Prove that every repeating decimal represents a rational number

by RoRi

February 29, 2016

Prove that every repeating decimal represents a rational number.

Proof. Let

$x = a_0.a_1 \ldots a_k a_1 \ldots a_k a_1 \ldots a_k \ldots$

be any repeating decimal. Let

$R = a_1 a_2 \ldots a_k.$

Then,

$\begin{align*} x &= a_0 + \sum_{n=1}^{\infty} \frac{R}{(10^k)^n} \\[9pt] &= a_0 + R\sum_{n=0}^{\infty} \frac{1}{(10^k)^n} - R \\[9pt] &= a_0 + R \left( \frac{1}{1 - \frac{1}{(10^k)^n}} \right) - R \\[9pt] &= (a_0 - R) + \frac{R \cdot 10^k}{10^k - 1}. \end{align*}$

But this is a rational number since $a_0 - R$ is an integer (since both $a_0$ and $R$ are integers) and $\frac{R \cdot 10^k}{10^k - 1}$ is a rational number (and the sum of an integer and a rational number is rational) $. \qquad \blacksquare$