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Home›Base Converter›Arbitrary Radix Gray Code Converter

Arbitrary Radix Gray Code Converter

Convert between binary and Gray code, plus explore arbitrary radix Gray codes with step-by-step derivation

Binary Number

Gray Code Conversion Principle

① Binary → Gray: MSB same, Gᵢ = Bᵢ XOR Bᵢ₋₁

② Gray → Binary: MSB same, Bᵢ = Gᵢ XOR Bᵢ₋₁

③ Successive values differ by only one bit

Gray code (reflected binary) ensures only one bit changes between consecutive numbers, minimizing transition errors.

⚠ Gray code is not a positional number system. The same sequence of bits represents different values depending on its position in the sequence.

What Is Gray Code?

Gray code, or reflected binary code, is a binary numeral system where consecutive values differ by exactly one bit, invented by Frank Gray in 1953.

Single Bit Change

000 → 001 → 011 → 010 → 110 → 111 → 101 → 100. Only one bit flips at each step.

Error Minimization

In mechanical encoders, single bit changes prevent large errors if a sensor misreads a transition.

Binary → Gray

G₀ = B₀, G₁ = B₁^B₀, G₂ = B₂^B₁, G₃ = B₃^B₂, etc. (^ = XOR)

Gray → Binary

B₀ = G₀, B₁ = G₁^B₀, B₂ = G₂^B₁, B₃ = G₃^B₂, etc. (^ = XOR)

💡 Teaching Example: Binary 1010 → Gray: 1 (same), 1^0=1, 0^1=1, 0^0=0 → 1110. Gray 1110 → Binary: 1, 1^1=0, 1^0=1, 0^1=1 → 1010.

Applications

Rotary Encoders Position Sensors Karnaugh Maps Error Correction Digital Logic

Frequently Asked Questions

What is Gray code?▼

Gray code, also known as reflected binary code, is a binary numeral system where two successive values differ in only one bit. This minimizes errors during state transitions.

How do you convert binary to Gray code?▼

The most significant bit (MSB) stays the same. Each subsequent bit is the XOR of the current binary bit and the previous binary bit. Formula: Gᵢ = Bᵢ XOR Bᵢ₋₁

How do you convert Gray code to binary?▼

The MSB stays the same. Each subsequent bit is the XOR of the current Gray bit and the previous binary bit. Formula: Bᵢ = Gᵢ XOR Bᵢ₋₁

What is Gray code used for?▼

Gray code is used in rotary encoders, digital communications, error correction, Karnaugh maps, position sensors, and anywhere minimal state change is important.

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