**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 = √(x^{2} + y^{2}) =|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.