Argand Plane & Polar Representation of Complex Number

Argand Plane

We all know that the pair of numbers (x,y) can be represented on an XY-plane, where x is called abscissa and y is called the ordinate. Similarly, we can represent complex numbers also on a plane called Argand plane or Complex plane. Similar to the X-axis and Y-axis in two dimensional geometry, there are two axes in Argand plane.

  • The axis which is horizontal is called Real axis
  • The axis which is vertical is called Imaginary axis

The complex number x+iy which corresponds to the ordered pair(x,y)is represented geometrically as the unique point (x,y) in an XY-plane.

For example,

The complex number, 2+3i corresponds to the ordered pair (2,3) geometrically.

Similarly, -3+2i corresponds to the ordered pair (-3,2).

  • Complex numbers in the form 0+ai, where a is any real number will lie on imaginary axis.
  • Complex numbers in the form a+0i, where ais any real number will lie on real axis.

It is obvious that the modulus of complex number x+iy, \( \sqrt{x^2+y^2} \) is the distance between the origin (0,0) and the point (x,y).

  • The conjugate of z = x+iy is z = x-iy which is represented as (x,-y) in the Argand plane. Point (x,-y) is the mirror image of the point (x,y)across the real axis in Argand plane.

Example: Find the distance between the complex number z = 3 – 4i and the origin in Argand plane.

Distance between the origin and z= 3 – 4i is equal to the modulus of z.

|z|=\( \sqrt{3^2+(-4)^2 }\)= \( \sqrt{9+16} \) = \(\sqrt{25}\)=5 units

Polar representation of complex numbers:

Let A represent the non-zero complex number x + iy. OA is the Directed line segment of length r and makes an angle θ with the positive direction of X-axis.

Ordered pair (r,θ) is called as the polar coordinates of the point A since point A is uniquely determined by (r,θ).The origin is called the pole and the positive X-axis is called the initial line.


x = r cosθ

y = r sinθ

We can write z = x + iy as z = r cosθ + ir sinθ = r(cosθ + i sinθ), which is called the polar form of complex number.

  • Here, r = |z| = \( \sqrt{x^2+y^2}\) is modulus of z and θ is known as the argument or amplitude of z denoted as arg z
  • For any non-zero complex number z, there corresponds one value of θ, in the interval [0,2π)
  • In any other interval of length 2π, for example consider the interval -π < θ ≤ π, then the value of θ is called the principal argument of z.

Example: Represent z = \( \sqrt{3} \) + i in the polar form

\( \sqrt{3} \) = r cosθ

1=r sinθ

r = |z| = \( \sqrt{3 + 1} \) = 2

sin θ = \( \frac 12 \)

cos θ = \( \frac {\sqrt{3}}{2}\)

Which gives,

θ = \( \frac {\pi}{6} \)

Therefore, polar form of z is,

z = \( 2~(cos~\left(\frac {\pi}{6} \right)~+~i~sin~\left(\frac{\pi}{6}\right)) \)