Polar Form of Complex Numbers
2026-02-28 12:55 Diff
https://brightchamps.com/en-us/math/numbers/polar-form-of-complex-numbers

The complex plane is a two-dimensional plane, with the horizontal axis representing the real part and the vertical axis representing the imaginary part of the complex number. Follow these steps for the graphical representation of complex numbers in polar form. 

Step 1: Find the value of the modulus
The value of the modulus can be calculated using the formula, \(r = \sqrt { a^2 + b^2}\).

Step 2: Find the value of θ
The value of θ is calculated using the formula, θ = tan-1(b/a)

Step 3: Find the polar form of the complex number
Now we express the value of r and θ in z = r (cosθ + i sinθ)

Step 4: Graphical representation
In the graph, the x-axis represents the real part and the y-axis represents the imaginary part. Mark the center (0, 0), in the direction of θ, move a distance r (modulus) from the origin at an angle θ, and mark the point. Then connect the point with the center.  
 

Example:

Graphically represent z = 3 + 3i

Find the value of modulus, \(r = \sqrt { a^2 + b^2} \)

Here, \(r = \sqrt { 3^2 + 3^2} \)

\(=\sqrt{ 9 +9} = {\sqrt {18}} = 3 \sqrt 2 \)

Finding the value of θ 

\(\theta = \tan^{-1}\left(\frac{b}{a}\right) \)

\(\tan^{-1}\left(\frac{3}{3}\right) = \tan^{-1}(1) = 45^\circ \)


Quadrant check: If the complex number has a negative part, the angle would be adjusted depending on the quadrant. Since both a and b are positive here, we don't need to adjust the angle. 
In polar form, z can be expressed as
 

\(z = 3\sqrt{2} (\cos 45^\circ + i \sin 45^\circ) \)

The argument θ of a complex number determines its direction in the complex plane. It's essential to adjust θ based on the quadrant in which the complex number lies:

Quadrant I (x > 0, y > 0): \(\theta = \tan^{-1}\left(\frac{y}{x}\right) \)


Quadrant II (x < 0, y > 0): \(\tan^{-1}\left(\frac{y}{x}\right) + \pi \)

Quadrant III (x < 0, y < 0): \(\tan^{-1}\left(\frac{y}{x}\right) - \pi \)

Quadrant IV (x > 0, y < 0): \(\tan^{-1}\left(\frac{y}{x}\right) \)