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Which rational numbers between 0 and 1 have finite decimal expansions?

  1. Oct 27, 2005 #1
    The question I have been given is essencially this:

    Give a brief description, with some explanation, of which rational numbers between 0 and 1 have finite decimal expansions? [An example is 3/40 = 0.075, however 2/3 = 0.66666666666............]

    I am truly :confused: Please help.

    I have looked on http://mathworld.wolfram.com/DecimalExpansion.html
    but I need more??? anyone???
     
  2. jcsd
  3. Oct 27, 2005 #2
    The question I have been given is essencially this:

    Give a brief description, with some explanation, of which rational numbers between 0 and 1 have finite decimal expansions? [An example is 3/40 = 0.075, however 2/3 = 0.66666666666............]

    I am truly stuck as there are a infinite number of rational numbers. I just need a general explanation, but obviously the more I put the better.... please help! :-)

    I have looked on http://mathworld.wolfram.com/DecimalExpansion.html
    but I need more??? anyone???
     
    Last edited: Oct 27, 2005
  4. Oct 27, 2005 #3
    After some quick testing, it looks like the pattern seems to be

    [tex]
    finite decimal = \frac{x}{2^a \times 5^b}
    [/tex]

    I don't know how to "prove" that, but it makes sense as 10 = 2 X 5
     
  5. Oct 27, 2005 #4

    shmoe

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    (1)-(4) of your link explains the condition you're after. Do you understand what's there?
     
  6. Oct 27, 2005 #5
    That page gives an easy to derive characterization, r = p/(2a5b) where either a or b could be 0 and p is a prime.
     
  7. Oct 28, 2005 #6
    Splitting hairs

    I think either of a and b is more precise, and p does not have to be prime.
     
    Last edited: Oct 28, 2005
  8. Oct 28, 2005 #7
    The usage of "and" is incorrect; I don't require that both a and b be 0 whenever one of them is 0. Also, the "either ... or" construction does not require the use of "of". See American Heritage.
     
  9. Oct 28, 2005 #8
    Stand corrected, sorry.
    Actually, I changed my post before seeing yours having realised my mistake.:redface:
     
  10. Oct 28, 2005 #9
    No problem, I just looked that up myself. :smile: You're right, p doesn't have to be prime.
     
  11. Oct 28, 2005 #10
    Hey, upon further research (and splitting the "splits"), I found that "either of a and b" is a legitimate expression:
    it does mean either a, or b, or both (like in OR operator in Boolean logic)!:approve:
    In [tex]r = \frac {p}{2^a 5^b}[/tex] either of three scenarious (a=0, b=0, both a=0 and b=0) is possible.
    Well, if both a and b equal 0, can we still call r a fraction?:cool:

    BTW, seems we abandoned Natasha in our semantic fencing.
    Her plea "I need more??? anyone???" went unanswered.
    On the other hand, what could be more the MathWorld?
     
    Last edited: Oct 28, 2005
  12. Oct 28, 2005 #11

    HallsofIvy

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    Is this becoming an English grammar forum now?
     
  13. Oct 28, 2005 #12

    Hurkyl

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    Please don't multiple post. I've merged your two threads.
     
  14. Oct 28, 2005 #13
    Sorry I am new to all this. Won't multiply post in the future! Promise
     
  15. Nov 5, 2005 #14
    Can anyone see something else?

    Can anyone see something else? :bugeye:
     
  16. Nov 7, 2005 #15

    HallsofIvy

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    Suppose x= 0.a1a2...an.

    What happens if you multiply both sides by 10n?
    Do you see how to write x as a fraction?
    What is the denominator?
    What happens when you reduce to lowest terms?
     
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