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Thread: 0.999=1

  1. #1 0.999=1 
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    As in, 0.999 repeating.
    I am aware this is hotly debated, but it is confirmed by mainstream mathematics that 1=0.999. I just wanted to see the argument on here. It seems like you guys might have fun with this.
    My grades are as good/bad as the next guy, but if you divide 1 by three, 0.333, then you multiply it back by three, you get 0.999. Therefore 1=0.999.

    Also, I have 1.999 equations to prove that 1=0.999:
    x=0.999
    10x=9.999
    10x-x=9.999-0.999
    9x=9
    x=1
    and
    x=0.999, which is what we started with.

    1/9=0.111
    (1/9)x9=0.111x9
    1=0.999

    Boom.


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  3. #2 Re: 0.999=1 
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    Quote Originally Posted by Pomegranate Cameron
    As in, 0.999 repeating.
    I am aware this is hotly debated, but it is confirmed by mainstream mathematics that 1=0.999. I just wanted to see the argument on here. It seems like you guys might have fun with this.
    My grades are as good/bad as the next guy, but if you divide 1 by three, 0.333, then you multiply it back by three, you get 0.999. Therefore 1=0.999.

    Also, I have 1.999 equations to prove that 1=0.999:
    x=0.999
    10x=9.999
    10x-x=9.999-0.999
    9x=9
    x=1
    and
    x=0.999, which is what we started with.

    1/9=0.111
    (1/9)x9=0.111x9
    1=0.999

    Boom.
    This is correct.

    It is not hotly debated, or debated at all.

    This pretty standard high school freshman stuff.


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  4. #3 Re: 0.999=1 
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    Quote Originally Posted by DrRocket
    This is correct.

    It is not hotly debated, or debated at all.

    This pretty standard high school freshman stuff.
    I am merely nit picking in saying this, but I'm fairly certain that I learned this in a class that would have typically been taken at some point after the freshmen year.

    I remember that it was a homework problem involving limits were we had to demonstrate this, and I wasn't aware of it before (point: it would have been possible to demonstrate it at an earlier time than when I first took note of this fact; I may have even forgotten).
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  5. #4 Re: 0.999=1 
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    Quote Originally Posted by C.Elrod
    I am merely nit picking in saying this, but I'm fairly certain that I learned this in a class that would have typically been taken at some point after the freshmen year.

    I remember that it was a homework problem involving limits were we had to demonstrate this, and I wasn't aware of it before (point: it would have been possible to demonstrate it at an earlier time than when I first took note of this fact; I may have even forgotten).
    There's a reason for this: infinitesimals quantities, originally conceived by Leibniz, were the basis for differential and integral calculus. However, since infinitesimals were not made rigorous by Leibniz, Newton, or any other such predecessors the theory of limits were used as a means.

    This is sure as hell not something one would learn in freshman mathematics; perhaps at the preliminary calculus or introductory calculus level, however.
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  6. #5 Re: 0.999=1 
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    Quote Originally Posted by Ellatha
    Quote Originally Posted by C.Elrod
    I am merely nit picking in saying this, but I'm fairly certain that I learned this in a class that would have typically been taken at some point after the freshmen year.

    I remember that it was a homework problem involving limits were we had to demonstrate this, and I wasn't aware of it before (point: it would have been possible to demonstrate it at an earlier time than when I first took note of this fact; I may have even forgotten).
    There's a reason for this: infinitesimals quantities, originally conceived by Leibniz, were the basis for differential and integral calculus. However, since infinitesimals were not made rigorous by Leibniz, Newton, or any other such predecessors the theory of limits were used as a means.

    This is sure as hell not something one would learn in freshman mathematics; perhaps at the preliminary calculus or introductory calculus level, however.

    This issue has NOTHING to do with infinitesimals.
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  7. #6  
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    Just to help other people understand why this has nothing to do with infinitesimals, let me offer this definition: the infinitesimal unit, often written as , is a number that is smaller than any positive number, but greater than 0. (Similar to how infinity is greater than any positive number.) is still 1 though, not .
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  8. #7 Re: 0.999=1 
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    Quote Originally Posted by DrRocket
    This issue has NOTHING to do with infinitesimals.
    .000...1 + .999... = 1


    Quote Originally Posted by DrRocket
    This pretty standard high school freshman stuff.
    You are high.
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  9. #8 Re: 0.999=1 
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    Quote Originally Posted by Ellatha
    Quote Originally Posted by DrRocket
    This issue has NOTHING to do with infinitesimals.
    .000...1 + .999... = 1
    .999... = 1 as is usually shown in freshman high school algebra. It is done as in the OP, which is a valid proof if one understands it.

