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.

Before considering division, it may be helpful to first consider what the distributive law implies about negative integers and multiplication.  In particular, if n is a positive integer and m is its additive inverse, then m+n equals zero, and mX+nX must equal (m+n)X which must then also equal zero.  Consequently, mX must be the additive inverse of nX.  If nX is positive, mX must be negative, and vice versa.
The quotient n/d is defined as being a value q, if one exists, such that qd=n.  If e=-d, then qe=q(-d)=-(qd)=-n.  Since qe=-n, q=(-n)/e.  Since we also know that q=n/d, that implies that n/d=(-n)/(-d) in cases where a suitable value d exists.
Note that if one defines a form of division that requires an integer quotient but allows a non-zero remainder, that form of division can satisfy either -(n/d)=(-n)/d or (n+d)/d=(n/d)+1, but not both.  The problem scenarios arise when the division has a remainder; if one only consider division to be defined in cases where there is no remainder, both equalities hold.
