Polar Form of Complex Numbers

A complex number in polar form is represented using its modulus and argument. The polar form of a complex number \(z = x + iy \) with coordinates (x, y) is \(z = r (\cos\theta + i \sin\theta) = r (\cos\theta + i \sin\theta) \), where r is the modulus and θ is the argument. 
 

• The modulus of a complex number is the distance from the origin in the complex plane. It represents the length of the vector from (0, 0) to (a, b).
  
  Using the Pythagorean theorem, the formula to find modulus is r = |z| = \(\sqrt { a^2 + b^2} \). 
   

• The argument is the angle θ, measured counterclockwise from the positive real axis, to the vector representing the complex number in the complex plane.
  
  It can be calculated using the formula \(\theta = \tan^{-1}\left(\frac{b}{a}\right) \) gives the principal value; adjust based on quadrant for θ in (-π, π). 

Euler’s formula is a simpler way to write the polar form, and it is the link between the exponential and trigonometric functions.

Euler’s formula states that:
 

\(e^{i\theta} = \cos\theta + i \sin\theta \)
 

where e is the base of the natural logarithm, i is the imaginary unit, and θ is the angle in radians. 
 

• The complex number in polar form is written as \(z = r (\cos\theta + i \sin\theta) \)

• Therefore, the polar form of a complex number is given by \(z = re^{i\theta}, \text{ as } e^{i\theta} = \cos\theta + i \sin\theta \)

To convert the complex number from rectangular form to polar form, we use the formulas \(r = \sqrt{a^2 + b^2}, \quad \theta = \tan^{-1}\left(\frac{b}{a}\right) \). To convert, follow these steps. 

Step 1: Calculate the modulus: \(r = \sqrt {(a² + b²)} \)

Step 2: Find the argument θ using \(\theta = \tan^{-1}\left(\frac{b}{a}\right) \)

Step 3: Express it in the form \(r (\cos\theta + i \sin\theta) \)

For example, to convert \(z = 1 + \sqrt{3}i \) in polar form, where a = 1 and b = √3

Finding the value of r

\(r = \sqrt {(a² + b²)} = \sqrt {1^2 + {(\sqrt {3})^2}} \)

= \(\sqrt { 1+3} = \sqrt 4 = 2 \)

Finding the value of θ:

\(\theta = \tan^{-1}\left(\frac{\sqrt{3}}{1}\right) = \tan^{-1}(\sqrt{3}) \)

So, 

In polar form, it can be expressed as \(z = r (\cos\theta + i \sin\theta) \)

\(z = 2 (\cos 60^\circ + i \sin 60^\circ) \)

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.  

For 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): \(\tan^{-1}\left(\frac{y}{x}\right) \)

Polar form is the efficient method for multiplying and dividing complex numbers. It is used in the fields of engineering, physics, and signal processing. 

Multiplication in Polar Form 

In polar form to multiply complex numbers, we first multiply the moduli and add the arguments.

For example,

\(z_1 = r_1 (\cos\theta_1 + i \sin\theta_1) \) and 

\(z_2 = r_2 (\cos\theta_2 + i \sin\theta_2) \)

\(z_1 \times z_2 = r_1 \times r_2 \big(\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)\big) \)

Division in Polar Form

To divide complex numbers in polar form we divide the moduli and subtract the arguments.
For example, \(z_1 = r_1 (\cos\theta_1 + i \sin\theta_1) \quad \text{and} \quad z_2 = r_2 (\cos\theta_2 + i \sin\theta_2) \)

\(\frac{z_1}{z_2} = \frac{r_1}{r_2} \big(\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)\big) \)

Powers and Roots in Polar Form

In this section, we will learn how to find the value of power and roots of complex numbers using De Moivre’s theorem. We will learn them in detail.

De Moivre’s theorem is a simple way to raise complex numbers to powers for the polar form. The formula is:

\(\big[r (\cos\theta + i \sin\theta)\big]^n = r^n \big(\cos n\theta + i \sin n\theta\big) \)
 

Finding the roots of a complex number: The nth root of a complex number uses the formula.  

\(Z_k ={ r^{1/n}\space [{cos \big( {{θ + 2kπ} \over n }\big ) + i sin \big ( {{θ + 2kπ} \over n } \big )}]}, k = 0, 1, ..., n-1\)
 

These tips and tricks given below will help you quickly convert between forms, handle angles correctly, and avoid common mistakes.

 

• Always calculate the magnitude first: Use \(r = \sqrt{x^2 + y^2} \)  before dealing with angles to avoid errors.
  
   
• Determine the correct quadrant for the angle: Adjust θ based on which quadrant the complex number lies in to get accurate polar form.
  
   
• Use addition/subtraction of angles for operations: For multiplication, add angles; for division, subtract angles, this simplifies calculations.
  
   
• Practice converting back and forth: Switching between rectangular and polar forms regularly strengthens understanding and speeds up problem-solving.
  
   
• Leverage Euler’s formula: Expressing z as \(e^{i\theta} \) can simplify powers and roots, making calculations faster and more elegant.

The polar form of complex numbers is used in fields like engineering, physics, and computer science. Some applications of the polar form of complex numbers are given below:
 

• In signal processing and communication, complex numbers are used to represent the signals in the frequency domain. It allows for efficient analysis and manipulation of signals. 
   

• In the navigation system, the polar form is used to specify the position or direction. It is used in ships, planes, or even GPS.
   

• In math, the polar form is used to perform basic operations such as multiplication and division. 
   

• To understand the geometric nature of complex numbers, the polar form is used. 
  
   
• In electrical engineering, the polar form of complex numbers is used to represent alternating current (AC) voltages and currents, making it easier to analyze circuits using phasers.