Question

Show that 0.357 is a rational number.

Answer 1

`0.357 = (357)/(1000)`.

Explanation:

A number is a rational number if and only if it is equal to the ratio between two integers. In other words, a number
`x` is a rational number if and only if there are integers (whole numbers)
`p` and
`q` (
`q \\e 0`) such that
`x = p / q`.

Thus, the
`0.357` in this question would be a rational number as long as there are integers
`p` and
`q` (
`q \\e 0`) such that
`p / q = 0.357`. Simply finding the right
`p\!` and
`q\1` would be sufficient for showing that
`0.357\!` is rational.

Since
`0.357` is a terminating decimal, one possible way to find
`p\!` and
`q\1` is to repeatedly multiply
`0.357\!` by
`10` until a whole number is reached:

`10 * 0.357 = 3.57`.

`10^(2) * 0.357 = 35.7`.

`10^(3) * 0.357 = 357`.

Thus,
`0.357 = (357) / (10^(3))`, such that
`p = 357` and
`q = 1000` (both are integers, and
`q \\e 0`) would ensure that
`0.357 = p / q`.

Therefore,
`0.357` is indeed a rational number.