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.

Division can be thought of (in an algorithmic sense) as repeated subtraction.  The question "what is 6/2" is exactly equivalent to the question "how many times must one subtract 2 from 6 to reach 0".
How many times must one subtract 2 from 18 to reach 0?  Of course it's 9.
How many times must one subtract -2 from 18 to reach 0? Of course it's -9.
BTW this is a good way to help kids understand why division by zero is a singularity. How many times must one subtract 0 from X to reach 0?  Any finite or inifinite number is not sufficient.
