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Question: Does it equal 1?
Yes   -37 (71.2%)
No   -15 (28.8%)
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Author Topic: 0.99999999.......  (Read 10077 times)
Franzl
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« Reply #50 on: November 12, 2009, 08:40:55 am »
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I have nothing to add to the thread, but I'd like to point out that if somone gets the answer "wrong" and it pisses you off and makes you think less of them, YOU ARE AN ASSHOLE!  Period, end of story.

Why would anyone think less of someone because they got mathematical theory wrong?

Personality and other qualities have nothing to do with mathematical ability.
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« Reply #51 on: November 13, 2009, 07:04:45 pm »
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Al is the only one in this entire thread making any sense.
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« Reply #52 on: November 15, 2009, 01:12:38 pm »
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Actually for all intents and purposes this IS mathematical fact. It's universally agreed upon by mathematicians, probably more so than evolution is agreed upon by those in that field.

With mathematics and the sciences it's generally safer to assume that things like this are not solid facts, only well supported theories, as there's always the potential that something else could come along and convincingly contradict the matter in question.  However I don't doubt that this matter agreed upon more by mathematicians than evolution is by biologists.  But still, for example, no matter how obvious it is in our day to day lives, gravity is still a theory, if you catch my drift.

Mathematics and science are fundamentally different in this respect. Scientific views change in light of new facts based on measurements and observations. Scientists views can differ based on interpretation of the data.

Mathematics is based on a set of definitions and axioms that lead to logical proofs. When I specify a set of definitions then I can draw inescapable conclusions. There may be other definitions, but within the confines of one set of definitions there is complete agreement as to the conclusions drawn form those definitions. At times there are mathematical conjectures not yet proved or disproved, but that is different from a difference in interpretation that one sees in science.

In this case of real numbers we are talking about proved statements. If you wish to claim use only of the definitions for rational numbers then your skepticism about this thread's subject is based on that different use of definitions. It doesn't affect the conclusions drawn for real numbers.

I think the point I was trying to make was that mathematics isn't a closed book subject with no scope for nothing new, but rather continually developing with new concepts being found and looked into and so on, obviously I explained it in a rather convoluted way.  As for any scepticism on my part regarding this subject, well put it this way, 0.999... being equal to 1 doesn't make much sense to me (ok, so it still makes none), but I accept it as being the case.

I do not dispute that 0.9999999... is indeed extremely close to one, but so say that it equals one make absolutely no sense whatsoever (but when has mathematics ever done that).  The difference will indeed be very (understatement) small, incalculable in fact, but no matter how much I look at it and think about it, it still strikes me as less than 1.  That said, if I don't need to be too precise when displaying the results of a calculation, I'll just treat it as one all the same (considering that when rounded you get 1), but I'll never be able to view it as equal to one though, that is, exactly equal.  I will look into this further.

OK, despite Libertas extremely nasty behavior....his math isn't wrong.

We agree that any number divided by itself equals 1, right? That can't be disputed! Wink So 3/3=1.

Alright, now if you had to write 1/3 as a decimal number, what would it be? Is it 0.33? or 0.(lots of 3s?). It's clear that it's an infinite number of 3s behind the decimal point, correct?

But now if you multiply that number by 3, what do you get? wouldn't that be 0.333.... x 3 = 0.999...?


To argue that 0.999... is not equal to 1, you would have to dispute that 3 x 1/3 is equal to 1 Smiley




Even though it has moments like this that just don't make any sense on the outside, I actually quite enjoy reading about mathematics.


On the outside, 0.999... and 1 look completely different to each other, and would be seen as not exactly but certainly nigh on equal to each other.  When I first go into the subject, I knew it wouldn't be consistent, but there are something that really do take the cake.  Makes me wonder what it'd be like if we used base 6 numbering instead, for one thing.

I can indeed see where you're coming from here, I'd put most certainly 3 multiplied by 1/3 as equal to 1, such as when you take a circle, slice it into three equal slices and put them together again you get 1 circle.  So, 3 by 1/3 = 1, no argument from me there.  Tough one.  Anyway, you do indeed have a point there, unlike Libertas, who just trolls.  0.999..., weird number, cosmetically unequal to 1, yet for the purposes of most equations the former's treated just like the latter.

