1's Complement Representation vs 2's Complement Representation

1's Complement and 2's Complement are two methods to represent signed (positive and negative) integers in binary form. Both methods allow computers to perform arithmetic operations with both positive and negative numbers.

• 1's Complement: Negative numbers are formed by inverting all bits of the positive number, but dual zero representations complicate arithmetic.
• 2's Complement: Negative numbers are formed by inverting bits and adding 1, giving a single zero and streamlined arithmetic.

What is 1's Complement?

The 1's complement of a binary number is obtained by flipping all the bits, i.e., converting every 0 to 1 and every 1 to 0.

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)

Advantages of 1's Complement

• Easy to compute by flipping bits.
• Suitable for basic arithmetic operations with a simpler hardware design.

Disadvantages of 1's Complement

• 1's complement has two representations of zero (0000 and 1111), which can cause complications in calculations.
• Addition requires an end-around carry, which can complicate arithmetic operations.
• Not as efficient in representing negative numbers compared to 2's complement.

What is 2's Complement?

2's complement is another binary number representation technique used widely in modern computers. To obtain the 2's complement of a binary number, you invert all the bits (similar to 1's complement) and add 1 to the least significant bit.

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)

Advantages of 2's Complement

• Only one representation of zero, which simplifies arithmetic operations.
• No need for an end-around carry when adding numbers; subtraction can be done by adding the negative.
• Negative numbers have the most significant bit as 1, providing a clear distinction.

Disadvantages of 2's Complement

• Requires an additional step of adding 1 after inverting the bits.
• The range of negative numbers is one larger than the range of positive numbers, which can affect precision. 

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: -2^n-1-1 to 2^n-1-1

Range of 2's complement for n bit number: -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.

1's Complement Representation vs 2's Complement Representation