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

1. Oct 27, 2005

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

2. Oct 27, 2005

### 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???

Last edited: Oct 27, 2005
3. Oct 27, 2005

### 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

4. Oct 27, 2005

### shmoe

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

5. Oct 27, 2005

### hypermorphism

That page gives an easy to derive characterization, r = p/(2a5b) where either a or b could be 0 and p is a prime.

6. Oct 28, 2005

### ivybond

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
7. Oct 28, 2005

### hypermorphism

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.

8. Oct 28, 2005

### ivybond

Stand corrected, sorry.
Actually, I changed my post before seeing yours having realised my mistake.

9. Oct 28, 2005

### hypermorphism

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

10. Oct 28, 2005

### 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?

Last edited: Oct 28, 2005
11. Oct 28, 2005

### HallsofIvy

Staff Emeritus
Is this becoming an English grammar forum now?

12. Oct 28, 2005

### Hurkyl

Staff Emeritus

13. Oct 28, 2005

### Natasha1

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

14. Nov 5, 2005

### Natasha1

Can anyone see something else?

Can anyone see something else?

15. Nov 7, 2005

### HallsofIvy

Staff Emeritus
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?