What is Binary Coded Decimal (BCD) and how does it differ from pure binary?

In pure binary, the entire number is converted to base-2 (e.g. decimal 25 is `11001`). In BCD, each individual decimal digit is encoded separately as a 4-bit binary nibble (e.g. 2 = `0010`, 5 = `0101`, so 25 is `0010 0101`).

Why are the binary values 1010 through 1111 invalid in BCD?

Since decimal digits only range from 0 to 9, binary values representing 10 to 15 (1010 to 1111) are unused 'pseudo-tetrades' in 8421 BCD and are flagged as errors.

Where is BCD used in electronics and computing?

BCD is widely used in digital clocks, pocket calculators, 7-segment LED displays, financial accounting processors, and industrial PLC controllers to eliminate binary floating-point rounding errors.

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BCD to Decimal & Decimal to BCD Converter Studio

Welcome to the BCD Encoding Studio. Convert binary coded decimals, standard integers, and pure binary in real time with step-by-step nibble breakdowns.

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Interactive Multidirectional BCD Console🔒 100% Client-Side

Binary Coded Decimal (BCD - 4 bits per digit)

Decimal Integer Value (Base-10)

Pure Binary Equivalent (Base-2)

Modifying any field automatically synchronizes the rest in real-time. Spaces are automatically handled.

Equivalence Outcomes

Step-by-Step Nibble Expansion

A walkthrough of character-to-binary mapping

Nibble 1 (1001)

Binary 1001 ➔ Digit: 9

Nibble 2 (0001)

Binary 0001 ➔ Digit: 1

Study Presets & Architecture Benchmarks

Click any preset to instantly load and visualize its BCD register representation

Complete BCD (8421) Digit Encoding Table (0 through 9)

The complete mapping of all 10 legal decimal numerals to their 4-bit natural BCD (NBCD) nibbles:

Decimal Digit	8421 BCD Nibble	Bit Weight Sum (8 × b3 + 4 × b2 + 2 × b1 + 1 × b0)	Pure Binary	7-Segment Display Segments
0	0000	0 + 0 + 0 + 0 = 0	0	a, b, c, d, e, f
1	0001	0 + 0 + 0 + 1 = 1	1	b, c
2	0010	0 + 0 + 2 + 0 = 2	10	a, b, d, e, g
3	0011	0 + 0 + 2 + 1 = 3	11	a, b, c, d, g
4	0100	0 + 4 + 0 + 0 = 4	100	b, c, f, g
5	0101	0 + 4 + 0 + 1 = 5	101	a, c, d, f, g
6	0110	0 + 4 + 2 + 0 = 6	110	a, c, d, e, f, g
7	0111	0 + 4 + 2 + 1 = 7	111	a, b, c
8	1000	8 + 0 + 0 + 0 = 8	1000	a, b, c, d, e, f, g
9	1001	8 + 0 + 0 + 1 = 9	1001	a, b, c, d, f, g

Understanding Binary Coded Decimal (BCD) Architecture

Explore the architectural rules, hardware representations, and logical properties of BCD (8421) and standard Base-10 systems.

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The 4-Bit BCD Digit Rule

Unlike pure binary which represents an entire integer as a single base-2 polynomial, BCD encodes each decimal character independently into exactly 4 bits (one nibble). Decimal 95 becomes 1001 0101 instead of pure binary 01011111.

Rule:1 Decimal Digit = 1 BCD Nibble (4 bits).

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Real-Time Clocks & Displays

Real-time clocks (RTC chips like the DS1307) and electronic digital displays rely on BCD. Because each nibble maps directly to a decimal digit, display controllers drive 7-segment LEDs without requiring computationally expensive division instructions.

Application: Embedded hardware counters, timers, and legacy microcontrollers.

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Financial & Banking Mainframes

Floating-point binary numbers (IEEE 754) cannot accurately represent certain decimal fractions like 0.1 (resulting in repeating binary decimals: 0.000110011...). Banking mainframes and COBOL systems use Packed BCD to avoid fractional rounding errors on money transactions.

Advantage: Exact decimal penny accounting with zero floating-point drift.

Invalid 8421 BCD States (Pseudo-Tetrades) & The +6 Correction Rule

Because 4 bits can represent 16 distinct values (2⁴ = 16), but the decimal system only uses digits 0 through 9, the binary patterns from 1010 to 1111 are strictly invalid in BCD:

Invalid 4-Bit Code	Binary Value	Hex Value	Status in BCD	DAA (+6) Microprocessor Adjustment
`1010`	10	0x0A	Invalid BCD state	Add `0110` (6) → Produces `0001 0000` (BCD 10) with Carry
`1011`	11	0x0B	Invalid BCD state	Add `0110` (6) → Produces `0001 0001` (BCD 11) with Carry
`1100`	12	0x0C	Invalid BCD state	Add `0110` (6) → Produces `0001 0010` (BCD 12) with Carry
`1101`	13	0x0D	Invalid BCD state	Add `0110` (6) → Produces `0001 0011` (BCD 13) with Carry
`1110`	14	0x0E	Invalid BCD state	Add `0110` (6) → Produces `0001 0100` (BCD 14) with Carry
`1111`	15	0x0F	Invalid BCD state	Add `0110` (6) → Produces `0001 0101` (BCD 15) with Carry

When microprocessors (such as x86 via DAA instruction, or 8051/Z80 CPUs) perform binary addition on BCD digits and generate a sum greater than 9, hardware circuits detect the half-carry flag (auxiliary carry) and automatically add 6 (0110) to shift the excess value into the next decade.

Packed BCD vs. Unpacked BCD Architecture

Digital engineers format BCD in memory using two distinct layouts:

Unpacked BCD (1 Digit per Byte)

Stores one decimal numeral per 8-bit byte. The upper 4 bits are zeroed (0000) or set to ASCII mask 0011 (hex 0x30 to 0x39).

Example Decimal 59:
Byte 0: 0000 0101 (Digit 5)
Byte 1: 0000 1001 (Digit 9)

Simpler for direct UART serial transmission and character displays, but wastes 50% of memory storage.

Packed BCD (2 Digits per Byte)

Packs two decimal numerals into a single 8-bit byte: high nibble stores the tens digit, low nibble stores the units digit.

Example Decimal 59:
Byte 0: 0101 1001 (High: 5, Low: 9)

Utilized in COBOL COMP-3 fields, IBM System/360 business computing, and CMOS RTC battery-backed memory to double storage density.

Frequently Asked Questions: Binary Coded Decimal

How do you convert BCD to decimal manually?

Separate the binary bitstream into 4-bit clusters (nibbles) starting from right to left. Then convert each 4-bit cluster into its decimal value (0 to 9) using the 8-4-2-1 weight formula. For example, given 0111 0010: 0111 = 4+2+1 = 7, and 0010 = 2, giving decimal 72.

Why is pure binary more storage efficient than BCD?

Pure binary uses every combination of bits to represent magnitude (an 8-bit byte can represent 0 to 255). Packed BCD uses an 8-bit byte to represent only two decimal digits (00 to 99), leaving 156 bit combinations unused (a ~17% efficiency loss). However, BCD eliminates conversion lag and decimal rounding errors.

What is the difference between Natural BCD (8421) and Excess-3 (XS-3) code?

Natural BCD (8421) weights bits directly by powers of two. Excess-3 code adds binary 3 (0011) to each BCD digit. Excess-3 is a non-weighted, self-complementing code where the 9's complement of a decimal digit is formed by inverting all bits, greatly simplifying subtraction in early computer arithmetic units.