# 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.

Then,

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