A few years ago I made an exploration of some simple concepts that led to some fun facts and a sequence of numbers I later learned is quite well-known. It may even rival the Fibonacci sequence in its fame.

I thought it might be nice to collect my notes and present my findings here. I thought I would present a small portion once a week, so that others can have a chance to give feedback and add their own insights.

My own personal discovery of the Pell numbers began with a simple desire to find good rational approximations of the square root of two. I thought of poor Pythagoras and his mystic cult of followers, called the Brotherhood, struggling with the irrationality of the square root of two in ancient Greece. I pictured them, in their togas, sandals and olive laurels, toiling in vain under the hot Greek sun with sharpened sticks in the wet coastal sand trying to find a right isosceles triangle having natural side lengths.

In their arsenal was one of the best known theorems from plane geometry, which incidentally happens to incorrectly bear the name of Pythagoras. The Brotherhood did reportedly independently discover this truth, yet later it was found to be known for centuries to the Babylonians and Indians, among others.

Pythagoras had had much success describing nature with natural numbers, most notably with his well-tempered musical scale, and it was shocking to him and the Brotherhood that something as simple as the diagonal of a unit square could not be described with rational numbers, i.e., that it would be "incommensurable."

One can easily understand that attitude from a naive perspective on number theory well over two thousand years ago. The square root of two is thought to be the first irrational number discovered.

Yet the zealous Brotherhood would never be able to get closer than one tantalizing unit away from the legs of the triangle being equal using natural numbers, and as they progressed, the numbers would have grown to overwhelming magnitudes very quickly. It must be a trap set by the petulant gods, they might have thought, to keep man in his place and in ignorance of the true inner workings of the universe. As it turns out, while this search will never reveal the impossible rational representation for the square root of two, it is far from a fruitless search.

Were it not for the irrationality of the square root of two, we would obviously be able to find a Pythagorean triple of the form $\displaystyle (a,a,c)$ where, of course $\displaystyle \frac{c}{a}=\sqrt{2}$. We will now confirm the irrationality of root two with (hopefully) no worries of being executed for divulging a mystical mathematical secret to the public.

Comments and questions to this topic should be posted here: