> > >
> > > I am trying to figure something out, and wondered if you guys might have some ideas. Here is the situation:
> > >
> > > You are at point A. You want to reach point B.
> > > Every time you move towards point B, you can only move half as far as the before. Would you ever reach point B?
> > > I am thinking that EVENTUALLY you'll get there. By the way, no loopholes, guys..you can't reach B from A in one move. I don't know why I come up with stuff like this, maybe my brain wants excercise. Does a problem like this seem familiar to anyone else? I really think I've seen it somewhere....
> >
> > Go 2/3rds of the way the first time, and half that the second.
> >
>
> Yeah. This is really similar to one of Zeno's (Zeno is one of those Greek philosopher type guys) many infinity paradoxes, but it only works if you go half the distance or less the first time, in which case you'll never reach the other side.
>
> Tom

Ha! This paradox was the intro to a section in calc I just had a test on. Another one went like this:
"In another version of Zeno's paradox, Achilles can run 10 times as fast as the tortoise, who has a 100-yard headstart. Achilles cannot catch the tortoise, says Zeno, because when Achilles runs 100 yards the tortoise will have moved 10 yards ahead, when Achilles runs another 10 yards, the tortoise will have moved 1 yard ahead, and so on."

In mathspeak, assuming a distance of 1 mile between point A and B, reaching point B would amount to evaluating the infinite sum
1/2 + 1/4 + 1/8 + 1/16 + . . . + 1/2n + . . .
It can be shown that this series converges (approaches a real, finite number, not infinity), and has a sum of 1. Incidentally, it can also be shown that in the above problem, Achilles will catch the tortoise after running exactly 111 and 1/9 yards.
That's the first answer.

The second concerns the assumptions made in the question. The question assumes you can keep dividing the remaining lengths in half an infinite number of times. The problem with that assumption is that, according to a lot of current theory, on the scale of the very small, length is quantized. You can't move an infinitely small distance. At some point, you will reach a limit where you will have to move some finite distance, or not at all. (This distance, of course, is many orders of magnitude too small for anyone to notice, but it does short-circuit Zeno's paradox)

Bal "I hope I got an A on the test" anthalus