Revisiting The Ant

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Revisiting The Ant - 2012/02/20

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So last time in The Ant And The Rubber Band we were talking about an infinitely patient ant walking on an infinitely stretchy rubber band. If you haven't already, you'll need to read that.

So here's what's happening:

  • At the start:
    • the band is 1 m long
    • the ant is at position 0

  • Just as we reach 1 minute:
    • the band is about to be stretched
    • the ant has got to the 1 cm mark
      • this is 1% of the length of the band

  • Just after 1 minute:
    • the band is stretched out to 2 m
    • the ant is carried along by the stretch:
      • it's still at 1%, so now it's at the 2 cm mark
        • relative to the ground

  • Just before 2 minutes:
    • the band is about to be stretched
    • the ant has waked an extra 1 cm to be at 3 cm
      • that's 3 cm of 2 m, or 1.5%

  • Just after 2 minutes:
    • The band is now 3 m
    • the ant is still 1.5%, so now it's at 4.5 cm

And so on.

If you continue to work methodically like this, and don't make any mistakes (so it's best to write a spreadsheet or a program or something) then you can work out where the ant is at any moment. The problem is, working methodically with the numbers doesn't always provide the insight, especially if you program a computer to do it.

Sometimes working the numbers for yourself, by hand, allows the insight to come.

And the insight is in the percentages. For the first minute the ant walks 1 cm, which is 1% of the length. In the second minute it walks 1 cm, which is 1/2 % of the length. Then it walks 1 cm which is 1/3 % of the length.

And so on.

So the distance travelled, as a percentage, after n minutes is given by the series:

  • 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + ... + 1/n

And this is unbounded. No matter what (positive) total you want, you'll get there, if only you can take enough terms to do so.

But you may need a lot of terms. Next time I'll explore how we know this is unbounded, what it looks like, and some surprising results that follow from it.

Actually I did use (in a throwaway comment) the fact that the rationals are dense in the reals. I said that whenever we have an interval, there will always be at least two rationals in it. If you have a list that is not dense in the reals (such as the integers, for example) then you need to worry about the case where we get an interval with none, or only one, further item in it.
I also said last time that I'd talk about not using the properties of the rationals in the post that Irrationals Exist. And I didn't. All I used was that I had a list of rationals, nothing else.

So in fact this works to show that if you have any list of reals then you can construct a real that's not on the list. If we start with, say, the natural numbers, or the reciprocals of the primes, or rationals whose denominators are powers of 5, or the numbers that arise as solutions of polynomials, then the same technique can show that there is a real missing in every interval.

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