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Proving the irrational

When writing about science we often have to fight against irrational ideas that seem to grow on people's minds like fungi. Yet early mathematicians had the opposite problem of requiring the irrational. This was the irrational in the literal sense - a number that is not made up of a ratio. According to myth, when one of Pythagoras' merry band discovered that the length of a diagonal of a square with sides of length one (the square root of 2) was not a rational number (a fraction that's the ratio of two whole numbers), he was drowned for spreading such a malicious concept.

What's interesting, as I describe in my book A Brief History of Infinity, is that there is a remarkably simple proof that 2 is irrational. It requires little more than an understanding of odd and even and goes something like this:

Let's assume 2 can be represented by a rational fraction - we'll call it top/bottom.

To keep things simple, we are assuming that top/bottom provides the simplest fraction you can get - there's nothing to cancel out, so it's like 1/2 rather than 2/4. So:

top/bottom =2

Square roots are a bit fiddly, so let's multiply each side of the equals sign by itself. This gives us

top2/bottom2 = 2

In traditional mathematical fashion, we can get rid of the division by multiplying both sides of the equation by bottom2, giving us:

top2 = 2 x bottom2

Next, the Greeks relied on their knowledge of odd and even numbers. They knew three things about odd and even numbers.
  1. A number that can be divided by 2 is even.
  2. If you multiply an odd number by an odd number, you get another odd number.
  3. If you multiply any number (odd or even) by an even number, you get an even number.
As the right-hand side of the equals sign is 2 x bottom2, it must be even - it's the outcome of multiplying by an even number, 2. So top2 also must be even. And that means top has to be even (because were it odd we would be multiplying two odd numbers together and would get an odd result).
Now here comes the twist. If top is even, then it can be divided by 2. So top2 can be divided by 4. And we know that top2 is the same as 2 x bottom2. If 2 x bottom2 can be divided by 4, then bottom2 can be divided by 2. So bottom2 (and hence bottom) is even. (Read that again if necessary - it makes sense.)

So both top and bottom are even. But if both are even, then top/bottom isn't the simplest fraction we could have, since we can divide both top and bottom by 2. Yet we started by saying that top/bottom was the simplest fraction we could have. We've reached an impossible contradictory situation - which means our original assumption that it was possible to represent 2 by a ratio was false.

Added: Thanks to Thony Christie for pointing out that it's thought the Pythagoreans first discovered that 5 was irrational - but because 2 is based on the diagonal of a unit square, I think it makes the simplest example.


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