View Poll Results: Is 0.999... exactly equal to one?

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  • Yes, they are equal.

    52 60.47%
  • No, they are not equal.

    34 39.53%
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Thread: Is 0.999... equal to one?

  1. #11
    Senior Member Coelacanth's Avatar
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    Default Re: Is 0.999... equal to one?

    @tom - there is a discussion of the symbol 0.000...1 here, under the "p-adic numbers" section.

    0.999... - Wikipedia, the free encyclopedia...

    I recognize that you are not claiming this symbol to be a number different from zero.

  2. #12
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    Default Re: Is 0.999... equal to one?

    Let x = 0.999.... then 10x = 9.999... and subtracting gives

    10x - x = 9.999... - 0.999...

    So 9x = 9

    You can argue as much as you want about possible solutions of 9x = 9, but in any reasonable number system (field) x = 1 is a solution (by definition of identity).

    Hence, x = 0.999... = 1

    You can't do maths by a poll, logic isn't governed by democracy!

  3. #13
    Senior Member Coelacanth's Avatar
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    Default Re: Is 0.999... equal to one?

    Quote Originally Posted by Max Caley View Post
    You can't do maths by a poll, logic isn't governed by democracy!
    Um, perhaps you have been in a class at some point where they administer an examination? I don't think the exam is given so the teacher can tally up the answers and declare the most popular one to be the correct one. This poll is supposed to be something along the lines of those exams, albeit without the consequences.

    My reason for the poll was stated in the original post:

    Quote Originally Posted by Homo bibiens View Post
    Let's see if this board does better than BAUT.
    At the moment, it's not looking good. If you would perhaps vote in line with the answer that you posted, the results would be slightly better, but I've noticed you didn't vote.
    Last edited by Coelacanth; 11-14-2010 at 11:39 AM.

  4. #14
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    Default Re: Is 0.999... equal to one?

    0.999... is exactly equal to 1-0.000...1

  5. #15
    Senior Member Coelacanth's Avatar
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    Default Re: Is 0.999... equal to one?

    Quote Originally Posted by Wayne Bruinekool View Post
    0.999... is exactly equal to 1-0.000...1
    0.000...1 is not a symbol that occurs in my number system, but in yours, is it equal to zero, or is it something different than zero?

  6. #16
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    Default Re: Is 0.999... equal to one?

    Quote Originally Posted by Homo bibiens View Post
    0.000...1 is not a symbol that occurs in my number system, but in yours, is it equal to zero, or is it something different than zero?
    I am very sorry that your system is deficient. 0.000...1 is perfectly rational, phrased within the definitions of 0, 1,(.), and (...). I would be glad to help you fix your system. Your discussion is about inadiquate deffinitions and unstated assumptions or misinterpretations of those definitions and assumptions. Clarify the definitions so that everyone is thinking and talking about the same thing and the problem will no longer exist. Even 1 + 1 = 2 is only true by mutual agreement. If one person disagrees, he is as right as everyone else. 0.000...1 is equal to 1-0.999.... Zero is just a little teeny, tiny bit different. It is very tiny, smaller than one person out of all who have ever lived upon the earth, but yet somehow significant to me.

    We could define zero as the limit of the sequence 0.000...1 and one as the limit of the sequence 0.999..., but it would be wise to remember that even though we have agreed to this, for discussiion purposes, 0.999... still exists in its own right apart from 1 and that the limit zero or the limit one is not the same as the integer zero or one. It is in fact a limit, even though we have made that feature understood, and dropped the word and/or symbol for limit for our conveneince.

  7. #17
    tom
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    Default Re: Is 0.999... equal to one?

    One other note ...
    There are two systems for a lack of better word although related they are different.

    There are integers and real numbers. Integers are quantized ( sp? ) while real numbers are not.

    In the "system" of integers there are not really limits ... as you are either exactly at 0 or exactly at 1.

    In the system of real numbers I would say that 0.0000 ... 1 = 0 just as 1/infinity = 0

    Quote Originally Posted by Wayne Bruinekool View Post
    I am very sorry that your system is deficient. 0.000...1 is perfectly rational, phrased within the definitions of 0, 1,(.), and (...). I would be glad to help you fix your system. Your discussion is about inadiquate deffinitions and unstated assumptions or misinterpretations of those definitions and assumptions. Clarify the definitions so that everyone is thinking and talking about the same thing and the problem will no longer exist. Even 1 + 1 = 2 is only true by mutual agreement. If one person disagrees, he is as right as everyone else. 0.000...1 is equal to 1-0.999.... Zero is just a little teeny, tiny bit different. It is very tiny, smaller than one person out of all who have ever lived upon the earth, but yet somehow significant to me.

    We could define zero as the limit of the sequence 0.000...1 and one as the limit of the sequence 0.999..., but it would be wise to remember that even though we have agreed to this, for discussiion purposes, 0.999... still exists in its own right apart from 1 and that the limit zero or the limit one is not the same as the integer zero or one. It is in fact a limit, even though we have made that feature understood, and dropped the word and/or symbol for limit for our conveneince.

  8. #18
    Senior Member roncj5's Avatar
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    Default Re: Is 0.999... equal to one?

    i would like to thank homo bibiens for asking this question.at first my answer was no.but that didn't sit well.this caused me to research this and learn something new.for this i thank you.this is why i love this forum.it provokes thought(at least in me).bravo!by the way,i would like to change my answer to yes it does,at least on paper anyway.
    Last edited by roncj5; 11-19-2010 at 02:55 PM.
    "the memories of a man in his old age are the deeds of a man in his prime"

  9. #19
    Senior Member Coelacanth's Avatar
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    Default Re: Is 0.999... equal to one?