    .000...1 is meaningless. So is .000..... 2 and so is your post



    Quote Originally Posted by Ellatha
    uote="DrRocket"]This pretty standard high school freshman stuff.
    You are high.[/quote]

    No. I am a real mathematician. You are not. Obviously.
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  10. #9  
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    A proof for the statement .999 = 1 is based on the following Archimedean property:

    1 - .999 = .0...1

    Because there are no nonzero infinitesimals, .999 is not an infinitesimal, therefore .999 is equal to 1.

    This is related to infinitesimals.

    C.Elrod was an International Baccalaureate student at a top 100 high school in the nation. He did not learn this property his freshman year.
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  11. #10  
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    Neither of my math teachers readily believed 0.999 = 1. Just saying.
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  12. #11  
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    Assuming you mean 0.99999..., you need better teachers. Seriously, it's trivial to show that it must be true.
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  13. #12  
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    Quote Originally Posted by MagiMaster
    Assuming you mean 0.99999..., you need better teachers. Seriously, it's trivial to show that it must be true.
    The proof in the OP is perfectly valid if one understands what is being said, but for those who don't like it, here is a proof in gory detail.







    Similarly



    So, if



    And




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  14. #13  
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    Folks,

    .9999...(recurring, if that's what it's called) does not equal 1.

    They are 2 completely different numbers, there's no question about it.

    I never heard of this supposesd equality before and i don't understand the Zigma notation anymore,
    but come on they're two completely different numbers,

    using your methodologies i'm sure you could equally prove that .999..... = .999....8
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  15. #14  
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    Quote Originally Posted by pedronaut
    They are 2 completely different numbers, there's no question about it.
    Which should I accept? The assertions of the mathematics community, or an anonymous, unfamiliar poster on the internet? Your objection appears to be based upon two things: you don't like the idea that they are the same number; you don't understand the proofs. That seems sufficient reason for me to ignore your claim.
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  16. #15  
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    They are two completely different numbers.
    The only way to equal them is to round one off. Or use some trickery.

    Also one isn't a real number and one is. They are two completely different concepts.

    True, my Math is very limited, but i have the ability to see/think logically.
    I have no doubt that there are plenty of big boys in the Math community that would take my side on this.
    A consensus is not a conclusiveness.
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  17. #16  
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    Quote Originally Posted by pedronaut
    but i have the ability to see/think logically
    No, you don't, or you'd realize that a mathematical proof is the strongest form of proof and that intuition is the weakest. Also, there's no one in the mathematical community that would take your side on this.

    Tell me then, what is 1 - 0.999999....?

    Seriously. I'm getting tired of the endless examples of "I don't understand/like this, so despite everything, it must not be true" I see on these boards. It's getting tedious.
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  18. #17  
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    Quote Originally Posted by pedronaut
    Folks,

    .9999...(recurring, if that's what it's called) does not equal 1.

    They are 2 completely different numbers, there's no question about it.

    I never heard of this supposesd equality before and i don't understand the Zigma notation anymore,
    but come on they're two completely different numbers,

    using your methodologies i'm sure you could equally prove that .999..... = .999....8
    You don't know what you are talking about.

    Any professional mathemtician (like me for instance) or any decent high school algebra student knows and can prove that 0.99999.... = 1. The proofs above are quite correct.
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  19. #18  
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    Quote Originally Posted by pedronaut
    They are two completely different numbers.
    The only way to equal them is to round one off. Or use some trickery.

    Also one isn't a real number and one is. They are two completely different concepts.

    True, my Math is very limited, but i have the ability to see/think logically.
    I have no doubt that there are plenty of big boys in the Math community that would take my side on this.
    A consensus is not a conclusiveness.
    Dude. This is not trickery. This is math. If you don't understand it, that's cool.

    Dr. Rocket is a PhD mathematician.

    Learn to understand before forming value judgements. It's a good thing.

