Converting Terminating Decimals Into Rational Numbers

Converting Terminating Decimals Into Rational Numbers

A decimal number has an integer part and a fractional part. For example, \(\text{10.589}\) has an integer part of \(\text{10}\) and a fractional part of \(\text{0.589}\) because \(10 + \text{0.589} = \text{10.589}\).

Each digit after the decimal point is a fraction with a denominator in increasing powers of \(\text{10}\).

For example:

• \(\text{0.1}\) is \(\cfrac{1}{\text{10}}\)

• \(\text{0.01}\) is \(\cfrac{1}{\text{100}}\)

• \(\text{0.001}\) is \(\cfrac{1}{\text{1 000}}\)

This means that

\begin{align*} \text{10.589} & = 10 + \cfrac{5}{10} + \cfrac{8}{100} + \cfrac{9}{\text{1 000}}\\ & = \cfrac{\text{10 000}}{\text{1 000}} + \cfrac{\text{500}}{\text{1 000}} + \cfrac{80}{\text{1 000}} + \cfrac{9}{\text{1 000}}\\ & = \cfrac{\text{10 589}}{\text{1 000}} \end{align*}