19 - irrationality previous next


By subsituting the formula φ = 1 / (1 + φ) back into itself over and over, we get:


If you ponder this picture, you'll see it says the same thing:

Since the picture never ends, neither does the fraction for φ. This gives an easy proof that φ is irrational!

Did the Pythagoreans know this proof? It would have been easy for them. Is this why the pentagram had a mystical significance for them? We don't know.



For some fascinating theories about what the Greeks knew of continued fraction expansions and irrational numbers, read David H. Fowler's The Mathematics of Plato's Academy: a New Reconstruction.

I created the above picture. I hereby release it into the public domain.