Two's Complement Calculator

Write the number in binary with the width you need, invert every bit, then add 1. For 50 in 8 bits, 0011 0010 inverts to 1100 1101, and adding 1 gives 1100 1110, which represents -50. The calculator shows each step.

For a non-negative number, write its ordinary binary value and pad to the bit width. For a negative number, write the absolute value in binary, invert the bits and add 1, or compute 2ⁿ plus the number. -50 in 8 bits is 256 − 50 = 206 = 1100 1110.

If the leading bit is 0, read it as an ordinary binary number. If it is 1, the number is negative: treat the leading bit as having weight -2ⁿ⁻¹ and add the rest. 1100 1110 is -128 + 64 + 8 + 4 + 2 = -50. Or invert, add 1 and put a minus sign.

Invert all of the bits and add 1. The same operation turns a positive number into its negative and a negative into its positive. 0000 0101 (5) becomes 1111 1011 (-5), and applying it again gives back 0000 0101. The calculator's negate mode does exactly this.

Two's complement is the standard way computers store signed integers. The leading bit is the sign bit, and an n-bit pattern represents numbers from -2ⁿ⁻¹ to 2ⁿ⁻¹ − 1. Negatives are found by inverting the bits of the positive and adding 1.

Addition and subtraction use the same hardware for signed and unsigned numbers, there is just one representation of zero and negation is simple. The sum of a number and its two's complement is 2ⁿ, which wraps to 0 in n bits.

An 8-bit signed integer runs from -128 to 127, which is 256 values. The positive end is 0111 1111 and the negative end is 1000 0000. For 4 bits the range is -8 to 7, for 16 bits it is -32,768 to 32,767 and for 32 bits it is -2,147,483,648 to 2,147,483,647.

Zero is its own two's complement. Invert 0000 0000 to get 1111 1111, add 1 and the carry falls off the end, leaving 0000 0000. There is only one zero in two's complement, which is one of its main advantages over older schemes.

In 8 bits, -128 (1000 0000) is its own negation: inverting gives 0111 1111 and adding 1 gives 1000 0000 again. The positive 128 does not fit in 8 bits. The calculator flags this case when you negate the pattern.

One's complement just inverts the bits. Two's complement inverts and adds 1. One's complement has two zeros (0000 and 1111) and needs an end-around carry in addition. Two's complement has one zero and plain addition works. The comparison table above shows the three signed formats.

In sign-magnitude, the leading bit is the sign and the rest is the absolute value, so +3 is 0011 and -3 is 1011, and there are two zeros. In two's complement -3 is 1101, and arithmetic needs no special handling for the sign.

To compute a - b, add a to the two's complement of b and ignore any carry out of the top bit. For 5 − 3, add 0000 0101 and 1111 1101 to get 0000 0010, which is 2. Subtraction is addition of the negative.

Add the bit patterns as ordinary unsigned binary and drop any carry beyond the top bit. 5 + (-3) is 0000 0101 + 1111 1101 = 1 0000 0010, which is 0000 0010, or 2 after dropping the carry. The addition table above shows more cases.

Overflow happens when the true result is outside the range for the bit width, so the pattern wraps to the wrong sign. In 8 bits, 100 + 28 = 128 does not fit, and the pattern 1000 0000 reads as -128. Two positives giving a negative, or two negatives giving a positive, signals overflow.

The sign bit is the leading, most significant bit. It is 0 for non-negative numbers and 1 for negative ones. In two's complement it has the weight -2ⁿ⁻¹, so 1000 0000 is -128 in 8 bits, not -0. Unlike sign-magnitude, it is part of the number and not just a flag.

Sign extension widens a number to more bits by copying the sign bit into the new high bits. The 8-bit pattern 1111 0110 (-10) becomes 1111 1111 1111 0110 in 16 bits, which is still -10. Padding with zeros would give 246.

