But now that you ask, here’s one explanation:

(It’s a bit tricky, so don’t panic if it puzzles you at first. And don’t worry if you can’t imagine reproducing it on a test, as you won’t be asked to. But it’s worth a look  especially if you find this introductory stuff too ‘elementary’.)


Any ratio of integers can be reduced to "lowest terms" by cancelling out common factors.
So if root 2 is rational, then it can be expressed as a ratio m/n where m and n have no common factors.
Now if , then , so  is even.
But the prime factorization of   includes just the same factors as in m (each repeated twice as often). So if  is even, then the factor 2 must occur in m.

So  is actually a multiple of 4. Thus  is four times some whole number.

I.e.  for some whole number k. So   is even and so n  must be even also.
But then m and n both have a factor of two and we already cancelled out all common factors!
Thus if we assume that  is a ratio of whole numbers then we get a contradiction,
and so it cannot be true.