We’ll use the Wikipedia picture of isosceles triangle ABC with hypotenuse and legs . Pythagoras tells us that for such a triangle, the ratio . For to be rational, and must be integers, which we assume.
Inside ABC, construct a smaller triangle CDF with smaller integer hypotenuse and legs . Repeating in an infinite descent, we must end in a contradiction regarding the integer sides. (Say at the ith iteration, and the next iteration will be in contradiction to that.)
Aside: This ‘proof without words’ was published by Tom Apostol in the American Mathematical Monthly in November 2000. It was shown to me by my supervisor at work, himself a mathematician. One wonders how the ancient Greeks missed this proof. Perhaps they knew of it, but the evidence vanished in Julius Caesar’s Alexandria fire.
Aside: If needed, details of the geometric construction are on Wikipedia, Square root of 2.
Aside: Conway and Guy in their Book of Numbers offer a paper-folding argument along the lines of this proof.
Aside: This result can be derived by algebraic reasoning as well. Assuming , integers, , then , a contradiction since implies , but by hypothesis is the least value.