Question

2's complement of the 2's complement number is

• A. Equal to 1's complement of the number
• B. Equal to 2's complement of the number
• C. Equal to the number itself
• D. Not possible to execute

Hint:

This "double-negative" property is why 2's complement is so efficient for computer hardware—the same logic used to find the negative of a number can also be used to find the original value from its negative.

Answer 1

The Correct Option is C

In digital systems, the 2's complement is used to represent signed integers. The 2's complement of a binary number is obtained by inverting all the bits (1's complement) and then adding 1 to the least significant bit (LSB). 1. Mathematical Definition: If $A$ is a binary number, then its 2's complement $A^*$ is: $$A^* = \text{NOT}(A) + 1$$

2. Applying the Operation Twice: Let's see what happens if we take the 2's complement of $A^*$: $$(A^*)^* = \text{NOT}(A^*) + 1$$ Substituting $A^*$: $$(A^*)^* = \text{NOT}(\text{NOT}(A) + 1) + 1$$

3. Logical Illustration: Let's take an example with the 4-bit number 5 ($0101_2$):

• Original Number: 0101

• 1st 2's Complement:

• Invert bits: 1010

• Add 1: $1010 + 1 = 1011$

• 2nd 2's Complement (of 1011):

• Invert bits: 0100

• Add 1: $0100 + 1 = \mathbf{0101}$

As demonstrated, performing the 2's complement operation twice returns the value to its original state. This is because the operation represents taking the negative of a number in signed arithmetic; and mathematically, $-(-A) = A$.