Dear Uncle Colin ((This question actually came from my brother, raising interesting questions about the family tree)),

Supposing $\pi$ were to start repeating – how long would it take to confirm that it actually was?

- Recurring Over Uncanny Numbers of Decimals

Dear ROUND ((Honestly, this isn’t a reflection of the number of mince pies he ate over Christmas)) ,

Great question! $\pi$, as I’m sure you know, is a number close to 3.1416 that represents the ratio between a circle’s area and its radius. One of the few things everyone knows about $\pi$ is that it goes on forever without repeating.

However, many fewer people know how we know that. Saying the decimal digits of $\pi$ never repeat is equivalent to saying it’s an irrational number – that is, you can’t write it as a fraction using integers on the top and the bottom. Any fraction you write as a decimal either terminates – like $\frac{1}{2} = 0.5$ or recurs – like $\frac{8}{13} = 0.\dot{6}1538\dot{4}$. Anything that doesn’t terminate or recur is irrational, and vice-versa.

So, if you wanted to prove that the digits of $\pi$ started repeating, you wouldn’t go along checking them and looking for patterns – you’d try to show that it was a fraction. There are many proofs that it isn’t a rational number, although I wouldn’t call any of them easy!

I hope that clears up the problem, ROUND!

-- Uncle Colin