What does it mean to divide imaginary numbers exactly?. So division implies quantity, and i doesn't have a defined quantitative value (not that i know of at least). So how can we go about dividing complex numbers? Can we say that (a+bi)/(a+bi)=1? If so, can we justify this by referring to some set of complex number properties?
 A: Note that $$(a+bi)(\frac {a-bi}{a^2+b^2})=1$$
That is every non-zero complex number has a multiplicative inverse.
Now we can define division as multiplication by inverse, that is $$\frac {z}{w} = z(w^{-1})$$
A: The difficulty is in our imagination. We are used to the real number line and think of complex numbers as a pair of real numbers. But this isn't quite right. Such pairs wouldn't allow the equation $i^2=(0,1)\cdot(0,1)=(-1,0)=-1.$ So complex numbers are more than two real numbers. If you like then they could be thought of as a complex numer line. But then we would have a linear ordering along the line, which complex numbers do not have. In any case, our 'real' world imgination leaks fundamental properties. Personally, I like to think of complex numbers as polynomials, where we identify $x^2=-1$ which leaves us with linear polynomials and the complex multiplication with $x$ instead of $i.$ It does exactly the same but it does not suggest to be a complex plane, or a number in some sense.
Division is $\dfrac{1}{ax+b}=\dfrac{ax-b}{(ax-b)(ax+b)}=\dfrac{1}{a^2x^2-b^2}\cdot(ax-b)=\dfrac{b-ax}{a^2+b^2}$ since $x^2=-1.$
This is the algebraic view on complex numbers and is probably insufficient to perform complex analysis. However, it satisfies all basic calculation rules and most of all, avoids misconceptions like $$-1=i^2=\sqrt{-1}\cdot\sqrt{-1}\neq\sqrt{(-1)^2}=\sqrt{1}=1.$$ So in case you want to understand complex arithmetics, then the polynomial approach can be helpful.
A: The other answers are correct, but there is a geometric meaning to complex numbers that adds a lot to our understanding.  Historically, they were thought to be little more than a curiosity until Caspar Wessel gave them a geometric meaning.  Then Gauss and Euler expanded on that, of course.
Here’s the visual: The real number line is horizontal.  Picture a second real number line at a right angle to it.  So now you have the Cartesian plane.  Any complex number a + bi is placed at the point (a,b).  i itself is thus 0 + 1i, so it is placed one unit above the origin.  Now, with a little trigonometry it’s possible to redefine the point with the quantities r and theta, where r is the point’s distance from the origin and theta is the angle between the ray through the point and the positive x-axis.  Here’s the geometric definition of complex multiplication: Given two points, you multiply their radii and add their angles.
That’s really all there is to it.  So then division is the inverse- divide their radii and subtract their angles.  It seems a bit arbitrary, but this definition is completely consistent with the algebraic definition, so it’s the only one that works.  Your statement that “division implies quantity” is a bit vague, but I see what you’re getting at.  A better way to say it is that if you can define a size for two objects (the formal term is a “norm”), then dividing those objects will also divide the norms.  However, not all mathematical domains have norms.
As always, 3 Blue 1 Brown is a must watch for building geometric intuition: https://youtu.be/5PcpBw5Hbwo
