Bitwise complementation: Is the 1’s complement of a binary number obtained by inverting every bit (changing all 0s to 1s and all 1s to 0s) across the entire width?

Introduction / Context:
Complement systems are used in arithmetic and error detection. The 1’s complement and 2’s complement are closely related but distinct operations. This question confirms the exact transformation for obtaining a 1’s complement from a given binary pattern.

Given Data / Assumptions:

• A fixed bit width is implied (for example, 8, 16, 32 bits).
• All bit positions are subject to the operation.
• No arithmetic carry is involved in 1’s complement itself.

Concept / Approach:
The 1’s complement is defined as a bitwise NOT operation across the entire width: every 0 becomes 1 and every 1 becomes 0. In contrast, 2’s complement is obtained by taking the 1’s complement and then adding 1 to the result, which represents the negative of a number in signed arithmetic. Understanding this distinction avoids mistakes in sign inversion and bit manipulations.

Step-by-Step Solution:

Verification / Alternative check:
Example: for 8-bit 0101 1000, 1’s complement = 1010 0111; performing +1 yields 2’s complement = 1010 1000, demonstrating the relation but confirming that inversion alone defines 1’s complement.

Why Other Options Are Wrong:
Inverting only the MSB or trailing zeros is not the definition; adding 1 is part of 2’s complement, not 1’s complement.

Common Pitfalls:
Forgetting to keep a fixed width (leading to incorrect inversion on the MSB); confusing 1’s complement with 2’s complement in signed math.

Final Answer:
Correct