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Complex numbers

A complex number $z=a+bi$ has a real and an imaginary part, where $i^2=-1$ . Euler's formula connects complex exponentials to sine and cosine.

z = a+bi,\quad i^2=-1

Cartesian form

|z| = \sqrt{a^2+b^2}

modulus (magnitude)

e^{i\theta} = \cos\theta + i\sin\theta

Euler's formula

Symbols

$a,\ b$	real part, imaginary part
$\bar z$	conjugate: $a-bi$

$|3+4i| = \sqrt{9+16}=5$ , and $e^{i\pi}=\cos\pi+i\sin\pi=-1$ .

i^2=-1

is all you need to remember - every other rule follows from ordinary algebra.