     Well what happens is that the difference is infinitely small. As we know, 1/infinity=0. Because of that, any two numbers where the distance in between them is infinitely small are equal, because an infinitely small number is just equal to zero.

Pretty much.
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« Reply #53 on: November 15, 2009, 01:13:18 pm »
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It's certainly very close to 1, and if I was using it in some mathematical equation and didn't need to be too precise, I'd put it down as 1, but in the end though 0.9999999... ≈ 1, at the same time 0.9999999... ≠ 1 either.

It's infinitely close to 1, however, making it mathematically equal.

Quite frankly I fail to see how it's equal to 1, indeed it's very close, but just by looking at it you can see that it's not exactly equal.

Fun isn't it? Smiley

f(x) = 1 / x², a lot of people would say, never touches the x-axis, but only gets closer and closer to it.

It does, though, in theory. Under the same principle as the original question here, the function is considered to be infinitely long, and thus it also becomes infinitely close to the x-axis Smiley

Still, 0.9999999... is not equal to 1.  Tongue

It is though Smiley

What's the difference between 0.9 and 1?
What's the difference between 0.99 and 1?
What's the difference between 0.999999999999 and 1?

If you truly assume an infinite number of 9s behind the decimal point....there can't be any difference between the two numbers.

To claim that the two numbers are not equal, you would have to assume a finite number of 9s behind the decimal point. But even 0.99999999999999999999999999999999999999999999 (and a million more 9s) is not equal to "0.99....".

I do not dispute that 0.9999999... is indeed extremely close to one, but so say that it equals one make absolutely no sense whatsoever (but when has mathematics ever done that).  The difference will indeed be very (understatement) small, incalculable in fact, but no matter how much I look at it and think about it, it still strikes me as less than 1.  That said, if I don't need to be too precise when displaying the results of a calculation, I'll just treat it as one all the same (considering that when rounded you get 1), but I'll never be able to view it as equal to one though, that is, exactly equal.  I will look into this further.

It's certainly very close to 1, and if I was using it in some mathematical equation and didn't need to be too precise, I'd put it down as 1, but in the end though 0.9999999... ≈ 1, at the same time 0.9999999... ≠ 1 either.

It's infinitely close to 1, however, making it mathematically equal.

Quite frankly I fail to see how it's equal to 1, indeed it's very close, but just by looking at it you can see that it's not exactly equal.

Fun isn't it? Smiley

f(x) = 1 / x², a lot of people would say, never touches the x-axis, but only gets closer and closer to it.

It does, though, in theory. Under the same principle as the original question here, the function is considered to be infinitely long, and thus it also becomes infinitely close to the x-axis Smiley

Still, 0.9999999... is not equal to 1.  Tongue

(1/3)=0.333333333....

(2/3)=0.666666666....

(3/3)=0.999999999....


Gotta wonder whether that excellent public school education you kept telling me about is to blame for an 18-year-old being ignorant of a basic mathematical fact...

Being a dick again?  You don't have a proper argument against public schooling so you resort to insulting those who support it.  Above, I'm having a quiet debate with Franzl, and though he may not agree with me on the point at hand, at least he's not acting like a git over it.  Besides, your description of 1 being equal 0.99999... as mathematical fact isn't quite true, considering that mathematics is not a "closed book" or so to speak, if anything it's closer to a mathematical theory- commonly accepted and is backed up by proof and evidence, but still open to debate and amendment pending the surfacing of any proof/evidence in favour or to the contrary.

Anyway, tell me, why do you act like an obnoxious arsehole towards people who have differing opinions to you?  Come back when you have a credible response outside of calling everyone who disagrees with you "brainwashed statist zombie sheeple".  I'm open to discussing things but no personal insults, and you've just gone made them part of your initial response.  I've personally had enough of your egotism, and that "holier than thou" attitude of yours.
You're the one who claimed to be the poster boy of public school education and said I was somehow "jealous" of your public schooling. And here you are arguing against basic mathematics you should have learned when you were ten years old.

