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!
