Exponential form

Every complex number $z$ is the sum of a real and an imaginary component, $z=a+bi$. If you consider complex numbers to be coordinates in the complex plane with the $x$-axis consisting of real numbers and the $y$-axis pure imaginary numbers, then any point $z=a+bi$ can be plotted at the point as $(a,b)$. We can convert $z$ into polar form and re-write it as $z=r(\cos\theta+i\sin\theta)=r cis\theta$, where $r=|z| = \sqrt{a^2 + b^2}$. By Euler's formula, which states that $e^{i\theta}=\cos\theta+i\sin\theta$, we can conveniently (yes, again!) rewrite $z$ as $z=re^{i\theta}$, which is the general exponential form of a complex number.

See also

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