Two’s-complement arithmetic – for arithmetic negation of a signed binary value, which operation should be applied?

Introduction / Context:
Modern digital systems represent signed integers using the two’s-complement scheme because it unifies addition and subtraction in one adder and avoids separate negative-zero encodings. Knowing how to produce the arithmetic negative of a value is essential for ALU design, HDL coding, and low-level debugging.

Given Data / Assumptions:

• Signed integers are encoded in two’s complement with a fixed word width.
• Arithmetic negation means producing −X from X.
• No saturation or overflow handling changes the basic mechanism.

Concept / Approach:
In two’s-complement arithmetic, arithmetic negation is achieved by computing the two’s complement of the bit pattern: NOT(X) + 1. This yields the unique encoding of −X for every nonzero X and maps 0 to 0. The adder used for addition can also produce subtraction by inverting the subtrahend and adding 1, exploiting exactly this property.

Step-by-Step Solution:

Verification / Alternative check:
Example (8-bit): X = 00010110 (22). Invert → 11101001, add 1 → 11101010 = −22. Adding X + (−X) yields 00000000 with a discarded carry out, confirming the identity.

Why Other Options Are Wrong:
“1’s-complement” is a bitwise NOT (forms negative only if followed by +1); alone it leaves a negative-zero issue and is not standard for arithmetic negation. “sign inversion only” (flip MSB) breaks magnitude and is not a valid encoding. “surrogate inversion” and “bit rotate right” are unrelated to arithmetic negation.

Common Pitfalls:
Forgetting to add 1 after bit inversion, or misinterpreting overflow flags when negating edge cases (e.g., the most negative number, where |min| exceeds |max| in fixed width).

Final Answer:
2's-complement