Home  18.013A  Chapter 1  Section 1.2 


Prove that rational numbers repeat the same finite sequence of digits endlessly, and that irrational numbers do not.
Solution:
When you divide $a$ by $b$ there are only $b$ possible remainders.
This means that after at most $b$ steps in your long division the remainder must repeat itself.
Since $a$ and $b$ are integers, once your division gets past the decimal point, you always bring down a 0 in your division procedure.
If you always bring down a 0, the result of your division beyon any point depends only on the remainder at that point.
Thus once a remainder is repeated past the decimal point, the sequence of remainders between the repeat will continue repeating endlessly.
For example,
$\frac{1}{11}$ is .909090 ... or .(09)*.
$\frac{1}{7}$ is .142857142857142857 ... or .(142857)*.
If the decimal expansion of a number $x$ repeats endlessly, as $.(z)*$ for some sequence $z$ that is $k$ digits long, we can write
which is a rational number. If you have problems following this manipulation do not worry. We will talk about it later.
