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Jeremy
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 Re: 0.999.   « Reply #100 on: Nov 7th, 2002, 5:56am »

Ok, i really suggest we drop this thread... i think if we haven't saved anyone by this time, no one is going to change their opinions. Hopefully we can wait till they get to a pre-calculus class, and then they can take it up with their math teacher.

And as for Guess Who's comment about 1/3 not being equal to .333... because if you do long division you will ALWAYS have a remainder

umm.... well duh. the 3's just keep going... the ellipse (...) means it's a repeating series. The three's never stop, so you keep having these remainder's of 3, and you have to keep dividing and getting another digit. Really the series wills stop as soon as you divide into a nice even number with no remainder (and that will never happen)
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towr
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 Re: 0.999.   « Reply #101 on: Nov 7th, 2002, 5:56am »

do we really want to accept that every integer (except 0) has 2 representations?
-2 -1.999...
-1 -0.999...
1 0.999...
2 1.999...

I would argue 0.999... isn't a valid number, but rather bad syntax
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Jeremy
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 Re: 0.999.   « Reply #102 on: Nov 7th, 2002, 5:59am »

and towr, there's already an infinite number of ways to write every number:

1=2/2=3/3=4/4=5/5=6/6=7/7=8/8...
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towr
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 Re: 0.999.   « Reply #103 on: Nov 7th, 2002, 6:07am »

5/5 is an expression, not a number.. You need to calculate it's value to get one..
A number is an atom, an equation is a set of operations on atoms..
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S. Owen
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 Re: 0.999.   « Reply #104 on: Nov 7th, 2002, 7:10am »

on Nov 7th, 2002, 5:56am, towr wrote:
 I would argue 0.999... isn't a valid number, but rather bad syntax

Yeah, it would be funny to think that there is more than one base-10 decimal representation for 1.

Agreed, the "trick" is that "0.999..." is an abuse of notation. A decimal representation needs to have a finite number of digits. We can still establish a meaning for "0.999...", and its value, 1, but it's not another decimal representation of 1.

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TimMann
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 Re: 0.999.   « Reply #105 on: Nov 7th, 2002, 10:17am »

on Nov 7th, 2002, 7:10am, S. Owen wrote:
 Agreed, the "trick" is that "0.999..." is an abuse of notation. A decimal representation needs to have a finite number of digits.

Oops, that's a mistake. If you mean that, you'd be saying that irrational numbers and rational numbers that aren't multiples of negative powers of 10 have no decimal representation at all. The only decimal representation of 1/3 is 0.333..., and the only decimal representation of pi is 3.14159...

It's more common to do the opposite and say that all decimal representations have an infinite number of digits; for example, 1 = 1.000... This is needed in Cantor's diagonalization proof that the cardinality of the set of reals is greater than that of the set of integers.

There's nothing wrong with either 1.000... or 0.999... as a decimal representation for 1. It's too bad that some numbers turn out to have two infinite decimal representations, but we're stuck with it. The best we can do is choose one of the two and declare it to be canonical.
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S. Owen
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 Re: 0.999.   « Reply #106 on: Nov 7th, 2002, 11:41am »

Yeah, let me retract that... I guess I was taking towr to mean 'are there really two finite decimal representations of 1', which "0.999..." isn't, but that's not terribly interesting.

Well maybe... does "0.333..." count as an exact decimal representation of 1/3? Certainly if it equals anything, it's 1/3. But I've held the line that "0.333..." isn't really a decimal value as much as a tricky expression of a limit.

The answer is probably a matter of context... as far as (finite) computers are concerned, there is no way to exactly represent 1/3 as a base-2 decimal, so the answer's no in that context. But a mathematician might usefully define "0.333..." as the base-10 decimal representation of 1/3 and have no problems.
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Icarus
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 Re: 0.999.   « Reply #107 on: Nov 7th, 2002, 4:15pm »

on Nov 7th, 2002, 11:41am, S. Owen wrote:
 Well maybe... does "0.333..." count as an exact decimal representation of 1/3? Certainly if it equals anything, it's 1/3. But I've held the line that "0.333..." isn't really a decimal value as much as a tricky expression of a limit.

0.333... is a tricky expression of a limit, but that does not stop it from being a "decimal value". All decimals are tricky expressions of mathematical operations. After all, what does 0.3333 (finite) mean but

3/10 + 3/100 + 3/1000 + 3/1000 ?

