#08 Les opérations arithmétique en hexadécimal

Introduction

This tutorial will guide you through performing arithmetic operations in hexadecimal and octal number systems. Understanding these operations is crucial for programming, computer science, and digital electronics, where these number systems are frequently used. By the end of this guide, you will know how to add, subtract, and multiply numbers in both hexadecimal and octal systems.

Step 1: Understanding Hexadecimal and Octal Systems

• Hexadecimal System: Base-16 number system using digits 0-9 and letters A-F (A=10, B=11, C=12, D=13, E=14, F=15).
• Octal System: Base-8 number system using digits 0-7.

Practical Tips

• Familiarize yourself with converting between decimal, hexadecimal, and octal systems to ease operations.
• Use a conversion calculator if necessary, especially when starting.

Step 2: Addition in Hexadecimal

1. Align the numbers vertically, similar to decimal addition.
2. Add each column from right to left, carrying over as needed.
   • Example:
     • 1A + 2C
       • A (10) + C (12) = 1A (22), write down A and carry 1.
       • 1 (carry) + 1 (1) + 2 (2) = 4.
       • Result: 46.
3. Common Pitfalls:
   • Remember that if the sum exceeds 15, convert it to hexadecimal (e.g., 16 becomes 10).

Step 3: Subtraction in Hexadecimal

1. Align the numbers vertically.
2. Subtract each column, borrowing from the next column if necessary.
   • Example:
     • 2E - 1A
       • E (14) - A (10) = 4.
       • 2 (2) - 1 (1) = 1.
       • Result: 14.
3. Common Pitfalls:
   • Be cautious when borrowing; the value being borrowed must be adjusted correctly.

Step 4: Multiplication in Hexadecimal

1. Multiply each digit of the first number by each digit of the second number, similar to decimal multiplication.
2. Shift the results according to their place value.
   • Example:
     • 3 * 4 = C.
     • 3 * A = 1E (1E is 30 in decimal).
3. Add the results, aligning them appropriately.
   • Result example:
     • 3 * 2A = 3 * (20 + A) = 60 + 3A = 60 + 3*10 = 60 + 30 = 90 (in hexadecimal).
4. Common Pitfalls:
   • Ensure proper place value alignment when adding partial products.

Step 5: Addition in Octal

1. Align the octal numbers vertically.
2. Add each column from right to left, carrying over if the sum exceeds 7.
   • Example:
     • 47 + 23
       • 7 + 3 = 12 (which is 14 in octal), write down 2 and carry 1.
       • 4 + 2 + 1 (carry) = 7.
       • Result: 72.
3. Common Pitfalls:
   • Similar to hexadecimal, converting sums over 7 is crucial.

Step 6: Subtraction in Octal

1. Align the numbers.
2. Subtract while borrowing as necessary.
   • Example:
     • 75 - 34
       • 5 - 4 = 1.
       • 7 - 3 = 4.
       • Result: 41.
3. Common Pitfalls:
   • Be careful when the top number is smaller than the bottom number in a column.

Step 7: Multiplication in Octal

1. Multiply each digit as you would in decimal.
2. Shift results according to their place value.
   • Example:
     • 7 * 3 = 21 (in octal, this is 25).
3. Add the shifted results.
4. Common Pitfalls:
   • Ensure you convert correctly when the product exceeds 7.

Conclusion

In this tutorial, you have learned to perform addition, subtraction, and multiplication in both hexadecimal and octal systems. Practice these operations with various numbers to strengthen your understanding. Consider exploring further topics, such as division in these number systems or conversions between them, to enhance your skills in arithmetic operations.