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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.

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