How do we reach anything when the distance to it can always be halved?

Question

I read an answer to this a while ago, but have forgotten! Can anyone remind me?

Answer 1

As mentioned above, this is Zeno's (or Xeno's) paradox. Suppose a rabbit is chasing a tortoise. The tortoise starts some distance (x) ahead of the rabbit. In the time that it takes the rabbit to move ahead x to ctach up with where the tortoise is now, the tortoise has now moved ahead to y (y less than x). In the time it takes for the rabbit to get to y, the tortoise has now moved on to z (z less than y). And so on. It seems the rabbit can never catch the tortoise because it keeps moving ahead in this manner.

The problem for Zeno is that the ancient greeks didn't have any maths available to help them deal with infinite sums. These days we can readily demonstrate that some infinite sums diverge (ie add up to infinity) - eg the sum of real counting numbers diverges. On the other hand some sums to infinity converge on a finite answer - eg the sum of the fractions 1/2+1/4+1/8+1/16+.... will never be larger than 1. The case of the rabbit and tortoise above is such a sum, it converges on a finite (and rather short!) time period, and the rabbit overtakes the tortoise in short order.


Hope this helps!
The Chicken

Answer 2

This is Zeno's paradox. The answer is that you can have an infinite number of terms that add up to a finite sum. Take a ruler one metre long. Make a mark at half a metre; that is, halfway along. Then mark 3/4 of the way along. Then 7/8 of the way along, and so on. Starting from zero, the distance between marks is 1/2, 1/4, 1/8, 1/16 etc. So as you travel along the ruler from one end to the other, the distance you travel is 1/2+1/4+1/8+1/16+1/32... and so on forever. But you can see that this infinite series adds up to 1.

Answer 3

Here is a mathematical answer to this question. All of space can be thought of an infinite amount of infinitelly small pieces. This is the basic idea behind integral calculus. Although this idea is a fiction, it does help to analyze any physical system. However, since the space intervals in this question are discrete, one must not even use integral calculus. Consider a person walking one mile. He first walks half, then half of half, etc. The distance in which he walks is expressed as an infinite series: 1/2 + 1/4 +1/8 + ... + 1/(2^n). This series converges to 1, which solves the problem. A way to visualize this is to draw a square. Draw a line that divides it in half. Then draw another line that divides one of the halves in half. This process can be repeated indefinately, but the sum of all the little squares still adds up to one square.

Answer 4

Go twice as fast. (hehehahhehe) Seriously, the description of a object approaching some point by halving the distance is fine, but just a description. Asymptotic analysis would extend this solution to the inevitable result of arrival. We observe objects arriving, so the 'paradox' is not something physical, just a flaw in the description.

However, the case could be made that the time it take the observer to measure half the distance becomes infinite. Hence, there will never be enough time to get there, no mater how fast. (giggle, hehehe)

Answer 5

If you travel at constant velocity in the correct direction the fact that the distance to your final destination can be halved is irrelevant as you are covering a constantly increasing amount of ground anyway and weather you want to halve it, quarter it or anything else the final destination is not actually moving anywhere.

Answer 6

Because their are an infinite amount of halves between two different obstacles we cannot reach anything,try dividing one by two...you will never attain a zero,we actually feel as though we touch objects because of repulsion between the objects particles and us.

technically I don't believe it because lets say you are in a race,if you half the distance between yourself and the person in front of you how will you ever overtake that person?I believe that we subtract,for instance if there are 4cm between two people and one of them goes to steps close to the next person every 5second,in 10seconds they will be together... It makes more sense don't you think?Especial because it means two objects can overtake each other.But I ain't a superbrian so don't go about the world saying Quantum is wrong.

Answer 7

i assume that r question is in reference to a philisofical question put foward suggesting that if u can only move half the distance towards an object will u ever reach the object.

the simple answer is one of which the definition is the definer of the definition, what i am refering to is infinate, infinate is defined as the number of halves that are required to reach the object.

so my answer is that u will reach the destination after going half the distance an infinate amount of times

Answer 8

If you made the distance as a line of photons you would have to figure out a way to halve a photon.Remember your counting structure has diminished.You can not use an external counting system.

Answer 9

Because no mater how often you reach the middle, thats as far as you can ever go. If you go 10 miles from "A" to "B"- you first reach 5 miles, then 2.5 miles, then 1.25 miles then 500 yards, then 250 yards, then 125 yards then 62.5 yards,,,,its essentially a repeating decimal in physcial form

Answer 10

because the time required also halves. This is called Zeno's paradox and is not actually a paradox because there is a solution.
If he had kept thinking along these lines he might have discovered calculus way ahead of Newton.