gweetle
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1 divided by 2 = 1/2
1/2 divided by 2 = 1/4
1/4 divided by 2 (now going to be a comma for the pleasure of the reader) = 1/8
1/8, = 1/16
1/16, = 1/32
1/32, = 1/64, however the distance the frog is from the wall, will never reach 0 :)
its marvelous really.
Think of it as two stars. If we can only observe them at a certain distance, it may seem as though they are right next to each other, or touching, and that's all that our current distance system can give us. In forty years or so, Johnny Pearpit can come along and invent a device that allows you to view things 300x closer than what we saw before, and now we realize that they are not in fact touching, but they are so close they look as if they're touching :).
But the mathematical solution to the paradox allows for the frog to take an infinite amount of steps in a finite amount of time. which is really cool, and proven. both by rigorous mathematical proof, and of course, by demonstration.
But, eventually, it would be impossible for him to hop halfway: He would overshoot it.
If you can give me a link to this proof, I will gladly read it.
Gweetie's link was a wiki page about geometric series in general, not the frog problem
there are a couple of solutions on this page, though i could show you the short proof that it takes a finite amount of time to take the infinite amount of steps which does indeed involve the geometric series.
I tried ctrl + f and searched for frog, found nothing.
Also searched for half and found a few things, but nothing relating to this problem in particular.
But Homer isn't attempting to go half the distance with each step.
Eventually, one step will step over the remaining distance, regardless of how small or large.
He wouldn't be traveling half the distance anymore.
This is like saying that Pi has a definite end because circles are closed.
and well pi definitely is a finite number, no question about it.
but take it like this:
say the frog is traveling at a speed v, and starts out at a distance d from the wall.
then the time (d/2)*v + (d/4)*v + (d/8)*v + ... = (vd)*(the infinite sum of the reciprocals of powers of 2)
vd is finite.
the infinite sum of the reciprocal of powers of 2 is finite.
if ab=c and both a and b are finite, then c must also be finite.
which means that t is finite.
the frog can and does reach the wall.
and both physics and mathematics back that up.
The only thing I can come up with to say is that, as a word problem, it makes no sense that the frog would touch the wall.
Maybe if the amount of time was infinite, i don't know.
I quit.
My brain hurts...
You win.