a{| |- ! colspan="2" style="background: #e4bc7e; font-size: 110%;" | Complex Number Identities and Formulas (info) |- ! colspan="2" style="background: #eee;" | Basic Definitions |- | align="right" style="padding-right: 2em;" | imaginary number || $ i=\sqrt{-1} \ $ |- | align="right" style="padding-right: 2em;" | electrical engineers' imaginary number || $ j=\sqrt{-1}\ $ |- | align="right" style="padding-right: 2em;" | (info) conjugate of a complex number || if $ z=a+ib $, for $ a,b\in {\mathbb R} $, then $ \bar{z}=a-ib $ |- | align="right" style="padding-right: 2em;" | (info) magnitude of a complex number || $ \| z \| = z \bar{z} $ |- | align="right" style="padding-right: 2em;" | (info) magnitude of a complex number || $ \| z \| = \sqrt{\left(Re(z)\right)^2+\left(Im(z)\right)^2} $ |- | align="right" style="padding-right: 2em;" | (info) magnitude of a complex number || $ \| a+ib \| = \sqrt{a^2+b^2} $, for $ a,b\in {\mathbb R} $ |- | align="right" style="padding-right: 2em;" | (info) magnitude of a complex number || $ \| r e^{i \theta} \| = r $, for $ r,\theta\in {\mathbb R} $ |- ! colspan="2" style="background: #eee;" | Complex Number Operations |- | align="right" style="padding-right: 2em;" |addition || $ (a+ib)+(c+id)=(a+c) + i (b+d) \ $ |- | align="right" style="padding-right: 2em;" |multiplication || $ (a+ib) (c+id)=(ac-bd) + i (ad+bc) \ $ |- | align="right" style="padding-right: 2em;" |division || $ \frac{a+ib} {c+id}=\frac{ac+bd} {c^2+d^2}+ i \frac{bc-ad} {c^2+d^2} \ $ |- | align="right" style="padding-right: 2em;" | exponentiation || $ i^n =\left\{ \begin{array}{ll}1,& \text{when }n\equiv 0\mod 4 \\ i,& \text{when }n\equiv 1\mod 4 \\-1,& \text{when }n\equiv 2\mod 4 \\-i,& \text{when }n\equiv 3\mod 4 \end{array} \right. \ $ |- ! colspan="2" style="background: #eee;" | Euler's Formula and Related Equalities (info) |- | align="right" style="padding-right: 2em;" | (info) Euler's formula || $ e^{iw_0t}=\cos w_0t+i\sin w_0t \ $ |- | align="right" style="padding-right: 2em;" | A really cute formula || $ e^{i\pi}=-1 \ $ |- | align="right" style="padding-right: 2em;" | Cosine function in terms of complex exponentials|| $ \cos\theta=\frac{e^{i\theta}+e^{-i\theta}}{2} $ |- | align="right" style="padding-right: 2em;" | Sine function in terms of complex exponentials||$ \sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2i} $ |- ! colspan="2" style="background: #eee;" | Other Formulas |- | align="right" style="padding-right: 2em;" | De Moivre's theorem ||$ \left(\cos x+i\sin x\right)^n=\cos\left(nx\right)+i\sin\left(nx\right).\, $ |- |}
