# Rational Numbers

**Rational numbers** are numbers that can be expressed as the ratio of two integers. Rational numbers follow the rules of arithmetic and all rational numbers can be reduced to the form $\frac{a}{b}$, where $b\neq0$ and $\gcd(a,b)=1$.

Rational numbers are often denoted by $\mathbb{Q}$. These numbers are a subset of the real numbers, which comprise the complete number line and are often denoted by $\mathbb{R}$. Real numbers that cannot be expressed as the ratio of two integers are called **irrational numbers**.

The decimal expansion of a rational number always terminates after a finite number of digits or repeats a sequence of finite digits over and over. E.g $2.5$ has a terminating decimal expansion. Thus it is a rational number.

$\Rightarrow$ Every integer is a rational number.

$\Rightarrow$ Every fraction where the denominator $\neq 0$ is a rational number.

Determine the rational representation of $0.\overline{238095}$. The line over $238095$ denotes that it is a repeating decimal of the form $0.238095238095238095\ldots$.

Let $S = 0.\overline{238095}$. Then $1000000S = 238095.\overline{238095}$, and taking the sum, we obtain

$\begin{aligned} - S & = -000000.\overline{238095} \\ 1000000S & = 238095.\overline{238095}\\ \hline \\ 999999S & = 238095. \end{aligned}$

Hence, $S = \frac {238095}{999999} = \frac {5}{21}$. $_\square$

**Cite as:**Rational Numbers.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/rational-numbers/