Complements; Codes; Signed Number Representations

Complements

Complements simplify subtraction and represent negative numbers.

6.1 1's Complement (Binary)

Flip every bit (0→1, 1→0).

• +5 = 00101 → -5 (5-bit) = 11010
• Used in older systems; has two representations of zero

6.2 2's Complement (Binary)

1's complement + 1 (add 1 to LSB).

• Most widely used in modern computers
• Unique zero representation
• Range for n bits: -2^(n-1) to +2^(n-1)-1

Example: -5 in 8-bit 2's complement:

1. +5 = 00000101
2. 1's comp = 11111010
3. +1 → 11111011

6.3 9's Complement (Decimal)

Subtract each digit from 9.

Used for decimal subtraction: (A - B) = A + 9's_comp(B) + 1 (end-around carry)

Example: 52520 - 38380 using 9's complement:

• 9's comp of 52520 (5 digits) = 99999 - 52520 = 47479
• Add with end-around carry to get result

6.4 10's Complement (Decimal)

9's complement + 1

Codes

7.1 Weighted vs Non-Weighted Codes

7.2 BCD (Binary Coded Decimal)

• Each decimal digit → 4-bit binary
• Weights: 8, 4, 2, 1
• Valid BCD: 0000 to 1001 only (0–9)
• Invalid: 1010–1111 (used for other codes)

Example: (254)₁₀ → BCD

BCD → Decimal: Group in 4-bit nibbles: (0010 0101 0100) → 254

BCD → Binary: First convert BCD to decimal, then decimal to binary.

7.3 Excess-3 Code (XS-3)

• Non-weighted code
• Each BCD code + 0011 (3)
• Used for BCD addition (avoids invalid states during carry)

Conversion:

• Decimal → BCD → Add 3 to each digit
• Excess-3 → Subtract 3 from each nibble → BCD → Decimal

Example: (5)₁₀ → 0101 (BCD) → 0101 + 0011 = 1000 (Excess-3)

7.4 Gray Code

• Non-weighted, not arithmetic
• Only one bit changes between consecutive values
• Used in: shaft encoders, K-maps, error reduction in transitions

Binary → Gray: MSB same; each next Gray bit = XOR of current and previous binary bit.

Gray → Binary: MSB same; each next binary bit = XOR of Gray bit and previous binary result.

Signed Number Representations

8.1 Unsigned Numbers

• No sign bit — all values positive
• n-bit range: 0 to 2ⁿ - 1

8.2 Sign-Magnitude

• MSB = sign (0=+, 1=-)
• Remaining bits = magnitude
• Two zeros: +0 and -0
• n-bit range: -(2^(n-1)-1) to +(2^(n-1)-1)

Example (6-bit): -31 = 1 11111, +31 = 0 11111

8.3 1's Complement

• Positive: normal binary
• Negative: 1's complement of magnitude
• Two zeros — ambiguous
• Same range as sign-magnitude

8.4 2's Complement (Standard)

• Positive: normal binary
• Negative: 2's complement of magnitude
• Single zero
• n-bit range: -2^(n-1) to +2^(n-1)-1

Example (6-bit): -32 = 100000, +31 = 011111

8085 and most microprocessors use 2's complement for signed arithmetic.