Simple mathematical problem (?)

People are trying too hard, I think, to figure out what "0.999. . ." "really means". It doesn't "really mean" anything, any more than "42" really means anything. Both are just a bunch of squiggles on a piece of paper. (Or patterns of light on a monitor. Whatever.)

They mean whatever we define them to mean.

One can reasonably ask, "Why did mathematicians choose the particular definition they did?" (Many of the supposed proofs in this thread should really be considered answers to this question, and not actual proofs in the mathematical sense.)

But how can one claim something is false, or even that there's some uncertainty about it, when it's true by definition?
 
patoco12, I don't dismiss any proofs at all, simple or otherwise. It was just that the proof using limit was among those expounded previously.

I disagree about discarding the reason for using limit calculation. As I said in a post long ago and far away, and as explained by an MIT professor in my Uni text book, it was used to find a value towards which a series converged. The limit calculation is used to calculate this value, not necessarily the sum of the sequence at any point. (I'm probably not explaining my understanding too well here.)

Simple example of where my own head's at (a dis/proof by induction, I think...):

The series {0.9, 0.99, 0.999, 0.999} has only four terms and yet it can be calculated with a fair amount of certainty already that the limit value of this series is 1. But this series definitely does NOT equal 1 at any point, I think you will agree.

Now extend this to 100 terms in the series, and the last term will be 0.9..{many 9's}..99 but again it is still not exactly equal to 1. But the limit value is still exactly 1 now with an extremely high degree of confidence. And so on to a million terms, two million, a billion... Never QUITE reaching 1.

My question is: At what point does this series CHANGE from not-being-1 to becoming exactly-1? How many terms does it take to do this trick?
 
Zep said:

My question is: At what point does this series CHANGE from not-being-1 to becoming exactly-1? How many terms does it take to do this trick?
Click to expand...

This is the missing link I think...

If we take your series and look at every member individually, then you are right -- it never becomes exactly 1. But with the number .9~, we don't have to "get there" because we are ALREADY there.

The number .9~ is always 1. Repeating decimals aren't a process or a calculation; they are representations of numbers.


P.S. I'd like to say that I've enjoyed this entire thread. I've read a lot of point of views that I've never thought of before. No matter what, that is a good thing.
 
69dodge: One can reasonably ask, "Why did mathematicians choose the particular definition they did?" (Many of the supposed proofs in this thread should really be considered answers to this question, and not actual proofs in the mathematical sense.)
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Agreed. I mentioned something similar earlier in the thread, that the proofs are really just proofs that the notations are equivalent. As you seem to be saying, it is a way of confirming that the various definitions are consistent.

Zep: Does the term "for all intents and purposes" mean something here? In which case, I would happily employ it thus:

0.9... for all intents and purposes = 1
Click to expand...
That sounds rather weak. Saying it that way sounds like there might be some context other than an "intent or purpose" where 0.999... is not equal 1, and that would be false.

For example, we don't say

2+2 for all intents and purposes = 4.

Mathematicians can make a much stronger statement than "for all intents and purposes". They can say

2+2 is unequivocally equal to 4.
0.999... is unequivocally equal to 1.


Edited to fix spellign error.
 
patoco12 said:
This is the missing link I think...

If we take your series and look at every member individually, then you are right -- it never becomes exactly 1. But with the number .9~, we don't have to "get there" because we are ALREADY there.

The number .9~ is always 1. Repeating decimals aren't a process or a calculation; they are representations of numbers.
Click to expand...
So the logical inference is that my series can be continued indefinitely, that is to infinity, and its properties will always hold true? In which case I have proven my point, since that series is the definition of 0.9 (rep). At least, that is the definition that is used in the proof-by-limit method.

If this is not so, my question stands. At what point does my series change from being not-quite-1 to exactly-1? It would appear the answer "at infinity" would be reasonable, but that would also logically imply "never" - infinity is a long way off!

By the way, my head is now imploding with the implications... :)
 
Zep said:
So the logical inference is that my series can be continued indefinitely, that is to infinity, and its properties will always hold true? In which case I have proven my point, since that series is the definition of 0.9 (rep). At least, that is the definition that is used in the proof-by-limit method.
Click to expand...

OK, now I see what you are trying to do.