Convert the hex digits to binary, giving four bits per digit. If the top bit is 1, the number is negative. 0xCE is 1100 1110, so it is -50 in 8 bits. 0x8000 in 16 bits is -32,768. Choose hex input in the calculator and the bit width.

Find the two's complement bit pattern, group it in fours from the right and replace each group by a hex digit. -50 in 8 bits is 1100 1110, which is 0xCE. The calculator shows the bit pattern and its hex form together.

In 4 bits, 1010 is -6. Invert it to get 0101 and add 1 to get 0110, which is +6. So the two's complement of 1010 is 0110. As unsigned numbers the same pattern is 10, and its complement mod 16 is 6.

In 4 bits, 1011 is -5. Inverting gives 0100, and adding 1 gives 0101, which is 5. So negating 1011 gives 0101. The calculator has a negate mode that does this for any pattern at 4, 8, 16 or 32 bits, with every step shown.

-1 is all ones, whatever the width: 1111 in 4 bits, 1111 1111 in 8 bits and 0xFFFFFFFF in 32 bits. Adding 1 to it gives zero with a carry that is dropped. It is a handy pattern to recognise.

There are 2ⁿ patterns. Half of them, 2ⁿ⁻¹, have the sign bit set and so are negative, while the other half includes zero, leaving 2ⁿ⁻¹ − 1 positives after zero takes its place. In 8 bits that gives -128 to 127, with 128 negatives and 127 positives.

Find the smallest n such that -2ⁿ⁻¹ ≤ value ≤ 2ⁿ⁻¹ − 1. For -50 you need 7 bits, since the 7-bit range is -64 to 63. The calculator offers 4, 8, 16 and 32 bits, and tells you when a value does not fit.

This calculator works with whole-number integers only. Fixed-point fractions use the same invert-and-add-one idea, applied with the binary point at a fixed position, but that is a separate topic and is not covered here. Floating-point numbers use a different format altogether.

Yes. The two's complement of x in n bits is 2ⁿ − x, as an unsigned number. Whether you call the pattern signed or unsigned depends on how you read it. The calculator shows both the signed and unsigned value for every pattern.

Yes. It gives the final bit pattern, the hex form and the signed and unsigned values, plus the steps. Try your own value in a conversion and compare it with your working. It handles 4, 8, 16 and 32-bit widths and exact values.

Write the absolute value in binary, invert every bit and add 1, using a fixed width. For -5 in 8 bits: 5 is 0000 0101, inverted 1111 1010, plus one is 1111 1011. The calculator does this and shows the steps for 4, 8, 16 and 32 bits.

If the leading bit is 1, subtract 2ⁿ from the unsigned value, or invert the bits, add 1 and put a minus sign in front. 1111 1011 is 251 unsigned, and 251 − 256 = -5. Choose the 'two's complement to decimal' mode in the calculator.

In 8 bits, 5 is 0000 0101. Inverting gives 1111 1010 and adding 1 gives 1111 1011, which is -5 (0xFB). In 4 bits, 5 is 0101 and its two's complement is 1011. The width decides how many leading ones appear.

In 8 bits, 1 is 0000 0001. Inverting gives 1111 1110 and adding 1 gives 1111 1111, which is -1 (0xFF). So the two's complement of 1 is all ones at any width, which is why -1 is the easiest negative number to recognise in a register or a hex dump.

In 8-bit two's complement it is negative: it is -128, the smallest value. The sign bit is 1, and its weight is -128, with all other bits zero. As an unsigned number the same pattern is 128, so whether it is positive or negative depends entirely on how the bits are read.

It depends on the width. In 8 bits 0xFF is -1, while in 16 bits it is 255, because the sign bit is a zero. The calculator needs the bit width and shows both the signed and unsigned values of any pattern.

Convert each hex digit to four bits, invert them all, add 1 and convert back. 0x32 is 0011 0010, inverted 1100 1101, plus 1 is 1100 1110 = 0xCE. Use the negate mode with hex input in the calculator and pick the right bit width first.