Plotting Complex Numbers on the Complex Plane

Complex Numbers and Their Graphical Representation

Introduction to Complex Numbers

Complex numbers are numbers of the form a + bi, where a is the real part and b is the imaginary part. They can be represented as points in a two-dimensional plane called the complex plane or Argand diagram.

• Real Part (a): Plotted on the horizontal axis (x-axis).

• Imaginary Part (b): Plotted on the vertical axis (y-axis).

Plotting Complex Numbers

To plot a complex number a + bi:

1. Locate a on the real (horizontal) axis.

2. Locate b on the imaginary (vertical) axis.

3. Plot the point at the intersection of these values.

Example Table: Identifying Real and Imaginary Parts

Complex Number	Real Part	Imaginary Part
2 + 3i	2	3
-4 + 0i	-4	0
0 + 5i	0	5

Graphical Example

• Example: Plot the complex number $2 + 3i$.

• Move 2 units along the real axis (right), then 3 units up along the imaginary axis.

• Mark the point (2, 3) on the complex plane.

Key Formula

• General form: $z = a + bi$

Applications

• Complex numbers are used in engineering, physics, and mathematics to represent quantities with both magnitude and direction.

• Plotting helps visualize operations such as addition, subtraction, and finding the modulus (distance from the origin):

Modulus formula:

$|z| = \sqrt{a^2 + b^2}$

Additional info: Understanding how to plot complex numbers is foundational for later topics such as trigonometric (polar) form, Euler's formula, and applications in signal processing.