Well, let's see, I never claimed to be the poster boy of anything despite what you've said there, and there you are being a jerk making the false claim that I'm arguing against basic mathematics, and I'd explain myself if I didn't think you were going to find a way to twist my words (as you've done in the past).  Believe it or not, I was never told anything about the intricacies of the number 0.999999..., now I don't know how they do things in America, but that was something that was never a priority over here.  Seriously, if you're going to act like an jerk-ass every time I say something, don't bother replying to me.  At least I have the decency to simply pass by things that you say that I happen to passionately disagree with, rather than try to start a flame war like you've been doing.
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« Reply #54 on: November 17, 2009, 12:10:07 am »
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When I read the first page of this thread, it seemed to be a very interesting discussion on limits. However, this thread devolved a bit....

Let's talk limits again people.

Just to get you in the mood....

ASYMPTOTES

END BEHAVIOR

Now let's talk limits!
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« Reply #55 on: December 20, 2009, 04:34:23 pm »
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Yes, there is a mathematical theorem to prove this.
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« Reply #56 on: December 20, 2009, 07:28:35 pm »
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Yes.

You will realize why if you know how to convert an infinitely repeating decimal into a fraction (yes, it is possible).

For example, suppose x = 0.99999999.... (the number that we are interested in now). We want to get rid of the repeating decimal.

Then 10x = 9.99999999....

Subtracting the first equation from the second we get 9x = 9 (now we no longer have a repeating decimal), so x = 9/9, or 1.

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« Reply #57 on: December 20, 2009, 07:46:08 pm »
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Also what is 0.999... + 0.999... ?

It can't be 1.999...8 because you can't have a digit after an infinite amount. So it would be 1.999.... In other words, 0.999... + 0.999... = 1 + 0.999.... Subtracted 0.999... from both sides, and you get 0.999... = 1.
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« Reply #58 on: December 21, 2009, 02:23:53 pm »
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Question for the math and number geeks out there.

Could 1 - .9999(repeating) = *, be regarded as true?
* defined as an infinitesimal

I assume you mean positive infinitesimal. I'm not a mathematician, even of the amateur variety, but here's how I would approach your question:

On the real number line, zero is the only infinitesimal number. Thus, we cannot define 0.999... to be such that 1 - 0.999... is positive infinitesimal, for the simple reason that positive infinitesimal does not exist.

The neglected hyperreal number line does recognize non-zero infinitesimals, both positive and negative. But here, we would seem to encounter the opposite problem: "1 - 0.999... is positive infinitesimal" is ambiguous, because there are an infinite number of positive infinitesimals. Thus, 0.999... would refer not to a hyperreal number, but to a range of hyperreal numbers—all hyperreals smaller than 1 for which the standard part is 1.

Anyway, that's my layman's take. Accept it or reject it.

It can't be 1.999...8 because you can't have a digit after an infinite amount.

As intuitive as that rule is, it strikes me as ultimately an arbitrary one.
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« Reply #59 on: December 21, 2009, 03:00:13 pm »
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Infinity means without end. So nothing can be after an infinite amount as that would indicate an ending to it. 1.999...98 means that there is a final 9 before the 8.
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« Reply #60 on: December 21, 2009, 03:24:29 pm »
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Question for the math and number geeks out there.

Could 1 - .9999(repeating) = *, be regarded as true?
* defined as an infinitesimal

I assume you mean positive infinitesimal. I'm not a mathematician, even of the amateur variety, but here's how I would approach your question:

On the real number line, zero is the only infinitesimal number. Thus, we cannot define 0.999... to be such that 1 - 0.999... is positive infinitesimal, for the simple reason that positive infinitesimal does not exist.

The neglected hyperreal number line does recognize non-zero infinitesimals, both positive and negative. But here, we would seem to encounter the opposite problem: "1 - 0.999... is positive infinitesimal" is ambiguous, because there are an infinite number of positive infinitesimals. Thus, 0.999... would refer not to a hyperreal number, but to a range of hyperreal numbers—all hyperreals smaller than 1 for which the standard part is 1.

Anyway, that's my layman's take. Accept it or reject it.

It can't be 1.999...8 because you can't have a digit after an infinite amount.

As intuitive as that rule is, it strikes me as ultimately an arbitrary one.

