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How can you reach a full distance if you always have to travel half of it?

You have to travel half a distance to reach the full distance, therefore, how is it possible to reach the full distance? eg, to reach 1 mile you have to travel half, 1/2 mile, to reach the remaining hale mile you have to travel 1/4 mile, to reach 1/4 1/8 mile and so on?


Laurie Tingley


One Response

  1. This is a variant of the classic Zeno’s paradox of Achilles and the Tortoise.

    At first glance, it seems that the logic of the argument is sound. But observe that if we extrapolate this paradox, then all movement is rendered impossible. Because if its impossible to travel 1 full mile, then its also impossible to travel half a mile, a quarter of a mile… and so on and so forth, by the same reasoning. And obviously we’re able to move, so there must be something wrong here.

    The answer lies in the fact that geometric series are infinite, but have finite sums. Observe that:

    ∑(1/2)^n = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ……. and so on

    is in fact equal to 1. Which intuitively, should make sense, because if you split up a finite distance into infinitely many pieces, then the sum of those pieces must still be that finite distance. So even though we’re traveling an infinite amount of subsets of the original distance, we’re doing so in smaller and smaller increments of time, and those time increments also converge to a finite number. Eg. if it takes us 2 minutes to travel half a mile, then it takes 1 minute to travel a quarter of a mile, 1/2 a minute to travel 1/8th of a mile, and so on… in total it will take us:

    2 + 1 + (1/2 + 1/4 + 1/8 + 1/16 +….) minutes

    The stuff in brackets is a geometric series, which we already know adds up to 1. So the infinite sum of time increments becomes 2 + 1 + 1 = 4 minutes. So it is possible to travel the full distance, and it would only take 4 minutes. 🙂

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