Lecture on Binary Numbers: One's Complement and Two's Complement

Introduction

• Topic: Concepts of binary numbers, focusing on one's complement and two's complement.
• Purpose: Understand the mathematical concepts first, then application.

One's Complement

• Definition: A binary number transformation.
• Process:
  • Take an existing binary number (e.g., 8-bit: 01010001).
  • Convert all 0s to 1s and all 1s to 0s.
  • Example transformation: 01010001 becomes 10101110.
• Use Case: Simplifies certain calculations in binary arithmetic.

Two's Complement

• Definition: Another binary number transformation, more complex than one's complement.
• Process:
  • Two-step Method:
    1. First, find the one's complement of the number.
    2. Add 1 to the resulting one's complement.
  • Example:
    • Binary number: 01010110.
    • One's complement: 10101001.
    • Add 1: 10101010.
  • Shortcut Method:
    • Cover the number from the left up to the first 1.
    • Perform the one's complement on the uncovered part.
    • Retain the covered part as is.
    • Check with the two-step method to verify.
• Example with Shortcut:
  • Number: 10101011.
  • Covering up to and including the first 1, transform uncovered portion.
  • Verified using two-step method.

Key Points

• One's Complement: Simple inversion of bits.
• Two's Complement: Involves an additional step of adding 1.
• Shortcut Method: Useful for quick computation and verification.
• Importance: Critical for binary arithmetic and computer science.

Conclusion

• These methods are mathematical concepts that provide a foundation for further applications.
• Understanding both methods helps troubleshoot and optimize binary-related tasks.
• Encourage practice and leveraging both methods for different scenarios.