1s Complement and 2s Complement of Binary Numbers | Signed Binary Number Representation

1s complement and 2s complement are way of representing the signed binary numbers.

In general, the binary number can be represented in two ways.

1. Unsigned Binary Numbers
2. Signed Binary Numbers

Unsigned Binary Numbers

Using unsigned binary number representation, only positive binary numbers can be represented. For n-bit unsigned binary numbers, all n-bits are used to represent the magnitude of the number.

For example, if we represent decimal 12 in 5- bit unsigned number form then (12)₁₀ = (01100)₂. Here all 5 bit are used to represent the magnitude of the number.

In unsigned binary number representation, using n-bits, we can represent the numbers from 0 to 2ⁿ – 1. For example, using 4 -bits we can represent the number from 0 to 15 in unsigned binary number representation.

Signed Binary Numbers

Using signed binary number representation both positive and negative numbers can be represented.

In signed binary number representation the most significant bit (MSB) of the number is a sign bit. For positive numbers, the sign bit is 0 and for negative number, the sign bit is 1.

There are three different ways the signed binary numbers can be represented.

1. Signed Magnitude Form
2. 1’s Complement Form
3. 2’s Complement Form

Sign Magnitude Representation

In sign-magnitude representation, the Most Significant bit of the number is a sign bit and the remaining bit represents the magnitude of the number in a true binary form. For example, if some signed number is represented in the 8-bit sign-magnitude form then MSB is a sign bit and the remaining 7 bits represent the magnitude of the number in a true binary form.

Here is the representation of + 34 and -34 in a 8-bit sign-magnitude form.

Since the magnitude of both numbers is the same, the first 7 bits in the representation are the same for both numbers. For +34, the MSB is 0, and for -34, the MSB or sign bit is 1.

In sign magnitude representations, there are two different representations for 0.

Using n-bits, the range of numbers that can be represented in Sign Magnitude Representation is from – (2^n-1 – 1) to (2^{n -1} – 1).

1’s Complement Representation

In 1’s complement representation, the representation of the positive number is same as the negative number. But the representation of the negative number is different.

For example, if we want to represent -34 in 8-bit 1’s complement form, then first write the positive number (+34). And invert all 1s in that number by 0s and 0s by 1s in that number. The corresponding inverted number represents the -34 in 1’s complement form. It is also called 1s complement of the number +34.

Here is another example which shows how to represent -60 in 8-bit 1’s complement form.

Using n-bits, the range of numbers that can be represented in 1’s complement form is from – (2^n-1 – 1) to (2^{n -1} – 1). For example, using 4-bits, it is possible to represent integers numbers from -7 to +7 in a 1’s complement form representation.

Similar to sign-magnitude form, there are two different representations of 0 in 1’s complement form representation.

2’s Complement Representation

In 2’s complement representation also, the representation of the positive number is same as1’s complement and sign-magnitude form.

But the representation of the negative number is different. For example, if we want to represent -34 in 2’s complement form then

1. Write the number corresponding to +34.
2. Starting from Least Significant Bit (LSB), just copy all the bits until the first 1 is encountered in the number.
3. After the first ‘1’ is encountered, invert all the 1s in the number with 0s and 0s in the number with 1s (including the sign bit)
4. The resultant number is 2’s complement representation of the number -34.

The same is shown below.

The second way of representing -34 in 2’s complement form is

1. Write the number corresponding to +34.
2. Find 1’s complement of +34
3. Add ‘1’ to the 1’s complement number
4. The resultant is 2’s complement representation of -34

The same is shown below.

For n-bit number N, its 2’s complement is (2ⁿ – N). For example, the 2’s complement of +34 in 8-bit form is (2⁸ – 34). In binary, it is 100000000 – 00100010 = 11011110. That is a third way of finding the 2’s complement.

Here is the representation of -60 in sign-magnitude form, 1’s complement, and 2’s complement form.

Using n-bits, the range of number which can be represented in 2’s complement form is from – (2^n-1 ) to 2^n-1 – 1. For example, using 4-bits, it is possible to represent numbers from -8 to +7. Unlike 1’s complement and sign magnitude form, there is a unique way of representing 0 in this 2’s complement form.

For information about signed binary number representation, check this video: