How do we know that Pi is infinite?

2nd January 2013

Asked by: Max Paris, Oxford

Many people know that the value of Pi is roughly 22 divided by 7, which is around 99.96 per cent accurate – plenty good enough for most practical purposes. But in 1768, the Swiss mathematician Johann Lambert revealed the remarkable fact that it’s impossible to use any such fractions to pin down the precise value of Pi, as it just goes on forever.

To prove it, he showed that Pi is not a ‘rational’ number – that is one the exact value of which is given by the ratio of two whole numbers. Rational numbers can be turned into decimal numbers that either stop after a few places (like 1/8 = 0.125) or just keep repeating after a certain number of places (such as 4/7 = 0.571428571… and so on). By showing that Pi is not a rational number, Lambert revealed that its decimal value neither stops nor cycles – but just carries on to infinity.