Two's complement calculator

What is two’s complement?

Two’s complement is the standard way computers store signed integers — numbers that can be positive or negative — using a fixed number of bits. Instead of setting aside a separate symbol for the minus sign, the negative numbers are encoded so that ordinary binary addition just works, with no special handling for the sign. Almost every modern CPU represents integers this way.

This calculator takes a decimal integer and a bit width (8, 16, or 32 bits) and shows its two’s complement pattern in both binary and hexadecimal.

How does it work?

For a chosen width of $w$ bits, each value is stored as an unsigned bit pattern:

• If the number $n$ is non-negative, its pattern is simply the binary of $n$, padded with leading zeros to $w$ bits.
• If the number $n$ is negative, its pattern is the binary of $2^w + n$.

Because $n$ is negative in the second case, $2^w + n$ is a positive value smaller than $2^w$, so it always fits in $w$ bits. The most significant (leftmost) bit ends up being $1$ for every negative number and $0$ for every non-negative one — that bit acts as the sign.

Formula

The result is then written with exactly $w$ binary digits (or $w/4$ hexadecimal digits).

Worked examples

Example 1: a positive number

Encode $n = 5$ in $8$ bits. Since $5 \ge 0$, the pattern is just $5$ in binary, padded to eight digits:

Example 2: negative one

Encode $n = -1$ in $8$ bits. Since $-1 < 0$, compute $2^8 + (-1) = 256 - 1 = 255$:

Negative one is always an unbroken row of ones, whatever the width.

Example 3: negative five

Encode $n = -5$ in $8$ bits. Compute $2^8 + (-5) = 256 - 5 = 251$:

Reference table (8-bit)

Notes

• An $8$-bit signed integer covers the range $-128$ to $127$; $16$ bits covers $-32{,}768$ to $32{,}767$; $32$ bits covers roughly $\pm 2.1$ billion. Values outside the chosen range wrap around modulo $2^w$.
• The leading bit is the sign: $0$ marks a non-negative number and $1$ marks a negative one.
• To convert an ordinary non-negative number to binary without a sign bit, use the decimal to binary converter, or the general number system converter for other bases.

FAQ

What is -1 in two’s complement?

In any width it is all ones: $11111111_2$ (0xFF) for 8 bits, $1111111111111111_2$ (0xFFFF) for 16 bits, and so on.

How do I get the two’s complement of a negative number by hand?

Write the size of the number in binary, flip every bit, then add one. For $-5$ in 8 bits: $5$ is $00000101$, flipping gives $11111010$, and adding one gives $11111011$ — the same result as $256 - 5 = 251$.

Why does the bit width matter?

The width fixes how many bits the pattern occupies and therefore the range of numbers you can store. The same decimal value produces a longer string of leading zeros or ones as the width grows.