Modulus-Amplitude form (Polar form) and Euler form (exponential form)

Modulus-Amplitude form (Polar form):

The modulus-amplitude form, also known as the polar form, of a complex number is given by z = r(cos θ + i sin θ), where r is the modulus and θ is the amplitude.
The modulus and amplitude can be calculated using the formulas r = |z| = √(x² + y²) and θ = arg(z) = tan⁻¹(y/x), where x and y are the real and imaginary parts of the complex number.
The polar form of a complex number is useful for performing operations such as multiplication and division, which are simpler in polar form than in rectangular form.
The polar form of a complex number is also useful for representing complex numbers graphically in the complex plane.
The polar form of a complex number can be converted to rectangular form using the formulas x = r cos θ and y = r sin θ, while the rectangular form can be converted to polar form using the formulas r = √(x² + y²) and θ = tan⁻¹(y/x).

The Euler form, also known as the exponential form, of a complex number is given by z = re^(iθ), where r is the modulus and θ is the amplitude.
The exponential form of a complex number can be obtained from its polar form using the formula e^(iθ) = cos θ + i sin θ, known as Euler's formula.
The exponential form of a complex number is useful for performing operations such as powers and roots, which are simpler in exponential form than in rectangular or polar form.
The exponential form of a complex number is also useful for representing complex numbers graphically in the complex plane, as it provides a natural way to interpret rotations and magnifications.
The exponential form of a complex number can be converted to polar form using the formulas r = |z| = e^(ln(r)) and θ = arg(z) = Im(ln(z)/i), while the polar form can be converted to exponential form using the formula z = re^(iθ).