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, $x^{2}+1=0$ is an entirly real expression but $x$ cannot be a real number. The solution is $x=\pm i$.

For this we will consider two complex numbers, $z_{1}=x_{1}+iy_{1}$ and $z_{2}=x_{2}+iy_{2}$, where $x_{1},x_{2},y_{1},y_{2}\in \mathbb {R}$ and $i^{2}=-1$.

Addition

$z_{1}+z_{2}=(x_{1}+x_{2})+i(y_{1}+y_{2})$

Multiplication

$z_{1}z_{2}=(x_{1}+iy_{1})(x_{2}+iy_{2})=x_{1}x_{2}+ix_{1}y_{2}+iy_{1}x_{2}+i^{2}y_{1}y_{2}=(x_{1}x_{2}-y_{1}y_{2})+i(x_{1}y_{2}+y_{1}x_{2})$

Division

${\frac {z_{1}}{z_{2}}}={\frac {x_{1}+iy_{1}}{x_{2}+iy_{2}}}={\frac {x_{1}+iy_{1}}{x_{2}+iy_{2}}}\cdot {\frac {x_{2}-iy_{2}}{x_{2}-iy_{2}}}={\frac {(x_{1}x_{2}+y_{1}y_{2})-i(x_{1}y_{2}-y_{1}x_{2})}{(x_{2})^{2}+(y_{2})^{2}}}$

Here we see two important number. The complex conjugate, ${\bar {z}}={\overline {x+iy}}=x-iy$ and the length of a complex number ${||z||}^{2}=z.{\bar {z}}=x^{2}+y^{2}.$