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-l