## 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.”

 Question: a)       What is $$\dfrac{-a}{-b}$$? b)      What is $$\dfrac{-8}{9}\times \dfrac{2}{-5}$$?

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