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[[Category:Formulas]]
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keywords: magnitude, conjugate, de Moivre, Euler
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<center><font size= 4>
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'''[[Collective_Table_of_Formulas|Collective Table of Formulas]]'''
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</font size>
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'''Complex Number Identities and Formulas'''
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click [[Collective_Table_of_Formulas|here]] for [[Collective_Table_of_Formulas|more formulas]]
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</center>
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----
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{|
 
{|
 
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|-  
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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:  #e4bc7e; font-size: 110%;" | [[more_on_complex_numbers|Complex Number]] Identities and Formulas [[more_on_complex_numbers|(info)]]
 
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! colspan="2" style="background: #eee;" | Basic Definitions
 
! colspan="2" style="background: #eee;" | Basic Definitions
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| align="right" style="padding-right: 2em;" | imaginary number || <math>i=\sqrt{-1} \ </math>
 
| align="right" style="padding-right: 2em;" | imaginary number || <math>i=\sqrt{-1} \ </math>
 
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|-
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| align="right" style="padding-right: 2em;" | electrical engineers' imaginary number || <math>j=\sqrt{-1}\ </math>
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| align="right" style="padding-right: 2em;" | [[ECE|electrical engineers]]' imaginary number || <math>j=\sqrt{-1}\ </math>
 
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| align="right" style="padding-right: 2em;" | [[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: 2em;" | [[more_on_complex_conjugate|(info)]] conjugate of a complex number || <math> \text{if}\ z=a+ib,\ \text{for}\ a,\ b \in {\mathbb R},\ \text{then} \ \bar{z}=a-ib </math>
 
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| align="right" style="padding-right: 2em;" | [[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: 2em;" | [[more_on_complex_magnitude|(info)]] magnitude of a complex number || <math>\| z \| = \sqrt{ z \bar{z} } </math>
 
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|-  
 
| align="right" style="padding-right: 2em;" | [[more_on_complex_magnitude|(info)]] magnitude of a complex number || <math> \| z \| =  \sqrt{\left(Re(z)\right)^2+\left(Im(z)\right)^2}</math>
 
| align="right" style="padding-right: 2em;" | [[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: 2em;" | [[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: 2em;" | [[more_on_complex_magnitude|(info)]] magnitude of a complex number || <math>\| a+ib \| = \sqrt{a^2+b^2},\ \text{for}\ a,b\in {\mathbb R}</math>
 
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| align="right" style="padding-right: 2em;" | [[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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| align="right" style="padding-right: 2em;" | [[more_on_complex_magnitude|(info)]] magnitude of a complex number || <math>\| r e^{i \theta} \| = r,\ \text{for}\ r,\theta\in {\mathbb R}</math>
 
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! colspan="2" style="background: #eee;" | Complex Number Operations
 
! colspan="2" style="background: #eee;" | Complex Number Operations
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| align="right" style="padding-right: 2em;" |multiplication || <math>(a+ib) (c+id)=(ac-bd) + i (ad+bc) \ </math>
 
| align="right" style="padding-right: 2em;" |multiplication || <math>(a+ib) (c+id)=(ac-bd) + i (ad+bc) \ </math>
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|-
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| align="right" style="padding-right: 2em;" |multiplication in polar form|| <math>\left( r_1 (\cos \theta_1 + i \sin \theta_1) \right) \left( r_2 (\cos \theta_2 + i \sin \theta_2) \right)= r_1 r_2 \left( \cos (\theta_1+\theta_2)+i \sin (\theta_1-\theta_2) \right)\ </math>
 
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| align="right" style="padding-right: 2em;" |division || <math>\frac{a+ib} {c+id}=\frac{ac+bd} {c^2+d^2}+ i \frac{bc-ad} {c^2+d^2} \ </math>
 
| align="right" style="padding-right: 2em;" |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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|-
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| align="right" style="padding-right: 2em;" |division in polar form|| <math>\frac{ r_1 (\cos \theta_1 + i \sin \theta_1)}{  r_2 (\cos \theta_2 + i \sin \theta_2) }= \frac{r_1}{ r_2} \left( \cos (\theta_1-\theta_2)+i \sin (\theta_1+\theta_2) \right)\ </math>
 
