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Revisiting The Ant - 2012/02/20

This page has been Tagged As Maths.

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

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

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.

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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