BCD e Gray Code

Not todos binary representations são pure base-2. BCD (Binary-Coded Decimal) e Gray code são specialized encodings esse solve specific problems in digital systems—from displaying numbers on calculators to reading rotary encoders sem errors.

Binary-Coded Decimal (BCD)

O que Is BCD?

BCD represents cada decimal digit as a 4-bit binary number:

Key Difference

BCD uses 4 bits per decimal digit; pure binary é mais compact but harder to display.

Por que Usar BCD?

Fácil Decimal Display

• Cada 4-bit group → one digit on 7-segment display
• No complex binary-to-decimal conversão needed
• Calculators e meters use este

Exact Decimal Values

• Financial calculations need exact decimal
• 0.10 é exact in BCD, approximate in binary floating-point
• Avoids rounding errors in currency

Human-Readable Storage

• Timestamps often stored in BCD
• Fácil to read in hex dumps
• BIOS real-time clocks use BCD

BCD Arithmetic

Addition Challenge

BCD addition needs adjustment quando result > 9:

• 0101 + 0100 = 1001 (5 + 4 = 9) ✓ Valid BCD
• 0111 + 0101 = 1100 (7 + 5 = 12) ✗ Invalid BCD
• Must add 6 (0110) to correct: 1100 + 0110 = 0001 0010 = 12

Por que Add 6?

BCD apenas uses values 0000-1001 (0-9). Values 1010-1111 (10-15) são invalid. Adding 6 carries into o próximo digit.

BCD Variants

Packed BCD

• Two digits per byte
• Mais efficient storage
• Exemplo: 42 = 0100 0010 in one byte

Unpacked BCD

• One digit per byte (upper 4 bits unused/zero)
• Easier processing
• Exemplo: 42 = 00000100 00000010 (two bytes)

Excess-3 BCD

• Add 3 to cada digit before encoding
• Simplifies alguns arithmetic circuits
• 0 = 0011, 9 = 1100

Gray Code

O que Is Gray Code?

Gray code é a binary sequence onde apenas one bit changes entre consecutive values:

Por que Gray Code Matters

O Problem Gray Code Solves

In regular binary, going from 7 (0111) to 8 (1000) changes 4 bits simultaneously.

• If sensors don't switch at exatamente o mesmo instant
• Momentary false readings occur
• Could read 0000, 0001, 0011, 0111, ou 1111 briefly

Gray Code Solução

• Apenas one bit changes at a time
• No ambiguous intermediate states
• Essential for position encoders

Gray Code Aplicações

Rotary Encoders

• Motor position sensing
• Volume knobs (digital)
• Robotics joint angles
• CNC machine positioning

Error Minimization

• Analog-to-digital conversão
• Genetic algorithms (crossover)
• Karnaugh maps (logic minimization)

Communication

• Alguns error-correcting codes
• Certain modulation schemes

Converting Binary to Gray Code

Method

1. Keep o mais significant bit (MSB) o mesmo
2. XOR cada bit com o bit to its left

Exemplo: Converter 1011 (binary) to Gray

• MSB: 1 (keep)
• 1 XOR 0 = 1
• 0 XOR 1 = 1
• 1 XOR 1 = 0
• Result: 1110 (Gray)

Formula

Gray[i] = Binary[i] XOR Binary[i+1]

Gray[MSB] = Binary[MSB]

Converting Gray Code to Binary

Method

1. Keep o MSB o mesmo
2. XOR cada Gray bit com o previous Binary bit

Exemplo: Converter 1110 (Gray) to Binary

• MSB: 1 (keep)
• 1 XOR 1 = 0
• 0 XOR 1 = 1
• 1 XOR 0 = 1
• Result: 1011 (binary)

Conclusão

BCD e Gray code são specialized binary encodings esse solve specific problems. BCD simplifies decimal display e ensures exact decimal arithmetic—important for calculators e financial systems. Gray code ensures apenas one bit changes entre consecutive values—essential for position encoders e error prevention. While neither é as comum as pure binary, understanding them reveals como diferente number representations serve diferente engenharia needs.

Artigos Relacionados

• Binary in Computing: Como Computers Think in 1s e 0s
• Programming Number Formats: Como Languages Handle Numbers