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.333333333333*3 = 1?

  1. Dec 5, 2004 #1
    How can 1/3 multiplied by 3 give us 1 if 1/3 is the repeating decimal .3 to an infinite number of decimal places? If you multiplied 3 * .3 with an infinite number of repeating 3's wouldnt you get .9 with an infinite number of 9's repeating? Why do we say that (1/3) = .3 repeating and when you multiply that times 3 you get 1 instead of .9 repeating?
     
  2. jcsd
  3. Dec 5, 2004 #2
  4. Dec 5, 2004 #3
    because .999999999999999999999999999999999 repated equals one when you apply calculus

    look up zeno's paradoxs it has similar ideas
     
  5. Dec 5, 2004 #4

    Math Is Hard

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    heh heh heh! here we go again! ... poor Hurkyl... poor Matt.. poor Tom...
    I feel sorry for them already.
     
  6. Dec 5, 2004 #5
    :cry: :cry: :cry: :cry: :cry: :cry: :cry: :cry:

    well at least people are thinking
     
  7. Dec 5, 2004 #6

    Gokul43201

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    Because the real number represented by the repeating decimal 0.999... is defined to be identical to the number represented by 1.

    Read this or try this thread.
     
  8. Dec 5, 2004 #7

    Zurtex

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    Just so you guys know I actually managed to convince some people on another forum that 0.9 recurring = 1 who were arguing otherwise. :biggrin:
     
  9. Dec 5, 2004 #8

    Tide

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    Can you find any number between [itex]0.\bar 3[/itex] and [itex]\frac{1}{3}[/itex] ? If not, then they must be the same! :-)
     
  10. Dec 5, 2004 #9
    Haha, my thoughts exactly!

    I will say it can be an indicator someone has been using their coconut to try and make sense of things but, well, not always (often turns ugly for no good reason).
     
  11. Dec 5, 2004 #10

    Janitor

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    Where has Organic been of late?
     
  12. Dec 6, 2004 #11
    What is the calculus behind it?
     
  13. Dec 6, 2004 #12

    Gokul43201

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    I believe what Tom had in mind was the limit of the infinite sum "0.9 + 0.09 + 0.009 + ..." which is just a geometric progression, and its sum to an infinite number of terms is simply 0.9/(1- 0.1) = 1.
     
  14. Dec 6, 2004 #13
    i like this one

    [tex]0.33333\bar{3}=a[/tex]

    [tex]3.33333\bar{3}=10a[/tex]

    [tex]10a-a=9a=3[/tex]

    [tex]a=\frac{3}{9}=\frac{1}{3}[/tex]

    [tex]0.333\bar{3}=\frac{1}{3}[/tex]
     
  15. Dec 6, 2004 #14

    matt grime

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    One thing I've never understood is why people argue against the fact that it is even remotely possible for two decimals to represent the same real number yet are perfectly happy to accept there are an infinite number of rational representations for some element in Q {1/2, 2/4. 3/6, ...} Surely two different ones for only a few numbers must be a fantastic improvement over infinitely many for all.
     
  16. Dec 6, 2004 #15

    Hurkyl

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    My best guess is that they don't internalize 1/2 as being a number, but rather an arithmetic expression. The thing that bugs me is how many think of 0.999... as some strange sort of varying number.
     
  17. Dec 6, 2004 #16

    russ_watters

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    Because of this, I don't understand why this should be that hard of an issue:
    Some people have tried to get around it by pulling new numbers out of their a--air (0.000...1), but if you can't answer the question using the real number system, game over. I'm currently in page 9 of a similar thread at BadAstronomy, where the answer to that question was "an infinitessimally small number." :rolleyes: It makes me wonder whether this is an honest argument.

    The one (sorta) legitimate concern I've seen is from engineers who think its a matter of precision: you have to stop somewhere to round it off and thats how you get 1.
     
    Last edited: Dec 6, 2004
  18. Dec 6, 2004 #17

    matt grime

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    It cannot be an infinitesimal number, and non-zero, in the sense of non-standard analysis, since 1/3 - 0.33... is real, and an infinitesimal, hence zero - even in nonstandard analysis 0.9.. and 1 are the same.
     
  19. Dec 6, 2004 #18
    We should have a sticky for this. Or maybe we should have a faq forum for the different topics. Btw you can try here
     
  20. Dec 6, 2004 #19
    Nah, people would still continue to post this 5 times a day. I recommend just changing the url to 0.9repeating=1.end_of.story_so.please.stop_asking.com and replace the PF banner with something similar.
     
    Last edited: Dec 6, 2004
  21. Dec 6, 2004 #20
    9 = 9.9... - .9... = 10*(.9...) - 1(.9...) = (10 - 1)*(.9...) = 9*(.9...)

    -> 9 = 9*(.9...)
    -> 1 = .9...

    Using this same method can be used to prove the geometric series (I think its the geometric series) which is where it ties into calculus...
     
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