TITLE: Why does multiplication act like scaling and rotation of a vector in the complex plane?
QUESTION [10 upvotes]: I regularly use the geometric analogy of multiplication by a complex number to represent a scaling and rotation of a vector in the complex plane. For a very simple example, i would point up along the Y axis and multiplying it by i again would be a 90 degree rotation resulting in something pointing in the -X direction.
The thing is, I no longer recall why this is true. It's not obvious to me any longer why multiplication is in any way connected to rotation (scaling seems fairly obvious) and I was unable to explain the logic behind this useful trick to a friend who asked why it worked.
Could I get a very clear explanation of this geometric interpretation of multiplication by complex numbers? I feel like it had to do with Euler's identity and the polar form of complex numbers but this math is quite a few years behind me.

REPLY [2 votes]: Here is a different way to get some intuition on why the multiplication rule of the complex number is as it is from a purely geometrical point of view.
The real numbers can be represented by line-segments (as in Euclidean geometry) and in this representation multiplication can be viewed geometrically as follows: the line-segment of length $ab$ is a line-segment which "stands in relation" to $a$ as $b$ "stands in relation" to $1$. Here "stands is relation" refers to the ratio of the lengths so in formulas this is just the statement that $\frac{ab}{a} = \frac{b}{1}$.
The same type of geometrical interpretation lies behind complex multiplication: $z_1z_2$ is the number that "stands in relation" to $z_2$ as $z_1$ "stands in relation" to $1$. 
Let's first define what we mean by "stands in relation" here. For this purpose we will represent the complex numbers as pairs (think vectors in the plane) of real numbers $z = (r,\theta)$ where $r$ is the length of the vector and $\theta$ is the angle relative to unit "$1$" (the $x$-axis). 
The relation of the number $z_1 = (r_1,\theta_1)$ to $1 = (1,0)$ is the following: we scale the length ($1$) by $r_1$ and rotate (from $\theta = 0$) by $\theta_1$. Likewise the relation of $z_1z_2$ relative to $z_2$ is: scale the length ($r_2$) by $r_1$ and rotate (from $\theta = \theta_2$) by $\theta_1$. This is illustrated below and we see that this leads to $z_1z_2 = (r_1r_2,\theta_1+\theta_2)$ which is just the complex multiplication rule in disguise.
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$
In fact this procedure, generalizing multiplication of scalars to two dimensional vectors in the way we did above, is a way to "discover" the complex numbers (or more technically a ring with the same algebraic structure as the complex numbers) without even having to define "$i$" as $\sqrt{-1}$ (making it a good formulation to convince people that there is nothing magical about "$i$").