In 1938, the physicist Frank Benford made an extraordinary discovery about numbers. He found that in many lists of numbers drawn from real data, the leading digit is far more likely to be a 1 than a 9. In fact, the distribution of first digits follows a logarithmic law. So the first digit is likely to be 1 about 30 per cent of time while the number 9 appears only five per cent of the time.
That’s an unsettling and counterintuitive discovery. Why aren’t numbers evenly distributed in such lists? One answer is that if numbers have this type of distribution then it must be scale invariant. So switching a data set measured in inches to one measured in centimetres should not change the distribution. If that’s the case, then the only form such a distribution can take is logarithmic.
But while this is a powerful argument, it does nothing to explan the existence of the distribution in the first place.