Context

Complex numbers, despite being a somewhat abstract concept and seemingly distant from our daily reality, are closely related to various aspects of the real world, especially in fields such as Engineering, Physics, and Computer Science.

In the field of Engineering, for example, complex numbers are used to represent impedances and admittances in Electrical Circuit Analysis. In Quantum Physics, complex numbers are used in the representation of quantum states.

The operation of product and division of complex numbers is a fundamental concept for understanding the practical applications of these numbers. Therefore, developing skills in this area is of utmost importance for potential future professionals in these fields.

Introduction

Despite the name 'complex', complex numbers are not as intimidating as they seem. They are composed of a real part and an imaginary part, the latter being marked by the symbol 'i', which represents the square root of -1.

They have a geometric representation, which allows them to be visualized in the plane called the 'complex plane' or 'Argand-Gauss plane'. This geometric representation facilitates the understanding of arithmetic operations with complex numbers, such as multiplication and division.

Therefore, our goal in this project is to deepen the understanding of the multiplication and division of complex numbers, as well as the visualization of these operations in the Argand-Gauss plane.

Practical Activity

Activity Title: Complex Operations: Products and Divisions in the Argand-Gauss Plane

Project Objective:

The objective of this project is to analyze the multiplication and division of complex numbers. Students will do this by creating a simple application in a graphic programming software, such as Processing, for example. In addition to learning about complex numbers, students will have the opportunity to improve their programming and teamwork skills.

Detailed Project Description:

The project will be carried out in groups of 3 to 5 students and is expected to last about a week and a half. The project will be based on the construction of a graphic application that represents the multiplication and division of complex numbers in the Argand-Gauss plane.

Students will have to develop functions that solve the product and division of complex numbers, and then create a graphical representation of these operations in the application. Each group will be responsible for both processes (product and division).

Required Materials:

1. Computer with internet access
2. Graphic programming software (we recommend Processing, which is free and available for all platforms)
3. Reference material on complex numbers and their operations.

Step-by-Step Guide for the Activity:

1. Theme Study: Start by reviewing the theory on complex numbers, multiplication, and division of complex numbers. Use the provided references and feel free to investigate further on your own.

2. App Planning: Divide tasks into smaller ones. Think about how the app interface will be, what functions each group member will create, and in what order tasks will be executed.

3. App Development: Start developing the functions for multiplication and division of complex numbers. Then, create the graphical interface and mechanisms for data input.

4. Testing and Improvements: Test the application and make necessary adjustments. Make sure that the product and division are being calculated correctly and that the graph adequately represents the results.

5. Documentation: Write a detailed report on the project, including the planning process, application development, challenges encountered and how they were resolved, and any other relevant details. This report should also include an analysis of the results obtained, detailing how the application graphically represents the operations of complex numbers.

Project Deliverables:

1. Graphic Application: The application should be able to perform multiplication and division between any two given complex numbers, as well as graphically represent the results in the Argand-Gauss plane.

2. Report: The report should include the four main topics - Introduction, Development, Conclusions, and Bibliography used, and should be written clearly and completely, in a way that fits and complements the activities carried out during the project. Its structure should consist of:

   • Introduction: Contextualizing the theme and explaining the relevance of complex numbers. Additionally, it should include a clear description of the project's objectives.
   • Development: Here, all project stages should be detailed, starting with planning, moving on to application development, and reaching the testing and improvement phase. It should also explain the main theoretical concepts involved and how the functions of multiplication and division of complex numbers were implemented.
   • Conclusion: In this section, students should summarize the main points of the work, explain the learnings obtained, and draw conclusions about the project.
   • Bibliography: List of all sources used for the theoretical basis of the project.