#Informatique S1 03 Représentation de l'information (partie 1)

Introduction

This tutorial aims to explain the various number systems used in computing, specifically binary, octal, decimal, and hexadecimal. Understanding these systems is crucial for anyone interested in computer science or programming, as they form the foundation of how information is represented and manipulated in digital devices.

Step 1: Understanding the Number Systems

Familiarize yourself with the four primary number systems used in computing:

• Binary: Base-2 system using digits 0 and 1.
• Octal: Base-8 system using digits 0 to 7.
• Decimal: Base-10 system using digits 0 to 9 (the standard system used in daily life).
• Hexadecimal: Base-16 system using digits 0 to 9 and letters A to F (where A=10, B=11, C=12, D=13, E=14, F=15).

Practical Tip

• Use a conversion chart to visualize how each system relates to one another, which will aid in understanding and conversions.

Step 2: Converting Between Number Systems

Learn how to convert numbers from one system to another. Here’s a brief guide for each conversion type:

Binary to Decimal

1. Write down the binary number.

2. Multiply each bit by 2 raised to the power of its position (starting from 0).

3. Sum all the results.

   Example:

   • Binary: 1011
   • Calculation: (1×2^3) + (0×2^2) + (1×2^1) + (1×2^0) = 8 + 0 + 2 + 1 = 11

Decimal to Binary

1. Divide the decimal number by 2.

2. Record the remainder.

3. Repeat the division with the quotient until it reaches 0.

4. The binary number is the remainders read in reverse order.

   Example:

   • Decimal: 11
   • Division: 11 ÷ 2 = 5 (remainder 1), 5 ÷ 2 = 2 (remainder 1), 2 ÷ 2 = 1 (remainder 0), 1 ÷ 2 = 0 (remainder 1)
   • Binary: 1011

Hexadecimal to Decimal

1. Write down the hexadecimal number.

2. Multiply each digit by 16 raised to the power of its position.

3. Sum all the results.

   Example:

   • Hexadecimal: 2A
   • Calculation: (2×16^1) + (10×16^0) = 32 + 10 = 42

Decimal to Hexadecimal

1. Divide the decimal number by 16.

2. Record the remainder.

3. Repeat until the quotient reaches 0.

4. The hexadecimal number is the remainders read in reverse order.

   Example:

   • Decimal: 42
   • Division: 42 ÷ 16 = 2 (remainder 10), 2 ÷ 16 = 0 (remainder 2)
   • Hexadecimal: 2A

Step 3: Additional Conversions

Explore conversions between other systems such as binary to octal and octal to hexadecimal. The process involves intermediate conversions to decimal if needed.

Binary to Octal

1. Group the binary number into sets of three, starting from the right.
2. Convert each group to its octal equivalent.

Octal to Binary

1. Convert each octal digit to its binary equivalent (3 bits).

Hexadecimal to Binary

1. Convert each hexadecimal digit to its binary equivalent (4 bits).

Binary to Hexadecimal

1. Group the binary number into sets of four, starting from the right.
2. Convert each group to its hexadecimal equivalent.

Conclusion

Understanding the different number systems and their conversions is essential in the field of computing. Mastering these conversions will enhance your programming skills and deepen your knowledge of how data is represented in computers. As a next step, practice converting various numbers between these systems using the methods outlined in this tutorial.