posted
anti_maven: While .999...=1 may be not generally well known or understood, perhaps even unintuitive, I find your assertion that it is an "exotic theory" troubling. To borrow an example from physics, the rules of thermodynamics and their full consequences are not easily understood by most people. However, they are fundamental and knowledge of them is in so many areas of modern industrial technology, such that calling them "exotic" is a doing them a disservice.

Similarly, .999...=1 is not easily understood and like thermodynamics, we do not need to fully understand it in order to make use of theories and technology that make use of it. However, it is far from "exotic." IIRC, understanding it is essential for the concept of limits, which is a pretty basic building block for many of the proofs in calculus. In a related field, statistics, this also leads to consequences such as that implied by Papa Moose's area question. Unintuitively, if you have a continuous probability function, the probability at a single point is actually zero. In plain English, your probability of dying at 75 is zero. Your probability at dying at 75.0000001 is also zero. But your probability of dying between 75 and 75.0000001 is non-zero.

Failure to grasp it can lead to weird consequences such as Zeno's paradox. If you truly believe that .999.. != 1 then consider this (from MathWorld):

quote: Imagine the great Greek hero Achilles starting a race with a turtle. Achilles is a fast runner, running 10 metres per second, while the turtle is slow and runs at one metre per second. Therefore Achilles agrees to give the turtle some advantage and the turtle starts 10 metres in front of Achilles. The ancient Greek philosopher Zeno found the following “paradox”.

If Achilles wants to get in front of the turtle he first has to run to where the turtle started. But in that time the turtle has bridged some distance, which Achilles now has to run in order to take up. But in this time again the turtle has gone for some distance and Achilles is still in behind of the turtle. This process continues forever and apparently Achilles cannot pass the turtle.

So unintuitive, I can grant you. But "exotic" theory? Hardly. Certainly not in the same sense that, say, quantum mechanics can be.
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posted
Since it is something that I (so not a math person at all) learned in high school, I have a difficult time thinking it is particularly "exotic".
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posted
Either that last proof is wrong, or I missed something. How can you numerically substitute the x in 99x to get 99, without assuming x=1?
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quote:Originally posted by DDDaysh: Ok... so I think I'm the only one who could end up in an overwhelmingly painful argument about whether or not .99999999999........ is equal to 1.

I got into a fight with my partner about this same thing a couple of weeks ago. She insisted that they were different, and there was nothing I could say to convince her otherwise. And she's a teacher, too. The argument started because she was complaining about a fellow teacher who was saying .9 repeating was the same as 1. <sigh>
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quote:My head just exploded. Fortunately, the pieces are immeasurable, and we will be able to construct two perfectly good heads from them. Once we scrape them off the computer screen . . .

ROFL @ the Banach Tarski reference! Now I'm going to cry that I want to use that in conversation but I don't think I'll have the opportunity to for a long while, unless I don't care if no one understands me.
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quote:Originally posted by mr_porteiro_head: I really like the algabraic proof from the wikipedia link.

The fraction proof is the one I usually use. It's simple and straightforward. Kind of obvious, actually.
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posted
I don't understand why math people still love Zeno's paradox so much. Okay, so it's unintuitive. It's also wrong.

The only thing Zeno's paradox proves is that it's possible to demonstrate mathematically things that are completely bogus in the real universe. Every day of our lives we see examples of things catching up to other things in motion, despite the fact that Zeno "proved" this is impossible. Every time a car passes you on the road, they prove Zeno wrong.

If Zeno had been right, then no one running away from anyone or anything would ever get caught, unless they were stupid enough to stop moving. Races would never be run, because whoever first got the lead position would be guaranteed to keep it. If someone shot at you, all you'd have to do to survive is start walking away; it doesn't matter how slowly you move, because as long as you keep moving, the bullet can never reach you.

It would be perverse to deny a lifetime's worth of empirical evidence and say that all the things we see don't really happen because you can mathematically prove they're impossible. Where the theories fail to match up with reality, it is the theory, not reality, that is in error.

So, given that everyone is trying to prove that 0.999...=1, that they really are equal in reality as well as mathematical theory, then I must ask how Zeno's paradox even relates.
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posted
Verily: you misunderstand. Zeno's paradox only arises due to a misunderstanding of mathematics. When mathematics is properly applied, the paradox is resolved.