But we don't build an infinite sequence that way. The sequence representing .9~ already has an infinite number of members. There is no point in asking "at what point", because we are already there.

I don't know how else to explain it. Maybe somebody can help me??
 
Zep,

Your use of the word "never" indicates to me that you're still thinking of 0.999... as a process that takes time. As soon as I add the third dot to the ellipsis, 0.999... is exactly equal to 1.
 
Originally posted by Zep
Simple example of where my own head's at (a dis/proof by induction, I think...):

The series {0.9, 0.99, 0.999, 0.999} has only four terms and yet it can be calculated with a fair amount of certainty already that the limit value of this series is 1. But this series definitely does NOT equal 1 at any point, I think you will agree.
Click to expand...
The concept of "limit" is not really useful when dealing with a sequence of finite length. In any case, the limit of that sequence is definitely not 1. The closest any term gets to 1 is 0.999. In order for the limit to be 1, there must be terms that are arbitrarily close to 1.

Also, I'm not sure what you mean by "a fair amount of certainty." This is not probability or statistics. There is no uncertainty. If a sequence has a limit, it definitely has one; if it doesn't, it definitely doesn't.
Now extend this to 100 terms in the series, and the last term will be 0.9..{many 9's}..99 but again it is still not exactly equal to 1. But the limit value is still exactly 1 now with an extremely high degree of confidence.
Click to expand...
Still no.
And so on to a million terms, two million, a billion... Never QUITE reaching 1.

My question is: At what point does this series CHANGE from not-being-1 to becoming exactly-1? How many terms does it take to do this trick?
Click to expand...
It takes an infinite number of terms. (Unless the last term is exactly 1, of course.)

The limit of the infinite sequence 0.9, 0.99, 0.999, 0.9999, . . . is 1, not because any of the terms actually is 1, but because the terms get arbitrarily close to 1. That's what "limit" means.
 
patoco12: I don't know how else to explain it. Maybe somebody can help me??
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OK, I'll give it a shot. Literally. :)

Zep: If this is not so, my question stands. At what point does my series change from being not-quite-1 to exactly-1? It would appear the answer "at infinity" would be reasonable, but that would also logically imply "never" - infinity is a long way off!
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Let's do this the way Zeno would do it. If I shoot an arrow at your head, first it goes 9/10 of the way. Then it goes an additional 9/100 of the way. Then it goes an additional 9/1000 of the way. And then it goes an additional 9/10000 of the way. If I keep describing the arrow's motion in this manner, the arrow will never hit you. Wanna try the experiment? OK, you stand over there, I'll shoot the arrow, and you let me know when the arrow changes from being "not quite there" to being exactly there? :)
 
Question: Why can you NOT get the same infinite-length number from a suitable series with infinite terms?

If you CAN, where has my proof fallen down?

Xouper, I didn't miss it at all - you explained much that I needed to know, and I hope you will forgive if I used the wrong terms, and try to critique my attempt at method instead.

BTW, I do understand the concept of the numbers "already being there." Did from the start. I'm trying to move it on with that in mind.
 
xouper said:
OK, I'll give it a shot. Literally. :)

Let's do this the way Zeno would do it. If I shoot an arrow at your head, first it goes 9/10 of the way. Then it goes an additional 9/100 of the way. Then it goes an additional 9/1000 of the way. And then it goes an additional 9/10000 of the way. If I keep describing the arrow's motion in this manner, the arrow will never hit you. Wanna try the experiment? OK, you stand over there, I'll shoot the arrow, and you let me know when the arrow changes from being "not quite there" to being exactly there? :)
Click to expand...
I DO understand! :)

However, if an arrow DID perform to that mathematical definition then I would have no hesitation at all as I would not be touched! :)
 
Zep: Question: Why can you NOT get the same infinite-length number from a suitable series with infinite terms?
Click to expand...
The notation 0.999... is simply shorthand for the infinite series (0.9 + 0.09 + 0.009 + ... ). They are different notations for the same exact thing. Does this answer your question?

Xouper, I didn't miss it at all - you explained much that I needed to know, and I hope you will forgive if I used the wrong terms, and try to critique my attempt at method instead.
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Well, I wish I knew a nice way to say this so please pardon my directness here, but in mathematics, sloppy use of jargon implies a sloppy understanding of the concepts. In other words, when you use the jargon incorrectly, it is difficult to tell whether your misunderstanding is with the jargon or with the underlying concepts.
 
