Category:Examples of Converting Decimal Expansion of Rational Number to Fraction

This category contains examples of use of Converting Decimal Expansion of Rational Number to Fraction.

Let $x \in \Q$ be a rational number.

Let the decimal expansion of $x$ be:

$0 \cdotp a_1 a_2 \ldots a_m \dot b_1 b_2 \ldots \dot b_n$

where $a_i: i \in \set {1, 2, \ldots, m}$ and $b_j: j \in \set {1, 2, \ldots, n}$ be the digits in the base $10$ expansion of $x$.

Then $x$ can be expressed as the following fraction:

$x = \dfrac {a_1 a_2 \ldots a_m b_1 b_2 \ldots b_n - a_1 a_2 \ldots a_m} {10^m \times \paren {10^n - 1} }$