Modulus-argument form of complex numbers

The modulus-argument form, also known as the polar form, represents complex numbers in terms of their magnitude and direction rather than their components.

The modulus of a complex number, denoted

, is the distance from the origin to the point representing the complex number in the complex plane. It is calculated using the Pythagorean theorem as

= sqrt(a^2 + b^2), where z = a + bi.

The argument of a complex number, represented by arg(z), is the angle that the line drawn from the origin to the point makes with the positive x-axis.
A complex number can hence be written in the form z = r(cos θ + i sin θ), where r is the modulus and θ is the argument.
The argument of a complex number is usually expressed in radians for mathematical convenience, but it may also be expressed in degrees.
When the argument is not between -π and π, it can be adjusted by adding or subtracting multiples of 2π to bring it into this range.
Modulus-argument form is particularly useful when multiplying or dividing complex numbers. When multiplying, the moduli multiply, and the arguments add. When dividing, the moduli divide, and the arguments subtract.
De Moivre’s Theorem links the modulus-argument and Cartesian forms. It states [(cos θ + i sin θ) ^n ] = cos nθ + i sin nθ.
Understanding the conversion between Cartesian and modulus-argument forms and applying de Moivre’s theorem accurately are necessary skills when working with complex numbers in this form.