# Decimals and Rational numbers

How do you prove:

Q1) Why is every rational number (say m/n, where m and n are both positive integers) either a terminating or a repeating decimal? Q2) Why is every repeating decimal (or terminating decimal) a rational number?

For Q1 do long division. Either you reach a remainder $0$ and the division terminates, or you don't. If you don't, the number of possible remainders is finite and you must repeat.
For Q2, if the repeating decimal has $n$ digits, multiply by $10^n$ and subtract the original. If it terminates with $n$ digits after the decimal point, just multiply by $10^n$
Every rational number not having a finite number of decimals is a repeating decimal because of the following reason. When you make the division of $m$ by $n$ you get, at most, $n$ different remainders. So, if the division doesn't end, after, at most, $n+1$ steps, you get the same remainder you have got in a previous step. From this, you get a periodic repetition of remainders.
As said by @Meelo the other part of the question has been answered in H0w t0 prove that periodic decimal numbers are rational? $a_1...a_k(b_1b_2..b_l)={m \over n}$