# Complex Numbers

This page is a sub-page of our page on Geometric Numbers in the line and the plane.

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The interactive simulations on this page can be navigated with the Free Viewer
of the Graphing Calculator.

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Related KMR pages:

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Other related sources of information:

• Map of Mathematics at the Quanta Magazine
•• Complex numbers as operators on the universe

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Representation: $\, [ \, z \, ]_{R_{ectangular}} = \left< \, x + i y \, \right>_{R_{ectangular}} \,$.

Representation: $\, [ \, z \, ]_{P_{olar}} = \left< \, r e^{i \theta} \, \right>_{P_{olar}} \,$.

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Basic arithmetic

$\, [ \, z_1 + z_2 \, ]_{R_{ectangular}} \, = \, \left< \, (x_1 + i y_1) + (x_2 + i y_2) \, \right>_{R_{ectangular}} = \,$

$\,\,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \left< \, (x_1 + x_2) + i (y_1 + y_2) \, \right>_{R_{ectangular}} \,$

$\, [ \, z_1 + z_2 \, ]_{P_{olar}} \, = \, \left< \, r_1 \cos {\theta}_1 + r_2 \cos {\theta}_2 + i ( r_1 \sin {\theta}_1 + r_2 \sin {\theta}_2 ) \, \right>_{P_{olar}} \, = \,$

$\,\,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \, \left< \, (r_1 (\cos {\theta}_1 + i \, \sin {\theta}_1) \, + \, r_2 (\cos {\theta}_2 + i \, \sin {\theta}_2) \, \right>_{P_{olar}} \, = \,$

$\,\,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \, \left< \, r_1 e^{i {\theta}_1} \, + \, r_2 e^{i {\theta}_2} \, \right>_{P_{olar}} \,$

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Representations of multiplication:

$\, [ \, z_1 \, z_2 \, ]_{R_{ectangular}} \, = \, \left< \, (x_1 + i y_1) \,(x_2 + i y_2) \,\right>_{R_{ectangular}} \, = \,$

$\,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \, = \left< \, (x_1 x_2 - y_1 y_2) + i (x_1 y_2 + x_2 y_1) \, \right>_{R_{ectangular}} \,$

$\, [ \, z_1 \, z_2 \, ]_{P_{olar}} = \, \left< \, r_1 e^{i {\theta}_1} \, r_2 e^{i {\theta}_2} \, \right>_{P_{olar}} \, = \,$

$\, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \, \left< \, r_1 r_2 e^{i({\theta}_1 + \, {\theta}_2)} \, \right>_{P_{olar}} \,$

Animation of complex multiplication with filled triangles:

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Applications of Complex Numbers

Electrical impedance:

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Planar electro-magnetic wave:

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Imaginary numbers are real (13 videos by Welch Labs):

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Euler’s formula with introductory group theory (3Blue1Brown on YouTube):

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Imaginary Numbers, Functions of Complex Variables: 3D animations