Operations on Complex Numbers

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

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

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

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

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

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

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

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

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

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

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

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

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

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

PAKMATH Knowledge Learning Point