Irrationality


When I talk of irrationality, I mean, of course, irrational numbers. Now these numbers are not irrational in the same way your girlfriend is, these numbers go on and on and on forever and ever and ever. Ok, maybe there are some similarities, but unlike your girlfriend these numbers never repeat themselves.

Perhaps I should clarify (the extended girlfriend comparison made things a little complicated). An irrational number is one that cannot be expressed as the ratio of two integers, a fraction if you will. So while numbers like a third (0.333…) and a ninety-ninth (0.0101…) extend infinitely, they can be written as 1/3 and 1/99 respectively.

Pi (π = 3.1415…) is an example of an irrational number. “Hang on”, I hear you cry. “You’ve only quoted 5 numbers there, how do we know there’s no repeating sequence?” Well, you can either trust me or check for yourself. No, go on, I’ll wait. Nothing? Still, this doesn’t really prove anything, does it? It could repeat after the millionth digit, or the billionth. Proving it never repeats by checking every digit is like trying to prove you can hold your breath for an hour. It’s a remarkable effort, but you’ll definitely die before you reach the end.

No, George, that wasn't a challenge.

To add to the confusion, many of you may have used the approximation pi ≈ 22/7 (333/106 if you were really clever), which seems to show pi as the ratio of two integers. However, this doesn’t quite cut it after a certain number of decimal places. There is a proof of the irrationality of pi, but it’s a bit complicated for this blog.

A much simpler proof involves √2.
Start by assuming √2 is rational.
Then √2 = p/q where p and q are integers and p/q is in its simplest form.
Squaring both sides gives 2 = p²/q²
Multiply both sides by q² to give 2q² = p²q²/q² then cancel
So 2q² = p² meaning p² is even
Therefore p is even (it cannot be odd, as this would make  p² odd) and can be written as p = 2m
Then 2q² = (2m)² = 4m²
This simplifies to  q² = 2m²
By the same argument q is even and can be written as q = 2n
Going back to the start then, √2 = 2m/2n = m/n
But we specified at the start that p/q was the simplest form, so we’ve found a contradiction. Therefore √2 is irrational.

Pretty neat, huh? This is a beautiful example of proof by contradiction, a method whereby you assume the opposite to be true and then show that it is logically inconsistent. It’s like assuming your friend isn’t stealing from you, finding all of your bank statements in his house and concluding that you are an extremely poor judge of character.