The ancient Greeks first proved the existence of irrational numbers by proving that is irrational. The proof is, as modern proofs of irrationality go, fairly simple. It is often the first example of a proof of irrationality that students see in, say, a course in real analysis.

Yesterday, though, I saw an even simpler proof of irrationality in Morgan’s *Real Analysis* [1] that I thought worth showing to my students in advanced calculus.

*Theorem*: The number is irrational.

*Proof*: First, because 5 and 10 are both greater than 1. Suppose that is rational. Then there exist positive integers *p, q* such that . Thus However, ends in 0, while ends in 5. This is a contradiction, and so is irrational.

**References**

- Frank Morgan,
*Real Analysis*, American Mathematical Society, 2005.

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Thanks, that is a nice proof.

Here is a very short proof that sqrt(2) is irrational. I believe it is due to John Conway, but I am not sure:

Suppose sqrt(2) is rational. Then there is a smallest positive integer n with the property that

n sqrt(2) is an integer. But then m=n sqrt(2) -n has the same property, and it’s smaller, contradiction.

Nice!

(Note for other readers: This proof uses the property that must satisfy , which is easy to prove.)

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