Complex Plane, Modulus, Argument, and Graphical Representation of a Complex Number Hello, my name is Mohammad Ali Nayeem, a student at DIU studying Software Engineering. In Math 102, we delve into various concepts related to complex numbers, including the complex plane, modulus, argument, and their graphical representation. Complex Plane The complex plane is a two-dimensional plane used to visually represent complex numbers. It consists of a horizontal axis (the real axis) and a vertical axis (the imaginary axis). Any complex number 𝑧 = 𝑎 + 𝑏 𝑖 z=a+bi can be plotted on this plane, where 𝑎 a is the real part and 𝑏 b is the imaginary part. Modulus The modulus (or absolute value) of a complex number 𝑧 = 𝑎 + 𝑏 𝑖 z=a+bi is the distance of the point representing 𝑧 z from the origin (0, 0) in the complex plane. It is denoted by ∣ 𝑧 ∣ ∣z∣ and calculated using the formula: ∣ 𝑧 ∣ = 𝑎 2 + 𝑏 2 ∣z∣= a 2 +b 2 ​ The modulus provides a measure of the magnitude of the complex number. Argument The argument of a complex number is the angle 𝜃 θ between the positive real axis and the line segment connecting the origin to the point representing the complex number. It is denoted by arg ⁡ ( 𝑧 ) arg(z) and is usually measured in radians. The argument can be found using the formula: 𝜃 = tan ⁡ − 1 ( 𝑏 𝑎 ) θ=tan −1 ( a b ​ ) The argument indicates the direction of the complex number in the complex plane. Graphical Representation To graphically represent a complex number 𝑧 = 𝑎 + 𝑏 𝑖 z=a+bi, plot the point ( 𝑎 , 𝑏 ) (a,b) on the complex plane. The modulus is the length of the line from the origin to this point, and the argument is the angle this line makes with the positive real axis. For example, the complex number 3 + 4 𝑖 3+4i can be plotted as the point (3, 4). Its modulus is: ∣ 3 + 4 𝑖 ∣ = 3 2 + 4 2 = 5 ∣3+4i∣= 3 2 +4 2 ​ =5 Its argument is: 𝜃 = tan ⁡ − 1 ( 4 3 ) ≈ 0.93 radians θ=tan −1 ( 3 4 ​ )≈0.93 radians Understanding these concepts helps in visualizing and manipulating complex numbers, which are fundamental in many areas of engineering and applied mathematics. Best regards, Mohammad Ali Nayeem DIU Software Engineering Student

1. Complex plane, Modulus, Argument,
Complex plane, Modulus, Argument,
Graphical representation of a
Graphical representation of a
complex number.
complex number.
P r e s e n t a t i o n
o n
Submitted To
Mr. Md. Mozammelul Haque
Lecturer
Department of Software Engineering
Daffodil International University

2. Section : A
ID : 232-35-016
Reduan Ahmad
Section : A
ID : 232-35-003
Abdullah Al Noman
Section : A
ID : 232-35-022
Mohammad Ali Nayeem
Section : A
ID : 222-35-1189
Sabbir Hossen
Section : A
ID : 232-35-001
Prionti Maliha

3. Learning Target:
Complex plane
Modulus
Argument
Graphical representation of
a complex number.

4. Number Line
Negative Numbers (−) Positive Numbers (+)

5. A plane for complex
numbers!
3 units along (the real axis),
4 units up (the imaginary axis).
Plot a complex number like 3 + 4i

6. Modulus
The modulus of a complex number is a fundamental concept in mathematics.
The modulus of a complex number is the distance of the complex number from the origin in
the the complex plane
The modulus of a complex number (z), denoted as |z|, is the non-negative value equal to the
square root of the sum of the squares of its real (x) and imaginary (y) parts, expressed as:
In this equation, 𝑥 and 𝑦 represent the real and imaginary components of the complex number
𝑧 respectively.
When a complex number (z) is plotted on a graph (the complex plane), the distance between
the coordinates of the complex number and the origin (0, 0) is called the modulus of the
complex number.
Modulus of a complex number is always non-negative

7. Examples
Now lets Find Modulus of |z|=3 + 4i

8. Arguments in the Complex Plane
complex plane with a vector representing the complex number 3 + 4i

9. Argument of Complex Numbers Formula
θ = tan⁻¹ (y/x)
θ = π - tan⁻¹ |y/x|
θ = π + tan⁻¹ |y/x|
θ = 2π - tan⁻¹ |y/x|

10. Calculating the Argument
Be mindful of the quadrant of the complex number to determine the correct value of θ.
Formula:
For z = x + yi, the argument θ is given by:
tan(θ) = y/x
θ = arctan(y/x) (using the inverse tangent function)

11. Examples
Identify: x = 3 and y = 4
Calculate: tan(θ) = 4/3
Solve: θ = arctan(4/3) (using
a calculator)
Result: θ ≈ 53.13° (first
quadrant)

12. Graphical
Representation of
Complex Numbers

13. We can represent complex numbers in the
complex plane.
We use the horizontal axis for the real part and
the vertical axis for the imaginary part.

14. Example :
The number 3+2𝑗 (where 𝑗=−1​
) is represented by:

15. The point A is the representation of the
complex number 3+2𝑗.
The horizontal axis is marked R (for the "real"
numbered-component), and the vertical axis
is marked j (for the imaginary component of
the complex number).

16. THANK YOU!