Can you perform addition and subtraction using only bitwise operations (e.g., XOR, AND, NOT) without using the + or - operators?

Introduction / Context:
Bitwise-only addition and subtraction are classic algorithmic exercises that illuminate how CPUs implement arithmetic at the gate level. Understanding this helps in interviews, cryptography, embedded programming, and when implementing custom numeric types.

Given Data / Assumptions:

• We want to compute a + b (or a - b) using bitwise operators.
• We have XOR (^), AND (&), NOT (~), and shifts (<<, >>).
• We assume fixed-width two’s-complement integers.

Concept / Approach:
The sum without carry is a ^ b. The carry bits are (a & b) << 1. Repeating this process by setting a = a ^ b and b = (old_a & old_b) << 1 and iterating until b becomes 0 yields the final sum in a. Subtraction a - b can be achieved by adding a to the two’s-complement of b, i.e., a + (~b + 1). This demonstrates that addition and subtraction can be built from bitwise operations alone.

Step-by-Step Solution:

Verification / Alternative check:
Test with small values (e.g., a = 5, b = 3). The method converges to 8 without ever using + directly for the final arithmetic (except in forming ~b + 1, which itself can be built with the same loop).

Why Other Options Are Wrong:

Common Pitfalls:
Forgetting to loop until carry is zero; mixing signed and unsigned without understanding overflow rules.

Final Answer:
Yes — use XOR for sum without carry and AND/shift for carry; iterate until carry is zero.