1 | DE Notes

Complements are used in digital systems to simplify subtraction operations and represent negative numbers. Instead of building separate subtractor circuits, computers use complement arithmetic to perform subtraction using addition hardware.

Types of Complements

For a number system with base r (radix), there are two types:

Complement Type	Formula	Binary Name
(r-1)'s complement	rⁿ - 1 - N	1's complement
r's complement	rⁿ - N	2's complement

Where n = number of digits, N = the given number.

1 0 0 1 1 ← Carry out = 1

End-around carry: 0011 + 1 = 0100 = 4₁₀ ✓

Example

**Example: 3 - 7 (negative result)**

A = 3 = 0011 B = 7 = 0111 1's complement of B = 1000

0 0 1 1 + 1 0 0 0

1 0 1 1 ← No carry out (result is negative)

1's complement of 1011 = 0100 = 4 Result = -4₁₀ ✓

- Method 1	Find 1's complement, then add 1
- Method 2	Starting from the right, keep all bits up to and including the first 1, then flip all remaining bits
### Method 1	Invert and Add 1

Original: 1 0 1 1 0 0 1's Complement: 0 1 0 0 1 1 Add 1: + 0 0 0 0 0 1

2's Complement: 0 1 0 1 0 0

Example

### Method 2: Shortcut

Original: 1 0 1 1 0 0 ↑ Keep from right until first 1: ...0 0 Flip the rest: 0 1 0 1 0 0

1. Single zero	Only 0000...0 represents zero (no -0 problem)
2. Range (n bits)	-2ⁿ⁻¹ to +(2ⁿ⁻¹ - 1)
3. Asymmetric range	One extra negative number
3. If carry out	discard carry, result is positive
4. If no carry	result is negative, take 2's complement
**Example	12 - 5 (using 5 bits)**

A = 12 = 01100 B = 5 = 00101 2's complement of B = 11011

0 1 1 0 0 + 1 1 0 1 1

1 0 0 1 1 1 ← Carry out (discard)

Result = 00111 = 7₁₀ ✓

Example

**Example: 5 - 12 (negative result)**

A = 5 = 00101 B = 12 = 01100 2's complement of B = 10100

0 0 1 0 1 + 1 0 1 0 0

1 1 0 0 1 ← No carry (negative)

2's complement of 11001 = 00111 = 7 Result = -7₁₀ ✓

Example

--- ## 9's and 10's Complement (Decimal) ### 9's Complement Subtract each digit from 9.

9's complement of 3456 = 9999 - 3456 = 6543 9's complement of 0870 = 9999 - 0870 = 9129

Example

### 10's Complement 9's complement + 1.

10's complement of 3456 = 6543 + 1 = 6544 10's complement of 2500 = 7499 + 1 = 7500

Example

--- ## Hardware Implementation

┌─────────────────────┐ B₃ ──────►│ │ B₂ ──────►│ XOR Gates │──► To Adder B₁ ──────►│ (Controlled │ B₀ ──────►│ Inverter) │ │ │ SUB ──────►│ Control Signal │──► Carry-In └─────────────────────┘

When SUB = 0: B passes unchanged, Cin = 0 (Addition) When SUB = 1: B is inverted, Cin = 1 (2's complement subtraction)

Feature	1's Complement	2's Complement
Method	Flip all bits	Flip + add 1
Zero representations	Two (+0, -0)	One
Range (n bits)	-(2ⁿ⁻¹-1) to +(2ⁿ⁻¹-1)	-2ⁿ⁻¹ to +(2ⁿ⁻¹-1)
Subtraction carry	End-around carry needed	Discard carry
Hardware	Slightly complex	Simpler (preferred)
Modern usage	Rare (checksums)	Universal

Types of Complements

For a number system with base r (radix), there are two types:

Complement Type	Formula	Binary Name
(r-1)'s complement	rⁿ - 1 - N	1's complement
r's complement	rⁿ - N	2's complement

Where n = number of digits, N = the given number.

1 0 0 1 1 ← Carry out = 1

End-around carry: 0011 + 1 = 0100 = 4₁₀ ✓

Example

**Example: 3 - 7 (negative result)**

A = 3 = 0011 B = 7 = 0111 1's complement of B = 1000

0 0 1 1 + 1 0 0 0

1 0 1 1 ← No carry out (result is negative)

1's complement of 1011 = 0100 = 4 Result = -4₁₀ ✓

- Method 1	Find 1's complement, then add 1
- Method 2	Starting from the right, keep all bits up to and including the first 1, then flip all remaining bits
### Method 1	Invert and Add 1

Original: 1 0 1 1 0 0 1's Complement: 0 1 0 0 1 1 Add 1: + 0 0 0 0 0 1

2's Complement: 0 1 0 1 0 0

Example

### Method 2: Shortcut

Original: 1 0 1 1 0 0 ↑ Keep from right until first 1: ...0 0 Flip the rest: 0 1 0 1 0 0

1. Single zero	Only 0000...0 represents zero (no -0 problem)
2. Range (n bits)	-2ⁿ⁻¹ to +(2ⁿ⁻¹ - 1)
3. Asymmetric range	One extra negative number
3. If carry out	discard carry, result is positive
4. If no carry	result is negative, take 2's complement
**Example	12 - 5 (using 5 bits)**

A = 12 = 01100 B = 5 = 00101 2's complement of B = 11011

0 1 1 0 0 + 1 1 0 1 1

1 0 0 1 1 1 ← Carry out (discard)

Result = 00111 = 7₁₀ ✓

Example

**Example: 5 - 12 (negative result)**

A = 5 = 00101 B = 12 = 01100 2's complement of B = 10100

0 0 1 0 1 + 1 0 1 0 0

1 1 0 0 1 ← No carry (negative)

2's complement of 11001 = 00111 = 7 Result = -7₁₀ ✓

Example

--- ## 9's and 10's Complement (Decimal) ### 9's Complement Subtract each digit from 9.

9's complement of 3456 = 9999 - 3456 = 6543 9's complement of 0870 = 9999 - 0870 = 9129

Example

### 10's Complement 9's complement + 1.

10's complement of 3456 = 6543 + 1 = 6544 10's complement of 2500 = 7499 + 1 = 7500

Example

--- ## Hardware Implementation

When SUB = 0: B passes unchanged, Cin = 0 (Addition) When SUB = 1: B is inverted, Cin = 1 (2's complement subtraction)

Feature	1's Complement	2's Complement
Method	Flip all bits	Flip + add 1
Zero representations	Two (+0, -0)	One
Range (n bits)	-(2ⁿ⁻¹-1) to +(2ⁿ⁻¹-1)	-2ⁿ⁻¹ to +(2ⁿ⁻¹-1)
Subtraction carry	End-around carry needed	Discard carry
Hardware	Slightly complex	Simpler (preferred)
Modern usage	Rare (checksums)	Universal

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Types of Complements

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1

Types of Complements

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