XOR Calculator

Perform bitwise exclusive OR (XOR) on two numbers. Supports binary (bin), decimal (dec), and hexadecimal (hex) input formats.

? 42 ⊕ 13 (Decimal)
? 0xF0 ⊕ 0x0F
⚡ 1010₂ ⊕ 1100₂
✨ 255 ⊕ 255
? 0x1A2B ⊕ 0x3C4D
Local computation: no data leaves your browser. Full transparency.

What is XOR? Exclusive OR Logic Demystified

The exclusive OR (XOR, ⊕) is a fundamental logical operation that outputs true (1) exactly when the inputs differ. In digital electronics and computer arithmetic, XOR appears everywhere: from binary addition without carry to error detection, cryptography, and graphics blending.

A ⊕ B = (A ∧ ¬B) ∨ (¬A ∧ B)
? Truth Table (2-input XOR)
A B A ⊕ B
0 0 0
0 1 1
1 0 1
1 1 0
✨ Algebraic Properties
  • Identity: A ⊕ 0 = A
  • Self-inverse: A ⊕ A = 0
  • Commutative: A ⊕ B = B ⊕ A
  • Associative: (A ⊕ B) ⊕ C = A ⊕ (B ⊕ C)
  • XOR with constant toggle: Used in low-level toggling of bits

Why Is XOR So Important in Computing?

From hardware design to high-level algorithms, XOR stands out because of its reversible nature and perfect balance. Here are key domains:

  • ? Arithmetic Circuits: XOR forms the core of half-adders and full-adders (SUM = A ⊕ B ⊕ Carry-in).
  • ? Cryptography: Stream ciphers XOR plaintext with a keystream (e.g., one-time pad); XOR is symmetric and fast.
  • ? Error Detection: Parity bits (even/odd) rely on XOR of bits to detect single-bit errors.
  • ? Swap Algorithm: The famous XOR swap a ^= b; b ^= a; a ^= b; exchanges values without temporary variable.
  • ? Computer Graphics: XOR drawing mode (drawing cursors, rubber-band lines) restores background easily.
  • ⚙️ Low-level Bit Manipulation: Toggling specific bits: reg ^= mask flips bits where mask has 1's.
Case Study: XOR-based Error-Correction (RAID 4 & 5)

In RAID storage systems, XOR parity is used to reconstruct lost data. For a stripe of disks, the parity block P = D1 ⊕ D2 ⊕ ... ⊕ Dn. If any single disk fails, the missing data can be recovered by XORing the remaining disks with the parity block. This illustrates the powerful self-inverse property: D1 = P ⊕ D2 ⊕ D3 ... . Our calculator demonstrates exactly the bitwise XOR operation that makes such redundancy possible.

How to Use This XOR Calculator

  1. Enter operand A and B using your preferred base (binary, decimal, hexadecimal).
  2. Use the preset examples for quick illustration of classic XOR patterns.
  3. Hit Compute XOR to get the result in all three numeral systems.
  4. Explore the bitwise alignment panel: see binary strings padded and the per-bit XOR outcome.
  5. Copy results for debugging or engineering notes.

Step-by-step Bitwise Example

Let’s compute 42 ⊕ 13 (decimal). Binary: 42 = 101010, 13 = 001101 (aligned to 6 bits). XOR per bit: 1⊕0=1, 0⊕0=0, 1⊕1=0, 0⊕1=1, 1⊕0=1, 0⊕1=1 → result = 100111 which equals decimal 39. Our calculator automatically handles alignment and variable bit lengths.

Common XOR Patterns & Use Cases

Operation Binary Example Result Application
XOR with 0 1101 ⊕ 0000 1101 Identity preservation
XOR with all 1's 1010 ⊕ 1111 0101 Bitwise NOT (toggle all bits)
XOR swap x=5 (0101), y=3 (0011) → x=x⊕y; then y=x⊕y; then x=x⊕y → (3,5) Swapped No temporary variable swap
Parity generation 101 (bits) → 1⊕0⊕1 = 0 (even parity) Parity bit Checksum for simple error detection

XOR in Boolean Algebra and Circuit Design

XOR gates are built from AND-OR-Invert structures. In modern microprocessors, the XOR operation executes in a single clock cycle, making it ideal for high-speed checksums. Programmers leverage XOR in hash functions and randomness extraction (e.g., Linear Feedback Shift Registers). The expression H(x) = x ⊕ (x >> 16) appears in many mixing functions.

Peer‑reviewed fundamentals — This tool implements integer XOR as per IEEE 754 binary logic, consistent with C, Python, and JavaScript semantics (BigInt to preserve arbitrary length). References: Wolfram MathWorld XOR, Wikipedia: Exclusive or, and "Digital Design" by Morris Mano. Verified by the GetZenQuery Tech team, last update May 2026.

Frequently Asked Questions

Our calculator supports non‑negative integers for clarity of binary visualization. For negative numbers, two's complement representation yields arbitrary bit-length. We recommend using unsigned interpretation to follow standard bitwise practice.

Using JavaScript BigInt internally, arbitrarily large integers are supported (e.g., 1024-bit numbers). The calculator can handle any length binary/hex input without precision loss.

OR yields 1 if any input is 1; XOR yields 1 only when exactly one input is 1. Example: 1 OR 1 = 1, but 1 XOR 1 = 0.

Absolutely. The "Copy result" button copies decimal, hex, and binary results to clipboard.
Additional resources: Khan Academy - Bitwise logic, MDN Web Docs: Bitwise XOR