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| align="right" style="padding-right: 2em;" | exponentiation || <math> 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  
 
| align="right" style="padding-right: 2em;" | exponentiation || <math> 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  
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| align="right" style="padding-right: 2em;" | De Moivre's theorem ||<math>\left(\cos x+i\sin x\right)^n=\cos\left(nx\right)+i\sin\left(nx\right).\,</math>
 
| align="right" style="padding-right: 2em;" | 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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|-
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| align="right" style="padding-right: 2em;" | Root of a complex number || <math>\left( r (\cos x+i\sin x) \right)^{\frac{1}{n}}=r^{\frac{1}{n}}
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\cos\left(\frac{x+2 k \pi}{n}\right)
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+i\sin\left(\frac{x+2 k \pi}{n} \right), k=0,1,\ldots, n-1.\,</math>
 
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|}
 
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[[Collective_Table_of_Formulas|Back to Collective Table]]  
 
[[Collective_Table_of_Formulas|Back to Collective Table]]  
 
[[Category:Formulas]]
 
[[Category:Formulas]]
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[[Category:complex numbers]]

Latest revision as of 11:57, 24 February 2015


keywords: magnitude, conjugate, de Moivre, Euler

Collective Table of Formulas

Complex Number Identities and Formulas

click here for more formulas

Complex Number Identities and Formulas (info)
Basic Definitions
imaginary number $ i=\sqrt{-1} \ $
electrical engineers' imaginary number $ j=\sqrt{-1}\ $
(info) conjugate of a complex number $ \text{if}\ z=a+ib,\ \text{for}\ a,\ b \in {\mathbb R},\ \text{then} \ \bar{z}=a-ib $
(info) magnitude of a complex number $ \| z \| = \sqrt{ z \bar{z} } $
(info) magnitude of a complex number $ \| z \| = \sqrt{\left(Re(z)\right)^2+\left(Im(z)\right)^2} $
(info) magnitude of a complex number $ \| a+ib \| = \sqrt{a^2+b^2},\ \text{for}\ a,b\in {\mathbb R} $
(info) magnitude of a complex number $ \| r e^{i \theta} \| = r,\ \text{for}\ r,\theta\in {\mathbb R} $
Complex Number Operations
addition $ (a+ib)+(c+id)=(a+c) + i (b+d) \ $
multiplication $ (a+ib) (c+id)=(ac-bd) + i (ad+bc) \ $
multiplication in polar form $ \left( r_1 (\cos \theta_1 + i \sin \theta_1) \right) \left( r_2 (\cos \theta_2 + i \sin \theta_2) \right)= r_1 r_2 \left( \cos (\theta_1+\theta_2)+i \sin (\theta_1-\theta_2) \right)\ $
division $ \frac{a+ib} {c+id}=\frac{ac+bd} {c^2+d^2}+ i \frac{bc-ad} {c^2+d^2} \ $
division in polar form $ \frac{ r_1 (\cos \theta_1 + i \sin \theta_1)}{ r_2 (\cos \theta_2 + i \sin \theta_2) }= \frac{r_1}{ r_2} \left( \cos (\theta_1-\theta_2)+i \sin (\theta_1+\theta_2) \right)\ $
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. \ $
Euler's Formula and Related Equalities (info)
(info) Euler's formula $ e^{iw_0t}=\cos w_0t+i\sin w_0t \ $
A really cute formula $ e^{i\pi}=-1 \ $
Cosine function in terms of complex exponentials $ \cos\theta=\frac{e^{i\theta}+e^{-i\theta}}{2} $
Sine function in terms of complex exponentials $ \sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2i} $
Other Formulas
De Moivre's theorem $ \left(\cos x+i\sin x\right)^n=\cos\left(nx\right)+i\sin\left(nx\right).\, $
Root of a complex number $ \left( r (\cos x+i\sin x) \right)^{\frac{1}{n}}=r^{\frac{1}{n}} \cos\left(\frac{x+2 k \pi}{n}\right) +i\sin\left(\frac{x+2 k \pi}{n} \right), k=0,1,\ldots, n-1.\, $

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Abstract algebra continues the conceptual developments of linear algebra, on an even grander scale.

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