Ones Complement Calculator
Convert between decimal and ones complement binary representation with step-by-step explanations
Input & Settings
Calculate ones complement by flipping all bits
Result
Calculation output and explanation
Ones Complement Information
How Ones Complement Works
- • Positive numbers: same as binary representation
- • Negative numbers: flip all bits of positive representation
- • Range for n bits: -(2^(n-1) - 1) to +(2^(n-1) - 1)
- • Two representations for zero: +0 and -0
- • Simple to implement in hardware
Example (4-bit)
Understanding One’s Complement Arithmetic
One’s complement is a mathematical operation on binary numbers and a binary signed number representation based on this operation. It’s one of the methods used in computer systems to represent negative numbers in binary form. The one’s complement of a binary number is obtained by flipping all the bits - changing every 0 to 1 and every 1 to 0. This system was commonly used in early computer architectures and provides a simple method for representing both positive and negative numbers using the same bit patterns. While largely superseded by two’s complement in modern systems, understanding one’s complement is crucial for comprehending the evolution of computer arithmetic and is still used in certain specialized applications and networking protocols.
How One’s Complement Works
Basic Principle
In one’s complement representation:
- Positive numbers: Represented in standard binary form
- Negative numbers: Represented by flipping all bits of the positive number
- Sign bit: The most significant bit indicates sign (0 = positive, 1 = negative)
- Zero representation: Has two forms: +0 (all zeros) and -0 (all ones)
4-bit One’s Complement Examples:
Range and Properties
For n-bit one’s complement:
Examples by bit width:
Note: One’s complement has two representations for zero, which can complicate arithmetic operations and comparisons.
Conversion Algorithms
Decimal to One’s Complement:
- If positive: convert to binary normally
- If negative: convert absolute value to binary
- If negative: flip all bits (one’s complement)
- Pad with leading zeros to desired width
One’s Complement to Decimal:
- Check the sign bit (MSB)
- If 0: convert binary to decimal normally
- If 1: flip all bits first
- If 1: convert result to decimal and negate
Tip: The one’s complement operation is its own inverse - applying it twice returns the original number.
Arithmetic Operations in One’s Complement
Addition
One’s complement addition requires special handling:
- Add the two numbers normally
- If there’s a carry out of the MSB, add 1 to the result
- This is called “end-around carry”
Subtraction
Subtraction is performed by adding the one’s complement:
- Take one’s complement of subtrahend
- Add to minuend using one’s complement addition
- Handle end-around carry if present
Complexity: The end-around carry requirement makes one’s complement arithmetic more complex than two’s complement, requiring additional hardware or software logic to handle properly.
Applications and Use Cases
Historical Applications
- Early Computers: Used in CDC 6600, UNIVAC 1107, and other mainframes
- Simple Hardware: Easy to implement with basic logic gates
- Symmetric Range: Equal positive and negative ranges (except for dual zero)
- Educational Tool: Helps understand complement arithmetic concepts
Modern Applications
- Networking: Internet checksum calculations (RFC 1071)
- Error Detection: Simple parity and checksum algorithms
- Digital Signal Processing: Certain specialized arithmetic units
- Academic Study: Computer science and digital logic courses
Advantages
- Simple Negation: Just flip all bits to negate
- Symmetric: Same magnitude range for positive and negative
- Hardware Simplicity: Easy to implement basic operations
- Self-Inverse: Applying twice returns original number
Disadvantages
- Dual Zero: Both +0 and -0 representations
- Complex Addition: Requires end-around carry handling
- Comparison Issues: Two zero values complicate equality tests
- Performance: Slower than two’s complement arithmetic
Comparison with Other Number Systems
| Aspect | One’s Complement | Two’s Complement | Sign-Magnitude |
|---|---|---|---|
| Range (n-bit) | -(2^(n-1)-1) to +(2^(n-1)-1) | -2^(n-1) to +(2^(n-1)-1) | -(2^(n-1)-1) to +(2^(n-1)-1) |
| Zero Representation | Two (+0, -0) | One (0) | Two (+0, -0) |
| Negation | Flip all bits | Flip bits + add 1 | Flip sign bit |
| Addition | End-around carry | Simple binary addition | Complex (sign handling) |
| Hardware Complexity | Medium | Low | High |
| Modern Usage | Rare (networking) | Universal | Floating-point only |
Worked Examples
Example 1: Converting -10 to 8-bit One’s Complement
Example 2: Converting 10110011 to Decimal
Technical Implementation Details
Hardware Implementation
One’s complement arithmetic requires specific hardware considerations:
- Inverter Gates: Simple NOT gates can compute one’s complement
- End-Around Carry: Addition units need extra logic for carry propagation
- Zero Detection: Must handle both positive and negative zero
- Comparison Logic: Requires special handling for dual zero representation
Simple 4-bit One’s Complement Adder Logic:
Software Implementation Considerations
When implementing one’s complement in software:
- Bit Manipulation: Use XOR with all-ones mask for complement
- Range Checking: Validate inputs are within representable range
- Zero Handling: Normalize -0 to +0 for consistency
- Overflow Detection: Check for arithmetic overflow conditions
Example C Code for One’s Complement:
// One's complement for 8-bit numbers
uint8_t ones_complement(uint8_t value) {
return ~value; // Bitwise NOT
}
// One's complement addition with end-around carry
uint8_t ones_add(uint8_t a, uint8_t b) {
uint16_t sum = a + b;
if (sum > 0xFF) { // Carry out occurred
sum = (sum & 0xFF) + 1; // Add carry
}
return (uint8_t)sum;
}Learning Resources and Further Study
Practice Exercises
- 1. Convert decimal numbers -15, +7, -3 to 8-bit one’s complement
- 2. Add 01001100 + 11010011 using one’s complement arithmetic
- 3. Find the range of representable numbers in 12-bit one’s complement
- 4. Compare storage efficiency of different signed number representations
Key Concepts to Master
- • Bit manipulation and complement operations
- • End-around carry in arithmetic operations
- • Dual zero representation and its implications
- • Comparison with two’s complement systems
- • Historical context and modern applications
Common Mistakes to Avoid
- • Forgetting end-around carry in addition
- • Confusing one’s and two’s complement
- • Not handling dual zero properly
- • Incorrect range calculations
- • Assuming modern systems use one’s complement
Related Topics
- • Two’s complement arithmetic
- • Sign-magnitude representation
- • Binary coded decimal (BCD)
- • Floating-point number systems
- • Computer architecture and ALU design