Fractions are Hard!

3.2 Fractions with negative numerators and denominators

Mathematically “\(-2\)” represents the opposite of “\(2\)”, in the sense that adding \(2\) and \(-2\) together gives zero.

(See Piles and Holes for my general approach to this.)


The usual rules of arithmetic also allow us to think of \(-2\) as \(-1\times 2\) if we prefer.


Suppose now we can extend our work with fractions to include negative numbers as numerators and denominators. (Sharing anti-pie among anti-boys?)


Here’s a question:


Are \(\dfrac{-3}{5}\) and \(\dfrac{3}{-5}\) and \(-\dfrac{3}{5}\) the same fraction or are they all different as numbers?


It is hard to answer this question with our sharing model or in any of our models. (Is \(\dfrac{-3}{5}\), the result of sharing three anti-pies to five boys, the same as the result of \(\dfrac{3}{-5}\), sharing three pies to five anti boys? And are these answers both three-fifths of an anti-pie?)


But if we believe that the Key Fraction Property should hold for all types of numbers, including negative ones, then we have


\(\dfrac{-3}{5}=\dfrac{-3\times \left(-1\right)}{5\times\left(-1\right)}=\dfrac{3}{-5}\).


And we also have by our Basic Multiplication Belief that


\(-\dfrac{3}{5}=\left(-1\right) \times \dfrac{3}{5}=\dfrac{\left(-1\right)\times 3}{5}=\dfrac{-3}{5}\).


This shows all three quantities should be deemed the same number.


People call writing \(\dfrac{-a}{b}\) as \(-\dfrac{a}{b}\), and writing \(\dfrac{a}{-b}\) as \(-\dfrac{a}{b}\), as “pulling out a negative sign.”



a)       What is \(\dfrac{-a}{-b}\)?

b)      What is \(\dfrac{-8}{9}\times \dfrac{2}{-5}\)?


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