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

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Natasha1
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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?
 
on Phys.org
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?
 
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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
 
Natasha1 said:
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?
That page gives an easy to derive characterization, r = p/(2a5b) where either a or b could be 0 and p is a prime.
 
Splitting hairs

I think either of a and b is more precise, and p does not have to be prime.
 
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ivybond said:
I think either of a and b is more precise.
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.
 
hypermorphism said:
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.
Stand corrected, sorry.
Actually, I changed my post before seeing yours having realized my mistake.:redface:
 
No problem, I just looked that up myself. :smile: You're right, p doesn't have to be prime.
 
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?
 
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Sorry I am new to all this. Won't multiply post in the future! Promise
 
Can anyone see something else?

Can anyone see something else? :bugeye:
 

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