Difference between 1’s Complement representation and 2’s Complement representation Technique

Prerequisite – Representation of Negative Binary Numbers

1’s complement of a binary number is another binary number obtained by toggling all bits in it, i.e., transforming the 0 bit to 1 and the 1 bit to 0. Examples:

Let numbers be stored using 4 bits

1's complement of 7 (0111) is 8 (1000)
1's complement of 12 (1100) is 3 (0011)

2’s complement of a binary number is 1 added to the 1’s complement of the binary number. Examples:

Let numbers be stored using 4 bits

2's complement of 7 (0111) is 9 (1001)
2's complement of 12 (1100) is 4 (0100)

  These representations are used for signed numbers.

The main difference between 1′ s complement and 2′ s complement is that 1′ s complement has two representations of 0 (zero) — 00000000, which is positive zero (+0), and 11111111, which is negative zero (-0); whereas in 2′ s complement, there is only one representation for zero — 00000000 (0) because if we add 1 to 11111111 (-1), we get 100000000, which is nine bits long. Since only eight bits are allowed, the left-most bit is discarded(or overflowed), leaving 00000000 (-0) which is the same as positive zero. This is the reason why 2′ s complement is generally used.

Another difference is that while adding numbers using 1′ s complement, we first do binary addition, then add in an end-around carry value. But, 2′ s complement has only one value for zero and doesn’t require carry values.

Range of 1’s complement for n bit number is from -2^n-1-1 to 2^n-1-1 whereas the range of 2’s complement for n bit is from -2^n-1 to 2^n-1-1.

There are 2^n-1 valid numbers in 1’s complement and 2ⁿ valid numbers in 2’s complement.

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Difference between 1’s Complement representation and 2’s Complement representation in tabular form:

Criteria	1’s Complement	2’s Complement
Definition	The 1’s complement of a binary number is obtained by inverting all its bits.	The 2’s complement of a binary number is obtained by adding 1 to the 1’s complement of the number.
Range of values that can be represented with n bits	From -2^(n-1) + 1 to 2^(n-1) – 1	From -2^(n-1) to 2^(n-1) – 1
Number of representations for zero	Can be represented in two ways (all 0s and all 1s).	Can be represented in only one way (all 0s).
Addition of positive and negative numbers	Same as unsigned binary addition.	Same as unsigned binary addition.
Subtraction of numbers	Subtract the smaller number from the larger one, then add a sign bit to the result.	Add the negative number to the positive one using binary addition.