How to Convert Complex Numbers to Polar Form with Formula and Solved Examples
The numbers that are expressed in the form of a+ib where ‘i’ is an imaginary number called iota and has the value of (√-1) are known as complex numbers. Let’s take, for example, 2 + 3i is a complex number, where 2 is known to be a real number and 3i is an imaginary number. Therefore, the combination of both the real number and the imaginary number is known as a complex number.
Examples of Complex Numbers and Real Numbers
Examples of real numbers - 2, -13, 0.89,√5, etc.
Examples of imaginary numbers are -4i, 1.2i, (√2)i, 3i/2, etc.
An imaginary number is usually represented by ‘i’ or ‘j’, which is equal to √-1. Therefore, the square of an imaginary number gives a negative value.
Complex numbers are used to represent periodic motions such as, alternating current, water waves, light waves, etc., which rely on the cosine or sine waves, etc.
Polar Form of a Complex Number
We can also represent any given complex number in its polar form. The form z equals a + ib is called the rectangular coordinate form of a complex number.
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The horizontal axis is said to be the real axis whereas the vertical axis is said to be the imaginary axis. We find the real components and complex components in terms of r and θ.
where r = length of the vector, θ = angle made with the real axis.
What is the Polar Form of any Complex Number?
The polar form of a complex number is one way to represent a complex number apart from the rectangular form. Usually, complex numbers can be represented, in the form of z equals x + iy where ‘i’ equals the imaginary number.
But in polar form, we represent complex numbers as the combination of modulus and argument.
What is Absolute Value?
The modulus of a complex number is also known as the absolute value. This polar form can be represented with the help of polar coordinates of real as well as imaginary numbers in the coordinate system.
Polar Form Formula of Complex Numbers
Let us consider the coordinates (x, y) as the coordinates of complex numbers x+iy. We can represent it in a cartesian plane, as given below:
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Here in the above diagram, the horizontal axis denotes the real axis, and the vertical axis denotes the imaginary axis. The real components and complex components of coordinates are found in terms of r and θ where r is the length of the vector, and θ is the angle made with the real axis.
Using the Pythagorean Theorem, we can write;
r2 equals x2 + y2
From trigonometric ratios, we know that;
Cos θ equals Adjacent side of the angle θ / Hypotenuse, can be written as Cos θ = x/r
Also, sin θ = Opposite side of the angle θ /Hypotenuse, can be written as Sin θ =y/r
Multiplying each side by r :
r cosθ = x and r sinθ = y
The rectangular form of any given complex number can be denoted by:
z equals x + iy
Now, substitute the values of variables x and y.
z equals x + iy = r (cosθ + i rsinθ)
In the case of any given complex number,
r signifies the absolute value or the modulus, angle θ as the argument of the complex number.
Absolute Value of a Complex Number
Given, z equals x + yi, a complex number, the absolute value of z can be defined as |z|= x2 + y2.
It is known to be the distance from the origin to the point (x, y).
Notice that the absolute value of a real number gives us the distance of the number from 0 which is the origin, while the absolute value of a complex number gives the distance of the number from the origin, with coordinates (0, 0).
Adding Complex Numbers in Polar Form
Suppose we have any two given two complex numbers, one in a rectangular form and one in polar form. Now, we need to add these two numbers and represent them in the polar form again.
Let 7∠50°, 3 + 5i are the two complex numbers.
First, we will convert 7∠50° into a rectangular form.
7∠50° equals x + iy
Hence, x = 7 cos 50° = 4.5
y = 7 sin 50° equals 5.36
So, 7∠50° equals 4.5 + i 5.36
Therefore, when we add any two given complex numbers, we get;
(3 + i5)+ (4.5 + i 5.36) = 7.5 + I 10.36
Again, to convert the resulting complex number in polar form, we need to find the modulus as well as the argument of the number. Hence,
Modulus is equal to;
r equals |z|=√(x2 + y2)
r equals √(7.52 + 10.362)
r equals 12.79
And the argument is equal to;
θ = tan-1(y/x)
θ = tan-1(10.36/7.5)
θ = 54.1°
Therefore, the required complex number is 12.79∠54.1°.
Solved Questions
Question 1) Add the complex numbers (5 + 7i) + (2.0 + 2.36i).
Solution) On adding the two complex numbers (5 + 2.0) + (7 + 2.36)i
7.0 + 9.36i.
Question 2) Add the complex numbers (2 + 5i) + (2 + 5i).
Solution) On adding the two complex numbers (2 + 2) + (5 + 5)i
4.0 + 10i.
FAQs on Polar Form of Complex Numbers Explained with Geometry
1. What is the polar form of a complex number?
The polar form of a complex number expresses a complex number using its magnitude and angle as z = r(cos θ + i sin θ) or z = r e^{iθ}. In this form:
- r is the modulus (distance from origin).
- θ is the argument (angle with positive x-axis).
2. How do you convert a complex number from rectangular form to polar form?
To convert z = a + bi into polar form, compute its modulus and argument, then write z = r(cos θ + i sin θ).
- Step 1: Find modulus r = √(a² + b²).
- Step 2: Find argument θ = tan⁻¹(b/a) (adjust quadrant if needed).
- Step 3: Substitute into polar form.
- r = √2
- θ = π/4
3. What is the formula for the modulus of a complex number?
The modulus of a complex number z = a + bi is |z| = √(a² + b²). It represents the distance from the origin to the point (a, b) in the Argand plane. For example, if z = 3 + 4i, then |z| = 5.
4. What is the argument of a complex number?
The argument of a complex number is the angle θ that the line joining the origin to the point (a, b) makes with the positive x-axis. It is calculated as θ = tan⁻¹(b/a), adjusted for the correct quadrant. The principal argument usually lies between −π and π.
5. How do you multiply complex numbers in polar form?
To multiply complex numbers in polar form, multiply the moduli and add the arguments. If z₁ = r₁e^{iθ₁} and z₂ = r₂e^{iθ₂}, then:
- z₁z₂ = r₁r₂ e^{i(θ₁ + θ₂)}
6. How do you divide complex numbers in polar form?
To divide complex numbers in polar form, divide the moduli and subtract the arguments. If z₁ = r₁e^{iθ₁} and z₂ = r₂e^{iθ₂}, then:
- z₁ / z₂ = (r₁ / r₂) e^{i(θ₁ − θ₂)}
7. What is Euler’s form of a complex number?
Euler’s form of a complex number is z = r e^{iθ}, derived from Euler’s formula e^{iθ} = cos θ + i sin θ. This exponential form is equivalent to trigonometric polar form and is widely used in advanced mathematics, engineering, and signal processing.
8. How do you find the polar form of a purely imaginary number?
To find the polar form of a purely imaginary number z = bi, compute its modulus and argument. For z = bi:
- r = |b|
- θ = π/2 if b > 0
- θ = −π/2 if b < 0
9. What is De Moivre’s theorem in polar form?
De Moivre’s theorem states that (r e^{iθ})ⁿ = rⁿ e^{inθ} for any integer n. This theorem is used to compute powers and roots of complex numbers easily in polar form. Example: (e^{iπ/3})² = e^{i2π/3}.
10. Why is polar form useful for complex numbers?
The polar form of complex numbers is useful because it simplifies multiplication, division, powers, and roots. Key advantages include:
- Multiplication → multiply moduli and add angles.
- Division → divide moduli and subtract angles.
- Powers → apply De Moivre’s theorem.
- Clear geometric interpretation in the complex plane.
