Question

Show that the decimal expansion of a rational number must repeat itself from some point onward.

Short answer

Hence, the given statement is shown.

Step-by-step solution

Definition of rational number

A number that can be expressed as an integer or the quotient of an integer divided by a nonzero integer

Explanation

Let us consider the rational number\(\frac{m}{n},n > 0\). We can always express\(m\)as\(m = q.n + r\), where\(0 \le r \le n - 1\).

If this rational number has some nonzero decimal expansion then,\(r > 0\).

Consider the following set of equations:

\({10^i} \cdot r = {q_i} \cdot n + {r_i}\forall i = 1,2, \ldots ,n + 1\)

Where\(0 \le {r_i} \le n - 1\) for \(i = 1,2, \ldots ,n\). By the pigeonhole principle, at least two of these \((n + 1)\) integers must be equal, i.e. \({r_i} = {r_j}\) for some\(i \ne j\), which means there has to be a repeat in the decimal expansion.