Nowhere.

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**.