Gray Code — What It Is and How to Convert Binary to Gray Code

What is Gray Code?

In normal binary counting, multiple bits can change simultaneously — going from 0111 (7) to 1000 (8), all four bits flip. This causes glitches in hardware circuits because bits don’t switch at exactly the same instant. Gray code solves this by ensuring only one bit changes between consecutive numbers.

🔧 Practice conversions: Number System Converter — convert binary numbers instantly, and Binary Arithmetic Calculator for XOR operations.

Gray code (also called reflected binary code) is a binary numeral system where two successive values differ in only one bit position.

Complete 4-Bit Gray Code Table

Decimal	Binary	Gray Code	Bits Changed
0	0000	0000	—
1	0001	0001	1 bit
2	0010	0011	1 bit
3	0011	0010	1 bit
4	0100	0110	1 bit
5	0101	0111	1 bit
6	0110	0101	1 bit
7	0111	0100	1 bit
8	1000	1100	1 bit
9	1001	1101	1 bit
10	1010	1111	1 bit
11	1011	1110	1 bit
12	1100	1010	1 bit
13	1101	1011	1 bit
14	1110	1001	1 bit
15	1111	1000	1 bit

Notice: in the Gray column, every adjacent row differs by exactly one bit. In the Binary column, row 7→8 changes all 4 bits — that’s what Gray code prevents.

Binary → Gray Code Conversion

Rules:

1. MSB (leftmost bit) stays the same
2. Each subsequent Gray bit = XOR of current binary bit with previous binary bit

XOR Truth Table:

A	B	A XOR B
0	0	0
0	1	1
1	0	1
1	1	0

Example 1: Convert Binary 1011 to Gray Code

Binary:  1   0   1   1
         ↓   ↓   ↓   ↓
Gray:    1  1⊕0 0⊕1 1⊕1
       = 1   1   1   0

Answer: 1110

Step by step:

• G₃ = B₃ = 1 (MSB stays)
• G₂ = B₃ ⊕ B₂ = 1 ⊕ 0 = 1
• G₁ = B₂ ⊕ B₁ = 0 ⊕ 1 = 1
• G₀ = B₁ ⊕ B₀ = 1 ⊕ 1 = 0

Example 2: Convert Binary 1101 to Gray Code

B: 1  1  0  1
G: 1  0  1  1

G₃ = 1
G₂ = 1⊕1 = 0
G₁ = 1⊕0 = 1
G₀ = 0⊕1 = 1

Answer: 1011

Example 3: Convert Binary 10110 to Gray Code

B: 1  0  1  1  0
G: 1  1  1  0  1

Answer: 11101

Gray Code → Binary Conversion

Rules:

1. MSB stays the same
2. Each subsequent Binary bit = XOR of current Gray bit with the previous Binary bit (already computed)

Example: Convert Gray 1110 to Binary

Gray:    1   1   1   0
Binary:  1   ?   ?   ?

B₃ = G₃ = 1
B₂ = G₂ ⊕ B₃ = 1 ⊕ 1 = 0
B₁ = G₁ ⊕ B₂ = 1 ⊕ 0 = 1
B₀ = G₀ ⊕ B₁ = 0 ⊕ 1 = 1

Answer: 1011 ✓ (matches our earlier conversion)

Worked Conversions Summary

Binary	Gray Code	Verification
0000	0000	MSB=0, 0⊕0=0, 0⊕0=0, 0⊕0=0
0101	0111	0, 0⊕1=1, 1⊕0=1, 0⊕1=1
1011	1110	1, 1⊕0=1, 0⊕1=1, 1⊕1=0
1100	1010	1, 1⊕1=0, 1⊕0=1, 0⊕0=0
1111	1000	1, 1⊕1=0, 1⊕1=0, 1⊕1=0

Why Gray Code is Used

1. Rotary encoders — shaft position sensors use Gray code discs to avoid ambiguous readings during transitions
2. Karnaugh maps (K-maps) — adjacent cells differ by one variable, which is Gray code ordering
3. Error minimization — only 1 bit can be wrong during a transition, limiting errors to ±1
4. Analog-to-digital converters — reduces glitches during conversion
5. Genetic algorithms — Gray coding prevents large jumps from single-bit mutations

Common Exam Questions

• “Convert binary 10110 to Gray code” (direct application)
• “Convert Gray code 11001 to binary” (reverse application)
• “Why is Gray code used in K-maps?” (theory)
• “Draw 3-bit Gray code sequence” (listing)
• “Design a circuit to convert binary to Gray” (XOR gates)

Related Tools

• Number System Converter
• Binary Arithmetic Calculator
• 1’s & 2’s Complement

Related Articles

• Binary Number System Explained
• Bitwise Operators Explained
• Negative Numbers in Binary

FAQ

Q: What is the advantage of Gray code over binary? A: Only one bit changes between consecutive numbers, preventing glitches in hardware transitions. In binary, 0111→1000 changes all 4 bits simultaneously — if they don’t switch at the exact same time, intermediate invalid values appear.

Q: How is XOR used in Gray code conversion? A: Binary→Gray: each Gray bit (except MSB) = XOR of adjacent binary bits. Gray→Binary: each binary bit = XOR of Gray bit with previously computed binary bit.

Q: Is Gray code weighted or unweighted? A: Gray code is unweighted — unlike binary where each position has a weight (1, 2, 4, 8…), Gray code positions don’t have fixed mathematical weights. You can’t directly compute decimal value from Gray code without first converting to binary.

Q: How many bits change between Gray code 7 and 8? A: Only 1 bit. Gray(7) = 0100, Gray(8) = 1100 — only the MSB changes. In binary, 0111→1000 changes all 4 bits.

Q: Where do K-maps use Gray code ordering? A: K-map column/row headers use Gray code sequence (00, 01, 11, 10) so that adjacent cells differ by exactly one variable — enabling grouping of minterms.