### Keeping the numberline dry

Although A Brief History of Infinity has been around a while, it is still one of my best selling titles and I probably get more letters and emails about it than anything else. I think it reflects the timeless fascination of infinity. Any road up, I thought I'd bring a little brightness to your Friday with a paradox of infinity that didn't make it into the book, though it does appear on the Popular Science website.

We start by thinking of the number line - let's say for simplicity, all the numbers from 0 upwards. So we've got a line, rather like the edge of a ruler, starting from zero and heading off to infinity, featuring all the numbers and fractions along its length.

Now we know the rational fractions (n/m where n and m are whole numbers) have the same cardinality as the counting numbers, thanks to a proof by Cantor (it's in the book). Simply put, this means you can match off each of the of the rational fractions with a positive integer. They are the same 'size' of infinite set. And we're going to use another set of fractions alongside them - the sequence 1/2, 1/4, 1/8, 1/16... It is simple enough to show that these also have the same cardinality: you can match one of these with each of the positive integers too. And we can prove that the sum of the whole series 1/2+1/4+1/8+1/16... is just 1. With these 'given's the fun begins.

Imagine we wanted to protect the whole number line from getting wet. What we are going to do is issue each rational fraction along the line an umbrella. The umbrella will be a simple T shape. The first umbrella we give out is 1/2 a unit of the number line across the T. The second umbrella is 1/4 of a unit of the number line across and so on. Once every rational fraction has an umbrella, then the whole number line is covered. The umbrella extends half its width in either direction - so, for instance, the first umbrella will cover all numbers for 1/4 of a unit to its left and 1/4 of a unit to its right. Note that this too is a rational fraction - and adding it to or subtracting it from the starting point (itself a rational fraction) will reach another rational fraction.

Okay so far? Each umbrella spans from its starting point to a rational fraction on either side of it. Now bearing in mind we've issued an umbrella to every rational fraction, the whole number line is covered, because there's at least a meeting of umbrellas and in most cases an overlap.

We've covered the whole line from 0 to infinity with our umbrellas. But, remember how wide the umbrellas were. Their widths form the infinite series 1/2+1/4+1/8... so with no overlaps, the maximum amount of the number line those umbrellas can cover is 1 unit - and with overlaps they will cover even less. A set of items with a width of just 1 covers a line that goes all the way to infinity.

Spooky!

### 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