BCD y Gray Code

Not todo binary representations son pure base-2. BCD (Binary-Coded Decimal) y Gray code son especializado encodings eso solve specific problems in digital sistemas—desde displaying numeros on calculators un/una reading rotary encoders without errors.

Binary-Coded Decimal (BCD)

What Is BCD?

BCD represents cada decimal digit as un/una 4-bit binary numero:

Key Difference

BCD uses 4 bits per decimal digit; pure binary es mas compact pero harder un/una mostrar.

Why Use BCD?

Easy Decimal Display

• Each 4-bit group → one digit on 7-segment mostrar
• No complex binary-un/una-decimal conversion needed
• Calculators y metros usar esto

Exact Decimal Values

• Financial calculos necesitar exacto decimal
• 0.10 es exacto in BCD, approximate in binary floating-point
• Avoids rounding errors in currency

Human-Readable Storage

• Timestamps often stored in BCD
• Easy un/una read in hex dumps
• BIOS real-time clocks usar BCD

BCD Arithmetic

Addition Challenge

BCD addition needs adjustment cuando resultado > 9:

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

Why Suma 6?

BCD solo uses valores 0000-1001 (0-9). Values 1010-1111 (10-15) son invalid. Adding 6 carries into el/la siguiente digit.

BCD Variants

Packed BCD

• Two digits per byte
• More efficient storage
• Ejemplo: 42 = 0100 0010 in one byte

Unpacked BCD

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

Excess-3 BCD

• Suma 3 un/una cada digit before encoding
• Simplifies algunos arithmetic circuits
• 0 = 0011, 9 = 1100

Gray Code

What Is Gray Code?

Gray code es un/una binary sequence donde solo one bit changes entre consecutive valores:

Why Gray Code Matters

El/La Problem Gray Code Solves

In regular binary, going desde 7 (0111) un/una 8 (1000) changes 4 bits simultaneously.

• If sensors don't switch at exactamente el/la mismo instant
• Momentary false readings occur
• Could read 0000, 0001, 0011, 0111, o 1111 briefly

Gray Code Solution

• Only one bit changes at un/una time
• No ambiguous intermediate states
• Essential for position encoders

Gray Code Aplicaciones

Rotary Encoders

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

Error Minimization

• Analog-un/una-digital conversion
• Genetic algorithms (crossover)
• Karnaugh maps (logic minimization)

Communication

• Some error-correcting codes
• Certain modulation schemes

Converting Binary un/una Gray Code

Method

1. Keep el/la la mayoria significant bit (MSB) el/la mismo
2. XOR cada bit with el/la bit un/una su left

Ejemplo: Convertir 1011 (binary) un/una 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 un/una Binary

Method

1. Keep el/la MSB el/la mismo
2. XOR cada Gray bit with el/la previous Binary bit

Ejemplo: Convertir 1110 (Gray) un/una Binary

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

Conclusion

BCD y Gray code son especializado binary encodings eso solve specific problems. BCD simplifies decimal mostrar y ensures exacto decimal arithmetic—importante for calculators y financial sistemas. Gray code ensures solo one bit changes entre consecutive valores—esencial for position encoders y error prevention. While neither es as comun as pure binary, understanding them reveals como diferente numero representations serve diferente engineering needs.

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