Binary to Gray Code Converter

What is Gray Code?

The Gray Code (also known as Reflected Binary Code or RBC) is an ordering of the binary numeral system such that two successive values differ in only one bit position. It was originally patented by Frank Gray at Bell Labs in 1953 as a method to prevent spurious or erroneous outputs from electro-mechanical switches.

In standard binary representation, incrementing a number can cause multiple bits to change simultaneously. For example: $$\text{Decimal } 3 \to 4 \implies 011_2 \to 100_2 \quad (\text{Three bits change!})$$ In a physical system (like a rotary encoder or mechanical switch), these bits do not transition at exactly the same microsecond, which can lead to temporary erroneous readings like $010_2$ or $111_2$. In Gray Code, only a single bit changes: $$\text{Decimal } 3 \to 4 \implies 010_{Gray} \to 110_{Gray} \quad (\text{Only one bit changes!})$$

Mathematical Formula: Binary to Gray Code

Converting a standard binary number to its Gray code equivalent is computationally simple and can be done using the bitwise Exclusive OR (XOR) operation.

For a binary string $B = b_{n-1}b_{n-2}\dots b_0$, the corresponding Gray code $G = g_{n-1}g_{n-2}\dots g_0$ is defined by: $$g_{n-1} = b_{n-1} \quad (\text{The most significant bit remains identical})$$ $$g_i = b_{i+1} \oplus b_i \quad \text{for } 0 \le i < n-1$$ Where $\oplus$ represents the Exclusive OR (XOR) logical operator.

In whole-number arithmetic or high-performance programming, the entire number can be converted instantly using a shift-and-XOR operation: $$G = B \oplus (B \gg 1)$$ Where $\gg$ is the bitwise right shift operator.

Example Step-by-Step Transition

Let's convert binary $1011_2$ (decimal 11) to Gray code:

1. Bit 3 (MSB): $g_3 = b_3 = 1$
2. Bit 2: $g_2 = b_3 \oplus b_2 = 1 \oplus 0 = 1$
3. Bit 1: $g_1 = b_2 \oplus b_1 = 0 \oplus 1 = 1$
4. Bit 0 (LSB): $g_0 = b_1 \oplus b_0 = 1 \oplus 1 = 0$

So, the Gray code of $1011_2$ is $1110_{Gray}$.