# 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/