Achilles and the Tortoise have, for some unknown reason, decided to race one another. Being the sporting type, Achilles decides to let the Tortoise have a strong lead - it’ll be a two mile race, but the Tortoise will get to start a mile in. Achilles runs much, much faster than the Tortoise, so he figures it shouldn’t make too much of a difference if the Tortoise starts out ahead.
Unfortunately for Achilles though, he hasn’t been thinking much about infinity.
As the gong goes off to signal the start of the race, the Tortoise and Achilles both begin to run - but how can Achilles ever pass the Tortoise? He first must cover half the distance between himself and the Tortoise’s starting point - but by then, the Tortoise will have moved ahead! Now he must cover half the distance between where the Tortoise was then, but again, the Tortoise has moved forward. Achilles keeps up his chase, each time able to only close half the distance between where he and the Tortoise were at last time we checked in - so how, as the Tortoise is always moving forward, is Achilles ever to pass the Tortoise? Think for a while, and we’ll work on some other problems!
Exercise 1: On a seemingly unrelated note, see if you can add the following:
1/2+1/4+1/8+1/16+1/32+...
As a hint, try drawing a line, splitting it in half, then half of the remainder, then half of the remainder of that, etc.
The “paradox” that Zeno proposed relies on the same question: Can we add up an infinite number of things to get a finite amount? The answer, of course, is yes - as Exercise 1 shows. In the same way, while Achilles has infinitely many times he will catch up to where the Tortoise has been, it doesn’t take him the same amount of time between each “catch up.”
Let’s run through how this works. There are two different ways of thinking about this.
Exercise 2:
(a) Let’s assume that Achilles can run a five minute mile and the Tortoise can run a ten hour mile. That puts Achilles speed at 12 miles an hour and the Tortoises speed at what?
(b)When two objects are both moving at different speeds, we can change our frame of reference to make the problem a little easier - let’s say Achilles ran at 12 miles an hour and the Tortoise was running at 6 miles an hour. We can change our frame of reference to be a MOVING frame - say a frame moving at 6 miles an hour in the same direction as both of our runners. Now the Tortoise (in this frame) is running at 0 miles an hour and Achilles is running at 6 miles an hour. How long does it take for Achilles to pass the Tortoise now? Remember that Achilles started out 1 mile behind the Tortoise
(c)At the REAL speed the Tortoise was traveling, how long does it take him to pass Achilles?
In that way of solving the problem, we seem to have avoided the paradox entirely - it is obvious if the Tortoise is not moving and Achilles is, he can catch up to the Tortoise! So what happened to the paradox?
The other way of thinking about the problem (and this is the way that hits the “paradox” part head on) is to remember that it doesn’t take Achilles equal amounts of TIME to catch up to each of the Tortoise’s previous stops. It is much harder to write DOWN this sum than it was for us to just change our frame of reference, but it clearly must take Achilles a finite amount of time to cross all of those increasingly small distances - one of those infinite sums, as above, that still sum to a finite number - because, as Exercise 2 showed, we can calculate the exact time that Achilles passed the Tortoise!
Lesson 5: Zeno's Paradox
7/29/09
This is a classic problem that has stumped many students over the years - it seems to say that the slow, ponderous Tortoise can always outrun Achilles, the fastest man on earth. Can you figure out what’s wrong?