    Really.
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  20. #19  
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    Excuse me while I embarrass myself further

    Following the beginner’s level solution in the Opening Post;

    x is deleted from 10x

    the .999..s are subtracted from one another to equal .000…
    This is where the trickery lies,
    The recurring element of x and 10x recur to two distinctly different infinities
    The mathematics community are not subtracting like from like
    Where x = .999… and 10x = 9.999…, x will always recur for one step further than 10x. (because the decimal point has shifted to the right)

    Infinite Black, is not equal to Infinite white

    It is lazy of the mathematics community to assume that these two distinctly different numbers/values (x and 10x) have their .999…’s recurring for equally infinite steps.
    And to subtract them so neatly from one another is a lie.
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  21. #20  
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    Quote Originally Posted by pedronaut
    Excuse me while I embarrass myself further

    Following the beginner’s level solution in the Opening Post;

    x is deleted from 10x

    the .999..s are subtracted from one another to equal .000…
    This is where the trickery lies,
    The recurring element of x and 10x recur to two distinctly different infinities
    The mathematics community are not subtracting like from like
    Where x = .999… and 10x = 9.999…, x will always recur for one step further than 10x. (because the decimal point has shifted to the right)

    Infinite Black, is not equal to Infinite white

    It is lazy of the mathematics community to assume that these two distinctly different numbers/values (x and 10x) have their .999…’s recurring for equally infinite steps. And to subtract them so neatly from one another is a lie.
    You are wrong.

    You are ignorant of mathematics.

    No one has assumed anything here. The equality of .999... and 1 has been rigorously proved.

    The "beginner's proof" in the OP is valid. Your objections are bogus.

    Since you have been shown and have rejected a rigorous demonstration of the FACT that .999.... = 1 on more than one occasion I must conclude that you are also more than a bit dim-witted.
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  22. #21  
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    DrRocket,
    Fair enough maybe i am a bit dim witted. It wouldn't be the first time i've been called an idiot.

    MagiMaster,
    you ask, what is 1 - 0.999999....?
    I assume you're being rhetorical and i'm supposed to fall down and say it is clearly Zero.

    Well it's not, It's probablly the closest thing to Zero, but never quite Zero.
    It is not a real number, it has no end, no definitiveness.

    Sorry i can't follow the calculus/Zigma notation, and sorry for wasting your time with my 'tedious' interest,

    But i must say for a bunch of learned Mathematicians you suck at illustrating this strangeness to a layperson.

    Bunch of Liars.
    Though it's not your fault, whoever you're parrotting lied to you first.

    (no hate intended)
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  23. #22  
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    Quote Originally Posted by pedronaut
    Well it's not, It's probablly the closest thing to Zero, but never quite Zero.
    OK. Prove it.
    Bunch of Liars.
    Though it's not your fault, whoever you're parrotting lied to you first.

    (no hate intended)
    So rather than correct your own ignorance, you would rather simply insult those who have taken the time to learn this stuff?
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  24. #23  
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    OK, in your high-fallutin Math-speak you can prove it.

    But stand back and look at it
    1 and .999... are not the same

    In my dreams i'd have the math-speak to prove it, or agree with the consensus.

    sorry to use insulting language, however i wasn't the first to throw an insult.

    As i said, i see the trickery to be assuming two different infinities to be equal.

    Perhaps i should move to the Philosophical section of the science form
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  25. #24  
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    If your philosophy is like your math, then it's bad philosophy.

    Usually, we like to identify "II" with "2". We accept that these are the same number even though they are different representations of that number.

    So it is with "0.999..." and "1".
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  26. #25  
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    does ?
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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    Yeah i dig that a third + a third + a third = 1
    and .333... + .333... + .333... = 0.999... or 1

    here the .333s recur for equal amount of steps after the decimal point

    9.999... (10x) & 0.999... (x) do not recurr for equal distance

    Two different Infinities, that's my story and i'm sticking to it for now
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  28. #27  
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    Quote Originally Posted by pedronaut
    Bunch of Liars.
    Though it's not your fault, whoever you're parrotting lied to you first.
    wrong fool.

    You have been presented with the truth, the whole truth and nothing but the truth, including the observation that you are dim-witted.

    The fact that you are too damn stupid to understand does not make liars of the people who have presented you with the facts.

