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

In summary, if x=0.a1a2...an then r=\frac{10n}{2^a5^b} where either a or b could be 0 and p is a prime.Splitting hairsI think either of a and b is more precise, and p does not have to be prime.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.The usage of "and" is incorrect; I don't require that both a and b be
  • #1
Natasha1
493
9
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?
 
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  • #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?
 
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  • #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
 
  • #4
(1)-(4) of your link explains the condition you're after. Do you understand what's there?
 
  • #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.
 
  • #6
Splitting hairs

I think either of a and b is more precise, and p does not have to be prime.
 
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  • #7
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.
 
  • #8
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:
 
  • #9
No problem, I just looked that up myself. :smile: You're right, p doesn't have to be prime.
 
  • #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?
 
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  • #11
Is this becoming an English grammar forum now?
 
  • #12
Please don't multiple post. I've merged your two threads.
 
  • #13
Sorry I am new to all this. Won't multiply post in the future! Promise
 
  • #14
Can anyone see something else?

Can anyone see something else? :bugeye:
 
  • #15
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?
 

1. What is a rational number?

A rational number is any number that can be expressed as a ratio of two integers (whole numbers), such as 1/2, 3/4, or -5/8.

2. Why is the range between 0 and 1 specified?

This range is specified because it includes all numbers that have finite decimal expansions, which is the focus of the question.

3. What is a finite decimal expansion?

A finite decimal expansion is a decimal representation of a number that has a finite number of digits after the decimal point, such as 0.25 or 0.75.

4. How do you determine which rational numbers between 0 and 1 have finite decimal expansions?

To determine which rational numbers between 0 and 1 have finite decimal expansions, you can use the decimal representation of the fraction and see if it has a finite number of digits after the decimal point. If it does, then it has a finite decimal expansion.

5. Are there rational numbers between 0 and 1 that do not have finite decimal expansions?

Yes, there are rational numbers between 0 and 1 that do not have finite decimal expansions. For example, 1/3 is a rational number between 0 and 1, but its decimal representation is 0.333... which has an infinite number of digits after the decimal point.

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