[Maths Class Notes] on Argument of Complex Numbers Pdf for Exam

In Mathematics, complex planes play an extremely important role. We also call it a z-plane which consists of lines that are mutually perpendicular known as axes. The real numbers are represented by the horizontal line and are therefore known as the real axis whereas the imaginary numbers are represented by the vertical line and are therefore known as an imaginary axis. We basically use complex planes to represent a geometric interpretation of Complex Numbers. It is just like the Cartesian plane which has both the real as well as imaginary parts of a Complex Number along with the X and Y axes. Complex Numbers are branched into two basic concepts i.e., the magnitude and Argument. But for now we will only focus on the Argument of Complex Numbers and learn its definition, formulas and properties. 

 

What is A Complex Number?

A Complex Number is written as a + ib, where “a” is a real number and “b” is an imaginary number. The Complex Number consists of a symbol “i” which satisfies the condition [i^{2}] = −1. Complex Numbers are referred to as the extension of one-dimensional number lines. In a complex plane, a Complex Number denoted by a + bi is usually represented in the form of the point (a, b). We have to note that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with absolutely no imaginary part is known as a real number.

What is the Argument of Complex Numbers? 

The Argument of a Complex Number is an angle that is inclined from the real axis towards the direction of the Complex Number which is represented on the Complex plane. We can denote it by “θ” or “φ” and can be measured in standard units “radians”.

 

In the diagram above, the complex number is denoted by the point P. The length OP is the magnitude or modulus of the Number, and the angle at which OP is inclined from the positive real axis is known as the Argument of the point P.

 

How to Find Arguments of A Complex Number?

There are few steps that need to be followed if we want to find the Argument of a complex number. These steps are given below: 

Step 1) First we have to find both real as well as imaginary parts from the Complex Number that is given to us and denote them x and y respectively. 

Step 2) Then we have to use the formula θ = [tan^{-1}](y/x) to substitute the values.

Step 3) If by solving the formula we get a standard value then we have to find the value of  θ or else we have to write it in the form of [tan^{-1}] itself.

Step 4) The final value along with the unit “radian” is the required value of the Complex Argument for the given Complex Number.

With this method you will now know how to find out the Argument of a Complex Number.

 

Argument of Complex Number Examples

1. Find the Argument of -1+i and 4-6i

Ans: We would first want to find the two Complex Numbers in the complex plane. This will make it easy for us to determine the quadrants where angels lie and get a rough idea of the size of each angle. 

 

For, z= –+i 

 

We can see that the Argument of z is a second quadrant angle and the tangent is the ratio of the imaginary part to the real part, in such a case −1. The tangent of the reference angle will thus be 1. Write the value of the second quadrant angle so that its reference angle can have a tangent equal to 1. If the reference angle contains a tangent which is equal to 1 then the value of reference angle will be π/4 and so the second quadrant angle is π − π/4 or 3π/4. 

 

For z = 4 − 6i: 

 

This time the Argument of z is a fourth quadrant angle. The reference angle has a tangent 6/4 or 3/2. None of the well known angles consist of tangents with value 3/2. Therefore, the reference angle is the inverse tangent of 3/2, i.e.[tan^{-1}](3/2). Hence the Argument being the fourth quadrant itself is 2π − [tan^{-1}](3/2).

 

In order to get a complete idea of the size of this Argument, we can use a calculator to compute 2π −[tan^{-1}](3/2) and see that it is approximately 5.3 (radians). In degrees this is about 303o.

 

Solved Example

1. Find the Argument of a complex number 2 + 2[sqrt{3i}].

Ans: Let z = 2 + 2–[sqrt{3i}].

The real part, x = 2 and the Imaginary part, y = 2[sqrt{3}]

We already know the formula to find the Argument of a complex number. That is 

arg (z) = [tan^{-1}](y/x)

arg (z) = [tan^{-1}](2[sqrt{3}]/2)

arg (z) = [tan^{-1}]([sqrt{3}])

arg (z) = [tan^{-1}](tan π/3)

arg (z) = π/3

Therefore, the Argument of the complex number is π/3 radian.

 

Complex Numbers

Complex Numbers are those numbers which are used in finding the square root of negative numbers. The Complex Numbers were first introduced by a Greek mathematician named Hero of Alexandria who tried to find the square root of negative numbers but wasn’t able to solve it. This was solved by an Italian mathematician called Gerolamo Cardano who found the negative roots of cubic and quadratic polynomial expressions using Complex Numbers. Complex Numbers have many uses in scientific research, fluid dynamics, quantum mechanics and signal processing.

A complex number can be defined as the sum of an imaginary number and a real number. It is written as a + ib which can be denoted by z. The a and b here are real numbers. The value of a is said to be the real part which is denoted by Re(z) and the b is said to be the imaginary part written as Im(z). The ib is also an imaginary number. For example 2 + 3i and -2 -5i are examples of Complex Numbers. The i written here is called iota which is used to represent the imaginary part of Complex Numbers. It is also helpful to find the square root of negative numbers.

What Does the Argument of Complex Numbers Mean?

When in geometrical representation, the line representing the complex number and origin makes an angle with the positive X-axis in an anticlockwise direction then this is said to be the argument of the complex numbers. This is represented by the inverse of tan of the imaginary number in the complex number which is divided by the real part of the complex number.

Argz(θ) = Tan−1(b/a)