Use and Conversion of the Cartesian form

A complex number is written in Cartesian form as z = x + iy, where x is the real part, y is the imaginary part, and i denotes the square root of -1.
This representation, also known as the rectangular form, allows for intuitive operations, formulating real and imaginary components separately.
Using real and imaginary components, Cartesian form straightforwardly handles addition and subtraction of complex numbers.

The polar form of a complex number uses the magnitude (r) and the angle (θ) of the complex number in the complex plane: z = r(cos θ + i sin θ).
To convert from the Cartesian form to polar form, use the equations: r = sqrt(x^2 + y^2) and θ = tan^-1(y/x).
The polar form offers a practical approach while multiplying or dividing complex numbers as it turns into simple multiplication or division of magnitude and addition or subtraction of angles.
Note that computer programming languages tend to use the function atan2(y, x) rather than atan(y/x) because atan2 correctly deals with x being zero and gives answers in the correct quadrant.

Euler’s form is a concise representation of the polar form: z = re^(iθ).
To convert from Cartesian form to Euler’s form, utilise the same conversions as for polar form: r = sqrt(x^2 + y^2) and θ = tan^-1(y/x).
Euler’s form is particularly useful in solving problems involving powers and roots of complex numbers, as exponentiation and root extraction become more straightforward operations.

The complex conjugate of a complex number in Cartesian form gives significant insights.
The complex conjugate of z = x + iy is z̅ = x - iy.
Multiplying a complex number by its complex conjugate simplifies the calculation and results in a real number equal to the square of its magnitude: z*z̅ = x^2 + y^2 = r^2.

A complex number in Cartesian form can be represented as a 2x2 real matrix, which can be helpful in visualising certain types of transformations.
The complex number z = x + iy has the matrix form [x -y; y x].