Exponential form of complex numbers

The exponential form of a complex number is a different way to express a complex number, instead of the Cartesian form (a+bi).
The general expression for the exponential form of a complex number is re^(iθ), where r is the modulus and θ is the argument of the complex number.
Euler’s formula connects exponential and trigonometric functions: e^(iθ) = cos(θ) + i*sin(θ).
The modulus of a complex number in exponential form, re^(iθ), is r, a real number. It corresponds to the distance of the complex number from the origin in the complex plane. It can’t be negative.
The argument of a complex number is essentially the angle it makes with the positive real axis, and is denoted with the Greek letter, theta (θ).
An important fact to remember is that the argument is not unique - you can add any integer multiple of a 2π to it and still get a valid argument for a given complex number.
You can convert a complex number from Cartesian to exponential form and vice versa. To do so, you need to use Pythagoras’ Theorem and trigonometry.
In the multiplication and division of complex numbers, the exponential form can help simplify calculations. When multiplying complex numbers, you simply multiply the moduli and add the arguments. When dividing complex numbers, you divide the moduli and subtract the arguments.
De Moivre’s Theorem simplifies the process of raising complex numbers to an integer power. According to this theorem, [ r(cos θ + i sin θ) ] ^ n = r^n * (cos(nθ) + i sin(nθ) ) where n is an integer.
Exponential form is highly beneficial when dealing with roots of complex numbers. It simplifies the process of finding all possible roots.