    Quote Originally Posted by Wayne Bruinekool View Post
    I am very sorry that your system is deficient. 0.000...1 is perfectly rational, phrased within the definitions of 0, 1,(.), and (...). I would be glad to help you fix your system.
    I am very sorry you disagree with yourself:

    Quote Originally Posted by Wayne Bruinekool View Post
    If one person disagrees, he is as right as everyone else.
    Self-contradiction seems to be a very common property of many of the arguments I have seen in defense of one side of this argument at other boards, although this particular one, I have not seen before.

    Quote Originally Posted by Wayne Bruinekool View Post
    Zero is just a little teeny, tiny bit different.
    Different from what you refer to as 0.000....1? If that is what you mean, then I decline your offer to "fix" my number system, I prefer the one I have right now.

    If 0.000...1 is different than zero, does it obey all the usual rules of arithmetic that the real numbers obey? (The reason I ask is related to the above discussion about self-contradiction; if someone invents a number system with contradictory properties, then I cannot agree that they are as right as everyone else.)

    Quote Originally Posted by Wayne Bruinekool View Post
    We could define zero as the limit of the sequence 0.000...1
    I am glad you say elsewhere "Clarify the definitions", because I think definitions are extremely important in this sort of thing. Can you clarify your definition of 0.000...1? Specifically, what is the "sequence" to which you refer? Is it 0, 0.0, 0.00, 0.000, 0.0000, etc., or 1, 0.1, 0.01, 0.001, 0.0001, etc., or something different from either of these?

    But either way, my usual definition of the real number zero is a particular equivalence class of Cauchy sequences. There are other ways of doing it, but since all totally ordered complete fields are isomorphic, they get us to the same place.

    Quote Originally Posted by Wayne Bruinekool View Post
    and one as the limit of the sequence 0.999..., but it would be wise to remember that even though we have agreed to this
    Same as above. This is not my definition of the real number one. I am not sure what your definition is.

    Quote Originally Posted by Wayne Bruinekool View Post
    for discussiion purposes, 0.999... still exists in its own right apart from 1 and that the limit zero or the limit one is not the same as the integer zero or one.
    Real numbers are usually defined as limits. (As per above, there is at least one other way to do it, but it gets you the same end result.) Are you arguing that the real number one is not equal to the integer one? I have no problem defining equality between those two, and as per one of your two opinions on the matter, I am as right as everyone else.

    Thank you once again for your kind offer, but the one hint about infinitesmal numbers causes me to want to stick with the number system that I have. If I have misunderstood Bruinekool-numbers, a very rigourous definition would be most helpful in setting things straight.
    Last edited by Coelacanth; 11-20-2010 at 03:12 AM.

  10. #20
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    Default Re: Is 0.999... equal to one?

    Quote Originally Posted by tom View Post
    Integers are quantized ( sp? ) while real numbers are not.
    Hi tom,

    Not familiar with the term "quantized" (although this doesn't mean it isn't used out there), but I wonder if you refer to being "countable"?

    If so, then another set of numbers that is countable is the rational numbers. These can be put in a one-to-one correspondence with the integers (simply stated, you can put the rational numbers in a list), which may seem rather counter-intuitive. The real numbers cannot be put in one-to-one correspondence with the integers (Cantor's proof).

    Quote Originally Posted by tom View Post
    In the "system" of integers there are not really limits ... as you are either exactly at 0 or exactly at 1.
    There is another interesting (well, interesting to me) property of the real numbers, which is that they are complete. You can have "Cauchy" sequences of rational numbers, that is sequences in which (roughly speaking) the differences between the elements keeps getting smaller and smaller, but the limit of the sequence does not exist as a rational number. For example, there is a sequence 3, 3.1, 3.14, 3.141, 3.1415, (maybe you can guess how it continues), each number in the sequence is rational, but the sequence does not converge to a rational limit. The ancient Greeks were aware that there was no rational number equal to the square root of two, although you could find rational numbers as close as you like. The system of real numbers is complete - every Cauchy sequence of real numbers has a limit that is itself a real number (and in fact one of the two usual ways of defining the real numbers does this explicity by construction).

    The real numbers have some unique properties - it can be shown that every complete (has the property above) totally ordered (for any two elements, a>b, a<b, or a=b) field (follows the common laws of arithmetic) is essentially equivalent to the real numbers. Consequently, if you want to have things like numbers infinitely small but still larger than zero, you need to make some sacrifices. Some of the common properties of the real numbers must get broken if you want to do this.

    Quote Originally Posted by tom View Post
    In the system of real numbers I would say that 0.0000 ... 1 = 0
    I don't have meaning assigned to the symbol "0.000...1". I asked someone else who used this symbol what it meant, and, well, you saw the result of that.

    The sequence, 0, 0.1, 0.01, 0.001, 0.0001, etc. (if that is what is meant by "0.000...1") does indeed have the limit zero.

    Quote Originally Posted by tom View Post
    just as 1/infinity = 0
    I avoid arithmetic with infinity like the plague, because there are different conventions for what it means As per the above, it is impossible to extend the real numbers to include an "infinity" without breaking some of the familiar properties of the real numbers.

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