### 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.

### Why I hate opera

If I'm honest, the title of this post is an exaggeration to make a point. I don't really hate opera. There are a couple of operas - notably Monteverdi's Incoranazione di Poppea and Purcell's Dido & Aeneas - that I quite like. But what I do find truly sickening is the reverence with which opera is treated, as if it were some particularly great art form. Nowhere was this more obvious than in ITV's recent gut-wrenchingly awful series Pop Star to Opera Star , where the likes of Alan Tichmarsh treated the real opera singers as if they were fragile pieces on Antiques Roadshow, and the music as if it were a gift of the gods. In my opinion - and I know not everyone agrees - opera is: Mediocre music Melodramatic plots Amateurishly hammy acting A forced and unpleasant singing style Ridiculously over-supported by public funds I won't even bother to go into any detail on the plots and the acting - this is just self-evident. But the other aspects need some ex

### Is 5x3 the same as 3x5?

The Internet has gone mildly bonkers over a child in America who was marked down in a test because when asked to work out 5x3 by repeated addition he/she used 5+5+5 instead of 3+3+3+3+3. Those who support the teacher say that 5x3 means 'five lots of 3' where the complainants say that 'times' is commutative (reversible) so the distinction is meaningless as 5x3 and 3x5 are indistinguishable. It's certainly true that not all mathematical operations are commutative. I think we are all comfortable that 5-3 is not the same as 3-5.  However. This not true of multiplication (of numbers). And so if there is to be any distinction, it has to be in the use of English to interpret the 'x' sign. Unfortunately, even here there is no logical way of coming up with a definitive answer. I suspect most primary school teachers would expands 'times' as 'lots of' as mentioned above. So we get 5 x 3 as '5 lots of 3'. Unfortunately that only wor

### Which idiot came up with percentage-based gradient signs

Rant warning: the contents of this post could sound like something produced by UKIP. I wish to make it clear that I do not in any way support or endorse that political party. In fact it gives me the creeps. Once upon a time, the signs for a steep hill on British roads displayed the gradient in a simple, easy-to-understand form. If the hill went up, say, one yard for every three yards forward it said '1 in 3'. Then some bureaucrat came along and decided that it would be a good idea to state the slope as a percentage. So now the sign for (say) a 1 in 10 slope says 10% (I think). That 'I think' is because the percentage-based slope is so unnatural. There are two ways we conventionally measure slopes. Either on X/Y coordiates (as in 1 in 4) or using degrees - say at a 15° angle. We don't measure them in percentages. It's easy to visualize a 1 in 3 slope, or a 30 degree angle. Much less obvious what a 33.333 recurring percent slope is. And what's a 100% slope