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What is Binary Coded Decimal (BCD)?

Binary Coded Decimal (BCD) is a class of binary encodings where each digit of a decimal number is represented by a fixed number of binary bits, usually four. Unlike standard binary representation where the entire decimal number is converted to base-2, BCD converts each decimal digit individually.

For example, to represent the decimal number 39 in standard binary vs. BCD:

Standard Binary: $39_{10} = 100111_2$
BCD (8421 representation): $3_{10} = 0011_2$ and $9_{10} = 1001_2 \implies 0011\ 1001_{BCD}$

The 8421 BCD Weighting System

The most common version of BCD is the 8421 encoding, where each bit position in a 4-bit group (tetrad) is assigned a specific mathematical weight:

Bit 3 (Most Significant Bit): Weight of $8$
Bit 2: Weight of $4$
Bit 1: Weight of $2$
Bit 0 (Least Significant Bit): Weight of $1$

For any 4-bit BCD group $b_3b_2b_1b_0$, the decimal value $D$ is calculated as: $$D = (b_3 \times 8) + (b_2 \times 4) + (b_1 \times 2) + (b_0 \times 1)$$ Where $D \le 9$. Note that binary values from $1010_2$ to $1111_2$ (decimal 10 to 15) are considered invalid BCD codes in standard 8421 BCD because they do not correspond to a single decimal digit.

How to Convert BCD to Decimal

Converting BCD back to a standard decimal number is an easy and straightforward process:

Group by Four: Separate the binary string into groups of 4 bits starting from the right. If the leftmost group has fewer than 4 bits, pad it with leading zeros.
Validate: Ensure that no 4-bit group represents a value greater than 9 ($1001_2$). If a group is greater than 9 (e.g., $1010_2$), it is invalid.
Convert Each Group: Translate each 4-bit binary group into its decimal digit using the 8421 weights.
Combine Digits: Write the decimal digits side-by-side to form the final decimal number.

Example Calculation

Let's convert the BCD sequence $0101\ 1000\ 0010$ to decimal:

First group: $0101 \implies (0 \times 8) + (1 \times 4) + (0 \times 2) + (1 \times 1) = 5$
Second group: $1000 \implies (1 \times 8) + (0 \times 4) + (0 \times 2) + (0 \times 1) = 8$
Third group: $0010 \implies (0 \times 8) + (0 \times 4) + (1 \times 2) + (0 \times 1) = 2$

Combining the results gives: 582.

Frequently Asked Questions

What is standard BCD vs. packed BCD?

Standard or unpacked BCD stores each BCD digit in a separate byte (usually leaving the upper 4 bits unused or padded with zeros). Packed BCD stores two BCD digits in a single byte (4 bits for the first digit and 4 bits for the second digit), which is much more storage-efficient.

Why are binary values 1010 to 1111 invalid in BCD?

BCD specifically represents decimal digits (0 through 9). Since 1010 to 1111 correspond to two-digit decimal numbers (10 to 15), they cannot represent a single decimal digit and are marked as invalid codes.

Where is BCD commonly used today?

BCD is commonly used in electronic systems where decimal calculations need to be displayed directly on 7-segment screens, such as digital clocks, calculators, voltmeter readouts, and financial calculations where rounding errors introduced by standard float conversions must be avoided.

What is the advantage of using BCD over binary?

The primary advantage is that BCD makes it extremely simple to convert and display numbers on hardware screens without requiring division/modulo logic. It also preserves decimal precision without the fractional representation errors typical of standard floating-point formats.