🧮 Binary Complement Arithmetic

This module explores how 1’s complement and 2’s complement systems handle addition, subtraction, and multiplication in binary. These systems allow negative numbers to be encoded and manipulated using only addition and bitwise logic.

📌 1. Complement Basics

➕ 2. Addition in Complement Systems

✅ 2’s Complement Addition

  A =  0101   (5)
  B =  1110   (-2 in 2's complement)
------------------
A + B =  0011   (3)

• No special handling needed.
• Overflow is ignored unless sign bit flips unexpectedly.

✅ 1’s Complement Addition (with end-around carry)

  A =  0101   (5)
  B =  1101   (-2 in 1's complement)
------------------
Sum =  10010  → drop overflow bit → 0010
Add carry: 0010 + 1 = 0011 (3)

Overflow is ignored because we specifically designed complements to ignore overflow, it is by design, check Derivation of Complement Forms to understand why it was designed like so

➖ 3. Subtraction via Complement Addition

✅ 2’s Complement Subtraction

To compute A - B, do A + (2’s complement of B):

  A =  0101   (5)
  B =  0011   (3)
  ~B + 1 = 1100 + 1 = 1101 (-3)
------------------
A - B = 0101 + 1101 = 10010 → drop overflow → 0010 (2)

✅ 1’s Complement Subtraction

Use A + (1’s complement of B) + 1, then apply end-around carry:

  A =  0101   (5)
  B =  0011   (3)
  ~B = 1100
------------------
Sum = 0101 + 1100 + 1 = 10010 → drop overflow → 0010
Add carry: 0010 + 1 = 0011 (2)

✖️ 4. Multiplication in Complement Systems

Multiplication is typically done using unsigned logic, then sign is handled separately.

✅ Example: 2’s Complement Multiplication

  A =  1110   (-2)
  B =  0011   (3)
------------------
Unsigned: 2 × 3 = 6
Sign: Negative × Positive = Negative
Result: 11111010 (−6 in 8-bit 2’s complement)

⚠️ Notes

• Multiplication requires sign extension.
• Most systems use Booth’s algorithm or signed multiplication logic.

🧩 Summary Table