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.
\[\dfrac{-1}{-13}=\dfrac{1}{\Box}=-\dfrac{1}{\Box}\]

#\dfrac{-1}{-13}=\dfrac{1}{13}=-\dfrac{1}{-13}#

The fraction #\dfrac{-1}{-13}# 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:
\[\dfrac{-1}{-13}=\dfrac{1}{13}=-\dfrac{1}{-13}\]

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