Complex rational numbers
Definition
- A complex rational number is a rational number in which the numerator is a rational number, or the denominator is a rational number, or both are. For example,
- \(\dfrac{\tfrac{2}{3}}{5}\): the numerator is \(\tfrac{2}{3}\) and the denominator is \(5\).
- \(\dfrac{4}{\tfrac{3}{5}}\): the numerator is \(4\) and the denominator is \(\tfrac{3}{5}\).
- \(\dfrac{\tfrac{2}{5}}{\tfrac{3}{4}}\): the numerator is \(\tfrac{2}{5}\) and the denominator is \(\tfrac{3}{4}\).
Converting to simple rational numbers
Every complex rational number can be expressed as a simple rational number with an integer numerator and an integer denominator. We simply perform division of the numerator by the denominator and evaluate the result. For example
\[\dfrac{\tfrac{2}{3}}{5} = \tfrac{2}{3}\div 5 = \tfrac{2}{3}\times \tfrac{1}{5} = \dfrac{2}{15}\] \[\dfrac{4}{\tfrac{3}{5}} = 4\div \tfrac{3}{5} = 4\times \tfrac{5}{3} = \dfrac{20}{3}\] \[\dfrac{\tfrac{2}{5}}{\tfrac{3}{4}} = \tfrac{2}{5}\div\tfrac{3}{4} = \tfrac{2}{5}\times\tfrac{4}{3} = \dfrac{8}{15}\]If either the numerator or the denominator evaluates to a rational number, we also have a complex rational number. For example
\[\dfrac{\tfrac{2}{3}+\tfrac{5}{3}}{\tfrac{1}{4}} = \dfrac{\tfrac{7}{3}}{\tfrac{1}{4}} = \tfrac{7}{3}\times\tfrac{4}{1} = \dfrac{28}{3}\]Complex rational forms
The same principle applies to expressions that have unknowns (i.e., symbolic placeholders).
\[\dfrac{\tfrac{a}{b}+\tfrac{c}{b}}{\tfrac{d}{b}}=\dfrac{\tfrac{a+c}{b}}{\tfrac{d}{b}}=\tfrac{a+c}{b}\times\tfrac{b}{d}=\tfrac{a+c}{d}\]