Little is known about the Pythagoreans, except that they believed that “all is number” and used the pentagram as their symbol. They are said to have discovered irrational numbers through the diagonal of the square, although others conjecture that it may instead have been through the golden ratio in the pentacle. In any event, here is a geometric proof that φ is irrational, based on the ratio of a regular pentagon's diagonal to its side.
Let's begin with the red pentagon. If φ were rational, we could write it as A/B, where A is the diagonal of the red pentagon and B is its side
(A > 0, B > 0, A and B are integers with no common divisor).
Then b = A − B is an integer, and in the pentagon with side b, the golden ratio, by its geometric definition, is B/b, with b < A and A − B < B… contradicting our assumption! This elementary argument is enough to prove that φ is irrational.
