Polar form of a complex number

Parametric equations

Any nonzero complex number can be written $z=r{e}^{i\theta }$, where $r=|z|$ and $\theta =\mathrm{Arg}(z)$. Polar form separates scaling from rotation. That separation is the key to complex multiplication and to Euler’s formula.

Interaction

• Modulus r: drag the slider to change the length of the vector
• Argument θ: rotate the vector, and the real and imaginary projections update together with the arc of $\theta$

What to notice

• $x=r\cos \theta$ and $y=r\sin \theta$ match rectangular coordinates with polar coordinates.
• The modulus $r$ is the distance from the origin, and the argument $\theta$ is the direction.
• The same $z$ can be read as the rectangular form $x+yi$ or the polar form $r{e}^{i\theta }$.

Related

• Geometric significance of complex arithmetic
• Euler’s formula, rotating

Further reading

• Polar coordinate system (Wikipedia)
• Argument (complex analysis) (Wikipedia)