What is two's complement?

Two's complement is a binary encoding for signed integers. The most significant bit (MSB) acts as a sign bit. To negate a number, invert all bits and add 1. It is used by virtually all modern CPUs because addition and subtraction work the same way for both positive and negative numbers.

What range fits in N bits?

An N-bit two's complement number can hold values from −2^(N−1) to 2^(N−1)−1. For 8 bits that is −128 to 127; for 16 bits, −32 768 to 32 767; for 32 bits, −2 147 483 648 to 2 147 483 647.

How do I convert a negative number to two's complement?

Take the absolute value, write it in binary padded to N bits, invert every bit (one's complement), then add 1 to the result. The MSB of a negative two's complement number is always 1.

How do I convert a two's complement binary string back to decimal?

If the MSB is 0 the number is positive and you read it normally. If the MSB is 1, invert all bits and add 1 to get the magnitude, then apply a minus sign.

Why does 10000000 equal −128 in 8-bit two's complement?

Inverting 10000000 gives 01111111, adding 1 gives 10000000 again — a carry overflow that wraps to 128. By convention this bit pattern represents the minimum value, −2^(N−1), which is −128 for 8 bits.

Twos Complement Calculator

Convert any integer to two's complement binary (and back) for 4, 8, 16, 32, or 64-bit widths.

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Summary

Two's complement is the standard way modern CPUs represent signed integers in binary. This calculator converts a decimal integer to its two's complement binary form for any chosen bit width, and converts two's complement binary strings back to their signed decimal value. It also shows the hexadecimal and octal equivalents and a step-by-step derivation.

How it works

Choose a bit width: 4, 8, 16, 32, or 64 bits.
Enter a decimal integer (or a binary string in "Binary to Decimal" mode).
The tool checks the value fits in the selected bit width's signed range.
For positive numbers, the result is the standard binary representation padded with leading zeros.
For negative numbers, the tool inverts all bits of the absolute value and adds 1.
Hex and octal equivalents are derived from the resulting bit pattern.

Use cases

Understand how negative numbers are encoded in assembly or C.
Verify signed integer overflow boundaries when writing embedded firmware.
Debug bit-manipulation code by checking the expected binary pattern.
Convert raw memory dumps to signed decimal values.
Teach or learn computer architecture and binary arithmetic.
Quickly compute the two's complement of a value for interview practice.

Frequently Asked Questions

Last updated: 2026-06-04 · Reviewed by Nham Vu

Twos Complement Calculator

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