Zeno's Paradox in the 1 = .999... article

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In today’s Straight Dope article at http://www.straightdope.com/columns/030711.html, Cecil writes:

If you’re still having trouble, consider another example involving a converging infinite series: Zeno’s paradox, proposed by the Greek philosopher Zeno in the fifth century BC. Suppose Achilles and a tortoise have a footrace. Achilles is ten times faster than the tortoise, but the tortoise has a ten-meter head start. In the time Achilles runs those ten meters, the tortoise crawls one meter. In the time Achilles runs that one meter, the tortoise plods another .1 meter. In the time Achilles runs that .1 meter, the tortoise lumbers ahead .01 meter. You get the picture. We seem to be reasoning ourselves to the conclusion that Achilles can never pass the tortoise.

I contend that it is impossible that Zeno created this paradox!

After all, the meter wasn’t invented until then 18th century.

[Lightguy]

This reminds me of a problem I posed to myself one lonely night driving home.
I happened to see a sign saying that home was 60 miles away, and I was going 60 mph. Then I changed roads and by the time I was 45 miles from home I was going 45 mph. So, I asked myself, what if that were a smooth and continuous progression? Would I ever get home? How long would it take?
This is why I gave up math in college, it makes my head hurt.

[Achernar]

No, you wouldn’t ever get home. However, I myself once worked out a similar problem.

If you stayed 1 MPH above your distance, you’d make it. For instance, you went 61 MPH when 60 miles away, and 46 MPH when 45 miles away, and finish going 1 MPH. It would take you ln(1 + x) hours, where x is your initial distance. So if you start out 60 miles away, it would take ln(61) = 4.1109 hours to get home.

[RalfMehta]

Zeno was right in a theoritical sense. He wrote this up to lpease mathematicians all over. And boy was he good. Even after centuries and millenia, it facinates mathematicians.

Practically however, we have seen that Achilles will catch up with the Tortoise. How does he do it? He cheats. At least theoritically speaking.

Achillis does not traverse or pass through every points on the race track. He jumps from one point to another which are not adjacent. We all do when we walk or run. Even that Tortoise does that. But Tortoise’ strides are smaller than Achilles and that is why Achilles will overtake Tortoise.

In Zeno’s mathematical world, both Achillis and Tortoise (and whoever cares to go there) will have to traverse every point on the track and they will prove Zeno right. Actually noone gets anywhere in the mathematical world. Perhaps that is why it is the most hated subject in school.

[moriah]

*Originally posted by RalfMehta *
Zeno was right in a theoritical sense. … Practically however, we have seen that Achilles will catch up with the Tortoise. How does he do it? He cheats. At least theoritically speaking. …Achillis does not traverse or pass through every points on the race track. He jumps from one point to another which are not adjacent…

Ah… no.

Zeno’s problem is that in addition to considering smaller and smaller distances, he’s also considering smaller and smaller moments of time. When he reduces distance to the limit of zero, he’s also bringing time to zero. When time becomes zero, there is no motion, just a snapshot of the event. Thus, Zeno is wrong in theory as well as practically.

**Actually noone gets anywhere in the mathematical world. Perhaps that is why it is the most hated subject in school. **

Ah… no. The math courses I took in school did very well in dealing with motion over time, even at a theoretical level. Zeno is not the mathematical paradigm of any modern mathematics course. And math was my favorite subject.

Peace.

In theory, this post was never posted.

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So what did Zeno use as the unit of distance when he wrote his Paradox? Cubits?

[emarkp]

Zeno also assumed space was continuous.

[jovan]

*Originally posted by tracer *
**So what did Zeno use as the unit of distance when he wrote his Paradox? Cubits? **

Zeno, according to Plato, wrote one book but it is lost now. We know his paradoxes through commentaries by other writers. The “Achilles” paradox shows up in Aristotle’s Physics. From book VI:

Zeno’s arguments about motion, which cause so much disquietude to those who try to solve the problems that they present, are four in number. The first asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal. This we have discussed above.

The second is the so-called ‘Achilles’, and it amounts to this, that in a race the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. This argument is the same in principle as that which depends on bisection, though it differs from it in that the spaces with which we successively have to deal are not divided into halves. The result of the argument is that the slower is not overtaken: but it proceeds along the same lines as the bisection-argument (for in both a division of the space in a certain way leads to the result that the goal is not reached, though the ‘Achilles’ goes further in that it affirms that even the quickest runner in legendary tradition must fail in his pursuit of the slowest), so that the solution must be the same. And the axiom that that which holds a lead is never overtaken is false: it is not overtaken, it is true, while it holds a lead: but it is overtaken nevertheless if it is granted that it traverses the finite distance prescribed. These then are two of his arguments.

As you can see, the telling of the paradox as a tale is entirely a modern invention as the “original” text is rather theoretical.

[Chronos]

Zeno also assumed space was continuous.

