Look here for a small mathematical problem that was concerning me somewhere in the high school.
It is called Zeeman’s problem. But i think that the problem was also posed by some antic philosophers.
The problem sounds as follows :
Suppose you want to pass accros the street. You pass a half. So it rests you another half to pass. You pass another half of this half and it rests you one quarter. You pass a half of this quarter and it rests you an eighth of the total distance. You pass a half of this eighth and it rests you a 16th part of the road.
It seems that you’ll never reach the destination and some car will hit you because you didn’t passed the road.
But how can we solve this interesting problem? It’s obvious that we’ll always have a small part to cross, even if it is very very small. The mathematics resolved this problem centuries ago (16 th century).
Look here for a Solution (there are of course others) :
We observe that we always have halves. We have :
1/2, 1/4, 1/8, 1/16, …, 1/n
So basically they are powers of 2 :
2^-1, 2^-2, 2^-3, 2^-4,…, 2^-n
Or :
1/2^1, 1/2^2, 1/2^3, 1/2^4,…, 1/2^n
If we add them arithmetically we see that it’s always missing a unit of a number power of 2 :
for example : 15/16+… . It is missing a 1/16.
If we have n=8. We’ll find that we have crossed 255/256. It rests 1/256 still to cross
But we want to find which is this small part when the n increase to infinity.
We must prove that when n increase to infinity this small part is 0. The part that it rests us can be written as 1/2^n.
And the mathematics has a way to say this :
lim 1/2^n = ?
n->oo
Well, the answer is very simple. The result is 0. The missing part is 0 if n increase to infinity.
So the answer is :
lim 1/2^n = 0
n->oo
And you have passed the road. You are safe now. No car will hurt you.
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