How is saying that any different from saying 0.333... means

[sum]i=1[supinfty] 3/10i ?

The only difference I see is the sophistication of the concepts, and that 0.333... has a second level of notational convention applied (the ellipsis). However the ellipsis and the summation are both well-defined concepts whose meanings allow us to say 0.333... = 1/3, exactly, and not by "abuse of notation".

One other thing. The only real numbers with more than one decimal expression are the non-zero rationals whose denominators are expressable as 2n5m. They have exactly two: the terminating one which ends in repeating 0s, and the one that ends in repeating 9s (with the last digit not 9 being 1 less that the equivalent digit in the terminating version).

And, it's not really a matter of whether we want it or not. The two notations are a byproduct of how decimals work. We could decide to throw the repeating 9s notations out, but then we would have to make exceptions every time we discuss decimals:

a * (0.bcdef...) is calculated by multiplying a by each digit in turn, summing and performing the carries except when this leads to repeating 9s, then you have to ...

I personally think it is much easier to live with double notations, where you only have to state the equivalence once, and then you're covered for life!
 « Last Edit: Aug 19th, 2003, 7:07pm by Icarus » IP Logged

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towr
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 Re: 0.999.   « Reply #108 on: Nov 7th, 2002, 11:43pm »

simplicity makes perfect..

If I answered 0.999... on a math-exam when the answer would be 1, it would be marked as wrong, or at least not right..
The simplest answer is the right one..

Would have been nice to try when I was in highschool though.. I could have driven my math-teacher mad..
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Icarus
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 Re: 0.999.   « Reply #109 on: Nov 8th, 2002, 3:45pm »

When I was teaching, I would have taken a point off for failure to simplify, but I certainly would not have called it wrong, or even "not right", just not the best form for the answer.
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towr
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 Re: 0.999.   « Reply #110 on: Nov 10th, 2002, 7:49am »

well.. a failure to simplify seems to qualify as not right imo.. In any case it's what I meant..

hmm.. Maybe I should teach my niece to try this later when she's in highschool, and how to argue the point its the same..
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 Re: 0.999.   « Reply #111 on: Nov 21st, 2002, 8:22am »

To all those claiming 0.999... <1 :
It never ceases to amaze me how many stupid idiots on  message boards have no idea about this problem. Go off and study how mathematics and analysis works before jumping in with your pathetic uneducated opinions.
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towr
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 Re: 0.999.   « Reply #112 on: Nov 21st, 2002, 9:16am »

It never ceases to amaze me what spectacular significant contributions some people can bring to a discussion..

You, however, certainly aren't one of them..
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 Re: 0.999.   « Reply #113 on: Nov 21st, 2002, 6:31pm »

towr : your posts really are laughable.

"well.. a failure to simplify seems to qualify as not right imo.. In any case it's what I meant.. "

So if the answer to a calculation is (x+4)(x+5) and I write x^2 + 9x + 20 then I'm wrong am I? Come on. I actually have a degree in mathematics and I know both answers are equally valid, no matter how you're marked in your "math exam". If the question asked to put something in its simplest terms, eg 4/8, then 1/2 or 0.5 would be correct answers and 2/4 would not.

"5/5 is an expression, not a number.. You need to calculate it's value to get one..  "

Of course its a number, you imbecile.
5/5 and 0.999... are equal real numbers more commonly known as 1. They are the same real number, we just choose to call it 1. 0.999... may be an abuse of notation but its perfectly well defined.

0.999... = lim n-> infinity of (9/10 + 9/100 + 9/1000 + ... + 9/(10^n))
Just look it up in any mathematics textbook.

For your information x/2, (x+3)(x+5) and exp((x^2)/2) are expressions.

"A number is an atom, an equation is a set of operations on atoms.. "
Nice defintion. No really. And your point?

"It never ceases to amaze me what spectacular significant contributions some people can bring to a discussion..
You, however, certainly aren't one of them.. "

I refer the right honourable cretin to the reply I gave a few moments ago.
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Jeremiah Smith
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 Re: 0.999.   « Reply #114 on: Nov 21st, 2002, 7:13pm »

It's about time this board got its very own pretentious asshole! I was getting sick and tired of how everyone was getting along so well!

