Use the modulus-argument form of a complex number

The modulus-argument form is also called the polar form of a complex number, given as z = r(cos θ + i sin θ), where r is the modulus of z, and θ is the argument of z.

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

and it signifies the magnitude of z, given by the formula

= √(x^2 + y^2).

The argument of a complex number z = x + iy denotes its angle in the complex plane and is often denoted as arg(z). The argument can be calculated using the formula arg(z) = tan^(-1)(y/x), where x is the real part and y is the imaginary part of z.
To find the argument of z, it is essential to consider the sign of the real and imaginary parts. This is because there are two angles that have the same tangent in the range 0 to 360 degrees.

The modulus-argument form of a complex number is useful for multiplication and division of complex numbers as it makes the operations simpler.
When multiplying two complex numbers in modulus-argument form, the resultant modulus is the product of their moduli, and the resultant argument is the sum of their arguments.
For dividing two complex numbers in modulus-argument form, the modulus of the answer is the modulus of the numerator divided by the modulus of the denominator, and the argument is the argument of the numerator minus the argument of the denominator.
The modulus-argument form also aids in raising a complex number to a power. If z = r(cos θ + i sin θ) is raised to the power of n, the modulus of the result is r^n, and the argument is nθ.

The modulus-argument form of a complex number gives insightful geometric interpretation. It explains the complex number in terms of its magnitude and direction in the complex plane.
This form is fundamental for deriving and understanding De Moivre’s theorem, used to calculate powers and roots of complex numbers.
The modulus-argument form also underpins the development of Euler’s formula, which creates a deep connection between complex analysis, trigonometry, and exponential functions.
It provides the groundwork for Fourier series and transformations, vital tools in solving differential equations, data analysis, and signal processing.