How do I find the two's complement of a number?

Write the positive value in binary at your bit width, invert every bit (one's complement), then add 1. For 5 in 8 bits: 00000101 → 11111010 → 11111011, which represents −5.

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

−128 to +127. The most significant bit is the sign bit: 0 for non-negative, 1 for negative.

Two's Complement Walkthrough (How Negatives Are Stored)

Q: Why is two's complement used?

Because addition and subtraction work with a single set of rules: adding a number and its two's complement gives zero (ignoring the carry out), so the hardware needs no special case for negatives.

Two's complement is how nearly every computer represents negative whole numbers. Once you know the invert-and-add-one rule, reading and writing signed binary is straightforward — and a base-N calculator confirms it instantly.

Why not just a minus sign?

Computers only store 0s and 1s, so the minus sign has to be encoded in the bits. The naive approach (a dedicated sign bit, "sign-magnitude") creates two zeros and needs special-case addition. Two's complement avoids both problems and lets one adder handle additions and subtractions.

The invert-and-add-one rule

Write the positive value in binary at your chosen bit width.
Invert every bit (the one's complement).
Add 1.

8-bit example: −5

+5            = 00000101
invert bits   = 11111010   (one's complement)
add 1         = 11111011   ← −5 in two's complement

So in 8-bit, −5 = 11111011. As an unsigned value that pattern is 251, and 251 − 256 = −5, which is exactly the signed interpretation.

Why it works

Add 5 and −5 and watch the bits cancel:

  00000101   (5)
+ 11111011   (−5)
  --------
1 00000000   → drop the carry → 00000000 = 0  ✔

The carry out of the top bit is discarded, leaving zero — no special handling required. That's the whole point of the scheme.

Sign bit & range

The most significant bit acts as the sign: 0 for non-negative, 1 for negative. With 8 bits the signed range is −128 to +127. (Double the width to 16 bits and it becomes −32768 to +32767.)

On the calculator

In BASE-N mode, negative results are shown in two's complement form, and the NOT operator gives the one's complement — add 1 to reach two's complement. Build operands with decimal to binary conversion and operate using bitwise operations.

Frequently asked questions

How do I find the two's complement?

Write the value in binary, invert all bits, then add 1. For 5 (8-bit): 00000101 → 11111011 = −5.

Why is two's complement used?

Adding a number to its two's complement yields zero (carry dropped), so one adder handles add and subtract.

What's the range of an 8-bit signed integer?

−128 to +127, with the top bit as the sign.

Verify two's complement

Enter a negative number in BASE-N mode and view it in binary.

Open the two's complement calculator →