Definition
The binarycoded decimal (BCD) is an encoding for decimal numbers in which each digit is represented by its own binary sequence.
Basics
In computing and electronic systems, binarycoded decimal (BCD)
is an encoding for decimal numbers in which each digit is represented
by its own binary sequence. Its main virtue is that it allows easy
conversion to decimal digits for printing or display and faster decimal
calculations. Its drawbacks are the increased complexity of circuits
needed to implement mathematical operations and a relatively
inefficient encoding – it occupies more space than a pure binary
representation. Even though the importance of BCD has diminished, it is still widely used in financial, commercial, and industrial applications.
In BCD, a digit is usually represented by four bits which, in general, represent the values/digits/characters 09. Other bit combinations are sometimes used for sign or other indications.
To BCDencode a decimal number using the common encoding, each decimal digit is stored in a fourbit nibble.
Decimal: 0 1 2 3 4 5 6 7 8 9
BCD: 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001
Thus, the BCD encoding for the number 127 would be:
0001 0010 0111
Since most computers store data in eightbit bytes, there are two common ways of storing fourbit BCD digits in those bytes:
 each digit is stored in one byte, and the other four bits are then set to all zeros, all ones (as in the EBCDIC code), or to 0011 (as in the ASCII code)
 two digits are stored in each byte.
Unlike binary encoded numbers, BCD encoded numbers can easily be
displayed by mapping each of the nibbles to a different character.
Converting a binary encoded number to decimal for display is much
harder involving integer multiplication or divide operations.
BCD in electronics
BCD is very common in electronic systems where a numeric value is to be displayed, especially in systems consisting solely of digital logic, and not containing a microprocessor.
By utilising BCD, the manipulation of numerical data for display can be
greatly simplified by treating each digit as a separate single
subcircuit. This matches much more closely the physical reality of
display hardware—a designer might choose to use a series of separate
identical 7segment displays
to build a metering circuit, for example. If the numeric quantity were
stored and manipulated as pure binary, interfacing to such a display
would require complex circuitry. Therefore, in cases where the
calculations are relatively simple working throughout with BCD can lead
to a simpler overall system than converting to 'pure' binary.
The same argument applies when hardware of this type uses an embedded microcontroller
or other small processor. Often, smaller code results when representing
numbers internally in BCD format, since a conversion from or to binary
representation can be expensive on such limited processors. For these
applications, some small processors feature BCD arithmetic modes, which
assist when writing routines that manipulate BCD quantities.
Packed BCD
A widely used variation of the twodigitsperbyte encoding is called packed BCD (or simply packed decimal), where numbers are stored with two decimal digits "packed" into one byte each, and the last digit (or nibble) is used as a sign indicator. The preferred sign values are 1100 (hex
C) for positive (+) and 1101 (hex D) for negative (−); other allowed
signs are 1010 (A) and 1110 (E) for positive and 1011 (B) for negative.
Some implementations also provide unsigned BCD values with a sign
nibble of 1111 (hex F). In packed BCD, the number +127 is represented
as the bytes 00010010 01111100 (hex 12 7C), and −127 as 00010010
01111101 (hex 12 7D).
Sign
Digit 
BCD
8 4 2 1 
Sign 
A 
1 0 1 0 
+ 
B 
1 0 1 1 
− 
C 
1 1 0 0 
+ (preferred) 
D 
1 1 0 1 
− (preferred) 
E 
1 1 1 0 
+ 
F 
1 1 1 1 
+ (unsigned) 
Packing fourbit digits and a sign into eightbit bytes means that an nbyte packed decimal value (where n typically ranges from 1 to 15) contains 2n−1 decimal digits (which is always an odd number of digits). In other words, d decimal digits require a packed decimal representation that is ½(d+1)
bytes wide. For example, a fourbyte packed decimal number holds seven
decimal digits plus a sign, and can represent values from ±0,000,000 to
±9,999,999. In contrast, a fourbyte binary two's complement integer can represent values from −2,147,483,648 to +2,147,483,647.
While packed BCD does not make optimal use of storage (about ^{1}/_{6} of the required memory is wasted), conversion to ASCII, EBCDIC, or the various encodings of Unicode
is still trivial, as no arithmetic operations are required. The extra
storage requirements are usually offset by the need for the accuracy
that fixedpoint decimal arithmetic provides. More dense packings of
BCD exist which avoid the storage penalty and also need no arithmetic
operations for common conversions.
Fixedpoint packed decimal
Fixedpoint decimal numbers are supported by some programming languages (such as COBOL and PL/1),
and provide an implicit decimal point in front of one of the digits.
