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:
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.
- A number that can be divided by 2 is even.
- If you multiply an odd number by an odd number, you get another odd number.
- If you multiply any number (odd or even) by an even number, you get an even number.
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.