#Informatique S1 04 représentation de l'information (partie 2)

Introduction

This tutorial provides a comprehensive guide on converting between different number systems: binary, octal, decimal, and hexadecimal. Understanding these conversions is essential in computing and digital electronics, as they form the basis of data representation. This guide will break down the conversion processes step-by-step, making it easier to grasp and apply in practical scenarios.

Step 1: Understanding the Number Systems

Before diving into conversions, it’s important to understand each number system:

• Binary (Base 2): Uses two digits, 0 and 1. It is the fundamental language of computers.
• Octal (Base 8): Uses eight digits, from 0 to 7. Commonly used in computing as a shorthand for binary.
• Decimal (Base 10): The standard number system for humans, using ten digits from 0 to 9.
• Hexadecimal (Base 16): Uses sixteen symbols, from 0 to 9 and A to F, providing a more compact representation of binary data.

Step 2: Converting Binary to Decimal

To convert a binary number to decimal, follow these steps:

1. Write down the binary number.
2. Assign powers of 2 to each digit, starting from the right (0 for the rightmost digit).
3. Multiply each binary digit by its corresponding power of 2.
4. Sum all the results.

Example:

• Binary: 1011
• Calculation:
  • 1 × 2³ = 8
  • 0 × 2² = 0
  • 1 × 2¹ = 2
  • 1 × 2⁰ = 1
• Total: 8 + 0 + 2 + 1 = 11 (Decimal)

Step 3: Converting Decimal to Binary

To convert a decimal number to binary:

1. Divide the decimal number by 2.
2. Record the remainder (0 or 1).
3. Continue dividing the quotient by 2 until you reach 0.
4. The binary representation is the remainders read in reverse order.

Example:

• Decimal: 13
• Steps:
  • 13 ÷ 2 = 6 remainder 1
  • 6 ÷ 2 = 3 remainder 0
  • 3 ÷ 2 = 1 remainder 1
  • 1 ÷ 2 = 0 remainder 1
• Binary: 1101

Step 4: Converting Binary to Octal

To convert binary to octal:

1. Group the binary digits into sets of three, starting from the right. Add leading zeros if necessary.
2. Convert each group to its octal equivalent.

Example:

• Binary: 101110
• Grouping: 0 101 110 (adding leading zero)
• Conversion:
  • 101 = 5
  • 110 = 6
• Octal: 56

Step 5: Converting Octal to Binary

To convert octal to binary:

1. Convert each octal digit to a 3-digit binary equivalent.

Example:

• Octal: 56
• Conversion:
  • 5 = 101
  • 6 = 110
• Binary: 101110

Step 6: Converting Decimal to Octal

To convert decimal to octal:

1. Divide the decimal number by 8.
2. Record the remainder.
3. Continue dividing the quotient by 8 until you reach 0.
4. Read the remainders in reverse order.

Example:

• Decimal: 20
• Steps:
  • 20 ÷ 8 = 2 remainder 4
  • 2 ÷ 8 = 0 remainder 2
• Octal: 24

Step 7: Converting Hexadecimal to Decimal

To convert hexadecimal to decimal:

1. Write down the hexadecimal number.
2. Assign powers of 16 to each digit, starting from the right.
3. Multiply each digit by its corresponding power of 16.
4. Sum all the results.

Example:

• Hexadecimal: 2A
• Calculation:
  • 2 × 16¹ = 32
  • A (10) × 16⁰ = 10
• Total: 32 + 10 = 42 (Decimal)

Conclusion

This tutorial outlined the essential steps for converting between binary, octal, decimal, and hexadecimal number systems. Mastering these conversions is crucial for anyone working in fields related to computer science and digital electronics. Practice these conversions with different numbers to reinforce your understanding. For further learning, explore coding applications that utilize these number systems, such as binary arithmetic in programming.