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My lastest posts can be
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Previous blog posts:
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Additionally, some earlier writings:
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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 DavidBedford -
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?

[[[>50
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?

More about that next time.

----

So I thought I'd reply to a few of the comments from my previous post
- IrrationalsExist. 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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!! Comments

I've decided no longer to include comments directly via the Disqus
(or any other) system.  Instead, I'd be more than delighted to get
emails from people who wish to make comments or engage in discussion.
Comments will then be integrated into the page as and when they are
appropriate.

If the number of emails/comments gets too large to handle then I
might return to a semi-automated system.  We'll see.

********<