Logic behind dividing negative numbers I've learnt in school that a positive number, when divided by a negative number, and vice-versa, will give us a negative number as a result.On the other hand, a negative number divided by a negative number will give us a positive number.
Given the following equations:


*

*$\frac{-18}{2} = -9$

*$\frac{18}{-2} = -9$

*$\frac{-18}{-2} = 9$


This is how I would think the logic behind the equation would be:


*

*If I have a debt of 18 dollars , owed to 2 people equally, I would owe each of them $9


*

*$\frac{-18}2 = -9$


*If I have 18 dollars, owed to 2 people equally, I would thus give them $9 each


*

*$\frac{18}{-2} = -9$



However, I can't seem to come to terms with a negative number dividing by a negative number giving me a positive number as a result. What would be the logic behind it?
Also, I think that I have the logic/reasoning for the 2nd example wrong, as it is exactly the same as the reasoning for the first example?Could someone give me a better example of the logic behind the 2nd example?
I would really appreciate it if anyone could enlighten me.

I caution against trying to model numeric relationships on real-world phenomena. To do so well, you must make sure that your mathematical model accurately captures the real world, which can be tricky and non-intuitive.
In this case, is a debt a negative asset? Is a negative debt an asset? Is negative owing an expectation of payment? So is $\frac{-18}{-2}$ a negative asset of $18 expected from two people? Clear as mud.
Instead, I recommend considering division as multiplication by a reciprocal (which I believe it always can be), so $\frac{-18}{-2}$ becomes $-18 \times \frac{1}{-2}$. If you factor out the $-1$ from $\frac{1}{-2}$, you get $-18 \times -1 \times \frac{1}{2}$. This becomes $18 \times \frac{1}{2}$, or $9$.
