How the two's complement calculator works
Choose a width and an encoding, then type a decimal number. The result card shows the bit pattern in groups of four with its hex form, and the bit grid marks the sign bit. For a negative number in two's complement, the working shows the size in binary, every bit flipped, and the 1 added, with a check that reading the pattern as unsigned and subtracting 2 to the power of the width gives your number back.
The “read four ways” table shows what the same bits mean as unsigned, two's complement, one's complement and sign-magnitude — a pattern never records its own encoding. To add or subtract signed values, use the binary calculator.
The two's complement formula
Example: −5 in 8 bits is 2⁸ − 5 = 251 = 11111011. Reading it back: −128 + 64 + 32 + 16 + 8 + 2 + 1 = −5.
Worked examples
Example 1 — encoding −100 in 8 bits
- Number:
- −100
- Width:
- 8 bits
- 100 in binary: 01100100
- Invert every bit: 10011011
- Add 1: 10011100
Two's complement: 10011100 (0x9C)
Check: 10011100 is 156 unsigned, and 156 − 256 = −100.
Example 2 — decoding a 16-bit pattern
- Pattern:
- 1111 1111 1000 0000
- Top bit is 1: negative
- Read as unsigned: 65,408
- Subtract 2¹⁶: 65,408 − 65,536
Value: −128
It is the 8-bit pattern 10000000 sign-extended to 16 bits by copying the 1 into the new top byte.
Integer ranges by width
| Width | Two's complement (signed) | Unsigned |
|---|---|---|
| 8-bit | −128 to 127 | 0 to 255 |
| 16-bit | −32,768 to 32,767 | 0 to 65,535 |
| 32-bit | −2,147,483,648 to 2,147,483,647 | 0 to 4,294,967,295 |
| 64-bit | −9,223,372,036,854,775,808 to 9,223,372,036,854,775,807 | 0 to 18,446,744,073,709,551,615 |
Key facts
- In n-bit two's complement, a negative number −x is stored as the unsigned value 2ⁿ − x.
- An 8-bit two's complement integer holds −128 to 127, and a 32-bit one holds −2,147,483,648 to 2,147,483,647.
- In two's complement, −1 is all ones at every width, and the most negative number is a 1 followed by all zeros.
- Since C++20, two's complement is the only signed integer representation the C++ standard allows.
Why computers use two's complement
With two's complement, the circuit that adds unsigned numbers also adds signed ones: 11111011 (−5) plus 00000011 (3) is 11111110, which is −2, with no special case for the sign. There is also only one zero, unlike one's complement and sign-magnitude. That is why every mainstream processor uses it, and why C++20 made it the only representation the language allows (source: cppreference). You need this when reading a memory or register dump, decoding a sensor value that arrives as raw bytes, debugging an integer overflow, or answering a computer architecture exam question.
Common two's complement mistakes
- Forgetting the width. −5 is 11111011 in 8 bits but 1111111111111011 in 16. Without a width a negative pattern is meaningless.
- Inverting without adding 1. That gives one's complement: 11111010 is −5 only in one's complement; read as two's complement it is −6.
- Padding with zeros. Widening a negative value means copying its sign bit (sign extension), not adding zeros on the left.
- Negating the minimum. −(−128) does not fit in 8 bits; inverting and adding 1 to 10000000 gives 10000000 again.
Frequently Asked Questions
What is −1 in two's complement?
−1 is all ones at every width: 11111111 in 8 bits, 16 ones in 16 bits, and so on. Invert 00000001 to get 11111110, then add 1 to get 11111111. Read as unsigned, the same pattern is 255.
How do you find the two's complement of a negative number?
Write the number's size in binary at the full width, flip every bit, then add 1. For −12 in 8 bits: 12 is 00001100, flipped it is 11110011, and adding 1 gives 11110100. Another way is to subtract the size from 2 to the power of the width: 256 − 12 = 244, which is 11110100.
How do you convert a two's complement number back to decimal?
If the top bit is 0, read it as ordinary binary. If the top bit is 1, the number is negative: read the bits as unsigned and subtract 2 to the power of the width. 11110100 is 244 unsigned, and 244 − 256 = −12.
Why can 8 bits hold −128 but only +127?
There are 256 patterns. Zero uses one, leaving 255 to split between negatives and positives, so one side gets an extra value. Two's complement gives it to the negatives: 10000000 is −128, and its positive counterpart +128 would need a ninth bit.
What happens when a two's complement number overflows?
Adding 1 to the largest value wraps to the smallest: in 8 bits, 127 + 1 gives 10000000, which reads as −128. The calculator reports a value outside the range as an overflow, shows the range, and shows the pattern a register would wrap it to.
What is the difference between one's and two's complement?
One's complement only flips the bits, so −5 is 11111010 and there are two zeros, 00000000 and 11111111. Two's complement flips the bits and adds 1, so −5 is 11111011, there is one zero, and ordinary binary addition gives the right answer for negative numbers too.
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Notes & Assumptions
- Inputs are whole numbers, worked exactly with arbitrary-precision integers at 8, 16, 32 or 64 bits.
- A value outside the range is reported as overflow rather than wrapped silently; for unsigned and two's complement the wrapped pattern is shown as well.
- Fractional and floating-point values are not encoded here; the IEEE 754 converter covers those.
This tool is for education and general calculation. Check how a particular language or processor handles overflow against its documentation.