Negative fractions

Numbers: Fractions

Negative fractions

The rules for fractions containing negative numbers are comparable to the rules for dividing negative numbers:

$\begin{array}{rclrcl} \dfrac{\green{\text{positive}}}{\green{\text{positive}}} &=&\green{\text{positive}}\\[5pt] \dfrac{\green{\text{positive}}}{\blue{\text{negative}}} &=&\blue{\text{negative}}\\[5pt] \dfrac{\blue{\text{negative}}}{\green{\text{positive}}} &=& \blue{ \text{negative}} \\[5pt] \dfrac{\blue{\text{negative}}}{\blue{\text{negative}}} &=&\green{\text{positive}} \\ \end{array}$

Examples

$\begin{array}{rcl|rcl}\require{color} \\\phantom{xxx}\\ \dfrac{\green2}{\green3} &=& \green{\dfrac{2}{3}} \\[10pt] \dfrac{\green2}{\blue{-3}} &=&\blue{-\dfrac{2}{3}}\\[10pt] \dfrac{\blue{-2}}{\green3} &=&\blue{-\dfrac{2}{3}}\\[10pt] \dfrac{\blue{-2}}{\blue{-3}} &=& \green{\dfrac{2}{3}} \\ \end{array}$

In general

In general, for positive integers $a$ and $b$ , the following is true:

$\begin{array}{l} \dfrac{\green a}{\blue{-b}} = \dfrac{\blue{-a}}{\green b} = \blue{-\dfrac{a}{b}} \\[10pt] \dfrac{\blue{-a}}{\blue{-b}} = \green{\dfrac{a}{b}} \end{array}$

Examples

$\begin{array}{l} \dfrac{\green2}{\blue{-3}} = \dfrac{\blue{-2}}{\green3} = \blue{-\dfrac{2}{3}} \\[10pt] \dfrac{\blue{-2}}{\blue{-3}} = \green{\dfrac{2}{3}} \end{array}$

Fill in the blanks with the correct numbers.
$\dfrac{-14}{-15}=\dfrac{14}{\Box}=-\dfrac{14}{\Box}$

$\dfrac{-14}{-15}=\dfrac{14}{15}=-\dfrac{14}{-15}$

The fraction $\dfrac{-14}{-15}$ is positive because two negative numbers divided by each other give a positive number. The other fractions should, therefore, also be positive. That means the number of minus signs in each fraction should be even. This gives:
$\dfrac{-14}{-15}=\dfrac{14}{15}=-\dfrac{14}{-15}$

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