How to divide by a complex number

When we divide a complex number $z_1$ by another one, say $z_2,$ we usually want to obtain the result in standard form. So how do we simplify a quotient

$\frac{z_1}{z_2} = \frac{x_1+i y_1}{x_2+i y_2}$

such that the result takes the form $x + i y$ ?

The answer is: We multiply the numerator and the denominator of the quotient by the complex conjugate of the denominator.

The complex conjugate $z^\star$ (sometimes written $\bar{z}$ of any complex number $z=x+i y$ is defined by a simple change of the sign in front of the imaginary part: $z^\star = x+i y$ . If we multiply a complex number by its own complex conjugate, we always obtain a real number:

$z \cdot z^\star = (x + i y)(x - i y)=x^2+ixy-ixy-i^2y^2=x^2+y^2$

This number is the square of the absolute value $|z|=\sqrt{x^2+y^2}$ (sometimes also called modulus or magnitude) of the complex number $z$ . The fact that this number is always real helps with the simplification of the above quotient. Multiplying the numerator and denominator with the complex conjugate of the denominator we obtain

$\frac{z_1}{z_2}=\frac{x_1+iy_1}{x_2+iy_2}\cdot \frac{x_2-iy_2}{x_2-iy_2}=\frac{(x_1+iy_1)(x_2-iy_2)}{x_2^2+y_2^2}=\frac{x_1 x_2 +y_1 y_2}{x_2^2+y_2^2}+i\frac{x_2 y_1 - x_1 y_2 }{x_2^2+y_2^2}.$

This result is now in standard form $x+iy$ where $\frac{x_1 x_2 +y_1 y_2}{x_2^2+y_2^2}$ is the real part and $\frac{x_2 y_1 - x_1 y_2 }{x_2^2+y_2^2}$ is the imaginary part.

Using exponential form and polar form

It is worth noticing that dividing by a complex number is easier if we write the numbers in exponential form or polar form. We recall that every complex number $z=x+iy$ can be written in exponential form $z=|z|e^{i\varphi}$ where $|z|$ is the absolute value and $\varphi$ is the argument of the complex number. If we plot the number $z$ in the complex plane with coordinates $x$ and $y$ , and imagine a straight line between the origin and $z$ , we obtain $\varphi$ as the angle of this line with the positive real axis.