Specifically, even if you can break a finite distance into infinitely many parts, the sum of the infinitely many pieces of time to travel those infinitely many parts can still be finite, meaning that people do catch up to other people and run into walls.

Zeno though that because there were an infinity of parts of the journey there would be an infinity of time required, which is not the case. Properly applying mathematics shows that there is no paradox, its only a misunderstanding of mathematics that causes confusion.

The reason Zeno's paradox is brought up is because the mathematical issue at its heart is a limit, and .999999 . . . is just another way of expressing a very similar limit (you could even make it the same limit by doing Zeno's paradox with 9/10 instead of 1/2).
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posted
fugu13: Bravo, I was going to type out the explanation, but you jumped in and posted a much simpler one.

Verily: To elaborate, the whole point is that Zeno's paradox is used to challenge your initial intuition. If you follow the initial intuition that .999... != 1 then there are weird consequences including the unintuitive Zeno's paradox. Your own belief that Zeno's paradox is wrong should nag and poke at your belief that .999.. != 1 until you realise that your intuition in that case is wrong.

(I'm using "your" in the generic sense)
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quote:Originally posted by mr_porteiro_head: I really like the algabraic proof from the wikipedia link.

The fraction proof is the one I usually use. It's simple and straightforward. Kind of obvious, actually.

I agree, but that can also be a weakness for some people.

It's so simple and obvious that some people will say "There's got to be a trick. You're missing something".

I actually had a conversation the other day with someone about the Monty Hall problem and when I proved to them the counterintuitive solution, they said exactly that.
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There's actually an interesting lecture by Richard Dawkins on this subject here. For those that are religious and worried about his reputation as an aggressive atheist, rest assured that this lecture contains no religious (or anti-religious) content.
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posted
m_p_h: which counterintuitive solution? There's a correct one and an incorrect one, and the incorrect one is significantly more popular.
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quote:Originally posted by Will B: Either that last proof is wrong, or I missed something. How can you numerically substitute the x in 99x to get 99, without assuming x=1?

The proof is badly phrased; I do not believe you are missing anything. Try it this way:

x = 0.999... 100x = 99.999... 100x - x = 99.999... - 0.999... 100x - x = 99 x = 1

QED. You will note that I never assume x=1, I assume x=0.999... and also that y.999... - 0.999... = y.
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quote:Originally posted by fugu13: m_p_h: which counterintuitive solution? There's a correct one and an incorrect one, and the incorrect one is significantly more popular.

OK, here's a thumbnail sketch of the proof I used:

If you don't switch, then the only way to win is to guess the right door intially. There's a 1:3 chance of doing that.

If you do switch, you will always lose if you choose the right door initially (1:3 chance), and you'll always win if you choose a goat initially (2:3 chance).

So, tell me, how is this incorrect?
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quote:Originally posted by mr_porteiro_head: I really like the algabraic proof from the wikipedia link.

The fraction proof is the one I usually use. It's simple and straightforward. Kind of obvious, actually.

I agree, but that can also be a weakness for some people.

It's so simple and obvious that some people will say "There's got to be a trick. You're missing something".

I actually had a conversation the other day with someone about the Monty Hall problem and when I proved to them the counterintuitive solution, they said exactly that.

I accept that the Monty Hall thing is true, but it feels wrong to me. While the .9repeating = 1 thing seems intuitively obvious to me.

A friend of mine in college once defined a natural-born mathematician and someone to whom e to the i(pi) is obviously -1.
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There's actually an interesting lecture by Richard Dawkins on this subject here. For those that are religious and worried about his reputation as an aggressive atheist, rest assured that this lecture contains no religious (or anti-religious) content.

I'm not surprised. The concept of infinity is very much connected to faith. So is the ability to accept and to use ideas that we can never entirely get our heads around. I can't really watch the lecture here at work. Is there a transcript anywhere?
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posted
m_p_h: You're mixing probabilities from before the goat door was opened with probabilities from after the door was opened, which is the fallacy. The door being opened changes the probabilities.

The prize is placed behind one door at random. The probability of it being behind any one door is exactly the same as the probability of it being placed behind any other door, absent further information.

Before the door is opened, the probability of it being behind your door (or any of the other doors) is 1/3, because its one of three equally likely possibilities.