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Icarus
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 Re: 0.999.   « Reply #115 on: Nov 21st, 2002, 8:20pm »

So, "Guest", You have a degree in mathematics! Very good! Now get in line, like the rest. I have a PhD in it myself, and there are several other mathematics degrees on this board, as well as degrees in related fields. The regulars on this board, including towr, are both intelligent and well-educated. You would be wise to consider their words carefully, even when you disagree. Unlike you, they know the value of differing points of view. Even those who are mistaken can bring new depth to your understanding, and suggest things you may not have considered. For instance, Kozo earlier in this thread talks about 0.99...(infinitely many 9s)...9. Of course, YOU know that this isn't a real number. But I ask, do you really know why it isn't? Have you ever considered the possibility of defining numbers that can be expressed in this fashion? Can it be done consistently? If so, how do they behave? How do they relate to the real numbers?
These are useful considerations, but you are so full of yourself I doubt you ever saw them. Just as you failed to understand towr's remarks because you would rather belittle them than think!

Now, if you have something useful to add this or any other thread, I would love to read it. But if you want to keep up with this self-aggrandizement, stop wasting our time.
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 Re: 0.999.   « Reply #116 on: Nov 22nd, 2002, 6:14am »

So, "Icarus", You have a PhD in mathematics! Very good! Then you will agree that everything I have said was mathematically correct if not the tone in which it was written.

"For instance, Kozo earlier in this thread talks about 0.99...(infinitely many 9s)...9. Of course, YOU know that this isn't a real number. But I ask, do you really know why it isn't?"

Take a never ending sequence of 9s and put a 9 on the end, except you can never reach the end. So that doesn't make sense, it is not well defined, hence the number he is trying to define is not a member of the reals. However we can talk about the limit as n tends to infinity of 0.99... (n 9s) ...9 which is just 0.999... . Similarly the limit of 0.999...(n 9s)...A  as n tends to infinity where A is a member of the set {0,1,2,3,4,5,6,7,8} is also equal to 1 and hence 0.999... by standard epsilon proofs.

Have you ever considered the possibility of defining numbers that can be expressed in this fashion?

Yes before, and when I read Kozo's post. See the above analysis. Because the last digit is on the end of a very long sequence whose length is tending to infinity, it has zero effect on the value of the number. So I ask of the usefulness of this idea. It doesn't work very well in the decimal system that we use but if you were to give definitions that would work better then I would be interested to know about them.
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Pietro K.C.
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 Re: 0.999.   « Reply #117 on: Nov 22nd, 2002, 8:38am »

Hauehauehauheauehauh!!!

Who would ever have thought that the "0.999... = 1?" question could bring such strong emotions to the fore? I don't believe this "Guest" is for real. I think it's just Wu messing with us, or maybe one of the forum regulars, just for the hell of it.

Anyways, as for the "mathematics degree" which you have, why don't you shove it in the Putnam section? If you got a degree just to know facts (which you are so anxious to show), you should have studied botany, or something like that. I would love to see a non-Googled solution of yours to "3 points in a circle", or "bright and dark stars". Thanks!
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FenderStratFatMan
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 Re: 0.999.   « Reply #118 on: Nov 22nd, 2002, 8:50am »

on Jul 29th, 2002, 6:40am, Kozo Morimoto wrote:
 OK, nobody seems to be getting the '#' besides me so I'll drop it...   How about this.   0.9 is pretty close to 1, an approximation of 1. You add 0.09 to make it 0.99 and its even closer to 1 than 0.9 but still not 1.  Repeat to infinity.  You get ever closer and closer to 1 from the low side, but you never get there?   Using the ideas given above, does it mean that 1.000 ... 001 with infinite zeroes between the two 1s make it equal to 1?  So Does it mean that 0.999... = 1.000...001 ?  How about 0.999...998 = 0.999... = 1.000...001?

Wow Kozo your using the same argument as my little brother. Anyway, the reason that would be wrong is because there is no way to write .0000000000000 infinite with one in the last place. Secondly 1 - .9 infinite is 0 try on a mathematical calculator. 1/.9999999999 = 1 and 1 + .99999999999 = 2. Explain any of those. In reality a complex equation is not needed to prove that .999999999 = 1.

-FenderMan
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Pietro K.C.
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 Re: 0.999.   « Reply #119 on: Nov 22nd, 2002, 9:24am »

Quote:
 Wow Kozo your using the same argument as my little brother

Come on, FenderMan, the point of the discussion is to try and clarify people's thoughts. Kozo's mistake is as valid as any other, and, as I see it, is pretty much the ONLY mistake you can plausibly make regarding this puzzle. This kind of judgment is really uncalled for. Kozo is being polite, the least we can do is return the favor.

