Polar Form of Complex Numbers

A complex number in polar form is represented using its modulus and argument. Whether you calculate it manually or use a polar form converter, the goal is to transform the number z = x + iy with coordinates (x, y) into the equation \(z = r(\cos \theta + i \sin \theta)\), where r is the modulus and \theta is the argument.

Examples
 

• \(z = 2(\cos 30^\circ + i \sin 30^\circ)\)
   
• \(z = 5\left(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}\right)\)
   
• \(z = \sqrt{2}(\cos 180^\circ + i \sin 180^\circ)\)
   
• \(z = 4\left(\cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2}\right)\)
   
• \(z = 10(\cos 300^\circ + i \sin 300^\circ)\)

• 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 understand polar form, we need to shift our perspective from a stationary (x, y) grid to a system based on distance and rotation.

1. Geometric Representation

The complex number appears visually as a vector in the complex plane.

• r (Modulus): The length of the line from the origin (the hypotenuse).
• \(\mathbf{\theta}\) (Argument): The angle of rotation measured from the positive x-axis.

2. Algebraic Representation

We use trig to turn the 'over and up' instructions (x, y) into 'distance and direction' (r, \(\theta\)).

We can substitute \(x = r \cos \theta\) and \(y = r \sin \theta\) into the equation z = x + iy.

\(z = (r \cos \theta) + i(r \sin \theta)\)

Which factors into the standard polar form:

\(z = r(\cos \theta + i \sin \theta)\)

Think of this as translating a location. When you convert from rectangular to polar coordinates, you are simply switching from a rectangular to a polar coordinate system. This process gives you the polar form of a complex number. To write this polar form of a complex number correctly, you need to figure out the distance (r) and the angle (\(\theta\)) from your original variables.

Step 1: Identify Coordinates

Determine the real (x) and imaginary (y) parts of your equation.

Step 2: Calculate the Modulus (r)

The distance from the origin can be calculated using the Pythagorean theorem:

\(r = \sqrt{x^2 + y^2}\)

Step 3: Find the Reference Angle (\alpha)

Calculate the acute angle by taking the inverse tangent of the absolute values.

\(\alpha = \tan^{-1}\left(\left|\frac{y}{x}\right|\right)\)

Step 4: Determine the True Argument (\theta)

Adjust the angle depending on the quadrant where (x, y) is located:
 

• Quadrant I (+, +): \(\theta = \alpha\)
• Quadrant II (-, +): \(\theta = 180^\circ - \alpha\ (or\ \pi - \alpha)\)
• Quadrant III (-, -): \(\theta = 180^\circ + \alpha\ (or\ \pi + \alpha)\)
• Quadrant IV (+, -): \(\theta = 360^\circ - \alpha\ (or\ 2\pi - \alpha)\)

Step 5: Write the Final Form

Insert your values into the standard polar equation:

\(z = r(\cos \theta + i \sin \theta)\)

Example: Convert z = 1 - i to polar form

1. Identify Coordinates:

x = 1, y = -1

2. Calculate Modulus (r):

\(r = \sqrt{1^2 + (-1)^2} = \sqrt{2}\)

3. Find Reference Angle:

\(\tan^{-1}\left(\left|\frac{-1}{1}\right|\right) = \tan^{-1}(1) = 45^\circ \text{ (or } \frac{\pi}{4})\)

4. Determine Argument (\theta):

Given that x is positive and y is negative (Quadrant IV),

\(\theta = 360^\circ - 45^\circ = 315^\circ \text{ (or } \frac{7\pi}{4})\)

5. Final Polar Form:

\(z = \sqrt{2}(\cos 315^\circ + i \sin 315^\circ)\)

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.  
 

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.

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.

Example:

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

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

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.  

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

Understanding how to map numbers on a spinning plane rather than a rigid grid can be challenging. Shifting perspective from x and y coordinates to distance and rotation is key to mastering this topic. To make the learning journey smoother, here are a few tips and tricks to help simplify the process.

• Visualize a Radar Screen: Instead of a standard grid, picture a radar screen or a clock face. This mental image helps learners grasp complex numbers' polar form naturally, seeing points as "how far from the center" and "which direction to point," rather than just coordinates.
   
• Walk the Path: Encourage physical visualization. Imagine walking three steps East and four steps North (in a rectangular pattern). Then, imagine pointing directly at that final spot and walking straight there. This physical comparison clarifies the definition of a complex number in polar form.
   
• Recall the Right Triangle: Remind students that the polar representation of complex numbers is essentially a hidden right triangle. The modulus (r) is simply the hypotenuse, and the argument (\(\theta\)) is the angle, connecting the concept back to the Pythagorean theorem they likely already trust.
   
• Do a "Quadrant Check" First: Before calculating, look at the signs. If x is negative and y is positive, the angle must be in the second quadrant. This quick estimate helps prevent common errors when converting the following complex number to its polar representation.
   
• Highlight the "Multiplication Superpower": Explain why we use polar form. Multiplying in rectangular form is messy algebra, but in polar form, you multiply the lengths and add the angles. Giving the method a clear purpose increases engagement.
   
• Use the 'CIS' Shortcut: Writing out the full \(r(\cos \theta + i \sin \theta)\) repeatedly can be intimidating and tedious. Distinct notation like \(r \, \text{cis} \, \theta\) (cosine plus i sine) can make the equations look cleaner and faster to write.
   
• Leverage Dynamic Tools: Static drawings can feel abstract. Using graphing software to drag a point around the plane and watching the radius and angle change in real-time provides immediate feedback on how the two forms relate.

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.