Contextualization

Complex numbers are an extension of the set of real numbers and are fundamental in mathematics. They have the form a + bi, where a and b are real numbers and i is the imaginary unit, which is the square root of -1. Complex numbers expand the concept of number beyond the real numbers we are familiar with.

Complex numbers were created to solve equations that did not have real solutions, such as the equation x² + 1 = 0, which cannot be solved with real numbers but can be solved with complex numbers. They are also used in many areas of science and engineering, including physics, electrical engineering, and economics.

The basic operations with complex numbers, which are addition, subtraction, multiplication, division, and exponentiation, are natural extensions of operations with real numbers. Learning to perform these operations is essential to understanding advanced mathematics, physics, and engineering.

Importance

Understanding complex numbers and knowing how to manipulate them is vital for many areas of knowledge. In physics, they are used to describe electromagnetic waves. In electrical engineering, they are used to analyze AC circuits. In economics, they are used to model complex economic systems. Learning complex numbers, therefore, not only enriches mathematical knowledge but also opens a door to a variety of practical applications.

Complex numbers also play an important role in understanding concepts such as multidimensional space, which are fundamental to learning machine learning and data science. This makes complex numbers a valuable tool for those wishing to pursue a career in these areas.

Practical Activity

Activity Title:

Building Electric Circuits with Complex Numbers

Project Objective:

The objective of the project is to relate the knowledge of complex numbers (acquired in mathematics) and electrical circuits (acquired in physics), so that students understand how these two topics can be used together in the real world and appreciate the interconnection between the fields. In addition, by working in groups, students will also develop collaboration and time management skills.

Detailed Project Description:

In this project, groups will build a simple electric circuit and calculate the impedance (complex resistance) of the circuit using complex numbers. Students will work with resistors, inductors, and capacitors, which have their resistances expressed as complex numbers.

Group work will be organized as follows:

• 1 student will be responsible for the theoretical part: studying and describing the theory of complex numbers and the laws of electrical circuits.
• 1-2 students will be responsible for building and measuring the actual electric circuit.
• 1-2 students will be responsible for calculating the circuit impedance using complex numbers.

Required Materials:

• Basic electric circuit construction kit (including resistors, capacitors, and inductors)
• Multimeter for current and voltage measurement
• Computer/calculator for complex number operations

Step by Step:

1. Start with the study of the theory of complex numbers. The student responsible for the theoretical part should review the addition, subtraction, multiplication, division, and exponentiation of complex numbers and explain this to the group members.

2. Study the theory of electrical circuits and how resistance, inductance, and capacitance are related and represented by complex numbers.

3. Design a simple circuit that includes resistors, inductors, and capacitors.

4. Build the physical circuit and make sure it works correctly.

5. Calculate the total impedance of the circuit using complex numbers and compare this impedance with the actual measurement using the multimeter.

6. Discuss the findings and try to understand why there may be discrepancies between theoretical calculation and practical measurement.

7. Prepare a final project report, including Introduction, Development, Conclusions, and Bibliography used.

Project Deliverables:

Students, as a group, must deliver a detailed final report, along with the constructed physical circuit. The report should include:

Introduction:

In this section, students should contextualize the application of complex numbers in electrical circuits and explain why it is important to understand these concepts. Students should explain the project's objective and the relevance of studying complex numbers.

Development:

In this section, students should detail the project process. Including the theory of complex numbers and electrical circuits, the design and construction of the circuit, the calculations performed, and the group discussions. They should also explain in detail the methodology used.

Conclusions:

Students should conclude their main results, highlighting what they learned from the project and discussing any discrepancies between theory and practice.

Bibliography:

Students should list all sources that were used to acquire knowledge of the subject, including books, websites, videos, and other materials.

The evaluation will be according to the correction criteria detailed in the next section, taking into account the accuracy of complex calculations, the quality of the constructed circuit, the demonstrated theoretical understanding, the quality of the report, and the ability to work in a team.