Thread: Proof repeated decimals are rational

1. Proof repeated decimals are rational

I've just been reviewing stuff on the number sets/lines, and I was looking at why repeated decimals are rational numbers. I knew this before, but have never seen a proof. I know there is one method you can use to prove specific examples:

e.g:

r = 4.27519191919...
100r = 427.519191919...
100r-r = 423.244 = 99r
r = 423244/99000
423244 and 99000 are both integers, so r is rational.

I was wondering whats the general proof of this for any repeating decimal....since this method requires that you pick an exmple.

Another thing I don't understand is how this is solved in the first place, since when you input it in the calculator, you only input the value to a certain number of deciml places, and it only computes the answer to a finite number of decimal places too, so you always get a terminating decimal which can be expressed as a rational number p/q where p,q are integers. For example, I got the above example from a book, but when I typed it in myself, the calculator gave a longer decimal expansion (the above answer, 423.244 is rounded).

2. Originally Posted by Greengoblin
I've just been reviewing stuff on the number sets/lines, and I was looking at why repeated decimals are rational numbers. I knew this before, but have never seen a proof. I know there is one method you can use to prove specific examples:

e.g:

r = 4.27519191919...
100r = 427.519191919...
100r-r = 423.244 = 99r
r = 423244/99000
423244 and 99000 are both integers, so r is rational.

I was wondering whats the general proof of this for any repeating decimal....since this method requires that you pick an exmple.

Another thing I don't understand is how this is solved in the first place, since when you input it in the calculator, you only input the value to a certain number of deciml places, and it only computes the answer to a finite number of decimal places too, so you always get a terminating decimal which can be expressed as a rational number p/q where p,q are integers. For example, I got the above example from a book, but when I typed it in myself, the calculator gave a longer decimal expansion (the above answer, 423.244 is rounded).
This is by no means a proof, but I would say that all repeating decimals may be represented as an infinite series. Which entails an infinite amount of rational numbers being added together. Thus rational.

3. To add more on Mathstud's post, take a look at this:
http://www.mathhelpforum.com/math-he...102-post5.html

4. Originally Posted by Mathstud28
This is by no means a proof, but I would say that all repeating decimals may be represented as an infinite series. Which entails an infinite amount of rational numbers being added together. Thus rational.
A repeating decimal can be represented as a rational and the sum of a rational geometric series, which is known has a rational sum.

This:

"Which entails an infinite amount of rational numbers being added together. Thus rational"

is nonesense, since every real can be represented as a countable sum of rationals.

RonL

5. Ahhh, thanks, I didn't think about using infinite series. Thanks alot.