Sum of Two Integers - Problem
You're given two integers a and b, and your task is to return their sum without using the addition (+) or subtraction (-) operators.
This is a classic bit manipulation problem that challenges you to think at the binary level. Instead of relying on built-in arithmetic operators, you'll need to simulate addition using bitwise operations - the same way computers actually perform addition at the hardware level!
Goal: Calculate a + b using only bitwise operators like AND (&), OR (|), XOR (^), and bit shifting.
Example: If a = 1 and b = 2, return 3. But remember - no + or - allowed!
Input & Output
example_1.py — Basic Addition
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Input:
a = 1, b = 2
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Output:
3
💡 Note:
1 + 2 = 3. In binary: 001 + 010 = 011. XOR gives 011, AND gives 000 (no carries), so result is 3.
example_2.py — Addition with Carries
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Input:
a = 3, b = 5
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Output:
8
💡 Note:
3 + 5 = 8. In binary: 011 + 101. XOR: 110, AND<<1: 010. Next: 110^010=100, (110&010)<<1=100. Next: 100^100=000, (100&100)<<1=1000. Result: 1000 = 8.
example_3.py — Negative Numbers
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Input:
a = -1, b = 1
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Output:
0
💡 Note:
-1 + 1 = 0. Using two's complement representation, the carries eventually cancel out to produce zero.
Visualization
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Understanding the Visualization
1
Convert to Binary
Express both numbers in binary format (001, 010)
2
XOR for Partial Sum
XOR operation gives us the sum ignoring any carry bits
3
AND+SHIFT for Carries
AND finds where carries occur, left shift moves them to next position
4
Repeat Until No Carries
Continue the process until carry becomes zero
Key Takeaway
🎯 Key Insight: This is exactly how your computer's ALU (Arithmetic Logic Unit) performs addition - no magic + operator, just clever bit manipulation!
Time & Space Complexity
Time Complexity
O(log(max(|a|, |b|)))
In worst case, we iterate once for each bit position, which is logarithmic in the maximum value
⚡ Linearithmic
Space Complexity
O(1)
Only using a constant amount of extra variables
✓ Linear Space
Constraints
- -1000 ≤ a, b ≤ 1000
- No use of + or - operators allowed
- Must handle negative numbers correctly using two's complement
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Explanation
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