TITLE: How can I visualize division of negative numbers
QUESTION [9 upvotes]: $$-4 / 4$$
I have trouble because I don't know what's happening. I can do positive because I can see in my head with $$10 / 2$$
$$2, 4, 6, 8, 10 = 5$$
But with negative numbers, I can't see anything happening, I can only guess or use a calculator. Especially when there is one negative and one or more positive. Can someone help me visualize the division of one or more negative numbers? Thanks.
Also: Can someone edit the tag sources for me; I cannot find the correct tag to use for this. I tried multiplication, but "my rank does not allow me to create new tags".

REPLY [7 votes]: Sure. Multiplication by a positive real number $x$ scales all points on the real number line; specifically it multiplies their distance from the origin by a factor of $x$. And multiplying by $-1$ flips the real line around so that it is in reverse. Now the magnitude of $a/b$ represents by how large of a factor that $b$ has to be scaled to reach the same magnitude as $a$, and the sign of $a/b$ represents whether or not we have to flip $b$ around the origin in the process of going to $a$.
In terms of additive increments in $\mathbb{Z}$ (the integers), we could say that adding $-n$ goes $n$ steps backwards instead of forwards. Hence the absolute value of $a/b$ represents the number of times you need to go $b$ steps to reach $a$, and the sign represents whether you go the natural direction that $b$ is already oriented in order to get to $a$ or if you have to flip the orientation and go in the opposite direction.