$$-4 / 4$$

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".

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    $\begingroup$ I assume by $2, 4, 6, 8, 10 = 5$ you mean that you prefer to think of multiplication as incremented addition. If this is the case you can do the same thing in the example you provided, for example you can think "what number, $x$, added to itself $4$-times is $-4$?" $(x+x+x+x) = -4$ and the answer clearly becomes "-1". Hope this helps $\endgroup$ – Deven Ware Sep 14 '11 at 23:38
  • $\begingroup$ There may not be a way to "see anything happening," but it is helpful to view division as the inverse of multiplication. For example, to say $-4/4 = x$ means that $-4 = 4 \times x$. If you're comfortable with multiplication of signed numbers, then this may work for you. You could ask "4 of what gives me a debt of 4?" Well, four debts of 1 do just that, so $-4 = 4 \times -1$, or $-4/4 = -1$. $\endgroup$ – Shaun Ault Sep 14 '11 at 23:39

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.


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