    No one, other than you, is simply parroting anything. You have been presented with rigorous proofs, and those presenting those proofs understand them thoroughly. It is rather the absolute height of arrogance for you, who clearly do not understand, to accuse a professional of parroting lies.
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  29. #28  
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    DrRocket
    Well thanks for explaining it so clearly, nice chatting with ya

    (sarcasm intended)
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  30. #29  
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    Quote Originally Posted by pedronaut
    DrRocket
    Well thanks for explaining it so clearly, nice chatting with ya

    (sarcasm intended)
    Dr. Rocket would be forgiven for being far ruder to you.
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  31. #30  
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    Quote Originally Posted by PhysBang
    Dr. Rocket would be forgiven for being far ruder to you.
    He seems to be mellowing in his old age. Perhaps its senility. :wink:
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  32. #31  
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    Quote Originally Posted by pedronaut
    Yeah i dig that a third + a third + a third = 1
    and .333... + .333... + .333... = 0.999... or 1

    here the .333s recur for equal amount of steps after the decimal point

    9.999... (10x) & 0.999 (x) do not recurr for equal distance

    Two different Infinities, that's my story and i'm sticking to it for now
    Your error lies in assuming that these two infinities are somehow different. Prove this and you may have a point, but since you somehow think proofs are "trickery," I don't really know what else to suggest.

    BTW, here the words "repeats infinitely" means that there is no end, and there is no place where the two are different. 0.33333... * 3 gives the same 0.99999... as 0.11111... * 9 and every other way of writing it.

    The only reason you won't accept the proofs is that you don't like the idea. You are definitely not thinking logically. (Pretty much exactly the opposite.)

    Oh yeah, and there is no such thing as "the closest thing to zero, but not zero" in the real numbers.
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    Quote Originally Posted by pedronaut
    Yeah i dig that a third + a third + a third = 1
    and .333... + .333... + .333... = 0.999... or 1

    here the .333s recur for equal amount of steps after the decimal point

    9.999... (10x) & 0.999 (x) do not recurr for equal distance

    Two different Infinities, that's my story and i'm sticking to it for now
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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  34. #33  
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    Thanks MagiMaster,
    Iappreciate your efforts and understand my skepticism must be annoying.
    I never heard of this equation until earlier this evening, browsing this thread.

    I see from Google i'm not the first to have trouble absorbing this.

    Put it this way;
    what number is closest to 1 but <1 ?
    There is no number and the answer is the limit to 1 or something like that.

    well i would have thought it was .999...
    it seems the consensus is there is no such thing as .999... the way i imagine it

    That's news to me
    First Santa Claus now .999...
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    .99999..... and .3333..... are the way they are because of the inability of the decimal system to adequately represent 1/3. The problem is in the system, not the concept. In base 3, .1 is the equivalent of .33333333...... in decimal, and it's clear that .1+.1+.1=1 in base 3. likewise, .333....+.333....+.333....=1. Since it appears to be .999...., its obvious .999....=1, and nothing else.
    Wise men speak because they have something to say; Fools, because they have to say something.
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  36. #35  
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    If you learned something new, then all's well. :-D
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    I still disagree, the Emperor has no clothes.

    Infinity.A ≠ Infinity.B

    I think the only consensus is that Mathematicians have chosen to interpret 0.999 as 1 rather than a value <1. Seems more like Democracy than Math.

    0.999… + 0.001 = 1
    0.999…∞ + 1/∞ = 1

    1/∞ ≠ 0
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    Excuse me while I embarrass myself further and Parot what I hinted at above and previously,

    I think the problem lies in interpreting infinity.

    0.333…
    0.333…
    +0.333…
    1.0
    Agreed. The .333s recur for equal distance.


    x = 0.999…
    10x – x =

    9.999…
    - 0.9999…
    9.0
    No. You are not subtracting like for like. You assume the .999s recur for equal distance.

    I believe these are not equal Infinities and only by convention we deem them equal to manage the limitations of the decimal system.
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  38. #37  
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    Lol! You are indeed a glutton for punishment.
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  39. #38  
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    Quote Originally Posted by pedronaut
    1/∞ ≠ 0
    ∞ doesn't seem to mean anything here. Can you explain how to use it in a way that is not inconsistent?
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    Quote Originally Posted by pedronaut
    I think the only consensus is that Mathematicians have chosen to interpret 0.999 as 1
    In fact they haven't

    0.999… + 0.001 = 1
    Nope.
    0.999…∞ + 1/∞ = 1

    1/∞ ≠ 0
    None of this makes any sense
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    Excuse me while I embarrass myself further
    x = 0.999…
    10x – x =

    9.999…
    - 0.9999…
    9.0
    Bang! Ouch my foot hurts. Who fired that gun?

