# Proposition: Multiplication of Complex Numbers Using Polar Coordinates

If $z=r\exp(i\phi)$ and $w=s\exp(i\psi)$ are two complex numbers, then their product can be written as $$z\cdot w=(r\cdot s)\cdot [\cos(\phi+\psi)+i\sin(\phi+\psi)].$$

### Geometric Construction of the Multiplication of Complex Numbers

The product $z\cdot w$ of two complex numbers $z$ and $w$ can be geometrically constructed as follows:

• Connect the points $z$ and $w$ with the origin of the complex plane and measure the lengths $r$ and $s$ of the created segments $(0,z)$ and $(0,w)$.
• Measure the angles $\phi$ and $\psi$ between the positive $x$-axis and the segments $(0,z)$ and $(0,w)$.
• Add the angles $\phi + \psi$ and multiply the lengths $r\cdot s.$
• Draw a circle with radius $r\cdot s$ about the origin of the complex plane and draw a ray at the angle $\phi+\psi$.
• The intersection point between the circle and the ray that is drawn that way is the product of both complex numbers.

Proofs: 1

Proofs: 1

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### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983