Suppose that the swift Achilles is having a race with a tortoise. Since the tortoise is much slower, she gets a head start. When the tortoise has reached a given point a, Achilles starts. But by the time Achilles reaches a, the tortoise has already moved beyond point a, to point b. And by the time Achilles reaches b the tortoise has already moved a little bit farther along, to point c. Since this process goes on indefinitely, Achilles can never catch up with the tortoise.

How can this be?

Taken from - http://members.aol.com/kiekeben/zeno.html

(In reply to Well by Lawrence)

Building on Lawrence's idea, you could write equations comparing the instances of Achilles catching up to the tortoise's previous position (a, b, c . . .) with the amount of time passed and the distance between the two. If you actually assigned numbers and did some figuring (which I haven't bothered to do here) you would find that the distance approaches, but never reaches, zero. This leads you to think that maybe Achilles never catches up. However, you would also find that the amount of time passed also approaches a certain number (depending on the speeds of the racers) which means that the description given only covers a certain amount of time, and only part of the distance of the race. Anything beyond that would exceed the limit of time. Therefore, it can't follow the pattern. At that limit of time, the distance reaches zero, and Achilles passes the tortoise.

An additional idea:

The distance between Achilles and the tortoise, as well as the amount of time passed, are, in this description, dependant on the number of times Achilles catches up with the tortoises previous position (a, b, c . . .) You can't calculate the distance or time for when this happens infinite times, but it will, because Achilles and the tortoise pass infinite points during the race. (I think this is just saying the above in a different way).

Anyway, I don't have much experience with this type of problem, and I hope that my explanations are sensible, as well as understandable.

Posted by Hal on 2006-04-22 23:35:13