Contextualization

In our daily lives, we are surrounded by systems that use different numerical bases. The most common one is decimal (base 10), which we use daily for counting, adding, subtracting, and performing other operations. However, many of the electronic devices we use, such as cell phones, computers, and calculators, operate based on binary systems (base 2) or hexadecimal (base 16). Therefore, having a solid understanding of how these numerical systems work can be extremely useful, not only for your mathematical education but also for understanding the functioning of many of the technologies we use daily.

The base 2 conversion, or binary conversion, is a fundamental process in mathematics and computer science. By understanding how these base changes work, we can better understand how information is processed by computers and various other digital devices. Converting numbers between different numerical bases is an important skill for anyone working with exact sciences or computing.

Introduction

As we begin to explore the world of mathematics, we learn that the numbers we use daily are written in base 10. This means that each position in a number represents ten times the value of the position to its right. In the base 2 system, each position represents two times the value of the position to its right. Each digit in a binary number can only be 0 or 1.

To convert a number from base 10 to base 2, we divide the number by 2 and record the remainder. We continue to do this until the number we are dividing is 0. The sequence of remainders, written in reverse, is the number in base 2.

To convert a number from base 2 to base 10, multiply each digit by the value of its position (starting at 0 on the right) raised to 2 and add up all these products.

Having understood the concept of numerical bases and how to perform base conversions, we are ready to delve deeper into these processes and practice these skills.

Practical Activity: "Decoding Binary: A Journey in Base Conversion"

Activity Title

"Decoding Binary: A Journey in Base Conversion"

Project Objective

To employ knowledge about base 2 conversion and apply it practically in encoding and decoding messages, to understand the importance of the binary system in the operation of computers and electronic devices.

Detailed Project Description

Students will work in groups of 3 to 5 members to explore the base 2 conversion in a way that involves Mathematics and Informatics. The activity will include creating secret encoded messages using binary and decoding messages from other groups.

Required Materials

• Computer with internet access
• Paper and pencil for drafts and calculations
• Calculator (optional)
• "Numerical Base Converter" app (optional)

Detailed Step-by-Step

1. Study and Research: Start the project with an in-depth study on base conversion, especially from base 10 to base 2 and vice versa, using the recommended resources.

2. Conversion Practice: After learning the theory, practice converting numbers to base 2 and back to base 10. Use examples at the beginning and then create your own numbers to convert.

3. Message Encoding: Once comfortable with the conversion, each group should create a secret encoded message. Since our alphabet has 26 letters, create a five-digit binary code (as 2^5 = 32, which is more than enough for our alphabet) to represent each letter (for example, 'A' = 00001, 'B' = 00010, etc.). Write your secret message and then convert each letter into binary.

4. Message Decoding: Exchange the encoded message with another group. Your task will be to decode their message, converting each binary sequence back to base 10 and then to the corresponding letter, using the code you established.

5. Project Report: At the end, each student must write a report documenting the project.

Project Deliverables

• Notebook with conversion calculations: To record the entire process of study, research, and conversion practice.
• Encoded message: A Portuguese phrase converted into binary code.
• Decoded message: The secret message from another group decoded and converted back to Portuguese.
• Project Report: Final document containing four sections - Introduction, Development, Conclusions, and Bibliography.

The report should explain the process of encoding and decoding messages, the theory behind it, the importance of base 2 conversion, and how it applies in the real world. Each section should have the following:

• Introduction: Contextualize Base 2 Conversion and its application in technology and communication. Describe the importance of this project and its objective.

• Development: Explain the theory of Base 2 Conversion. Detail the process of encoding and decoding messages. Describe the methodology used and the main difficulties encountered. Present the results, showing the encoded and decoded messages.

• Conclusions: Summarize the main points of the work, emphasizing the knowledge and skills acquired. Discuss the impact of this learning on the understanding of technology and Mathematics.

• Bibliography: List the resources used in the project (books, videos, websites, etc.).

This activity will require effort, research, calculations, and creativity, but upon completion, you will have a better understanding of Base 2 Conversion and how it is applied in real life.