In terms of real number analysis, an infinitesimal is not a number, but instead is symbolic of a quantity that is as close to zero as one needs. It's a concept essential to changing from explicit statements of limits to the notation of calculus. Since it's not a real number it can't be the result of the subtraction of two real numbers.

As I described earlier in the thread, we often confuse real numbers with rational numbers when they are quite different objects. Part of the confusion is that there is a subset of real numbers that are equivalent to the rational numbers. Rational numbers can be expressed as a fraction of integers, and a mathematical proof of the type involving multiplication and division relies on the behavior of rational numbers.

First it is useful to note that rational numbers often use two different symbols for the same value. For instance, 1/2 is the same value as 3/6. This doesn't seem to bother
most people, so let's recognize that number systems can have duplicate ways of expressing a value.

Real numbers are constructed as infinite series of rational numbers that converge to a specific value. That value may be a rational number like 1 or 8/3 or it may be an irrational number like pi or the square root of 2.

There is more than one way to write a sequence of numbers that converge to the same value. For instance 5, 4, 3, 2, 1, 1, 1, 1,  ... and 0, 1/2, 2/3, 3/4, 4/5, 5/6, ... both converge to 1. If I go out enough steps both can get as close to 1 as I like. The fact that the first sequence converges to 1 in a finite number of steps doesn't change the fact the second sequence also converges to 1. In terms of convergence they are equivalent and the sequences represent the same real number 1.

Decimal notation is one way the write a representation of the infinite series, so for instance pi = 3.14159 ...  can represent the sequence 3, 31/10, 314/100, 3141/1000, 31415/10000, 314159/100000, ... . When I use 1 as a real number I really mean 1.0000 ..., because that indicates a particular sequence. The sequence represented by 0.9999 ... converges to the same value just not in a finite number of steps in the sequence.

Where the equivalence of 1 and 0.9999... often bothers people is the expectation that two distinct decimal representations each represent different numbers. It's true that the integers each have a unique symbol, but I showed earlier that the rational numbers do not have unique symbols to represent them. The real numbers are based on sequences, and I've already shown that two sequences can converge to the same value. One should accept that real numbers in their decimal representations, like rational numbers, also may have duplicate representations of the same value.

/lecture (it's Christmas break after all Smiley )
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« Reply #61 on: December 21, 2009, 03:36:37 pm »
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So zero is not considered "infinitesimal" for purposes of the real number line? I stand corrected.

But how does your reasoning transfer over to the hyperreal number line? Granted, the series 9/10, 90/100, 900/1,000, 9,000/10,000, ... converges to 1, but isn't the entire question whether we should define the series in terms of the real number it converges to?

BRTD,

It seems to me that there can be a "last" 9 in 0.999..., so long as an infinite number of 9's precedes it.
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« Reply #62 on: December 21, 2009, 04:45:07 pm »
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So zero is not considered "infinitesimal" for purposes of the real number line? I stand corrected.

But how does your reasoning transfer over to the hyperreal number line? Granted, the series 9/10, 90/100, 900/1,000, 9,000/10,000, ... converges to 1, but isn't the entire question whether we should define the series in terms of the real number it converges to?

BRTD,

It seems to me that there can be a "last" 9 in 0.999..., so long as an infinite number of 9's precedes it.

For this discussion I'm avoiding the hyperreals of nonstandard analysis. At this point they have mostly been a convenient shorthand to reach conventional proofs of real analysis. I wouldn't confuse them with numbers that have specific values, which I find to be the essence of this thread.

To your comment to BRTD, there cannot not be a "last" 9. The nature of an infinite series is that it never ends, therefore it is meaningless to say there is a last value in the sequence. It would be just as meaningless to ask for the largest integer. There is always another one higher.
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« Reply #63 on: December 21, 2009, 05:51:37 pm »
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Looks like we only disagree on my reply to BRTD, then. As I see it, speaking of a "last" 9 does not imply an end to the sequence if it is preceded by an infinite succession of 9's. It is figurative rather than literal. (Of course, for consistency it would be necessary to write 0.99, with the bar over the first 9. Multiplication of that value by two could then be "intuitively" paired with 0.98, with the bar over the first nine.)
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« Reply #64 on: December 21, 2009, 06:15:18 pm »
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Looks like we only disagree on my reply to BRTD, then. As I see it, speaking of a "last" 9 does not imply an end to the sequence if it is preceded by an infinite succession of 9's. It is figurative rather than literal. (Of course, for consistency it would be necessary to write 0.99, with the bar over the first 9. Multiplication of that value by two could then be "intuitively" paired with 0.98, with the bar over the first nine.)

