Argand Plane

Polar Representation of Complex Numbers

You would be already aware of the Cartesian or the XY – plane or the Z - plane, that is used to represent any given pair of a point graphically. It has two mutually perpendicular axes namely the x-axis and the y-axis. Hence any ordered pair (x, y) can be located on this XY – plane. This Cartesian plane is used for locating any given pair of the real numbered points.

Argand plane is also a kind of the XY – plane or the Cartesian plane, however, here you consider the x-axis as the real axis and the y-axis as the imaginary axis. And thus the argand plane is used for locating the complex numbers in a graphical manner. In this article, we will learn about the argand plane and the polar representation of complex numbers in detail.

Argand Diagram

Any complex number in the form of z = a + ib can be treated as the ordered pair (a, b) and can be represented accordingly on the argand plane.

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Consider the following example:

Consider z = 2 + 3i

You have Re(z) = 2 and Im(z) = 3

The distance of any given point from the origin is known as its Modulus and is denoted by |z| for any given complex number z.

If the complex number is purely a real number, that is, its imaginary part is zero or (b = 0), then on the argand plane, it is purely located on the real axis either towards the right or the left of the origin depending on the sign of the given number. This means that any given point on the real axis would be in the form of  z = a + i0.

While for the purely imaginary number, that is, its real part is zero or (a = 0), on the argand plane it is located on the imaginary axis either toward the upwards or downwards of the origin depending on the sign of the number’s imaginary part. This means that any point on the imaginary axis would be in the form of z = 0 + ib.

The amplitude of any given complex number refers to the angle that is made by the complex number on the argand plane from the plane’s positive real axis.

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Polar Form of Complex Numbers

Let us now learn about the polar form of complex numbers and how to represent a complex number in polar form.

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Consider that A represents the non-zero complex number x + iy. Here, OA is the directed line segment having the length r which makes an angle θ along the positive direction of the x-axis.

The ordered pair (r, θ) is referred to as the polar coordinates of the point A because point A is uniquely determined by (r, θ). The origin is known as the pole and the positive x-axis is referred to as the initial line.

Then you would get,

x = r cosθ

y = r sinθ

You can write this as

z = x + iy as z = r cosθ + ir sinθ = r(cosθ + i sinθ) 

This is referred to as the polar form of the given complex number.

Here, 

r = |z| = \[\sqrt{(x^{2}+y^{2})}\] is known as the modulus of z and θ is called the argument or the amplitude of z which is denoted as arg z

For any given non-zero complex number z, there is one value of θ in the interval (0, 2π).

In any given other interval having the length 2π, for example, consider the interval   -π < θ  ≤  π the value of the θ is known as the principal argument of z.

Solved Examples

Let us understand the argand plane and the polar form of complex numbers with some solved examples.

Example 1

Find the modulus and the amplitude for the given complex number z = -1- i.

Solution:

The modulus is given by |z| = \[\sqrt{(x^{2}+y^{2})}\].

For z=a+ib, find the value of the acute angle of which is \[\theta = tan^{-1}|\frac{y}{x}|\].

Then look for the values of (a, b).

If (a, b) lies in the first quadrant of the plane, Argument = θ.

If (a, b) lies in the second quadrant of the plane, Argument = π - θ .

If (a, b) lies in the third quadrant of the plane, Argument = -π + θ.

If (a, b) lies in the fourth quadrant of the plane, Argument = − θ.

Hence, |z| = \[\sqrt{2}\]

The acute angle is given by

 \[\theta = tan^{-1}|\frac{y}{x}|\].

Hence, θ = π/4

Hence, arg =  -3π/4

Example 2

Represent the equation z = √3 + i in the polar form.

Solution:

√3 = r sin  θ

= 1 = r sin θ

r = |z| = \[\sqrt{3+1}\]

= 2

sin θ =  \[\frac{1}{2}\] and cos θ =\[\frac{\sqrt{3}}{2}\]

This gives θ = π/6

Hence, the polar form of z is given by

Z = 2 ( cos (π/6) + i sin (π/6) )