How does two's complement encode negative numbers?

For a negative value in W bits, the stored pattern equals 2^W plus the value. Equivalently, you invert all bits of the magnitude and add one. Positive values are stored as their plain binary.

What is the range of an 8-bit signed integer?

An 8-bit two's-complement integer represents -128 to +127. In general, W bits cover -2^(W-1) to 2^(W-1)-1, which is why the negative range extends one further than the positive.

Why is two's complement used everywhere?

Because the same binary adder hardware correctly handles signed and unsigned arithmetic, and there is only one representation of zero. This simplicity is why nearly all modern CPUs use it.

What does the most significant bit mean?

In two's complement the most significant bit is the sign bit: 0 means non-negative and 1 means negative. But unlike sign-magnitude, the remaining bits are not a plain magnitude when the sign bit is set.

Is my input transmitted anywhere?

No. The conversion uses BigInt arithmetic locally in your browser, so it is exact up to 64 bits and nothing leaves your device.

Two's Complement Converter

Two’s complement is the dominant way computers store signed integers. It lets a single set of adder circuits handle both signed and unsigned arithmetic and gives a unique representation of zero. This converter shows the exact bit pattern for any signed integer at common widths.

How it works

For a width of W bits the signed range is -2^(W-1) to 2^(W-1) - 1. To encode a value:

If the value is non-negative, write it in plain binary, zero-padded to W bits.
If the value is negative, the stored pattern is 2^W + value. A practical recipe: take the magnitude in binary, invert every bit, then add 1.

Decoding reverses this: if the top (sign) bit is 1, subtract 2^W from the unsigned reading to recover the negative value.

Example

Encode -42 in 8 bits. The magnitude 42 is 00101010. Invert the bits to get 11010101, then add 1 to reach 11010110. That is 0xD6, and read as unsigned it is 214, which equals 256 - 42 as expected.

Notes

The tool uses BigInt internally, so 64-bit values are computed exactly without floating-point rounding. If you enter a number outside the signed range for the chosen width, it tells you the valid range rather than silently wrapping.