The trick is that there is not even an end to the figurative sequence. The appearance is deceptive because it looks like a string of 9's, but that is just notation. The real sequence is 9/10, 99/100, 999/1000, ... . In its real form, there is not even a last element that would make sense figuratively.
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« Reply #65 on: December 22, 2009, 01:25:05 pm »
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To your comment to BRTD, there cannot not be a "last" 9. The nature of an infinite series is that it never ends, therefore it is meaningless to say there is a last value in the sequence. It would be just as meaningless to ask for the largest integer. There is always another one higher.

That's my point really. If you have an infinite number of 9s, you can't have an 8 after them.
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« Reply #66 on: December 22, 2009, 04:49:45 pm »
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To your comment to BRTD, there cannot not be a "last" 9. The nature of an infinite series is that it never ends, therefore it is meaningless to say there is a last value in the sequence. It would be just as meaningless to ask for the largest integer. There is always another one higher.

That's my point really. If you have an infinite number of 9s, you can't have an 8 after them.

     As muon suggests, if you could have that then you could have a largest integer after an infinite number of smaller ones. Maybe it's not totally counterintuitive, but formally defining it would be impossible.
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« Reply #67 on: December 24, 2009, 09:23:52 am »
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If you needed ten billion dollars, and had $9,999,999,999.00 do you have ten billion dollars? Smiley

I think the answer "infinitely close to one in the base ten system" might be the best.

One problem I could see is looking at a situation where X has to be a number equal to or greater than one, X >/= 1.  0.9999999999... would not be greater than or equal to x in that case.
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« Reply #68 on: December 24, 2009, 03:59:31 pm »
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If you needed ten billion dollars, and had $9,999,999,999.00 do you have ten billion dollars? Smiley

I think the answer "infinitely close to one in the base ten system" might be the best.

One problem I could see is looking at a situation where X has to be a number equal to or greater than one, X >/= 1.  0.9999999999... would not be greater than or equal to x in that case.

     It would be more like if you had $9,999,999,999.99999.... The problem with your example is that you are short by an amount that is defined as $1, a non-zero value. In the topic's question, 0.99999999... is short by an infinitely small amount, which is mathematically defined as 0. You see what I mean?
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« Reply #69 on: December 24, 2009, 07:13:44 pm »
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If you needed ten billion dollars, and had $9,999,999,999.00 do you have ten billion dollars? Smiley

I think the answer "infinitely close to one in the base ten system" might be the best.

One problem I could see is looking at a situation where X has to be a number equal to or greater than one, X >/= 1.  0.9999999999... would not be greater than or equal to x in that case.

     It would be more like if you had $9,999,999,999.99999.... The problem with your example is that you are short by an amount that is defined as $1, a non-zero value. In the topic's question, 0.99999999... is short by an infinitely small amount, which is mathematically defined as 0. You see what I mean?

I question if yo can really define this as zero, anymore than can define it as 42. 

For example: X is defined the amount of molecules of a substance needed for a reaction, and the amount of X needed is 1 unit.  0.999999999... units would not be enough.

So long as X isn't a minimum amount, the definition would work.  As soon as it becomes a minimum value, there is a definitional problem.
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« Reply #70 on: December 24, 2009, 09:23:41 pm »
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If you needed ten billion dollars, and had $9,999,999,999.00 do you have ten billion dollars? Smiley

I think the answer "infinitely close to one in the base ten system" might be the best.

One problem I could see is looking at a situation where X has to be a number equal to or greater than one, X >/= 1.  0.9999999999... would not be greater than or equal to x in that case.

     It would be more like if you had $9,999,999,999.99999.... The problem with your example is that you are short by an amount that is defined as $1, a non-zero value. In the topic's question, 0.99999999... is short by an infinitely small amount, which is mathematically defined as 0. You see what I mean?

I question if yo can really define this as zero, anymore than can define it as 42. 

