Polar form of a Complex number

Let z = x+iy denote the point P(x,y) in the xy plane.

Then from the figure, we have

z = r{cosq+isinq} is called the polar form of the complex number.

x + iy (which is in the cartesian form) where   is called the modulus of the complex number z denoted by |z| and q = tan^-1 y/x is called the amplitude or argument of the complex number z denoted by amp(z) or arg(z).

The value of q is such that -p < q £  p, is called the principal value of the amplitude.

The general value of the amplitude is q + 2np, where n is positive or negative integer and is the principal value of the amplitude.

To find the principal value of (amplitude) from the equation cos q = x/r and sin q = y/r, the following table is useful.