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

[Natasha1]

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 Please help.

I have looked on http://mathworld.wolfram.com/DecimalExpansion.html
but I need more? anyone?

 

[Natasha1]

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?

 

[ktoz]

After some quick testing, it looks like the pattern seems to be

$$finite decimal = \frac{x}{2^a \times 5^b}$$

I don't know how to "prove" that, but it makes sense as 10 = 2 X 5

 

[shmoe]

(1)-(4) of your link explains the condition you're after. Do you understand what's there?

 

[hypermorphism]

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/(2^a5^b) where either a or b could be 0 and p is a prime.

 

[ivybond]

Splitting hairs

I think either of a and b is more precise, and p does not have to be prime.

 

[hypermorphism]

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.

 

[ivybond]

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.

 

[hypermorphism]

No problem, I just looked that up myself. You're right, p doesn't have to be prime.

 

[ivybond]

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)!
In $$r = \frac {p}{2^a 5^b}$$ 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?

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?

 

[HallsofIvy]

Is this becoming an English grammar forum now?

 

[Hurkyl]

Please don't multiple post. I've merged your two threads.

 

[Natasha1]

Sorry I am new to all this. Won't multiply post in the future! Promise

 

[Natasha1]

Can anyone see something else?

Can anyone see something else?

 

[HallsofIvy]

Suppose x= 0.a₁a₂...a_n.

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?