Ask Uncle Colin: How do we know the digits of $\pi$ don’t recur?

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions -- and to show off your skills at coming up with clever acronyms. Send your questions to and Uncle Colin will do what he can.

Dear Uncle Colin1,

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

- Recurring Over Uncanny Numbers of Decimals

Dear ROUND2 ,

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


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

  1. This question actually came from my brother, raising interesting questions about the family tree []
  2. Honestly, this isn't a reflection of the number of mince pies he ate over Christmas []


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