Problem: Show that is an irrational number (can’t be expressed as a fraction of integers).
Solution: Suppose to the contrary that for integers , and that this representation is fully reduced, so that . Consider the isosceles right triangle with side length and hypotenuse length , as in the picture on the left. Indeed, by the Pythagorean theorem, the length of the hypotenuse is , since .
Swinging a -leg to the hypotenuse, as shown, we see that the hypotenuse can be split into parts , and hence is an integer. Call the point where the and parts meet . If we extend a perpendicular line from to the other leg, as shown, we get a second, smaller isosceles right triangle. Since the segments and are symmetrically aligned (they are tangents to the same circle from the same point), they too have length equal to . Finally, we may write the hypotenuse of the smaller triangle as , which is also an integer.
So the lengths of the sides of the smaller triangle are integers, but by triangle similarity, the hypotenuse to side-length ratios are equal: , and obviously from the picture the latter numerator and denominator are smaller numbers. Hence, was not in lowest terms, a contradiction. This implies that cannot be rational.
This proof is a prime example of the cooperation of two different fields of mathematics. We just translated a purely number-theoretical problem into a problem about triangle similarity, and used our result there to solve our original problem. This technique is widely used all over higher-level mathematics, even between things as seemingly unrelated as topological curves and groups. Finally, we leave it as an exercise to the reader to extend this proof to a proof that whenever is not a perfect square, then is irrational. The proof is quite similar, but strays from nice isosceles right triangles