Polar representation of a complex number
Let the point P represent the nonzero complex number z = x + iy. Let the directed line segment OP be of length r and θ be the angle which OP makes with the positive direction of x-axis (Fig 5.4). We may note that the point P is uniquely determined by the ordered pair of real numbers (r, θ), called the polar coordinates of the point P. We consider the origin as the pole and the positive direction of the x axis as the initial line.
We have, x = r cos θ, y = r sin θ and therefore, z = r (cos θ + i sin θ). The latter is said to be the polar form of the complex number. Here r = √(x2 + y2) =|z| is the modus of z and θ is called the argument (or amplitude) of z which is denoted by arg z. For any complex number z ≠ 0, there corresponds only one value of θ in 0 ≤ θ < 2π. However, any other interval of length 2π, for example – π < θ ≤ π, can be such an interval.We shall take the value of θ such that – π < θ ≤ π, called principal argument of z and is denoted by arg z, unless specified otherwise. (Figs. 5.5 and 5.6)
Example Represent the complex number z = 1 + i √3 in the polar form.