346
edits
Changes
Jump to navigation
Jump to search
m
Arithmetic with complex number (view source)
Revision as of 02:35, June 22, 2008
, 02:35, June 22, 2008There are more I will do them later
The motivating factor for the use of [[complex number]]s is that they form an [[algebraically closed]] [[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>.
For this we will consider two complex numbers, <math>z_{1}=x_{1}+iy_{1}</math> and <math>z_{2}=x_{1}+iy_{1}</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 cojugate''', <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]]
For this we will consider two complex numbers, <math>z_{1}=x_{1}+iy_{1}</math> and <math>z_{2}=x_{1}+iy_{1}</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 cojugate''', <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]]