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Halving and Halving Again - Zeno's Paradox

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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?

Thanks,
Chris
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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

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

Hi Chris,

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.
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/
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Associated Topics:
High School Calculus
High School History/Biography
High School Sequences, Series

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