2 Complement Calculator

Introduction

I’m Leo Park, Quantitative Analyst & Math Modeler. Task: implement and interpret a 2's complement calculator that flips bits and adds one. I’ll keep assumptions explicit and steps reproducible.

Inputs and Validation

• Input: binary (string) containing only 0/1; no sign, no spaces.
• Assumption: fixed bit-width equals input length; output is zero-padded back to this width.
• Invalid if regex ^[01]+$ fails.

Method: Exact Operations

We follow the given spec formulas, naming every symbol:

• inverted = invert(binary)
• complement = binaryToDecimal(inverted) + 1
• complementBinary = decimalToBinary(complement)

Operational details:

• invert(binary): map each character: 0 → 1, 1 → 0, preserving length n.
• binaryToDecimal(inverted): parse base-2 to an integer.
• Add 1 in base-10 (equivalently in base-2).
• decimalToBinary(complement): convert integer back to base-2; then left-pad with zeros to length n.

Edge Cases and Constraints

• All-zeros input (e.g., 0000): inverted = 1111; +1 produces 1 0000 in binary; after trimming to width n via left-padding rule, result becomes 0000 if you strictly pad to n by discarding overflow. The provided UI pads to original length; any extra carry beyond width is not kept.
• All-ones input: inverted becomes all-zeros; +1 yields 1; zero-pad to n.
• Large strings: time O(n); memory O(n). Parsing uses integer arithmetic; in JS, binary strings longer than ~53 bits may exceed safe integer if converted directly. Here, spec uses decimal parse, so for very long widths a pure bitwise add-1 routine is safer. The provided UI still uses parseInt; acceptable for typical UI ranges.

Worked Example (US formatting)

Example 1 (given): binary = 1101

1. inverted = 0010
2. complement = binaryToDecimal(0010) + 1 = 2 + 1 = 3
3. complementBinary = decimalToBinary(3) = 11; zero-pad to 4 bits → 0011

Example 2: binary = 1000 (n = 4)

1. inverted = 0111
2. complement = binaryToDecimal(0111) + 1 = 7 + 1 = 8
3. complementBinary = decimalToBinary(8) = 1000; pad to 4 → 1000

Cost perspective: If you processed 12,345 inputs at $0.002 each, the total would display as $24.69 using en-US formatting.

Sanity Checks and Common Mistakes

• Sanity: For any nonzero binary x, x + two'sComplement(x) (mod 2^n) = 0. Quick test at n bits.
• Don’t forget zero-padding to original length; dropping it changes width and numeric interpretation.
• Avoid mixing sign conventions; input here is unsigned bit pattern only.

Implementation Notes

• Precision/rounding: none; purely integer operations.
• Convergence: not applicable; deterministic.
• If extremely long inputs are needed, replace decimal parse with bitwise carry addition to avoid overflow.

Conclusion

Procedure is minimal and exact: invert bits, add one, zero-pad to original width. Interpret results strictly at the input bit-length. Validate by modulo sum check and handle overflow by truncating to n bits in display.