This diagram is a graphical representation of a complex number. Using Cartesian axes, we can represent the complex number in space. While doing this, the real part of the complex number is plotted on the horizontal axis (x-axis), and the imaginary part is plotted on the vertical axis (y-axis). Thus, it can be visualized with coordinates a and b.

3.5.1 Polar Form of a Complex Number

A complex number can be expressed in polar form by using its distance from the origin, denoted as r, and the positive real axis angle θ. The polar form of a complex number is given as:

z = r (cos θ + i sin θ)

Here:

• r = √(a2 + b2) represents the magnitude of the complex number.
• θ = tan-1 (b/a) is the argument of the complex number, measured in radians, and the angle θ is determined in the complex plane with respect to the real axis.

The polar form is useful for operations like multiplication and division. The multiplication and division of complex numbers in polar form are easier and more straightforward. It will be helpful to examine the multiplication and division of complex numbers from the perspective of their arguments.

Let’s define the complex numbers and proceed with their operations:

z1 = 5 + 45° and z2 = 30°

Multiplication:
(5 × 3) (45° + 30°) = 15 ∠ 75°

Division:
(5 / 3) (45° - 30°) = 5 / 3 ∠ 15°