Second, a calculator will tell you all sorts of things, including absurdities like:

2/3 = 0.666666667
pi = 3.141562654
1 - 0.9999999999 (terminating) = 0.

Don't trust them too much!
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Guest
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 Re: 0.999.   « Reply #120 on: Nov 22nd, 2002, 1:32pm »

"Pietro" :

To be fair, I can't fault any of your analysis on this thread so far.

"I don't believe this "Guest" is for real." Really?

"Anyways, as for the "mathematics degree" which you have, why don't you shove it in the Putnam section?"
I might do that. I mentioned the "mathematics degree" to add to the argument proving towr and his assumption wrong. What are your qualifications?

"If you got a degree just to know facts (which you are so anxious to show), you should have studied botany, or something like that."
I imagine botanists would be quite annoyed about that comment. What are you trying to say?

I would love to see a non-Googled solution of yours to "3 points in a circle", or "bright and dark stars". Thanks!

I will tackle those problems I choose to, not anything laid down as a challenge. As anyone who has done a degree will tell you, it allows one to specialise to a certain extent. I reserve that right here. Analysis is one of my preferred subjects, real and complex. A particularly nice result attributed to Picard is that supposing a complex function f has an isolated essential singularity at a, then in any punctured disc centred at a, f actually assumes every complex value in the plane (except possibly one). [Source : Introduction to Complex Analysis, H. A. Priestly]

Now this is not "anxiousness" to show facts for "self-aggrandizement", it's actually a beautiful and somewhat surprising result. What is pathetic is this : people with zero concept of mathematical definitions of very simple ideas (e.g. numbers) completely disregarding the complexity and beauty of mathematics by a) arguing strongly only with their intuition b) being pretty sure they are right, and c) not listening to reason or being willing to learn.
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James Fingas
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 Re: 0.999.   « Reply #121 on: Nov 22nd, 2002, 2:02pm »

Fellow Forum-goers:

I would really hate this thread, which has had a lot of good (and also some not-so-good) discussion on it, to devolve into a flame-war.

Guest, I found your post on Nov. 21st to be rude. If people do not know about mathematics, then there is no better place for them to raise their questions and learn from the answers than in a forum like this.

The question is simple enough that people can make up their own ideas, but complex enough to allow some serious mathematical formalization. Although I do not have a degree in math, I appreciate some of the insights that I've seen in this particular thread, shared with all of us by the people who know what they're talking about. If I never ask a stupid question, how can I come to a deeper understanding of how math works?
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Icarus
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 Re: 0.999.   « Reply #122 on: Nov 22nd, 2002, 4:00pm »

Yes, you can define numbers with infinitely many nines, and then some more. They are not Real numbers, so your argument against them, which relies on concepts of the Reals, is fallacious. Consider sequences from the countable ordinals into the 10 digits for the definition. They also do not form a field, or even a group, but they do have some interesting behaviors of their own. (Topologically, I believe this is still the long line, though the construction is different.)

Yes, the actual math you presented was correct. However your belief that you have any idea what towr was talking about is not. This is evident, since nothing mathematical you said disagrees with what towr's points. If you will read towr's comments again with a less haughty disposition, you will discover that he never argued that 0.999... != 1. His only point was that 0.999... should not be considered an actual number at all. I argue that such a definition would create far more problems than the minor one it solves (two representations for terminating decimal numbers). This does not mean that his point is wrong. There is no "right" or "wrong" on this one, it's a matter of convention.

Now, if you have read James' post and understand the truth of what he is saying, and will behave yourself better in the future, I would be more than happy to see you contribute to these forums. But insulting everyone who disagrees with you and being condescending is childish.
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jon g
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 Re: 0.999.   « Reply #123 on: Nov 23rd, 2002, 1:52am »

just drop it, you all have gone too far. Why don't you answer a real question like what is the meaning of life or is there a god?
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Jeremiah Smith
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 Re: 0.999.   « Reply #124 on: Nov 23rd, 2002, 11:53am »

on Nov 23rd, 2002, 1:52am, jon g wrote:
 just drop it, you all have gone too far. Why don't you answer a real question like what is the meaning of life or is there a god?

Simple. Those questions haven't been added to William's puzzle pages yet.
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