Two’s (2’s) Complement Calculator

How to Use the Calculator?

To use the calculator, simply enter your number, choose whether it is decimal, hex, or binary, and then select the bit length you want to use. The calculator will automatically display the two’s complement value. The equivalent values in decimal, hex, and octal will also be shown.

What Is Two’s Complement?

In ordinary decimal numbers, we already know how to show negatives: we simply place a minus sign in front, like $-5$ or $-12$. But computers do not store numbers with a little minus sign the way we do in notebooks. Inside a computer, everything must be represented using only $0$s and $1$s.

So the computer needs a smart system for representing both positive and negative integers using the same kind of binary digits. That system is called two’s complement.

How to Find the Two’s Complement

To find the two’s complement, follow this procedure:

1. Write the given number in binary using the correct bit width.
2. Invert every bit by changing zeroes to ones and ones to zero.
3. Add $1$ to the result.

Suppose we want to find two’s complement of $5$ in 8-bit. First write $5$ in binary:

$$ 5 = 00000101 $$

Now invert all bits:

$$ 00000101 \rightarrow 11111010 $$

Then add $1$:

$$ 11111010 + 1 = 11111011 $$

So $5$’s two’s complement in 8-bit is:

$$ 11111011 = -5$$

Understanding Bit Length

While learning about two’s complement numbers, it is extremely important to understand that the number of bits matters. A 4-bit number and an 8-bit number are not interpreted in the same way, even if some of the digits look similar.

For example, $1111$ in 4-bit two’s complement form means $-1$. But $00001111$ in 8-bit two’s complement form means $15$. Even though both numbers end with the same four digits, they do not represent the same value. This shows why bit length must always be considered when reading signed binary numbers.

This is because, in an ordinary 4-bit unsigned binary number, each bit represents a positive power of two. For example, the decimal number $12$ is written as $1100$ in binary, because:

$$ 1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 0 \cdot 2^0 = 12 $$

In a 4-bit two’s complement number, however, the place values are slightly different. The rightmost three positions still represent $2^0$, $2^1$, and $2^2$, but the leftmost bit represents $-2^3$ instead of $+2^3$. This means that a 4-bit two’s complement number can be interpreted by adding signed place values instead of only positive ones.

So the 4-bit signed binary number $1100$ can be read as:

$$ 1 \cdot (-2^3) + 1 \cdot 2^2 + 0 \cdot 2^1 + 0 \cdot 2^0 = -4 $$

This is why $1100$ does not mean $12$ in 4-bit two’s complement. In this system, it represents $-4$. That is one of the most important ideas in signed binary: the same pattern of bits can represent different values depending on whether the number is unsigned or written in two’s complement form.

The Range of Values in Two’s Complement

Because the leftmost bit is used in a special way, a fixed number of bits can represent only a certain range of signed integers. In an $n$-bit two’s complement system, the smallest number is $-2^{n-1}$ and the largest number is $2^{n-1} – 1$.

$$ \text{Range for } n\text{-bit two’s complement} = -2^{n-1} \text{ to } 2^{n-1}-1 $$

For a 4-bit system, that becomes:

$$ -2^{4-1} \text{ to } 2^{4-1}-1 = -8 \text{ to } 7 $$

So the 4-bit two’s complement numbers represent values from $-8$ to $7$. That gives 16 total values, which matches the 16 possible 4-bit patterns.

For an 8-bit system, the range is from $-128$ to $127$. For a 16-bit system, it is from $-32768$ to $32767$.

This range is not perfectly balanced around zero, because there is one more negative number than positive number. That happens because zero takes up one of the non-negative spots. This detail is normal in two’s complement, and it is one reason why the most negative number behaves a little specially.

Building a 4-Bit Two’s Complement Table

A very useful way to understand the system is to look at all 4-bit combinations and their meanings. In 4-bit two’s complement, the values are arranged like this:

$$\begin{aligned} 0000 &= 0 \\ 0001 &= 1 \\ 0010 &= 2 \\ 0011 &= 3 \\ 0100 &= 4 \\ 0101 &= 5 \\ 0110 &= 6 \\ 0111 &= 7 \end{aligned}$$

Then the pattern continues, but now it wraps into negative numbers:

$$\begin{aligned} 1000 &= -8 \\ 1001 &= -7 \\ 1010 &= -6 \\ 1011 &= -5 \\ 1100 &= -4 \\ 1101 &= -3 \\ 1110 &= -2 \\ 1111 &= -1 \end{aligned}$$

It is worth spending time looking at this pattern. Notice that the positive numbers begin with $0$ and the negative numbers begin with $1$. Also notice that $1111$ means $-1$, not $15$, because we are interpreting the bits as signed 4-bit two’s complement, not unsigned binary. This is why context matters.

If you count upward from $0111$, which is $7$, the next pattern is $1000$, which is not $8$ but $-8$. This is the wraparound effect.

So think of two’s complement as a circular number system. If a binary number gets too large, it wraps around. In an unsigned 4-bit system, the numbers go from $0000 = 0$ up to $1111 = 15$. But in a 4-bit two’s complement system, the same 16 bit patterns are shared between positive and negative values. The system wraps so that some patterns represent negative numbers. This wrapping is one of the key ideas that helps make arithmetic work smoothly.

8-Bit Two’s Complement Table

Here is the full list of 8-bit decimal numbers and their corresponding two’s complement for your quick reference.