Technically, 9.9999 repeating does not equal one, but nobody cares enough to argue. The frog would never touch the wall by hopping half the distance, but, eventually, it will be impossible to hop half the distance.
Except anon was saying that 99.99999999999(repeating)% of 1 is equal to 1, when actuality it is not.
Sidenote: In my earlier comment, I said 9.99999 when I meant 0.999999. Oops
This is Zeno's Paradox. The main issue with this argument is that infinity is used as a distance, and not a numerical value. True, there are an infinite number of distances between the frog and where he wants to go, but that does not make the distance itself infinite. The distance still has a set value.
infinte division by 2 say its 2 feet away and it goes half way so thats 1 foot then half of one foot is 6in then 3in then 1.5 in and so on. its a limit approacing infinity or an asymptote. its a thing you learn in precalc
well the term asymptote is what you learn. and yeah its obvious to some people but this guy didnt get the joke so i tried to explain the things in the joke.
between frog and wall is a set distance. the frog hops half that distance. a new distance has been created, which he hops half way through. in theory, he will never touch the wall, since he will allways be half the distance from frog to wall.
google Zeno's problem, and then google a picture of an asymptote, it'll kinda give you the best gist of what's going on without you having to be all "tl;dr, goddamn ************ science all up in dis bitch!" *ahem* I paraphrase, of course. Enjoy!
just think about it. imagine standing 5 units away from a wall, then walking halfway to the wall. now you're 2.5 units away. walk half the new distance, and now you're 1.25 units away. you could keep doing this and never reach the wall, since a/2 can't equal 0 unless a = 0.
Physics says that once you reach the Plank length however, that you've reached the wall in reality.
The shark explained it terribly, let me help, I'm an octopus.
For the frog to reach the wall from where he is, he must first travel half that distance, but to travel that remaining half way, he must travel half of that, so he's now 3/4 of the way there. Then half of that, making him 7/8, then half of that, so he's 15/16 of the way there, and so on, meaning he can never reach the wall.
He gets closer and closer and closer, but each time must still half the distance before moving on.
And since we can't divide down to zero, just smaller and smaller (but positive) values, he can never actually reach the wall.
<< The way this tends to 0 but never actually touches it, can be represented thusly on a graph.
That line is called an asymptote and will go on forever (in both directions and in this case with a mirror image, but that's just because I was too lazy to get a better diagram), getting closer and closer to zero but never touching it.
This assumes the universe is infinitely granular, however the Planck constant is theorised to be the smallest traversable distance and as such one would eventually reach this level of minuteness in which there is no longer a half step to take. thus arriving at the wall.