A golden proof

Here’s a gorgeous bit of maths that I promised the other week to non-theoblogger J. Rhombohedral Hematite.

The Pythagorean dream of a world of harmonious geometric ratios was famously shattered by the discovery that the square root of two is irrational (that is, it cannot be expressed as the ratio of two whole numbers). However, I find the usual proof of this statement rather tedious: more “Oh, right, I see. OK.” than “Eureka!”

So here’s a much more fun way to prove that irrational numbers exist, courtesy of Timothy Gowers in his Mathematics: A Very Short Introduction (see pp.43-45). It’s a proof that the golden ratio is irrational. (And note that I’ve simplified one or two points from Gowers’ account, so any handwaving is mine rather than his, OK, mathmos? ;-) )

Here’s a golden rectangle, which can be divided into a square (shown in blue) and a smaller rectangle (shown in pink) whose sides have the same ratio as the larger rectangle. This ratio is the golden ratio, and it is widely regarded as aesthetically pleasing, as well as cropping up in many surprising places in mathematics (see Wiki for details):

So how do we prove that this golden ratio is irrational?

First, we need to think about what happens if we repeat the process of division outlined above. We can keep cutting squares off to produce a smaller rectangle whose sides are in the golden ratio, and clearly this process will never end:

Now, if the golden ratio is rational, then it can be expressed as the ratio of two whole numbers. Let’s call them p and q, and then let’s draw a rectangle whose sides are lengths p and q. This means the rectangle can be divided into p x q squares, as shown:

Now let’s start dividing this up into squares and rectangles as before. At each stage, we’ll be removing a whole number of little squares. But this means the process will soon come to a juddering halt, as can be seen in the diagram, where that smallest 2 x 1 rectangle in the top right cannot be divided any more.

So, as Gowers summarises, we have shown the following:

  1. If the ratio of the sides of the rectangle is the golden ratio, then one can continue cutting off squares for ever.
  2. If the ratio of the sides of the rectangle is p/q for some pair of whole numbers p and q, then one cannot continue cutting off squares for ever.

So in other words, p/q cannot be the golden ratio, whatever values of p and q we choose. In other words, the golden ratio is irrational.

Gowers suggests that:

If you think very hard about the above proof, you will eventually realize that it is not as different from the proof of the irrationality of √2 as it might at first appear. Nevertheless, the way it is presented is certainly different – and for many people more appealing.

To my mind, what makes this more appealing is that you can see, visually, what is going on in the proof. It gives you a more concrete idea of what a number’s being irrational means. Your mileage may vary, however…