Square Root of 2

If you want to know how I spend the few quite moments of my day when I’m not sitting in my office or chasing around a rowdy 2-year old, you’re about to find out.  I recently came across an article that said the following (I’m paraphrasing):

Since the days of the Babylonians it’s been known that square root of 2 – which I’ll shorthand as SQR(2) from here on out – has a lot of numbers in the decimal places.  And even if they didn’t know specifically that it was irrational, with their available mathematical knowledge, they came up with an excellent approximation.  It wasn’t long thereafter, 1500 years or so, that the Pythagorean School of Greek Mathematicians came up with a more robust proof of it’s irrationality.  Fast forward to today, and any high school algebra student who wants to can fully grasp the proof in under 5 minutes.

At the time I came across the mention of the Babylonians and the irrationality of SQR(2), I didn’t know the actual proof.  And so, just for fun – because yes, this is what I do for fun  – I decided to try and derive the proof myself.

Knowing only that a rational number is defined as “any number that can be expressed as the ratio of two integers with no common factors,” I first tried to prove that SQR(2) is irrational.  It did not go well.  My notebook slowly filled with equations and ratios, a multitude of poorly drawn triangles and even more poorly drawn circles.  And letters…there were so many letters, that I couldn’t even keep things straight myself.

Finally, 30 minutes in, on my 4th sheet of scratch paper, this happened:

Yes, I dispatched a fighter jet to blow up my math.

Fast forward 48 hours, and I had a thought.  In the world of logic it’s equally valid to prove something is true as it is to prove that something can’t be false.  In the case of proving a number is irrational, it would therefore be equally valid to prove that it could not be rational.  From there, things become pretty simple.  starting with the basic premise that the SQR(2) could be expressed as the ratio of two integers – x and y – with no common factors, away I went.

I’ll spare you the rest of the math, but what it boils down to is this: both x and y must be even, which means they must share the common factor 2.  This is an internal contradiction because rational numbers are definable as the ratio of two integers sharing no factors.  Therefore, SQR(2) is irrational.

I know…FUN, RIGHT?!

Well, at least I had fun.  In retrospect, the entire process was made more satisfying by the initial failure, and the fact that the solution was ultimately unearthed by taking a step back and coming at the problem from a different angle.  My colleagues and I get to work on problems like these a lot, and though our puzzles tend to be confined to the investment space and a bit more complex, seeking out creative and innovative solutions to investment/portfolio management/financial/retirement puzzles is gratifying work…

…even when we occasionally have to call in the fighter jets and start over.