 |
 |
 |
| Home Page |
| Guest Book |
| Chat Room |
| Contact Me |
| Family Pages |
| Paul Bates's |
| Mathematics |
| Puzzles |
| Lists |
| My Books |
| Links |
Zeno's Paradoxes - The Achilles
The Problem
In his most famous paradox Zeno proves that anything that is moving can never catch up with anything that is moving slower than it.
The slower when running will never be overtaken by the quicker; for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must necessarily always be some distance ahead.'
Let's look at what he says.
First assume that a hare is chasing a tortoise. The tortoise starts some distance ahead - lets say 10 metres. They both start running at the same time. The hare runs at a speed of 10 metres every second and the tortoise runs at a speed of 1 metre every second. After 1 second the hare has got to the tortoise's starting position, but the tortoise has moved 1 metre in the same time, so the hare has not caught up yet. The tortoise is now 1 metre ahead, but by the time the hare travels 1 metre the tortoise has traveled 10cm. The tortoise is now 10 cm ahead, but by the time the hare travels 10cm the tortoise has traveled 1cm. The tortoise is now 1 cm ahead. I could keep going on indefinitely but you should see that the tortoise is always a bit further on when the hare gets to where he was.
Notes
Everybody knows in the real world Zeno's Statement is wrong. After all, if it were true the world would be a very different place. For one, the current hunting debate would not exist, as there would be no hunting. No one would chase a fox if they could never catch it. Although the statement is clearly not correct, Zeno's argument baffled the Greek philosophers - as well as the philosophers that came after them. Zeno's riddle plagued mathematicians for nearly two thousand years.
The Solution
It is the infinite that lies at the heart of Zeno's Paradox. Zeno had taken continuous motion and divided it into infinite steps. The Greeks believed that because there were infinite steps the race would go on forever, even though the steps got smaller and smaller. The ancient Greeks didn't have the mathematical knowledge to deal with the infinite, but modern mathematics does. The infinite must be approached very carefully, but it can be mastered.
It is sometimes possible to add an infinite number of terms together and get a finite result, but to do so the terms must approach zero. (This is a necessary, but not a sufficient condition. If the terms go to zero too slowly then the sum will not converge to a finite number).
Lets take another look at the problem. To make the numbers easier to work with lets assume that the hare runs twice as fast as the tortoise and that it starts 1 metre away. The logic still works. When the hare has traveled 1 metre the tortoise will have traveled a ½ metre. When the hare travels that ½ metre the tortoise will have traveled a ¼ metre further etc.
Let's look at the distances the hare runs in each step
Distance Traveled (m)
1 1 1/2 0.5 1/4 0.25 1/8 0.125 1/16 0.0625 1/32 0.03125 1/64 0.015625 1/128 0.0078125 1/256 0.00390625 1/512 0.001953125 1/1024 0.0009765625 You can keep adding rows to this table and the number will not reach 0, but it will get closer and closer. This is true until you add an infinite number of rows. The distance then approaches:
1/infinity 0 Modern mathematicians say that the terms have a limit.
If after infinite steps the the step size is zero, the journey has a destination. Once the journey has a destination it is easy to ask how long it will take to get there. This is something the Greeks could not do as they could not deal with infinity and therefore their race had no destination.
Now lets add the distance the hare runs
Total distance traveled (m)
1 1 1 + 1/2 1.5 1 + 3/4 1.75 1 + 7/8 1.875 1 + 15/16 1.9375 1 + 31/32 1.96875 1 + 63/64 1.984375 1 + 64/128 1.9921875 1 + 255/256 1.99609375 1 + 511/512 1.998046875 1 + 1023/1024 1.9990234375 The numbers 1+(1/2)+(1/4)+(1/8)+(1/16)+(1/32)+....+(1/2n)+... approach 2 as the limit. In the same way that the steps the hare runs get smaller and smaller, and closer and closer to zero the sum of these steps get closer and closer to 2. How do we know this? Well, lets start off with 2, and subtract the terms of the sum one by one. To start with we take 2 and subtract 1, which is of course,1. next subtract 1/2, leaving 1/2. Now take off 1/4, leaving 1/4, Next take off 1/8, leaving 1/8. We can see we are back to the familiar sequence, and as we know 1,1/2,1/4,1/8 etc has a limit of zero subtracting these sums from 2 will leave 0. So the hare will catch up the tortoise in 2 metres
Now if we go back to our original step sizes of 10,1,0.1,0.01,0.001 the terms again approach 0 and the sum of the terms approach 11.11111 (the .11 is recurring i.e. goes on for ever). So the hare will catch up with the tortoise in 1 + 1/9 metres (0.111111 = 1/9). There exists the possibility for some confusion here - you may believe that as the 1's go on forever so does the race, but don't be fooled. 0.11111111 recurring is a real number - it is 1/9 (Type 1 ÷ 9 into a calculator to check this). although a metre is not exactly divisible by 1/9 (it is 11.1111111cm) this is only a man made problem. We have 100cm in a metre simply because it is easy to work with, we can quite easily split the metre into any number of sub units, as do other measurement standards (the foot contains 12 inches for example). If we choose 90 sub units and give them a name - lets call them nics, but any word will do. (we define the nic as 1 ninetieth of a metre) We can now see that the hare will catch up the tortoise in 11.111111 metres or exactly 11 metres and 10 nics. (1 metre = 90 nics : (1/9)x90 = 10)
Now we know the distance we can also work out how long it will last. Since the race is 11 and 1/9 of a metre in length, and the hare is running at 10 metres per second (22.37mph - That's one fast hare) it is easy to work out that the hare will catch the tortoise in 1 and 1/9 of a second. (see previous paragraph regarding this 1/9 of a sec).
Footnote
The Greeks couldn't do this neat mathematical trick. They didn't have the concept of the limit. They didn't believe in 0 and they couldn't handle the infinite. This is the biggest failure in Greek mathematics, and is the only thing that stopped them discovering calculus.
If you understand the reasoning behind the answer to this paradox you are on the road to understanding calculus. And you thought math's was hard!
Other Explanations
If you don't understand the above answer fully then consider this.
The distance the hare travels in each step is 1/10 of the previous step distance. If you believe that you can keep on dividing a number by 10 forever Zeno must be correct. But we know he is not correct so there must be a problem somewhere. The mathematical world and the real world don't agree, as we know the hare will catch the tortoise. Maybe the error is our assumption that we can keep dividing a number by 10. We must change this assumption and state that in the real world (unlike in the mathematical world) space cannot be divided into an infinite number of steps. If this is true there is a smallest distance one can travel. Let's look at what this means. If the distance between A and B is the smallest distance that can be traveled and you were standing at A and moved to B you would never be anywhere in between. This sounds almost as strange as our original statement that a slower object could not be overtaken by a faster one, but Quantum Mechanics (The study of the very small) seams to backup this idea. In Quantum Mechanics there is a smallest distance you can travel. It a distance around 1019 (10000000000000000000) times smaller than a proton. Distance's this small are smaller than atoms by as much as atoms are smaller than stars.
Links To Other Websites
Further Reading
 |
|
|||||
|
|
|||||
|
This page was last updated on 20-07-2005. |
You can please some of the people all of the time, all of the people some of time, but you can't please all of the people all of the time