Respuesta :
Complex numbers comprise of two parts, namely: real and imaginary parts.
A typical complex number is given as
[tex]z=x+iy[/tex]Where x is the real part and y is the imaginary part
Now, how do we perform division for complex numbers?
Given two complex numbers Z₁ and Z₂, such that
[tex]\begin{gathered} z_1=x+iy \\ z_2=x-iy \\ \end{gathered}[/tex]Dividing Z₁ by Z₂ gives
[tex]\begin{gathered} \frac{z_1}{z_2}=\frac{x+iy}{x-iy} \\ \end{gathered}[/tex]We multiply the fraction by the conjugate of the denominator. The conjugate of Z₂ (denominator) is
[tex]x+iy[/tex]Thus, we have
[tex]\frac{z_1}{z_2}=\frac{x+iy}{x-iy}\times\frac{x+iy}{x+iy}=\frac{(x+iy)(x+iy)}{(x-iy)(x+iy)}=\frac{(x^2+ixy+ixy+i^2y^2)}{x^2-ixy+ixy-i^2y^2}[/tex]Collecting like terms, we have
[tex]\frac{x^2+2ixy+i^2y^2}{x^2-i^2y^2}[/tex]But i²= -1. Thus,
[tex]\frac{z_1}{z_2}=\frac{x^2+2ixy+i^2y^2}{x^2-i^2y^2}=\frac{x^2+2ixy-y^2}{x^2+y^2}=\frac{(x^2-y^2)+i2xy}{x^2+y^2}[/tex]