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Halving and Halving Again - Zeno's Paradox

Date: 8/22/96 at 4:27:1
From: Anonymous
Subject: Halving the half, etc.

I have a question about taking something and cutting it in half and 
cutting the half in half...and so on.  Since we can never really get 
to zero by reducing something by halves, does that mean that we are 
floating on air?  I mean, if you jump and you fall and reduce the 
distance you jumped by half, does that mean that you will never touch 
the ground?


Date: 8/22/96 at 11:53:2
From: Doctor Anthony
Subject: Re: Halving the half, etc.

This is the problem in Zeno's (5th century BC) Paradox where Achilles 
and the tortoise had a race.  Achilles could run ten times as fast as 
the tortoise, but the tortoise had a hundred yard start.  

Achilles runs the hundred yards, but the tortoise is now 10 yards 

Achilles runs the 10 yards, but the tortoise is now 1 yard ahead.  

Achilles runs the yard, but the tortoise is now 1/10 yard ahead.  

Achilles runs the 1/10th yard, but the tortoise is now 1/100 yard 
ahead, and so on.  

Zeno's question to his colleagues (which they were unable to settle 
satisfactorily) was how can Achilles overtake the tortoise?

In our enlightened times we are able to resolve this problem because 
we have the concept of the limit of an infinite series.  It is easy to 
show that 

100 + 10 + 1 + 1/10 + 1/100 + 1/1000 + .....
= 111.111111....
= 111 + 1/9  yards (exactly)

So however many fractions (or decimal places) you continue the 
calculation, its value cannot exceed 111 and 1/9 yard.  This is where 
Achilles overtakes the tortoise, and there is no paradox or 
contradiction.  The ancient Greeks did not have our ideas about 
limits, so in their logic the problem could not be solved.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 8/22/96 at 11:59:51
From: Doctor Mike
Subject: Re: Halving the half, etc.

Hi Chris,
People have been thinking about this question of yours for thousands 
of years!

Zeno of Elea was a Greek philosopher(c. 500 B.C.) whose name is the 
earliest associated with the paradox of infinity. He devised several
paradoxes of endlessness, the simpliest of which involved a fabled 
hero of the Trojan War:

If Achilles is to run from point A to B, he must first travel half the 
distance, then half again, and so on. Taking the distance from A to B 
as one, the distance Achilles must travel is the series 
1/2 + 1/4 + 1/8......
Because there is an infinity of terms in this series, Achilles can 
never reach his goal.
The problem with Zeno's paradox is that Zeno was uncomfortable with 
adding infinitely many numbers. In fact, his basic argument was: if 
you add infinitely many numbers, then - no matter what those numbers 
are - you must get infinity.  If that was true, it would take forever 
for Achilles to run from A to B, and this clearly was false.
The modern approach to mathematics and science resolves this paradox 
by using the idea of limits.  If you temporarily stop worrying about 
the infinite number of steps to end this series of distances, and 
instead concentrate of the start, you can see a tendency or trend.  
Adding the first 2 gives 3/4; adding the first 3 gives 7/8; adding the 
first 4 gives 15/16 , etc.  So look at the sequence :

       1     3     7     15     31     63
      --- , --- , --- , ---- , ---- , ---- , ...
       2     4     8     16     32     64   
Mathematicians now look at that and say what is happening is clearly 
that the results are getting closer and closer to one.  So we say that 
that ultimate limiting value *IS* the sum of all the infinite number 
of fractions.
But, you might say, just because a mathematician thinks this looks 
like a nice theory, is it really true?  I say "YES!", because this 
theory is proven out every time someone runs from first base to second 
base, or whenever a rock is dropped and actually hits the ground.  
Nature is actually doing the infinite number of calculations in a 
finite amount of time and getting a finite answer.  This is another of 
many ways where if you look at the mathematics of numbers in just the 
right way, then it can help explain how nature really works.
I hope this helps.

-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
Associated Topics:
High School Calculus
High School History/Biography
High School Sequences, Series

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