After the door with a goat is opened, the probability of it being behind your door (or the other door) is 1/2, because its one of two equally likely possibilities (the third having been ruled out).

Consider it this way. Number the doors one, two, and three. Say you pick door one, and door three is opened. By your logic the probability it is behind door two is higher than the probability it is behind door one, so you switch.

But if you had chosen door two and door three were revealed, your logic would assert the probability it was behind door one is higher (and of course you'd switch if that were so, so you switch).

So for your logic to be correct, somehow which door you pick has to influence which door the prize was placed behind. This would be quite a feat, since the prize is placed behind a door before you pick one!
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quote:So for your logic to be correct, somehow which door you pick has to influence which door the prize was placed behind.

I don't have the time to get into a big argument about this, but my logic does not depend on that at all. In fact, it depends on which door I pick to not influence which door the prize was placed behind.
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posted
Let me make it simpler. If you had picked the other (unopened) door first, you'd be arguing that you were more likely to win if you switched to your current door. Both arguments cannot be correct, since the event of you picking a door is independent of the location of the prize.
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posted
Fugu, here are the nine possibilities, which are equally likely, since the door you pick and the prize door are mutually exclusive events (the door the host picks is NOT a mutually exclusive event and is irrelevant to the analysis):

Prize Door / Door you pick / better to switch? 1 1 NO 1 2 Yes 1 3 Yes 2 1 Yes 2 2 NO 2 3 Yes 3 1 Yes 3 2 Yes 3 3 NO

As you can see, 2/3 of the time it is better to switch.
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posted
fugu, You've got that all wrong and mph has it correct. The probability after the door is opened is 2/3 that it is behind the other door and 1/3 behind your door.

You are not taking into account that the door opened is dependent on the door you chose. It's opening this door, and not where the prize is put behind that introduces the dependence that changes the probabilities. Also, the options aren't really win, lose, and lose, but rather win, lose one way, and lose another way.

Three results, G1, G2, and P1

Picking any one of them is equally likely

G1 -> take away G2, other is P1 G2 -> take away G1, other is P1 P1 -> take away either G1 or G2, the other remains

Only in the case where you initially pick P1, which we can all agree has a 1/3 probability, should you not switch.
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Let's stop with the imprecise language such as "other door". Let's talk about doors A, B, and C.

Let's also say that the prize is behind door A.

First, let's assume I never switch:

If I pick A, I win. If I pick B, I lose. If I pick C, I lose.

Now, let's assume I always switch:

If I choose A, he'll open B or C. I'll then pick either B or C and then I lose.

If I choose B, he'll open C. I'll then pick A and win.

If I choose C, he'll open B. I'll then pick A and win.

----

This can be repeated for when the prize is behind B and when it's behind C with equivalent results.

edit: Um, I hadn't realized that there were already two explinations of this. I wasn't meaning to dogpile you.
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quote:Let me make it simpler. If you had picked the other (unopened) door first, you'd be arguing that you were more likely to win if you switched to your current door. Both arguments cannot be correct, since the event of you picking a door is independent of the location of the prize.

This is fallacious reasoning, because, as I pointed out, picking a door changes what other door would be opened.
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posted
There's a fun simulator here which talleys the results and you can that as you go through more iterations, the percentages go toward 33% and 67%.

Not a proof, but for some people it's more convincing than a proof (for instance, the person who didn't believe my proof the other day, which started this tangent).

Unfortunately, I couldn't get it to work in firefox, so you'll probably have to use IE.
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posted
It took me a while to convince myself of the Monty Hall problem.

Here's what flipped the switch for me:

When you pick the door, there is a 1/3 chance that you picked the car.

When the host reveals the goat, there is STILL a 1/3 chance that you picked the car. No matter what door you picked, he will still show a goat, so it doesn't affect your probability of having selected the car.

That part is intuitive, and essential to understanding it, or at least it was to me.
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KoM, Indiana, that's an interesting proof. I get it now. I like that it can be done without using limits.
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posted
Well, actually, I'm still using two properties of limits, namely that they can be multiplied and subtracted in a reasonably intuitive fashion. Still, at least those properties really are fairly intuitive, although I'm just waiting for someone to object to the operation 100*0.999...=99.999....

In case you didn't notice, that last sentence does in fact have a period at the end. It just sort of fades into the scenery. It's a stealth period!
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