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°.
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.