15
$\begingroup$

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?

$\endgroup$
18
$\begingroup$

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

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – Justin L. Oct 16 '10 at 4:35
12
$\begingroup$

Add the angles and multiply the lengths.

$\endgroup$
5
$\begingroup$

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).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.