# Operations on Complex Numbers

##### Let      , ${y}_{1}$ ) =$\fn_cm&space;x_1+iy_1$

$\fn_cm&space;Z_2=(x_2,y_2)=x_2+iy_2$

##### Addition:  The sum of two complex numbers is

$\fn_cm&space;Z_1+Z_2=(x_1,y_1)+(x_2,y_2)$

$\fn_cm&space;=(x_1+x_2,y_1+y_2)$

$\fn_cm&space;=(x_1+x_2)+i(y_1+y_2)$

##### Subtraction:  The difference of two complex numbers is

$\fn_cm&space;Z_1-Z_2=(x_1,y_1)-(x_2,y_2)$

$\fn_cm&space;=(x_1-x_2),(y_1-y_2)$

$\fn_cm&space;=(x_1-x_2)+i(y_1-y_2)$

##### Multiplication:  The product of two complex numbers is

$\fn_cm&space;Z_1&space;\times&space;Z_2=&space;(x_1,y_1)&space;\times&space;(x_2,y_2)$

=  $\fn_cm&space;(x_1+iy_1)&space;\times&space;(x_2+iy_2)$

$\fn_cm&space;=(x_1x_2-y_1y_2,x_1y_2+x_2y_1)$

$\fn_cm&space;=(x_1x_2-y_1y_2)+i(x_1y_2+x_2y_1)$

##### Division:   The Quotient of two complex Numbers is

$\fn_cm&space;\frac{Z_1}{Z_2}=\frac{x_1+iy_1}{x_2+iy_2}=\frac{(x_1,y_1)}{(x_2,y_2)}.\frac{(x_2,-y_2)}{(x_2,-y_2)}$

=  $\fn_cm&space;\fn_cm&space;(\frac{x_1x_2+y_1y_2}{x_2^2+y_2^2},\frac{x_2y_1-x_1y_2}{x_2^2+y_2^2})$

$\fn_cm&space;\fn_cm&space;\frac{(x_1x_2+y_1y_2)+i(x_2y_1-x_1y_2)}{x_2^2+y_2^2}$

##### Remark: A set of complex numbers form a field.