Polar Form:

Any complex number z = a + bi can be thought of as a point (a,b) or vector in the complex plane in which the horizontal axis is the real part of z, and the vertical axis is an imaginary part of z. Therefore, a and b are the Cartesian or rectangular coordinates. As a vector can be presented by either it's rectangular or polar coordinates, a complex number can be given also by its polar coordinates r and θ, here r is the magnitude of the vector and θ is an angle.

To convert from the polar coordinates into the rectangular coordinates:

a = r cos θ

b = r sin θ

To convert from the rectangular into polar coordinates:

Therefore, a complex number z = a + bi can be written as r cos θ + (r sin θ )i, or z = r (cos θ  + i sin θ )

As e^{i θ} = cos θ + i sin θ , a complex number can also be written as z = re^{i θ}. In MATLAB, r can be found by using the abs function, and there is a special built-in function to find θ, known as angle.

>> z1 = 3 + 4i;

r = abs(z1)

r =

   5

>> theta = angle(z1)

theta =

   0.9273

>> r*exp(i*theta)

ans =

   3.0000 + 4.0000i