Step-by-Step Guide to Converting Complex Numbers to Polar Form
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 Made Easy
1. What is the polar form of a complex number?
The polar form represents a complex number using its distance from the origin (modulus, r) and the angle it makes with the positive x-axis (argument, θ). The standard polar form formula is z = r(cosθ + i sinθ), where 'z' is the complex number and 'i' is the imaginary unit (√-1). This is an alternative to the rectangular form z = x + iy.
2. How do you convert a complex number from rectangular (x + iy) to polar form?
To convert a complex number z = x + iy to its polar form r(cosθ + i sinθ), you need to find the modulus 'r' and the argument 'θ'. The steps are:
- Calculate the modulus (r) using the formula: r = √(x² + y²).
- Calculate the argument (θ) using the formula: θ = tan⁻¹(y/x), ensuring you place the angle in the correct quadrant based on the signs of x and y.
- Substitute these values of r and θ into the polar formula z = r(cosθ + i sinθ).
3. What do the modulus and argument of a complex number represent graphically?
In the Argand plane (complex plane), the modulus (r) represents the distance of the point (x, y) from the origin (0, 0). It is always a non-negative real number. The argument (θ) represents the angle that the line segment connecting the origin to the point (x, y) makes with the positive real (x) axis, measured counter-clockwise.
4. What is the main difference between representing a complex number in Cartesian and polar form?
The main difference lies in the coordinates used.
- The Cartesian form (x + iy) uses horizontal (real part 'x') and vertical (imaginary part 'y') distances to locate the point on the Argand plane. It is ideal for addition and subtraction.
- The polar form (r(cosθ + i sinθ)) uses a distance from the origin (modulus 'r') and an angle (argument 'θ'). This form is particularly useful for multiplication and division of complex numbers, as well as finding powers and roots using De Moivre's Theorem.
5. Why is the polar form of complex numbers useful in real-world applications?
The polar form is extremely useful in fields involving rotation and periodic phenomena. Its main advantage is simplifying multiplication and division. Key applications include:
- Electrical Engineering: Analysing AC circuits, where voltage and current have both magnitude (modulus) and phase shifts (argument).
- Physics and Engineering: Describing wave motions, oscillations, and rotations, such as in signal processing and mechanics.
- Computer Graphics: Performing rotations of objects on a 2D plane.
6. How do you write a purely imaginary number like 4i in polar form?
For the complex number z = 4i, the rectangular coordinates are x = 0 and y = 4.
1. First, find the modulus: r = √(0² + 4²) = √16 = 4.
2. Next, find the argument. Since the point (0, 4) lies on the positive imaginary axis, the angle with the positive real axis is θ = π/2 radians or 90°.
3. Therefore, the polar form is z = 4(cos(π/2) + i sin(π/2)).
7. Can the modulus (r) of a complex number be negative?
No, the modulus of a complex number cannot be negative. The modulus 'r' is defined as the distance from the origin to the point representing the complex number in the Argand plane. Since distance is a scalar quantity that measures a physical length, it is always non-negative (r ≥ 0).
8. Is the argument (θ) of a complex number unique?
No, the argument θ of a complex number is not unique. Because angles repeat every 360° (or 2π radians), if θ is an argument, then θ + 2nπ (where 'n' is any integer) is also a valid argument. To ensure a unique value, we often use the Principal Argument, which is restricted to the interval -π < θ ≤ π, as per the CBSE Class 11 syllabus for 2025-26.
9. How is the conjugate of a complex number represented in polar form?
If a complex number 'z' has the polar form z = r(cosθ + i sinθ), its conjugate, denoted as z̄, is found by negating the imaginary part. In polar form, this is equivalent to negating the angle. Therefore, the conjugate is z̄ = r(cos(-θ) + i sin(-θ)). Since cos(-θ) = cos(θ) and sin(-θ) = -sin(θ), this simplifies to z̄ = r(cosθ - i sinθ). The modulus remains the same, but the argument becomes -θ.
