Question

Write the decimal expansion of `1/7`. Hence, write the decimal expansions of `2/7, 3/7, 4/7, 5/7` and `6/7`.

Answer 1

The decimal expansion of `1/7` is:

`1/7 = 0.bar(142857)`

This means the digits “142857” repeat indefinitely.

Hence, the decimal expansions of the other fractions with denominator 7 are:

`2/7 = 0.bar(285714)`

`3/7 = 0.bar(428571)`

`4/7 = 0.bar(571428)`

`5/7 = 0.bar(714285)`

`6/7 = 0.bar(857142)`

Each of these decimal expansions is a repeating cycle of six digits, each a rotation of the digits in the repeating block for `1/7` .