Unit III — Binary Arithmetic: Multiplication & Division

Binary multiplication and division mirror their decimal counterparts but are simplified because the only digits are 0 and 1, making the multiplication table trivial.

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Binary Multiplication Rules

Key insight: Binary multiplication is essentially AND operation plus shifting.

• Multiplying by 0 gives 0.
• Multiplying by 1 leaves the number unchanged.
• Each partial product is shifted left by one position for each bit position of the multiplier.

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Binary Multiplication Examples

Example 1: Multiply (1011)₂ × (101)₂

        1 0 1 1   (11)
      ×   1 0 1   (5)
      ---------
        1 0 1 1   ← 1011 × 1 (bit 0, no shift)
      0 0 0 0 0   ← 1011 × 0 (bit 1, shift left 1)
    1 0 1 1 0 0   ← 1011 × 1 (bit 2, shift left 2)
    ---------
    1 1 0 1 1 1   (55)

Step-by-step addition of partial products:

• 1011 + 000000 + 101100 = 110111₂ = 55 ✓ (11 × 5 = 55)

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Example 2: Multiply (110)₂ × (11)₂

      1 1 0   (6)
    ×   1 1   (3)
    -------
      1 1 0   ← 110 × 1 (bit 0)
    1 1 0 0   ← 110 × 1 (bit 1, shift left 1)
    -------
   1 0 0 1 0  (18)

Result: (10010)₂ = 18 ✓ (6 × 3 = 18)

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Binary Division

Binary division follows long division — the same process as decimal but with only 0 and 1.

Steps:

1. Take bits from the dividend from left to right.
2. If the current partial dividend ≥ divisor → quotient bit = 1; subtract divisor.
3. If the current partial dividend < divisor → quotient bit = 0; bring down next bit.

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Example: Divide (110110)₂ ÷ (110)₂ (i.e., 54 ÷ 6)

Dividend: 110110
Divisor:  110

110 goes into 110 → 1, remainder 000
Bring down 1 → 0001
110 does not go into 0001 → 0
Bring down 1 → 00011
110 does not go into 00011 → 0
Bring down 0 → 000110
110 goes into 110 → 1, remainder 000

Quotient: 1001  (=9)
Remainder: 0

Verification: 9 × 6 = 54 ✓

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Summary Table

Key Takeaway: Binary multiplication reduces to AND operations and left shifts — much simpler than decimal multiplication. Binary division is long division with only two possible quotient bits (0 or 1). Always verify results by converting back to decimal.