Modulus-Argument Form of a Complex Number

The modulus-argument form is one of the ways to represent a complex number. In this form, a complex number is defined by its magnitude (modulus) and direction (argument).
A complex number is fundamentally described as z = x + yi, where ‘x’ is the real part, ‘y’ is the imaginary part, and ‘i’ is the square root of -1.

The modulus of a complex number z = x + yi is denoted as

and is found by taking the square root of the sum of the squares of ‘x’ and ‘y’. Mathematically, it’s represented as **

= √(x² + y²)**.

Modulus indicates the distance of the complex number from the origin on the Argand diagram and is always a positive quantity.

The argument of a complex number z = x + yi is usually denoted by ‘arg(z)’ and represents the angle formed by the line joining the point (x, y) to the origin, with respect to the positive real axis.
If ‘θ’ is the argument of z, then tan(θ) = y / x. To find θ, use the trigonometric function, arctan or inverse tan i.e., θ = arctan (y / x).
Be aware of the quadrants. For example, if the complex number lies in the second or third quadrant, add π (or 180°) to your arctan result to obtain the correct argument.

The modulus-argument form or polar form of a complex number is z = r(cos θ + i sin θ), where ‘r’ is the modulus of ‘z’ and ‘θ’ is the argument of ‘z’.
This form makes multiplication and division of complex numbers simpler, and aids in understanding geometric transformations such as rotation.

An Argand diagram is a plot on a plane consisting of a horizontal real axis and a vertical imaginary axis. It’s used to graph complex numbers as points in the plane.
Complex numbers in the modulus-argument form can be represented graphically on the Argand diagram with the point (x, y) corresponding to the complex number x + yi, the distance from the origin representing the modulus, and the angle from the positive real axis representing the argument.