Some More Infinite Series Jive

[kpux]

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By definition actually we know that 0.9999....<1.

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This is wrong. There is no point on the real number line between these two values.

[GrekeHaus]

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By definition actually we know that 0.9999....<1. This is an expression of the smallest number possible less than 1 in fact.

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One of the properties of the real numbers (and the rational numbers) is that for any two numbers, there is a number in between them. For example, if our numbers are r and s, then the number (r+s)/2 is between them. As a corrollary to this, for any number, there is no largest number less than it. If a number x < 1, then there exists a number between them (i.e. (1+x)/2).

If you don't believe this, you can use a method similar to the one described above to prove it.

Let x = 0.9999...
then 10x = 9.9999....

The subtract the two lines

10x = 9.9999...
- x = 0.9999...

Gives 9x = 9, so x=1.

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Getting back to the original problem, we also know for a fact that the object cannot possibly reach Point B. As the series progresses, we'll get closer and closer, but at no point can it be shown that we will reach it. Where x is a given point in the series, as the distance traversed during the last second, we will always be x distance away.

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After a finite ammount of time you won't be there. The definition of an infinite sum just says that the sum of the terms approaches that number as the number of terms gets arbitrarilly large. The problem is that you're thinking in finite terms.

[K C]

Requiring a number between two numbers is merely a convention as I see it, and it's going to lead us to a reductio ad absurdum anyway, since we'll have to justify the mumber in between as well, and the one in between these, and so on.

As for the equation you put forth, that's certainly the best justification for the two numbers to be identical, yet we end up with two distinct numbers being equal. The main question here is whether it would be correct to assume 10x is 9.999... It may seem so, but perhaps not. In order for this to be possible, we're moving the decimal one place to the left. Can we really do that with an infinite series of decimals though? It appears to me at least that the decimal series of 10x is going to be infinitely smaller than that of x due to all this, at least conceptually, which is all we're dealing with here anyway.

Now as far as the problem goes the way I posed it actually presents it as a finite problem, as looking into at what point in time can it be said that Point B is reached. Given any x where x is a point in the series we do know that it will not.

In a sense it's the failure of this problem which sheds the most light on Zeno's paradox. Instead of the original discussion imagine this taking place instead, taking it from the point where Achilles is conceding defeat to the Tortoise:

Wait a minute Achilles - under these assumptions the tortise will never make it to 12 meters, let alone the finish line.

Achilles: Duh?

You cover ten times as much distance as he does in a given timeframe. So each advance by him represents 1/10 the previous advance he's made. This mumber keeps decreasing by 1/10 each time. Since he covers 1 meter the first time, he only covers 1.1 meters the second, 1.11 the third, and so on. The decimal of 1 keeps increasing but he'll never even get past this, so he cannot possibly win the race under this conception.

Achilles: Hmm I think I see what you mean. (Looking over at the tortoise) Hey, you tricked me, you little *&^%$%

The tortoise meekly retreats into his shell...

KC

[jason_t]

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"Basically, you can add up an infinite number of positive real numbers and have it come out to a finite sum. For simplicity, let's say the distance A|B = 1. In this example, 1/2 + 1/4 + 1/8 + ... + 1/(2^n) + ... = 1."

This isn't true at all actually. You'll never get to 1 that way, and this gave me an idea for a variation of the paradox.


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The notation

1/2 + 1/4 + 1/8 + ... + 1/(2^n) + ... = 1

stands for

lim n->infinity sum from k = 1 to n of (1/2)^k = 1.

Anyone who tells you anything different is wrong.

[jason_t]

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Requiring a number between two numbers is merely a convention as I see it

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It's not a convention. It's a consequence of basic properties of real numbers that follow from the axioms governing the natural numbers, the construction leading to the real numbers, and basic principles of logic. Or, if one wants to accept the real numbers as basic objects, it is a consequence of the axioms governing the real numbers that for any two real numbers x < y there is a real number z with x < z < y. Quite simply, one can take z = (x+y)/2.

Again, it's not a convention.

[jason_t]

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By definition actually we know that 0.9999....<1. This is an expression of the smallest number possible less than 1 in fact. If we do come up with some equations that seem to conflict with this, it is the equation that has been disproven as correct since this cannot possibly be true under this most basic mathematical premise that it is competing with.

The only premise that we need to justify this in fact is 1>0.x, where x can represent any decimal. This is a convention in fact not a postulation. It doesn't really matter how many decimal places we go here, including an infinite amount of them.

Similarly, divisions such as 2/3 don't work out to 0.67, even though it does give you a result that is as close to this as possible. To get to 0.67 or 0.667 or whatever we need to round the number up. We're not entitled to round up in these examples though.


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This is absurdly wrong, as others have already elaborated. But I want to reiterate and elaborate: the statement under discussion here is not a convention. Mathematics doesn't work by making conventions about its facts. Mathematics works by assuming some basic, reasonable principles and deducing facts using the principles of logic.

Convention: 32F is the freezing point of water.
Fact: There is no real number between .999... and 1.

[FNHinVA]

I have nothing to add to this discussion. I proved that in the "Space, Time & Stephen Hawking Jive" thread I initiated (in a momentary lapse in judgement).

I will make an observation, though.

Watching Math Mystics argue among themselves to the point of describing another's view as "absurdly wrong" is infinitely amusing.

That said, I will run for cover.

[jason_t]

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I have nothing to add to this discussion. I proved that in the "Space, Time & Stephen Hawking Jive" thread I initiated (in a momentary lapse in judgement).

I will make an observation, though.

Watching Math Mystics argue among themselves to the point of describing another's view as "absurdly wrong" is infinitely amusing.

That said, I will run for cover.

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Your attempt to add to this discussion is akin to the following:

Person A: Flushes beat straights because the event (five cards are of the same suit) is less likely than the event (five cards are in sequence).
Person B: It's just a convention that that is true.
Person A: That's absurdly wrong. It's not a convention. It's a fact because the probability of the two events can be calculated exactly and it turns out that flushes are less likely than straights.
You: Person A, you can't call his view "absurdly wrong." It's his view, his opinion, his way of looking at things.

This isn't art, this isn't philosophy. These aren't opinions. They aren't open to debate. They are right or they are wrong. The OP is wrong.

[FNHinVA]

Your off-point response to my observation is also amusing.

[PairTheBoard]

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I have nothing to add to this discussion. I proved that in the "Space, Time & Stephen Hawking Jive" thread I initiated (in a momentary lapse in judgement).

I will make an observation, though.

Watching Math Mystics argue among themselves to the point of describing another's view as "absurdly wrong" is infinitely amusing.

That said, I will run for cover.

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What I'm watching is someone who doesn't know math trying to argue math with someone who does know math. I think the "Math Mystics" are posting on these Forums:

http://p205.ezboard.com/fembracingthemoonfrm38

PairTheBoard