Night thoughts

Strange what comes to you when you're staring at the ceiling and trying to go to sleep.

I don't sleep easily, and never have. One of the downsides to this is near-constant mild fatigue. One of the few upsides is that it gives you lots of time to think about random things that the Real World (tm) keeps your mind from wandering on to. A little light prompting is required, of course, and that night it was a trailer on the BBC World Service about a programme exploring the nature of infinity, from the point of view of mathematics. Infinity, it said, is a concept and not a number, and as a result isn't quantifiable, and, arguably, doesn't exist.

Fascinating and heady stuff for 01:59 BST, I'm sure you'll all agree.

So, there I was, pondering the nature of infinity through the six electromechanical chimes (five short, one long) of the BBC hour signal.

If infinity doesn't exist (thought I to myself), then it's nothing, and since nothing is zero then ∞=0.

... how profound, wizard!

Then my mind meandered a little further, well off the beaten track and into the forest of the far beyond.

Remember those two rules we were taught in primary school or thereabouts, by our maths teachers and our calculators?

1. The result of dividing any number by itself is 1; n/n=1.
2. The result of dividing any number by zero is infinity; n/0=-E-, said our calculators; 'Cannot divide by zero' says the computer. Rubbish, of course you can, but you just can't work with the result.

Anyway, back to a sleepless night in my lair.

If you divide infinity by 1, I thought, then the obvious answer is infinity: ∞/1=∞, logically. Similarly, 1/1=1, and o/1=0. On that I agreed with myself.

What if you divide by infinity? Well, the Hitchhiker's Guide to the Galaxy says that, given as infinity is such a large number, any number divided by it is going to be as near to zero as makes no odds, so we get 1/∞=0, 0/∞=0 and ∞/∞=0. Which doesn't make sense, really, since we know from statement 1 above that any number divided by itself yields a quotient of 1; so ∞/∞=1.

That's treating infinity as a number. Treat it as a concept, as per the World Service, and we get ∞/∞ → 0/0= ...

... and there my mind stopped, caught in a paradox. What do we get?

Any number divided by itself yields a quotient of 1. (Check.) But any number divided by 0 yields a quotient of infinity. (Check.) And any number divided by infinity yields a quotient of 0. (Check.)

So if ∞/0=∞ and ∞/∞=0 (or 1), does that mean that 0=1=∞?