Contextualization

Mathematics is vast and deep, consisting of several branches of study, where each one is interconnected and complementary. Complex numbers represent one of these branches, and in a peculiar way, they emerged not from a new discovery, but as a need to explain mathematical phenomena that could not be addressed through already known numbers. Initially, they were treated with some skepticism, but they soon became an integral and indispensable part of Mathematics, with applications in several scientific and technological fields.

A complex number consists of a real part and an imaginary part. It is represented in the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit (i² = -1). The equality between complex numbers, in turn, is something simple, but crucial. Two complex numbers are equal only when their real and imaginary parts are equal.

Application and Relevance

At first glance, complex numbers seem like an odd concept distant from practical reality. However, they have amazing practical applications in several areas that directly impact our lives. In Electronics, for example, they are used for the analysis of AC circuits. In Physics, they have a central role in the formulation of Quantum Mechanics. They are also widely used in Engineering, Computer Sciences, Image Processing, among others.

By learning about complex numbers and about the equality between these numbers, you will be opening a door to a deeper understanding of many areas of knowledge. You will also be developing valuable skills, such as the ability to think abstractly and to solve complex problems.

To assist in the study of the topic, we indicate the following resources:

• Article about complex numbers in Wikipedia: Complex Numbers
• Khan Academy: Free online course on Advanced Mathematics, which includes a module on complex numbers: Khan Academy
• Video on the application of complex numbers in electric circuits: Electric Circuits

Practical Activity: "Creating Complex Art with Complex Numbers"

Project Goal

The main goal of this activity is to apply the concept of equality in complex numbers and its graphical representation to create complex art.

Students will use the concepts of Mathematics (complex numbers) and Arts (graphical representation and aesthetics), in collaboration and teamwork, to create a representative image using complex numbers.

Detailed Project Description

This project will be carried out by groups of 3 to 5 students. Each group must work on creating a representative image, using the representation of complex numbers on the complex plane (or Argand-Gauss plane).

Required Materials:

• Graph plotting software (We recommend the open-source software Python with the library Matplotlib, but other software such as Geogebra or even online graph plotting software can also be used).
• Textbook or support material on complex numbers.
• Computer with internet access.

Step by Step

1. Theoretical Research (3 hours): After forming the groups, students must research the basic theory of complex numbers, focusing on the concept of equality. They must understand the graphical representation of complex numbers and how those numbers are plotted.

2. Brainstorming (2 hours): After understanding the concept, the groups must hold a brainstorming session to choose which representative image they will create. The image can be anything - an object, an animal, a logo, etc.

3. Creating the Complex Number Matrix (5 hours): Based on the chosen image, the group must then work on creating a matrix of complex numbers that, when plotted, represents the image.

4. Coding and Plotting (2 hours): Using graph plotting software, the group must then work on coding to plot the complex numbers in the matrix.

5. Adjustments and Refining the Image (3 hours): After the first plotting, the group must make the necessary adjustments to refine the image.

6. Preparing the Final Report (5 hours): Finally, the group must prepare a report detailing the entire process: the theoretical research, the choice of image, the creation of the complex number matrix, the plotting, and the final adjustments.

Project Deliverables

• Code used to plot the image.
• Final image obtained by plotting the complex numbers.
• Detailed report, containing:
  • Introduction: Description of the concept of complex numbers and their applicability in the real world. Must also include the explanation about basic equality in complex numbers.
  • Development: Detailed explanation of the activity carried out, followed by the methodology used. Students must explain how they chose the image, how they assembled the matrix of complex numbers, and how they managed to plot the image. The adjustments made must also be explained.
  • Conclusions: Students must revisit the main points of the report, explaining what they learned during the project and what conclusions they were able to draw.
  • Bibliography: The group must indicate what the main sources of information they used for the development of the project were.

This multidisciplinary project, in addition to working on important mathematical concepts, stimulates creativity, teamwork, problem solving and communication, essential skills for students' education.