Arithmetic with 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. In simpler language any algebraic operation that is performed on a complex number gives a complex number. This is not true for real numbers, 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>