Zeno’s Paradox: Achilles and the Tortoise

Diogenis_Laertii_De_Vitis_(1627)_-_Zenon_of_Elea_or_Zenon_of_Citium
An engraving of either Zeno of Elea or Zeno of Citium. Zeno of Elea (490 -430 BCE) was a pre-Socratic philosopher known for his paradoxes

During the 5th century BC, the great Philosopher Zeno of Elea asks the following problem: Let’s say that Achilles, one of the greatest warriors in Greek mythology, challenges a Tortoise to a foot race. Achilles, fast and overconfident gives the tortoise a 100 meter head start. However when Achilles finishes the 100 meter gap, the tortoise is now a smaller distance in front of Achilles. To catch up, Achilles needs to bridge this new gap, however when he does the tortoise is again a smaller distance ahead of him. Each time Achilles completes the gap, the Tortoise will put a smaller gap between them, resulting in an infinite number of gaps of decreasing size between Achilles and the Tortoise. So according to Zeno, how can Achilles reach the tortoise if there is an infinite number of gaps between them?

With a bit of math, Zeno’s paradox can easily be resolved. The key is that the number of gaps does not matter, what does is the fact that each gap is smaller than the previous gap. This means that eventually the gap will be so small that it will be virtually zero, giving us a finite distance. Let’s assume that each gap is half the distance of the gap before it. For example when Achilles had finishes running 100 meters the tortoise will be 50 meters ahead and when Achilles finishes 50 the tortoise will be 25 meters ahead. This will give us the infinite series:

100 + 50 + 25 + 12.5 +6.25 ...

Which becomes the convergent geometric series:

\sum\limits_{i=0}^n 100 * (1/2)^i = \frac{1}{1-1/2}= 100*2

Then with a little arithmetic the total distance becomes 200 meters, a distance that the Greek hero can easily run.

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