Changes

The motivating factor for the use of [[complex number]]s is that they form an [[algebraically closed]] [[Field_(mathematics)|field]]. In simpler language any algebraic operation that is proformed 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>.

'''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>.

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>.

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>

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]]

[[Category:Complex analysis]]