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:
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).
To add more on Mathstud's post, take a look at 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.