104 LearnersLast updated on July 5th, 2025
Polar Form of Complex Numbers
In the polar form of a complex number, it is the distance from the origin and the angle it makes with the positive real axis. In rectangular form, a complex number is written as z = a + bi. The modulus and argument are used to represent the complex number in polar form. It is represented in the equation of z = r(cos θ + i sin θ), where r is the modulus and θ is the argument. In this article, we will discuss the polar form, its modulus and argument, and their applications.
What is a Polar Form?
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 (cosθ+ i r sinθ) = r (cosθ + i sinθ), where r is the modulus and θ is the argument.
The Formula for Modulus and 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 (r) = |z| = √(a2 + b2).
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 θ = tan-1 (b/a), in the range (-π/2, π/2). The argument depends on the quadrant and can range from -π to π, based on the quadrant adjustment.
Euler’s Formula Connection
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 = eiθ = cosθ + i sinθ, 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 (cosθ + i sinθ),
Therefore, the polar form of a complex number is given by z = reiθ, as eiθ = cosθ + i sinθ.
Converting from Rectangular Form to Polar Form of a Complex Number.
To convert the complex number from rectangular form to polar form, we use the formulas r = √(a2 + b2), θ = tan-1 (b/a). To convert, follow these steps.
Step 1: Calculate the modulus: r = √a2 + b2
Step 2: Find the argument θ
Step 3: Express it in the form r(cosθ + i sinθ)
For example, to convert z = 1 + √3i in polar form, where a = 1 and b = √3
Finding the value of r
r = √a2 + b2 = √(12 + (√3)2)
= √1 + 3 = √4 = 2
Finding the value of θ
θ = tan-1(√3/1) = tan-1(√3)
So,
In polar form, it can be expressed as z = r(cosθ + i sinθ)
z = 2(cos 60° + i sin 60°)
Graphical Representation of a Complex Number in Polar Form
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 = √a2 + b².
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 = √a² + b²
Here, r = √3² + 3²
= √9 + 9 = √18 = 3√2
Finding the value of θ
θ = tan-1(b/a)
= tan-1(3/3) = tan-1(1) = 45°
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√2(cos 45° + i sin 45°)
Multiplication and Division in Polar Form
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, z1 = r1(cosθ1 + i sinθ1) and z2 = r2(cosθ2 + i sinθ2)
z1 × z2 = r1 × r2(cos(θ1 + θ2) + i sin(θ1 + θ2))
Division in Polar Form
To divide complex numbers in polar form we divide the moduli and subtract the arguments. For example z1 = r1(cosθ1 + i sinθ1) and z2 = r2(cosθ2 + i sinθ2)
z1/z2 = r1/r2((cos(θ1 - θ2) + i sin(θ1 - θ2))
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. Let's 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 [r (cosθ + i sinθ)]n = rn (cos nθ + i sin nθ).
Finding the roots of a complex number: The nth root of a complex number uses the formula.
Zk = r1/n (cos θ + 2kπ/n + i sin θ + 2kπ/n), where k = 0, 1, 2, … n-1.
Real-world Applications of Polar Form of Complex Numbers
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.
Common Mistakes and How to Avoid Them in Polar Form of Complex Numbers
Students usually make errors when working with complex numbers in polar form. So let’s learn about some mistakes and the ways to avoid them.
Mistake 1
Errors in conversion from rectangular to polar form
When converting a complex number to polar form, students tend to make errors, especially when finding the value of θ. To avoid such errors, students should remember that first, find the modulus, and then the argument.
Mistake 2
Incorrect calculation of modulus
A common mistake students make when finding the value of the modulus is adding the components directly. Students should remember that r = √a2 + b2.
Mistake 3
Confusing the signs in argument calculations
Confusing the signs in the argument calculations is common among students. So always check in which quadrant the complex number is.
Mistake 4
Misapplying Euler’s formula
When using Euler’s formula, students make errors by confusing an incorrect exponent, that is eiθ with eiθ. Using this can lead to errors, so students should always remember that Euler’s formula is eiθ = cosθ + i sinθ.
Mistake 5
Subtracting instead of adding in the multiplication of polar form
When multiplying polar form, students sometimes multiply the moduli and then subtract the arguments, which is wrong. In multiplication, we first multiply the moduli and add the arguments.
Solved Examples of Polar Form of Complex Numbers
Problem 1
Convert 1 + i to polar form
In polar form 1 + i is √2 (cosπ/4 + i sinπ/4)
Explanation
To convert a rectangular form to a polar form, we first find the value of r and θ.
r = √a2 + b2
Here, a = 1 and b = 1
Therefore, r = √12 + 12
= √2
The argument θ = tan-1(b/a)
= tan-1(1) = π/4
In polar form, it can be expressed as √2 (cos π/4 + i sin π/4)
Problem 2
Convert 3(cosπ/6 + i sinπ/6) to rectangular form
In rectangular form 3(cosπ/6 + i sinπ/6) can be expressed as 3(√3/2) + i(3/2)
Explanation
The value of cosπ/6 = √3/2
The value of sinπ/6 = 1/2
Multiplying by 3:
3 (cosπ/6) = 3 × √3/2
=3√3/2
3 (sinπ/6) = 3 × ½
=3/2
So, in rectangular form, it can be expressed as 3(√3/2) + i(3/2)
Problem 3
Find the product of 3√2 (cosπ/4 + i sinπ/4) and √2 (cos(-π/4) + i sin(-π/4)
The product of 3√2 (cosπ/4 + i sinπ/4) and √2 (cos(-π/4) + i sin(-π/4) is 6.
Explanation
When multiplying the polar form, we first multiply the moduli and then add the arguments.
Multiplying the moduli: 3√2 × √2 = 3 ×2 = 6
Adding the argument: π/4 + (-π/4) = 0
That is 6 (cos0 + i sin0) = 6 (1 + 0i)
= 6
Problem 4
Find the quotient of √2 (cosπ/4 + i sinπ/4) by 2 (cosπ/6 + i sinπ/6)
The quotient of √2 (cosπ/4 + i sinπ/4) by 2 (cosπ/6 + i sinπ/6) is √2/2 (cosπ/12 + i sinπ/12)
Explanation
When dividing the polar form, first we divide the moduli and then subtract the arguments.
Dividing the moduli: √2/2
Subtracting the argument: π/4 - π/6
= 3π/12 - 2π/12
= π/12
So, the quotient is √2/2 (cosπ/12 + i sinπ/12)
Problem 5
Convert -6 to polar form.
-6 in polar form can be expressed as 6 (cosπ + i sinπ)
Explanation
The number -6 lies on the negative real axis
So, r = |-6| = 6
θ = π, as the number is on the negative real axis
So, in polar form, it can be expressed as 6(cosπ + i sinπ)
FAQs on Polar Form of Complex Numbers
1.What is the polar form of a complex number?
2.What is the rectangular form of a complex number?
3.What is Euler’s formula?
4.How to multiply two complex numbers?
5.What is r in polar form?
6.How can children in Singapore use numbers in everyday life to understand Polar Form of Complex Numbers ?
7.What are some fun ways kids in Singapore can practice Polar Form of Complex Numbers with numbers?
8.What role do numbers and Polar Form of Complex Numbers play in helping children in Singapore develop problem-solving skills?
9.How can families in Singapore create number-rich environments to improve Polar Form of Complex Numbers skills?
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Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
