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Signed Numbers: 2's Complement Representation

Digital Logic Systems: From Zero to Hero

4. Signed Numbers: 2's Complement Representation

In digital systems, we need a way to represent negative numbers. The 2's Complement method is the standard approach because it simplifies subtraction operations.

1's Complement

The 1's complement of a binary number is found by inverting all the bits (0 becomes 1, and 1 becomes 0).

  • Example: $5_{10} = 0101_2$. 1's Complement is $1010_2$.

2's Complement

The 2's complement is found by taking the 1's complement and adding 1 to the result.

Steps to find -N using 2's Complement (8 bits):

  1. Start with the positive number N (e.g., $5_{10} = 00000101_2$).
  2. Find the 1's Complement: $11111010_2$.
  3. Add 1: $11111010 + 1 = 11111011_2$.

Thus, $-5_{10}$ is represented as $11111011_2$.

Key Feature: The Sign Bit

In 2's complement representation, the Most Significant Bit (MSB) determines the sign:

  • 0: Positive number.
  • 1: Negative number.

This system allows microprocessors to treat both addition and subtraction as simple addition, simplifying hardware design.