Two's Complement Calculator
Calculate one's complement and two's complement of binary numbers. Understand negative number representation in binary.
Valid characters: 0 and 1 only
Select the bit width for representation
What is Two's Complement?
Two's complement is a mathematical operation on binary numbers and the most common method of representing signed (positive and negative) integers in computers. In this system, positive numbers are represented normally in binary, while negative numbers are represented by taking the two's complement of their positive counterpart.
The two's complement is calculated by inverting all the bits (one's complement) and then adding 1 to the result. For example, to represent -5 in 8-bit two's complement: Start with +5 (00000101), invert to get 11111010, add 1 to get 11111011. This elegant system eliminates the need for separate addition and subtraction circuits in computer processors.
Our free two's complement calculator makes these conversions instant and accurate. Choose your bit width (4-bit, 8-bit, 16-bit, or 32-bit), enter any binary number, and get the one's complement, two's complement, and both signed and unsigned decimal representations. Perfect for computer science students, programmers, and anyone working with low-level binary operations.
How to Calculate Two's Complement
Understanding the two's complement calculation process:
Two-Step Method:
- Step 1: One's Complement - Invert all bits (change 0 to 1 and 1 to 0)
- Step 2: Add One - Add 1 to the one's complement result
Example: Calculate Two's Complement of 00001010 (8-bit)
Original Binary: 00001010 (decimal 10) Step 1: One's Complement (flip all bits) 0 → 1, 0 → 1, 0 → 1, 0 → 1, 1 → 0, 0 → 1, 1 → 0, 0 → 1 Result: 11110101 Step 2: Add 1 11110101 + 1 ---------- 11110110 Two's Complement: 11110110 Interpretation: - As unsigned: 246 (decimal) - As signed (two's complement): -10 (decimal) This represents -10 in 8-bit two's complement notation.
The MSB (Most Significant Bit) serves as the sign bit: 0 = positive, 1 = negative. For other binary operations, check our Binary Calculator tool.
Two's Complement Range Table
The range of numbers that can be represented in two's complement varies by bit width:
| Bit Width | Minimum Value | Maximum Value | Total Values |
|---|---|---|---|
| 4-bit | -8 | +7 | 16 |
| 8-bit | -128 | +127 | 256 |
| 16-bit | -32,768 | +32,767 | 65,536 |
| 32-bit | -2,147,483,648 | +2,147,483,647 | 4,294,967,296 |
| 64-bit | -9,223,372,036,854,775,808 | +9,223,372,036,854,775,807 | 18,446,744,073,709,551,616 |
Formula: For n-bit two's complement, range is from -2(n-1) to 2(n-1) - 1
Why Use Two's Complement?
Two's complement is the standard for representing signed integers because of these advantages:
➕ Single Addition Circuit
Computers can use the same circuitry for both addition and subtraction (just add the negative number).
🔢 Unique Zero
Only one representation of zero (unlike one's complement which has +0 and -0).
⚡ Fast Operations
Arithmetic operations are efficient without special case handling for signs.
🖥️ Universal Standard
All modern CPUs use two's complement for integer representation.
📊 Range Efficiency
Uses all available bit patterns efficiently (no wasted combinations).
🔧 Simple Negation
To negate a number, just take its two's complement—no complex logic needed.
Common Two's Complement Examples
Here are practical two's complement examples in 8-bit representation:
Decimal -1
Binary: 11111111 (all bits set)
Process: +1 (00000001) → flip (11111110) → +1 = 11111111
Decimal -128 (8-bit minimum)
Binary: 10000000
Most negative number in 8-bit two's complement
Decimal +127 (8-bit maximum)
Binary: 01111111
Most positive number in 8-bit two's complement
Decimal -10
Binary: 11110110
Process: +10 (00001010) → flip (11110101) → +1 = 11110110
Decimal 0
Binary: 00000000
Only one representation of zero (unique property)
Benefits of Using Our Calculator
ProURLMonitor's free two's complement calculator offers multiple advantages:
- ✓Instant Calculation: Calculate one's and two's complement in milliseconds.
- ✓Multiple Bit Widths: Support for 4-bit, 8-bit, 16-bit, and 32-bit representations.
- ✓Step-by-Step Process: See exactly how one's and two's complement are calculated.
- ✓Dual Interpretation: View both unsigned and signed (two's complement) decimal values.
- ✓Input Validation: Automatically checks for valid binary input and bit width constraints.
- ✓Educational Tool: Perfect for learning computer architecture and digital systems.
- ✓100% Free: No registration, downloads, or hidden costs. Use unlimited times.
- ✓Mobile Friendly: Works on all devices—desktop, tablet, and smartphone.
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📚 Frequently Asked Questions (FAQs)
Q: What is the two's complement of 00001010?
A: Flip bits to get 11110101, then add 1 to get 11110110. This represents -10 in signed notation.
Q: How do I know if a binary number is negative in two's complement?
A: Check the most significant bit (MSB). If it's 1, the number is negative. If it's 0, the number is positive.
Q: Why does two's complement need to add 1 after flipping bits?
A: Adding 1 ensures that a number plus its two's complement equals zero, which is essential for subtraction to work as addition of negative numbers.
Q: What's the difference between one's and two's complement?
A: One's complement just flips all bits. Two's complement flips all bits AND adds 1. Two's complement is preferred because it has only one zero and simplifies arithmetic.
Q: Can I represent -128 in 8-bit two's complement?
A: Yes! -128 is represented as 10000000. Interestingly, you cannot represent +128 in 8-bit two's complement (range is -128 to +127).
Q: Why is the range asymmetric (e.g., -128 to +127)?
A: Because zero uses a positive bit pattern (00000000), leaving one extra negative value. This is why minimum is -2^(n-1) and maximum is 2^(n-1) - 1.
Q: How do computers perform subtraction using two's complement?
A: To compute A - B, computers calculate A + (-B), where -B is the two's complement of B. This allows using the same addition circuit for both operations.
Q: What happens if I overflow in two's complement?
A: Overflow occurs when the result exceeds the representable range. For example, in 8-bit: 127 + 1 wraps to -128. Most systems have overflow flags to detect this.
🚀 Start Calculating Two's Complement Now!
Use our free two's complement calculator to understand negative binary number representation. Perfect for computer science students, embedded systems programmers, digital logic designers, and anyone learning computer architecture. Calculate one's complement, two's complement, and see both signed and unsigned interpretations instantly.
No registration required. Unlimited calculations. Completely free forever!
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