For example: X is defined the amount of molecules of a substance needed for a reaction, and the amount of X needed is 1 unit.  0.999999999... units would not be enough.

So long as X isn't a minimum amount, the definition would work.  As soon as it becomes a minimum value, there is a definitional problem.

     If you figure that any number can be subdivided into infinitely small units, then you run into the problem of Xeno's Paradoxes, in that the runner can never catch the tortoise in that he must catch up to the tortoise at an infinite number of points before he can actually reach the tortoise; in other words, the graph of the distance between the runner & the tortoise approaches the x-axis, but never reaches it.

     As such we have the concept of limits. Experience dictates to use that the runner will eventually pass the tortoise, but for him to do so requires that the graph of the distance reaches the x-axis, or otherwise stated that he catches up to the tortoise. The idea is that x != 0 within a finite number of terms, but it does equal zero in an infinite number of terms. In application to this problem, 0.9999999... is defined by the series 9/10 + 99/100 + 999/1000..., which converges to 1 at inifinity.

     That's my shot at it, though muon2 can explain it better than I can.
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« Reply #71 on: December 24, 2009, 09:26:46 pm »
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If you needed ten billion dollars, and had $9,999,999,999.00 do you have ten billion dollars? Smiley

I think the answer "infinitely close to one in the base ten system" might be the best.

One problem I could see is looking at a situation where X has to be a number equal to or greater than one, X >/= 1.  0.9999999999... would not be greater than or equal to x in that case.

     It would be more like if you had $9,999,999,999.99999.... The problem with your example is that you are short by an amount that is defined as $1, a non-zero value. In the topic's question, 0.99999999... is short by an infinitely small amount, which is mathematically defined as 0. You see what I mean?

I question if yo can really define this as zero, anymore than can define it as 42. 

For example: X is defined the amount of molecules of a substance needed for a reaction, and the amount of X needed is 1 unit.  0.999999999... units would not be enough.

So long as X isn't a minimum amount, the definition would work.  As soon as it becomes a minimum value, there is a definitional problem.

0.9999999999.... is 1, and it's statistically significant. Wink
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« Reply #72 on: December 24, 2009, 10:23:52 pm »
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If you needed ten billion dollars, and had $9,999,999,999.00 do you have ten billion dollars? Smiley

I think the answer "infinitely close to one in the base ten system" might be the best.

One problem I could see is looking at a situation where X has to be a number equal to or greater than one, X >/= 1.  0.9999999999... would not be greater than or equal to x in that case.

     It would be more like if you had $9,999,999,999.99999.... The problem with your example is that you are short by an amount that is defined as $1, a non-zero value. In the topic's question, 0.99999999... is short by an infinitely small amount, which is mathematically defined as 0. You see what I mean?

I question if yo can really define this as zero, anymore than can define it as 42. 

For example: X is defined the amount of molecules of a substance needed for a reaction, and the amount of X needed is 1 unit.  0.999999999... units would not be enough.

So long as X isn't a minimum amount, the definition would work.  As soon as it becomes a minimum value, there is a definitional problem.

Your example is consistent with finite mathematics, but there is a definitional difference when dealing with real algebra. The root problem is in considering decimal expansions of a value using the ellipsis notation. That ellipsis really doesn't work in finite math, but allows one to take rational expressions like 1/3 and write it as a decimal value 0.333 ... . Real numbers which include rational and irrational numbers are defined in terms of infinite sequences and are incompatible with the definitions of finite math. In a sense one is converting the rational value 1/3 into a real analog 0.333 ... .

The ancient Greeks recognized that finite math and rational expressions could not explain all the numeric values they knew. Though they could prove it to be so, their number system lacked the tools to express these irrational values. The Arabic number system eventually provided those tools, which led to decimal expansions of all real numbers, rational and irrational. Yet even with this powerful tool there are cases where a finite rational expression like 1 will do fine, even though 0.999 ... is equal as a real number.

