Complex plane and Polar representation of complex number

In earlier classes of coordinate geometry we have already learnt how to plot point (x, y) in two dimensional plane.

But is it possible to plot complex number in any plane?
The answer is Yes, we can plot the complex number in a special two dimensional graph known as Argand or Complex Plane.

What is Argand or Complex plane?

Complex/Argand Plane is made up of two axis.
The x axis is for real numbers and y axis is known for imaginary numbers.
With the help of both these axis we can represent the complex number easily.

Let us consider below complex number as an example
z = 4 + 5i

Here
4 is a real number
5 is the imaginary number

Now let us plot this complex number in Argand plane

In the above figure you can easily see that:
–> the horizontal axis is known as the real axis
–> the vertical axis is known as imaginary axis

The point z is the representation of complex number z = 4+5i
Also, the distance OZ represent the Modulus of complex number |z|
where |z| = \sqrt{4^{2} +5^{2}}

Let us understand another example for our understanding.
Here we will consider two complex number z and its conjugate

Z= 6 + 8i
Q= 6 – 8i

Complex number Q is conjugate of number Z because
1. Real Number is same (i.e. 6)
2. Imaginary number is of opposite sign (8 & -8)

Now we will draw both the complex number on Argand Plane

==> You can see above in the figure how point Z and Q are plotted in the complex plane.
==> Both the points are conjugate, hence appear mirror image to each other.
==> Also distance of point Z & Q from point O is same [OZ = OQ]

Polar Representation of complex Number

If x & y coordinates of the complex plane is given we can represent the complex number as:
z = a + ib

But if the Modulus and angle of complex number is given then we need to represent the complex number in different manner

In the above argand plane, complex number P is shown.
Here two datas are provided to you
1. Modulus of complex number |z| = r
2. Argument of complex number: Angle made by complex number with x axis ( \theta )

These two data are called Polar Coordinate of point P. These point will help us to derive the required complex number.

Using simple trigonometry we know that;
x = r cos\theta
y = r sin\theta

After knowing the values of x and y, we can write complex number as
z = x + iy
z = r cos\theta+ i rsin\theta

z = r (cos\theta+ i sin\theta)
This is the formula you have to use in order to derive the complex number using Polar Coordinates of any point

Where r is the modulus of the complex number
r=\sqrt{x^{2} +y^{2}}

Representation of \theta

There are two ways in which \theta can be represented in polar coordinates
(a) 0 <\theta < 2\pi
(b) -\pi <\theta < \pi

In case (a), we will represent angle as following

In case (b), we will represent angle as following

Questions for Class 11 Math

(01) Represent the following complex number in polar form
z=1+i\sqrt{3}

Solution
z=1+i\sqrt{3}
This is rectangular representation of the complex number, we have to change it in polar form.

Let us represent this number into two dimensional plane

In order to represent the number in polar form we have to find the value of r and \theta\\\ \\ we\ know\ that\\ \\ r=\sqrt{x^{2} +y^{2}}\\ \\ r=\ \sqrt{1+3}\\ \\ r= 2\\\ \\ from\ the\ above\ graph\ we\ know\ that\\ \\ x=rcos\theta \ and\ y=rsin\theta \\ \\ 1=2cos\theta \ and\ \sqrt{3} =2\ sin\theta \\ \\ cos\theta =\frac{1}{2} \ \ and\ sin\theta =\frac{\sqrt{3}}{2}\\\ \\ hence\ we\ got\ \theta =60\ degree\

The polar representation of the above complex number is
z=\ 2\left( cos\frac{\pi }{3} +i\ sin\frac{\pi }{3}\right)

(02) Represent the following number into polar form
\frac{-16}{1+i\sqrt{3}}

Solution
\frac{-16}{1+i\sqrt{3}} \ \times \frac{1-i\sqrt{3}}{1-i\sqrt{3}}\\\ \\ \frac{-16+i\ 16\sqrt{3}}{1+3}\\\ \\ \frac{-16+i\ 16\sqrt{3}}{4}\\\ \\ -4+i\ 4\sqrt{3}

This is the rectangular representation of complex number. Let us draw the graphical representation in complex plane

The point z is the location of the complex number -4+i\ 4\sqrt{3}

The polar form of complex number is written as:
z=r( cos\theta +i\ sin\theta )\\\ \\ where\\ \\ r\ =\sqrt{x^{2} +y^{2}}\\\ \\ r\ =\sqrt{-4^{2} +\left( 4\sqrt{3}\right)^{2}}\\\ \\ r\ =\sqrt{16+48}\\\ \\ r=8\\\ \\ \\\ \\ we\ also\ know\ that:\\ \\ x\ =\ rcos\theta \\ \\ -4=8\ cos\theta \\ \\ cos\theta =\frac{-1}{2}\\ \\ \theta =\ 120\ degree\ =\frac{2\pi }{3}\\\ \\ \\\ \\ hence\ the\ polar\ form\ of\ complex\ number\ will\ be\\ \\ z\ =\ 8\left( cos\frac{2\pi }{3} \ +\ i\ sin\frac{2\pi }{3}\right)