    What is the number by which you need to multiply 10 such that this number subtracted from the result is 9?

    Hint: 10 times 1 = 10 and 10 - 1 = 9, so that 10x - x = 9 implies x = 1. Is there any other number that satisfies this?

    This sort of nonsense occurs all too frequently on sites like this, and I am thinking of following convention and locking this thread.

    Thoughts?
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  41. #40  
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    Wanna borrow some Loctite?
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    PhysBang,
    I am saying that 1 divided by Infinity does not equal Zero,
    The limit is Zero
    It is never actually Zero.
    ____________________________
    Guitarist'

    RE: "Quote:
    0.999… + 0.001 = 1
    Nope."

    sorry I meant to write;
    0.999 + 0.001 = 1
    ___________
    RE: "Quote:
    0.999…∞ + 1/∞ = 1

    1/∞ ≠ 0
    None of this makes any sense "

    In English I am saying;
    (0.999.. recurring infinitely) + (1 divided by Infinity) = 1

    & 1 divided by Infinity ≠ 0
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    We all know that three thirds equals 1
    0.333... is the decimal representation of a third

    0.333... X 3 ≠0.999...
    It equals 1

    If we begin with 0.999...
    And you ask me what it is
    I say it is <1
    Its limit is 1 but it is never actually 1

    I think 0.999... should do what it says on the tin and represent <1
    That's my point

    I'm not entirely wrong. The decimal system fails here.
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    Quote Originally Posted by pedronaut
    PhysBang,
    I am saying that 1 divided by Infinity does not equal Zero,
    The limit is Zero
    It is never actually Zero.
    But what does any of that crap you wrote even mean? Show your work!
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    PhysBang,
    I'm probablly misusing the word limit. I don't know.

    A simple question
    1/10 = a tenth
    1/100 = a hundreth

    what is 1 divided by Infinity!
    i would believe it is greater than Zero

    I don't know,
    Is there a consensus that 1 divided by infinity is equal to zero.
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  45. #44  
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    Lucky me. Lucky mud.
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    Cat's Cradle.
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    Quote Originally Posted by pedronaut
    PhysBang,
    I'm probablly misusing the word limit. I don't know.

    A simple question
    1/10 = a tenth
    1/100 = a hundreth

    what is 1 divided by Infinity!
    i would believe it is greater than Zero

    I don't know,
    Is there a consensus that 1 divided by infinity is equal to zero.
    no.
    does not exist, and likewise does not exist. The two terms are without utility and meaning.
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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    Also, the limit of as is in fact 0. Limits are the only way to make infinity meaningful in the context of the real (and most other) numbers.
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  48. #47  
    Forum Bachelors Degree x(x-y)'s Avatar
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    So, by the same logic, 1.999 = 2, am I right? Just wondering, this kind of topic fascinates me (we haven't done anything on it really and I'm 17 doing A-Level Mathematics!). So, with the following proof, you can show that any n.999 = n + 1?

    Proof:
    x = 1.999
    10x = 19.999
    10x - x = 19.999-1.999
    9x = 18
    x = 2
    "Nature doesn't care what we call it, she just does it anyway" - R. Feynman
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  49. #48  
    Forum Radioactive Isotope MagiMaster's Avatar
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    Yeah. That works. You could also say that 1.9999... = 1 + 0.9999... = 1 + 1 = 2
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  50. #49  
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    Quote Originally Posted by pedronaut
    Put it this way;
    what number is closest to 1 but <1 ?
    There is no number and the answer is the limit to 1 or something like that.

    well i would have thought it was .999...
    Ok, let's assume 0.999... is the closest number >1.

    Then :

    0.999...+0.999... < 0.999...+1 < 1+1

    and finally (0.999...+0.999...)/2 < (.999...+1)/2 < (1+1)/2

    Or in other words, 0.999...< (0.999...+1)/2 < 1.

    So your definition was wrong. 0.999... can't be the closest number to 1 that is not 1. In fact, no number can be the "closest" to another in the real numbers.
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