# The Ant And The Rubber Band

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## The Infinitely Patient Ant and the Infinitely Stretchy Rubber Band

I was giving my talk on "Infinity" the other day, and I mentioned, almost in passing, a problem I got some time ago from David Bedford - the question of the infinitely patient ant and the infinitely stretchy rubber band. It goes like this:

There's a 1 metre long rubber band, and an ant, standing on it at one end. The ant starts walking along it at a speed of 1 cm/min. Not very fast, but steady. However, every minute the rubber band is stretched (uniformly and instantaneously) to be one metre longer. The question is this: Will the ant ever get to the far end?

 Of course, it's not always an extra 1 cm that the ant gets "for free" from the stretch of the rubber band. This is the hard bit! Just how do you work out what's really happening! If you're struggling, make a model, draw a picture, don't just stare into space and try to visualise it because you won't. Well, probably won't. This is the entire challenge in maths, trying to understand what's happening, and just staring at it won't help. You have to play with the problem.
So we'll think about this a little. After one minute the ant has gone 1 cm, but then the band gets stretched out an extra metre to 2 metres. Ah, but the ant benefits from this, and is now an entire extra 1 cm from the starting point! Perhaps not a lot of help, really, when it's just been given an extra entire metre to go. Still it's now 2 cm from its starting point.

Then it walks another cm, so now it's 3 whole centimetres from its starting point. Then the rubber band gets stretched out another metre to a total of three metres long, again, dragging the ant a little "for free". (Tip: Think hard about where it is now)

And so it continues. An extra cm of walking, then the end-point moved by another metre. Will it ever get there?

So I thought I'd reply to a few of the comments from my previous post - Irrationals Exist. Someone was checking on what exactly I had shown. and had I shown that an irrational existed in every interval.

Yes. Between any two reals (or algebraics or rationals or integers) there exists an irrational. That's what I proved.

Someone else (Thanks Jorge-Nuno!) said I'd used the reals to show that the reals existed! What I actually did (and of course he knows it) is to say that if we start with the rationals, we can produce a sequence that should have a limit, but for which no rational can be that limit. After all, we want our number line to be "complete", and if we only have rationals, it isn't.

Another observation is that I never really used any of the properties of the rationals. I'll say more about that next time.

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