The physicist had been invited into a game to create whole numbers by using exactly four 2’s and the normal arithmetic operations, for example:

1 = \frac{2 + 2}{2 + 2}

2 = 2^{2 - \left(2 \div 2\right)}

4 = 2^2 \div 2 + 2

8 = 2^{2^{2}} \div 2

While four 2’s have to be used, and not any other numerals, it’s permitted to use the 2’s stupidly, as every one of my examples here does. Dirac went off and worked out a scheme for producing any positive integer from them. Now, if all goes well, Dirac’s answer should be behind this cut and it hasn’t been spoiled in the reader or the mails sent out to people reading it.

The answer made me slap my forehead and cry “of course”, if that helps you work it out before you look.