The definitional problem you cite exists, but is solved by adopting the definitions of real algebra. If you confine your math to the finite and rational you can avoid the problem. But like the ancient Greeks, you'll be in a bind if you want to express irrational numbers in a manner consistent with rational numbers, eg. to be able to express both 1/3 and the square root of 2 in decimal notation. The definition that takes care of that inconsistency creates the question raised by this thread.
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« Reply #73 on: December 24, 2009, 10:50:01 pm »
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     If you figure that any number can be subdivided into infinitely small units, then you run into the problem of Xeno's Paradoxes, in that the runner can never catch the tortoise in that he must catch up to the tortoise at an infinite number of points before he can actually reach the tortoise; in other words, the graph of the distance between the runner & the tortoise approaches the x-axis, but never reaches it.

     As such we have the concept of limits. Experience dictates to use that the runner will eventually pass the tortoise, but for him to do so requires that the graph of the distance reaches the x-axis, or otherwise stated that he catches up to the tortoise. The idea is that x != 0 within a finite number of terms, but it does equal zero in an infinite number of terms. In application to this problem, 0.9999999... is defined by the series 9/10 + 99/100 + 999/1000..., which converges to 1 at inifinity.

     That's my shot at it, though muon2 can explain it better than I can.

Actually, this is not a Zeno's Paradox situation.  It is how we use numbers.

We can say that for something to be X, it must have a greater than or equal to value than a specific number.

To oversimplify this, and reveal my lack of understand nuclear fission, at a specific temperature, pressure, shape and I'm sure some other things, let's say that substance Z has a critical mass of 1kg for a chain reaction.  Exactly one kg.  We could say that  the critical mass of Z is k.

You can say Zk >= 1.

Under your definition, you are saying, 0.999... equals 1.  1 = 0.999...  

If Zk >= 1, and 1 = 0.999 then we see that Zk does not equal 1 but an infinitesimal amount smaller than one.

Logically, Zk can be equal to 1.  This statement is then true:  Zk = 1.

You claim 1 can equal  an infinitesimal amount smaller than one.  1 = 0.999...

Therefor Z, which has a critical mass of one can have a critical mass of an infinitesimal amount smaller than one.  Zk = 0.999...  That statement does not follow Zk = 1 or Zk > = 1.

In other words, when 1 is a "threshold number," i.e. you need at least 1, 1 cannot equal an infinitesimal amount smaller than one.

The problem is, we can use a number as a threshold number.  In physics you use threshold numbers with things like critical mass.  In terms of legislative procedure, you use it with things like a 2/3 vote.

I think it is more of a definitional problem.
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« Reply #74 on: December 24, 2009, 10:59:40 pm »
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Your example is consistent with finite mathematics, but there is a definitional difference when dealing with real algebra. The root problem is in considering decimal expansions of a value using the ellipsis notation. That ellipsis really doesn't work in finite math, but allows one to take rational expressions like 1/3 and write it as a decimal value 0.333 ... . Real numbers which include rational and irrational numbers are defined in terms of infinite sequences and are incompatible with the definitions of finite math. In a sense one is converting the rational value 1/3 into a real analog 0.333 ... .

The ancient Greeks recognized that finite math and rational expressions could not explain all the numeric values they knew. Though they could prove it to be so, their number system lacked the tools to express these irrational values. The Arabic number system eventually provided those tools, which led to decimal expansions of all real numbers, rational and irrational. Yet even with this powerful tool there are cases where a finite rational expression like 1 will do fine, even though 0.999 ... is equal as a real number.

The definitional problem you cite exists, but is solved by adopting the definitions of real algebra. If you confine your math to the finite and rational you can avoid the problem. But like the ancient Greeks, you'll be in a bind if you want to express irrational numbers in a manner consistent with rational numbers, eg. to be able to express both 1/3 and the square root of 2 in decimal notation. The definition that takes care of that inconsistency creates the question raised by this thread.


No, I'm saying that 1 = 1, when one is used to represent a "threshold number."  In case where the numeral "1" (or any other numeral, including 1/3 or square root of 2) is used to represent a concept involving a minimum amount, as it can be, "1" cannot be an infinitesimal amount smaller than one.

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J. J.

"Actually, .. now that you mention it...." 
- Londo Molari

"Every government are parliaments of whores.
The trouble is, in a democracy the whores are us." - P. J. O'Rourke

"Wa sala, wa lala."

(Zulu for, "You snooze, you lose.")
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