Halving and Halving Again - Zeno's ParadoxDate: 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 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 ahead. 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/ |
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