Arithmetic with 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 proformed on a complex number gives a complex number. This is not true for real numbers, for example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^{2}+1=0} is an entirly real expression but Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{z}=\overline{x+iy}=x-iy} and the length of a complex number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {||z||}^{2}=z.\bar{z}=x^{2}+y^{2}.}