View source for ComplexNumberFormulas - Rhea

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