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Category Unit Converters

BCD Input (binary digits)

0 bits 0 nibbles

Decimal to BCD (reverse check)

Decimal Result

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Nibble Breakdown

#	Nibble	Calculation	Digit	Status

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About

Binary-Coded Decimal (BCD) encodes each decimal digit independently into a fixed-length 4-bit binary nibble using the 8421 weighting scheme. Unlike standard binary, BCD does not treat the entire number as a single base-2 value. A misread nibble or an invalid code (any value from 1010 to 1111) produces a fault in hardware decoders and financial computing systems that rely on exact decimal representation. This converter validates each nibble against the legal BCD range 0000 - 1001, flags invalid codes with their bit position, and outputs the concatenated decimal result. It assumes unsigned, integer BCD with no sign nibble. For packed BCD formats used in COBOL, IBM mainframes, and fiscal registers, enter consecutive nibbles without separators.

Formulas

Each 4-bit BCD nibble b₃b₂b₁b₀ converts to its decimal digit d using positional bit weights:

d = b₃ × 8 + b₂ × 4 + b₁ × 2 + b₀ × 1

For a multi-digit BCD number consisting of n nibbles, the full decimal value D is formed by concatenation, not arithmetic summation:

D = d₁ ⋅ 10ⁿ⁻¹ + d₂ ⋅ 10ⁿ⁻² + … + d_n ⋅ 10⁰

Validity constraint for each nibble: d ≤ 9. Any nibble yielding d ≥ 10 is an illegal BCD code.

Where b₃, b₂, b₁, b₀ = individual bits of the nibble (most significant to least significant). d = decoded decimal digit (0 - 9). n = total number of nibbles. D = final decimal output.

Reference Data

Decimal Digit	BCD (8421)	Bit Weight Calculation
0	0000	0×8 + 0×4 + 0×2 + 0×1 = 0
1	0001	0×8 + 0×4 + 0×2 + 1×1 = 1
2	0010	0×8 + 0×4 + 1×2 + 0×1 = 2
3	0011	0×8 + 0×4 + 1×2 + 1×1 = 3
4	0100	0×8 + 1×4 + 0×2 + 0×1 = 4
5	0101	0×8 + 1×4 + 0×2 + 1×1 = 5
6	0110	0×8 + 1×4 + 1×2 + 0×1 = 6
7	0111	0×8 + 1×4 + 1×2 + 1×1 = 7
8	1000	1×8 + 0×4 + 0×2 + 0×1 = 8
9	1001	1×8 + 0×4 + 0×2 + 1×1 = 9
Values below are invalid in standard 8421 BCD
INVALID	1010	Decimal 10 - exceeds single digit range
INVALID	1011	Decimal 11 - exceeds single digit range
INVALID	1100	Decimal 12 - exceeds single digit range
INVALID	1101	Decimal 13 - exceeds single digit range
INVALID	1110	Decimal 14 - exceeds single digit range
INVALID	1111	Decimal 15 - exceeds single digit range

Frequently Asked Questions

Nibble values from 1010 (10) through 1111 (15) are undefined in standard 8421 BCD. Hardware BCD decoders (such as the 74LS49 or CD4511) treat these as don't-care states, often producing garbled segment outputs. This converter flags each invalid nibble by position and halts conversion to prevent silent data corruption.

In standard binary, the decimal number 59 is encoded as a single 6-bit value 111011. In BCD, it occupies 8 bits: 0101 (digit 5) concatenated with 1001 (digit 9). BCD wastes approximately 17% of bit space because 6 of the 16 possible 4-bit codes are unused. The trade-off is exact decimal fidelity, which is critical in financial arithmetic where floating-point rounding is unacceptable.

Standard unsigned BCD does not. Signed BCD conventions exist (IBM mainframes use a trailing nibble: 1100 for positive, 1101 for negative), but these are vendor-specific. BCD fixed-point formats place an implied decimal point between nibbles. This converter handles unsigned integer BCD only. For signed packed BCD, strip the sign nibble before conversion and apply the sign manually.

IEEE 754 binary floating-point cannot represent 0.1 exactly. It stores an approximation: 0.1000000000000000055511151231257827021181583404541015625. Over millions of transactions, rounding errors accumulate into real monetary discrepancies. BCD stores each decimal digit exactly, so $0.10 remains precisely 0.10. IBM System/360 and its successors include hardware BCD arithmetic instructions for this reason.

This converter left-pads the input with leading zeros to form complete 4-bit nibbles. For example, an input of 10001 (5 bits) is treated as 0001 0001, yielding decimal 11. This matches the convention that leading zeros do not change the value. If you specifically need to flag incomplete nibbles as an error, ensure your source data is nibble-aligned before pasting.

Excess-3 adds 3 (0011) to each standard BCD code. Digit 0 becomes 0011, digit 9 becomes 1100. This self-complementing property simplifies subtraction in hardware because the 9's complement of an Excess-3 digit is its bitwise NOT. This converter uses the 8421 standard. To convert Excess-3 input, subtract 3 from each nibble's decimal value before interpreting it.