Zep: However, if an arrow DID perform to that mathematical definition then I would have no hesitation at all as I would not be touched! :)
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So - since the arrow does in fact touch you, there must be something wrong with analyzing the number the way Zeno would have you analyze it. :)
 
Zep, not to put words into your mouth, but I suspect the following is the "(dis)proof" you're looking for:

Proof that .999... is not equal to 1, by induction:

(Base case): Clearly .9 (just the one 9 following the decimal point) is not equal to one.

(Inductive step): If .999...9 (n 9's) is not equal to 1, then certainly .999...99 (n+1 9's) is also not equal to 1.

Therefore, by the principle of induction, .999... (infinitely many 9's) is not equal to one. QED

Is this a fair statement of your argument?

Unfortunately, there's a major fundamental flaw in this argument, which may or may not be obvious, depending on your mathematical background. Can you find it?

(I'll save pointing out the flaw for a later post, or if someone else wants to point it out, go right ahead).
 
In an earlier post I think I mentioned in passing the construction of the reals from the rationals, using Cauchy sequences of rationals. I think familiarity with this construction should aid in understanding why .999... = 1 exactly. The following link doesn't give much in the way of details, but may provide some illumination for those still confused:

http://en2.wikipedia.org/wiki/Construction_of_real_numbers
 
Cabbage said:
Zep, not to put words into your mouth, but I suspect the following is the "(dis)proof" you're looking for:

Proof that .999... is not equal to 1, by induction:

(Base case): Clearly .9 (just the one 9 following the decimal point) is not equal to one.

(Inductive step): If .999...9 (n 9's) is not equal to 1, then certainly .999...99 (n+1 9's) is also not equal to 1.

Therefore, by the principle of induction, .999... (infinitely many 9's) is not equal to one. QED

Is this a fair statement of your argument?

Unfortunately, there's a major fundamental flaw in this argument, which may or may not be obvious, depending on your mathematical background. Can you find it?

(I'll save pointing out the flaw for a later post, or if someone else wants to point it out, go right ahead).
Click to expand...


Here is the flaw...

You've proven that .999...9 is not equal to 1 for any finite n greater than 0 -- not for infinity.
 
patoco12 said:
Here is the flaw...

You've proven that .999...9 is not equal to 1 for any finite n greater than 0 -- not for infinity.
Click to expand...
Or, in other words, the flaw is the unstated erroneous assumption that, at some point, n + 1 = infinity.

BillyJoe
 
patoco12 said:



Here is the flaw...

You've proven that .999...9 is not equal to 1 for any finite n greater than 0 -- not for infinity.
Click to expand...
Absolutely right, of course.

I was going to illustrate the flaw with an analogous, and obviously flawed, argument:

The set {0} is finite.

If the set {0,1,2,...,n} is finite, then adding a single element to get {0,1,2,...,n,n+1} is surely still finite.

Therefore, by the principle of induction, the set of natural numbers {0,1,2,3,...} is finite. QED.

And which is obviously false.

I wanted to draw an anology between this example and a question Zep asked earlier:
At what point does this series CHANGE from not-being-1 to becoming exactly-1? How many terms does it take to do this trick?
Click to expand...
This is directly analogous to the question: At what point does the sequence of sets {0}, {0,1}, {0,1,2},... CHANGE from being finite to being infinite? How many terms does it take to do this trick?

And I think the answer should be clear. If I stop the sequence of sets {0}, {0,1}, {0,1,2},... at any point, I'm still left with merely a finite set. Only the "full" set {0,1,2,3,...} is infinite.

Similarly, if I stop the decimal expansion of .999... at any point, I'm left with a number strictly less than one. Only when there are infinitely many 9's do I get a number that is exactly equal to 1.
 
Igopogo said:

"1/3 = .3 rec
Multiply both by 3 to get
3/3 = .9 rec
3/3 is equal to 1
Therefore
1 = .9 rec"

I think this assumption is flawed:

3(.333…)=.999...
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Well I disagree but I'll put it another way.

What is .9 rec divided by 3? If it's anything other than .3 rec I'd be interested to see the proof.
 
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