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

Great proof. Any idea who proved it? Must be some Greek mathematician….

Hippamus probably

Awesome proof. This is a great blog.