Pythagoras the murderer!

Picture the scene. It's around 500 BC and it’s a lovely Summer day off the coast of Greece. Out on the water though there is a kerfuffle. A man is fighting for his life after getting thrown over the side of a boat and left to die. His name is Hippasus of Metapontum and he had revealed a basic piece of mathematical logic.

Let us dial back a bit and talk about Pythagoras. Most people remember Pythagoras from school and even if they can’t remember what his theorem is about most remember it’s something to do with triangles. (for a nice visual recap see here). Pythagoras was much more than a developer of mathematical theorems though, he was also the leader of a full on religious cult which believed in vegetarianism and that numbers were divine.

One of the central doctrines of the Pythagorean cult was that all numbers could be written as fractions. Hippasus was pretty certain he had seen numbers that didn’t fit this pattern and mentioned this to Pythagoras who, according to some legends, suggested a little ride in his boat and, well, you now know the rest.

So how can we can use (and maybe Hippasus did too) Pythagoras theorem, to show how some numbers can not be written as fractions (in modern terminology these are called irrational numbers) Imagine a simple square, each side 1cm in length. How long is the square's diagonal? Using Pythagoras theorem (a squared + b squared = c squared) Then the diagonal is 1 squared + 1 squared so the length of the diagonal is the square root of two.

Now, if you have a calculator at hand find the square root of two. The display will say 1.414213562. This number in of itself wasn't a problem for the Pythagoreans. They just had to figure out what two whole numbers made a fraction that produced this number. There was just one little problem no matter how hard they tried the Pythagorean cult could not find two whole numbers whose ratio produced this number. The only (correct) conclusion was….there is NO ratio that will produce √2. the decimal will go on for ever and ever. It is, in other words, an irrational number.

The maths bit

We can use what is called a proof by contradiction to show √2 is irrational. Firstly, let's suppose √2 is not an irrational number (i.e a rational number).  Then two whole numbers as a fraction will produce this number so we can show this as √2  = a/b where a and b are whole numbers, and b is  not zero.

We additionally assume that this a/b is simplified to the lowest terms, in order for a/b to be in its simplest terms, both a and b must be not be even. One or both must be odd. Otherwise, you could simplify. So if √2  = a/b it follows that:

2 = a²/b² (Square both sides) or

  

a² = 2 * b². (Multiply both sides by  b² )

So, the square of a is an even number since it is two times something. From this we deduce that a is an even number. Why? Well, it can't be odd; if a is odd, then a * a would be odd too. (Odd number times odd number is always odd. I can prove this if anyone wishes, it is very simple to do)

If a is an even number, then a is 2 times some other whole number, or a = 2n where n is this other number.

Now, If we substitute a = 2n into the original equation 2 = a²/b², this is what we get:

2	=	(2n)2/b2
2	=	4n2/b2
2*b2	=	4n2
b2	=	2n2.

This means b² is even, (because b squared = 2 times a number squared) from which follows again that b itself is an even number.

And that’s it. We have a contradiction. For why I hear you ask?  Because we started this logical process saying that a/b is simplified to its lowest terms and hence either a or b must be odd. It turns out though that if the √2 is rational then a and b would both be even. So logically √2 cannot be rational.

QED!

Anyhoo. That’s not the main purpose of today. Pythagoras is a dastardly murderer and his ideas are still corrupting the young 2,500 years after his death!