Question

Show that 0.357 is a rational number.

Answer 1

Answer:

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

Step-by-step 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 $q$ ($q \ne 0$) 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 $q$ ($q \ne 0$) 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:

$10 \times 0.357 = 3.57$.

$10^{2} \times 0.357 = 35.7$.

$10^{3} \times 0.357 = 357$.

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

Therefore, $0.357$ is indeed a rational number.