Plus Advent Calendar Door #2: Who will come second?

Will he rocket to the lead? (Most excellent tortoise suit by Katie Bradley.)

Achilles and a tortoise are competing in a 100m sprint. Confident in his victory, Archilles lets the tortoise start 10m ahead of him. The race starts, Achilles zooms off and the tortoise starts bumbling along. When Achilles has reached the point A from where the tortoise started, it has crawled along by a small distance to point B. In a flash Achilles reaches B, but the tortoise is already at point C. When he reaches C, the tortoise is at D. When he's at D, the tortoise is at E. And so on. He's never going to catch up with the tortoise, so he has no chance of winning the race.

Something's wrong here, but what? Let's assume that Achilles is ten times faster than the tortoise and that both are moving at constant speed. In the times it takes Achilles to travel the first 10m to point A, the tortoise, being ten times slower, has only moved by 1m to point B. By the time Achilles has travelled 1m to point B, the tortoise has crawled along by 0.1m to point C. And so on. After n such steps the tortoise has travelled $$ 1+1/10+1/100+1/1000+ \ldots +1/10^{(n-1)} \mbox{metres}. $$ And this is where the flaw of the argument lies. The tortoise will never cover the 90m it has to run using steps like these, no matter how many of them it takes. In fact, the distance covered in this way will never exceed $10/9=1.111\ldots$m. This is because the geometric progression $$ 1+1/10+1/100+1/1000+\ldots $$ converges to $10/9$. Since the tortoise is travelling at constant speed, it covers this distance in a finite time, and it's precisely when it's done that that Achilles overtakes it.

This problem is known as one of Zeno's paradoxes after the ancient Greek philosopher Zeno, who used paradoxes like this one to argue that motion is just an illusion.

Find out more about Zeno's paradoxes and infinite series on Plus.

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