As do we. There is no reason to believe otherwise, and it doesn’t change the final answer. Yes, there is some notion (note: not a theory) floating around in physics that space might be discreet, but that idea seems not to be self-consistent, and there is no experimental evidence for it whatsoever.

But that doesn’t matter. Mathematics is quite capable of dealing with Zeno’s Paradox in continuous spaces. The basic idea of the limit has existed since at least Archimedes’ time, and has been fully developed into calculus (which addresses any conceivable variation of Zeno’s paradox you might care to name) for hundreds of years, since Newton and Liebnitz.

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*Originally posted by jovan *
Zeno, according to Plato, wrote one book but it is lost now.

Rassin’-frassin’ Alexandria library fire. :mad:

[Scotty_Rudell]

All this hand waving and theory is making my eyes water. Here’s a simple mathematical proof that 0.999~ = 1.

(1) let x = 0.999~
(2) then 10x = 9.999~
(3) subtracting (2) from (1), noticing that all 9s to the right of the decimal all cancel out, since they repeat infinitely on both numbers, we see that 9x = 9.000~
(4) or, 9x = 9 exactly
(5) dividing by 9, x = 1
(6) substituting (5) into (1), 0.999~ = 1

[RM_Mentock]

*Originally posted by Scotty Rudell *
All this hand waving and theory is making my eyes water.

I was wondering why you were talking about .999…

[Exapno_Mapcase]

*Originally posted by tracer *
**So what did Zeno use as the unit of distance when he wrote his Paradox? Cubits? **

As far as I can tell, he didn’t use units at all, just fractions. (Half the distance to is a fraction, not a unit.)

[amazonian]

Ok, I’m a little intimidated to say anything cause im not as bright as all you ppl responding to this 0.999~=1 thingy.

Maybe we(humans) are just not smart enough and simply cannot understand math in its fullest. OR math is too ambiguous, and it cannot satisfy all of the world’s problems, dealing with numbers and whatnot. Let’s face it, math is man made, just like time. None of it really exists, its just a way for us to explain things to ourselves and others. Maybe math is faulty? Don’t get me wrong I love math, but maybe the whole idea of everything working out and being rational cannot exist. Math is kind of a synonym for perfect. Maybe humans got ourselves in this rut of trying to explain everything, and we just can’t. I dunno, I’d like to know if anyone agrees or disagrees and why. I have always asked myself this question. Just curious.

Anywho, just like to know ur thoughts.

[CurtC]

Godel, is that you?

[Shade]

lol

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My Gödel is killing me!

[Dr.Cobweb]

*Originally posted by Chronos *
Yes, there is some notion (note: not a theory) floating around in physics that space might be discreet . . . .

Well, if space is discrete then it’s certainly being discreet about it. One might even say coy. It apparently hasn’t revealed any evidence of being discrete so far, anyway.

But personally, I wouldn’t be at all surprised if it turned out that space-time is quantized; it would certainly be in keeping with what we know about mass-energy, and the universe does seem to love symmetry. In any case, I suspect it would be easier to prove that space-time is granular (if it is) – than to prove that it is NOT discrete (if it isn’t).

[Darrell]

*Originally posted by amazonian *
**
Maybe we(humans) are just not smart enough and simply cannot understand math in its fullest. OR math is too ambiguous, and it cannot satisfy all of the world’s problems, dealing with numbers and whatnot.
**

Well, it’s not that math is ambiguous since math has a reputation for being very nonambiguous.

Mathematics, contrary to the beliefs of some lay people, does not imply anything (not one ioda–zilch–nada) about the physical universe. But many real world problems are modelled with mathematics. It is impoirtant to realize that, looking at it this way, mathematics is just a tool used to ultimately accomplish something else. A means to an end. Except possibly to mathematicians themselves, who are many times more interested in the actual mathematics. There exists areas of mathematics with no currently known real world application.

**
Let’s face it, math is man made, just like time. None of it really exists, its just a way for us to explain things to ourselves and others. Maybe math is faulty?
**

This is an age old debate (whether math was invented or ‘discovered’). OTOH, time seems to be an actual property of the physical universe. But let’s not delve too deeply into these matters here.

Personally, I subscribe to the philosophy that mathematics is a purely man-made invention. It may very well have been invented out of necessity of knowing certain things about the world, but mathematics itself is not part of the world. It’s just a language, developed by people, that is used to describe our world (amongst other uses).

But this doesn’t mean math is faulty. It just means, it may happen (err…will happen) from time to time, that a person may incorrectly apply some mathematical model to a physical problem.

**

Don’t get me wrong I love math, but maybe the whole idea of everything working out and being rational cannot exist.
**

The whole idea (of math making everything “work out”) is not supposed to exist. What’s really amazing, is many, many things actually do “work out” with an appropriate mathematical model.

Regards,
Darrell