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Line 1: − {{merge|complex number}}+ − '''Arithmetic with complex number''' refers to simple operations mathematicians perform on a [[complex number]]. The motivating factor for the use of complex numbers is that they form an [[algebraically closed]] [[Field_(mathematics)|field]]. In simpler language any [[algebraic]] [[operation]] that is performed on a complex number gives a complex number. This is not true for [[real number]]s, for example, <math>x^{2}+1=0</math> is an entirly real expression but <math>x</math> cannot be a real number. The solution is <math>x=\pm i</math>. − − For this we will consider two complex numbers, <math>z_{1}=x_{1}+iy_{1}</math> and <math>z_{2}=x_{2}+iy_{2}</math>, where <math>x_{1},x_{2},y_{1},y_{2}\in\mathbb{R}</math> and <math>i^{2}=-1</math>. − − ====Addition==== − <math>z_{1}+z_{2}=(x_{1}+x_{2})+i(y_{1}+y_{2})</math> − − − ====Multiplication==== − <math>z_{1}z_{2}=(x_{1}+iy_{1})(x_{2}+iy_{2})=x_{1}x_{2}+ix_{1}y_{2}+iy_{1}x_{2}+i^{2}y_{1}y_{2}=(x_{1}x_{2}-y_{1}y_{2})+i(x_{1}y_{2}+y_{1}x_{2})</math> − − − ====Division==== − <math>\frac{z_{1}}{z_{2}}=\frac{x_{1}+iy_{1}}{x_{2}+iy_{2}}=\frac{x_{1}+iy_{1}}{x_{2}+iy_{2}}\cdot\frac{x_{2}-iy_{2}}{x_{2}-iy_{2}}=\frac{(x_{1}x_{2}+y_{1}y_{2})-i(x_{1}y_{2}-y_{1}x_{2})}{(x_{2})^{2}+(y_{2})^2}</math> − − − Here we see two important number. The [[complex conjugate]], <math>\bar{z}=\overline{x+iy}=x-iy</math> and the [[length]] of a complex number <math>{||z||}^{2}=z.\bar{z}=x^{2}+y^{2}.</math> − − [[Category:Complex analysis]]
Arithmetic with complex number (view source)
Revision as of 16:25, September 13, 2011
, 16:25, September 13, 2011Reverted edits by Shjt (talk) to last revision by PatrickD
#REDIRECT [[complex number]]