Two's Complement - Representing Negative Numbers

The Problem with Simple Sign Bit:

You might think: "Just use the leftmost bit as a sign (0=positive, 1=negative)." This causes problems: two representations of zero (±0), addition doesn't work without special logic.

Two's Complement Solution:

A clever system where the MSB has a negative place value. For 8-bit:

  Place values: -128  64  32  16   8   4   2   1
  
  Example: 10000101
  = (-128) + 4 + 1 = -123
  
  Range for 8-bit two's complement: -128 to +127
  

How to Convert to Two's Complement (Make Negative):

Method: Flip all bits, then add 1

Example: Convert +5 to -5

  Step 1: Start with +5:     00000101
  Step 2: Flip all bits:     11111010
  Step 3: Add 1:             11111011 = -5
  
  Check: -128 + 64 + 32 + 16 + 8 + 2 + 1 = -5 ✓
  

Why Two's Complement is Brilliant:

• Addition works the same for positive and negative numbers!
• Only one representation of zero (00000000)
• Easy to detect sign: if MSB = 1, it's negative
• Subtraction = add the negative (3 - 5 = 3 + (-5))

Example: Add -5 + 3 (Should = -2)

    11111011  (-5)
  + 00000011  (+3)
  ----------
   ¹¹¹¹¹¹¹¹ ← carries (overflow bit discarded)
  1 11111110  (-2)
  ↑
  Overflow bit is ignored in two's complement addition
  
  Check: 11111110 = -128 + 64 + 32 + 16 + 8 + 4 + 2 = -2 ✓