Exponential form

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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.

So $z=re^{i\theta}$ looks like: [asy] import markers; pair A,B,C,D,E; A=(0,1); B=(0,0); C=(1,0); D=(1/2,sqrt(3)/2); E=(1/2,0); draw(A--B--C); label("Im",2*A/3,W); label("Re",2*C/3,S); dot("$z$",D,NE); draw(B--D,blue); draw(E--D,red); draw(B--E,green); label("$r$",(B+D)/2,p=blue,NW); label("$\sin\theta$",(E+D)/2,p=red,E); label("$\cos\theta$",(B+E)/2,p=green,S); markangle("$\theta$",radius=13,C,B,D,ArcArrow,orange); [/asy]

See also

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