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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?

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...and (of course) in Needhams Visual Complex Analysis, in particular Figure [6] on page 9:… – Hans Lundmark Oct 16 '10 at 13:50
I found this to be a nice visualization: – Servaes Jun 26 at 21:58
up vote 10 down vote accepted

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 -

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As soon as you said $r_1 e^{i\theta_1} * r_2 e^{i\theta_2} $, it all made sense. Thanks :) Even looking at my crudely drawn vectors it looks to be exactly that. – Justin L. Oct 16 '10 at 4:35

Add the angles and multiply the lengths.

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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).

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