Negative mixed fractions

Question

I'm comfortable with fractions like $\frac{-3}{8}$ being the same as $\frac{3}{-8}$ (though I'd think the latter anachronistic and would in any case probably prefer to write either of those two as $-\frac{3}{8}$ ), and of course I'm comfortable with improper fractions like $\frac{-8}{3}$ being the mixed number $-2\frac{2}{3}$.

However, if I'm trying to teach a computer how to handle negative vulgar fractions, I should also consider the remaining cases, so how should I interpret:

1. $2\frac{-2}{3}$
2. $2\frac{2}{-3}$
3. $2\frac{-2}{-3}$
4. $-2\frac{-2}{3}$
5. $-2\frac{2}{-3}$
6. $-2\frac{-2}{-3}$

I'm considering the logical approach, by inference from $2\frac{2}{3}$ and $-2\frac{2}{3}$, so I'd get:

1. $1\frac{1}{3}$
2. $1\frac{1}{3}$
3. $1\frac{2}{3}$
4. $-1\frac{1}{3}$
5. $-1\frac{1}{3}$
6. $-2\frac{2}{3}$

Answer 1

Section 3.8 of Charles McKeague's Pre-Algebra: A Text/Workbook has the key information, with worked examples.

$$3\frac{2}{3} - 1\frac{1}{6}$$ is implicitly bracketed as $$\left(3\frac{2}{3}\right) - \left(1\frac{1}{6}\right)$$ so the answer in this example is $$2\frac{1}{2}$$

Applying that the the examples in the original question is then straightforward, and the proposed interpretations are correct.

Edit: Answers at Primary/Elementary Pedagogy: What is the rationale for the absent '+' in mixed fractions? also indicate the reason why we write a mixed number as $1\frac{1}{2}$ and not as $1+\frac{1}{2}$.

Answer 2

Recall that when $a,b,c \in \mathbb Z$, $$a\dfrac bc= a + \dfrac bc$$

If $\frac bc <0$, then $\left|\frac bc\right|$ is subtracted from $a$.

So, for example $$-2 \frac{-2}3 = -2 - \frac 23 = -\left(2 + \frac 23\right)$$

whereas $$-2\frac{-2}{-3} = -2 + \frac 23 = -\left(1+\frac 13\right)$$