Geometric interpretation of the multiplication of complex numbers?

Question

I've always been taught that one way to look at complex numbers is as a Cartesian space, where the real part is the $x$ component and the imaginary part is the $y$ component.

In this sense, these complex numbers are like vectors, and they can be added geometrically like normal vectors can.

However, is there a geometric interpretation for the multiplication of two complex numbers?

I tried out two test ones, $3+i$ and $-2+3i$, which multiply to $-9+7i$. But no geometrical significance seems to be found.

Is there a geometric significance for the multiplication of complex numbers?

Answer 1

Suppose we multiply the complex numbers $z_1$ and $z_2$. If these numbers are written in the polar form as $r_1 e^{i \theta_1}$ and $r_2 e^{i \theta_2}$, the product will be $r_1 r_2 e^{i (\theta_1 + \theta_2)}$. Equivalently, we are stretching the first complex number $z_1$ by a factor equal to the magnitude of the second complex number $z_2$ and then rotating the stretched $z_1$ counter-clockwise by an angle $\theta_2$ to arrive at the product. There are several websites that expand upon this intuition with graphics and more explanation. See this site for example - http://www.suitcaseofdreams.net/Geometric_multiplication.htm

Answer 2

Add the angles and multiply the lengths.

Answer 3

Yes, there is a simple geometric meaning, but you need to convert to the polar form of the complex numbers to see it clearly. $3+i$ has magnitude $\sqrt{10}$ and angle about $18^\circ$; $-2+3i$ has magnitude $\sqrt{13}$ and angle about $124^\circ$. Multiplication of the complex numbers multiplies the two magnitudes, resulting in $\sqrt{130}$, and adds the two angles, $142^\circ$. In other words, you can view the second number as scaling and rotating the first (or the first scaling and rotating the second).