Operations on Complex Numbers

Let      Z1=( x1y1 ) =\fn_cm x_1+iy_1

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

Addition:  The sum of two complex numbers is 

                 \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)

Subtraction:  The difference of two complex numbers is

                  \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)

Multiplication:  The product of two complex numbers is

                \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)

Division:   The Quotient of two complex Numbers is

              \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}

Remark: A set of complex numbers form a field.


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