]> Exercise 1.4

Exercise 1.4

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,

$1 11$ is .909090 ... or .(09)*.

$1 7$ 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

$x = z * 10 − k * ( 1 + 10 − ( k + 1 ) + 10 − 2 ( k + 2 ) + ⋯ ) = z 10 k ( 1 − 10 − ( k + 1 ) )$

which is a rational number. If you have problems following this manipulation do not worry. We will talk about it later.