For example, a packed decimal value encoded with the bytes 12 34 56 7C
represents the fixedpoint value +1,234.567 when the implied decimal
point is located between the 4th and 5th digits.
Higherdensity encodings
If a decimal digit requires four bits, then three decimal digits require 12 bits. However, since 2^{10}>10^{3}, if three decimal digits are encoded together then only 10 bits are needed. Two such encodings are ChenHo encoding and Densely Packed Decimal.
The latter has the advantage that subsets of the encoding encode two
digits in the optimal 7 bits and one digit in 4 bits, as in regular BCD.
Zoned decimal
Some implementations (notably IBM mainframe systems) support zoned decimal
numeric representations. Each decimal digit is stored in one byte, with
the lower four bits encoding the digit in BCD form. The upper four
bits, called the "zone" bits, are usually set to a fixed value so that
the byte holds a character value corresponding to the digit. EBCDIC
systems use a zone value of 1111 (hex F); this yields bytes in the
range F0 to F9 (hex), which are the EBCDIC codes for the characters "0"
through "9". Similarly, ASCII systems use a zone value of 0011 (hex 3),
giving character codes 30 to 39 (hex).
For signed zoned decimal values, the rightmost (least significant)
zone nibble holds the sign digit, which is the same set of values that
are used for signed packed decimal numbers (see above). Thus a zoned
decimal value encoded as the hex bytes F1 F2 D3 represents the signed
decimal value −123.
Fixedpoint zone decimal
Some languages (such as COBOL and PL/1)
directly support fixedpoint zoned decimal values, assiging an implicit
decimal point at some location between the decimal digits of a number.
For example, given a sixbyte signed zoned decimal value with an
implied decimal point to the right of the 4th digit, the hex bytes F1
F2 F7 F9 F5 C0 represent the value +1,279.50.
IBM and BCD
IBM used the terms binarycoded decimal and BCD for sixbit alphameric
codes that represented numbers, uppercase letters and special
characters. Some variation of BCD was used in most early IBM computers,
including the IBM 1620, IBM 1400 series, and nonDecimal Architecture members of the IBM 700/7000 series. With the introduction of System/360, IBM replaced BCD with 8bit EBCDIC.
Bit positions in BCD were usually labelled B, A, 8, 4, 2 and 1. For encoding digits, B and A were zero. The letter A was encoded (B,A,1).
In the 1620 BCD alphamerics were encoded using digit pairs,
with the "zone" in the even digit and the "digit" in the odd digit.
Input/Output translation hardware converted between the internal digit
pairs and the external standard sixbit BCD codes.
In the Decimal Architecture IBM 7070, IBM 7072, and IBM 7074 alphamerics were encoded using digit pairs (using twooutoffive code in the digits, not
BCD) of the 10digit word, with the "zone" in the left digit and the
"digit" in the right digit. Input/Output translation hardware converted
between the internal digit pairs and the external standard sixbit BCD
codes.
Today, BCD is still heavily used in IBM processors and databases, such as IBM DB2, mainframes and Power6.
In these products, the BCD is usually zoned BCD (as in EBCDIC or
ASCII), Packed BCD, or 'pure' BCD encoding. All of these are used in
within hardware registers and processing units and in software.
Addition with BCD
To perform addition in BCD, you can first addup in binary format, and then perform the conversion
to BCD afterwards. This conversion involves adding 6 to each group of
four digits that has a value of greaterthan 9. For example:
 9+5=14 = [1001] + [0101] = [1110] in binary.
However, in BCD, we cannot have a value greaterthan 9 (1001) pernibble. To correct this, one adds 6 to that group:
 [0000 1110] + [0000 0110] = [0001 0100]
which gives us two nibbles, [0001] and [0100] which correspond to
"1" and "4" respectively. This gives us the 14 in BCD which is the
correct result.
Background
The binarycoded decimal scheme described in this article is the
most common encoding, but there are many others. The method here can be
referred to as Simple BinaryCoded Decimal (SBCD) or BCD 8421. In the headers to the table, the '8 4 2 1' indicates the four bit weights; note that in the 5^{th} column two of the weights are negative.
The following table represents decimal digits from 0 to 9 in various BCD systems:
Digit 
BCD
8 4 2 1 
Excess3
or Stibitz Code 
BCD 2 4 2 1
or Aiken Code 
BCD
8 4 −2 −1 
IBM 702 IBM 705
IBM 7080 IBM 1401
8 4 2 1 
0 
0000 
0011 
0000 
0000 
1010 
1 
0001 
0100 
0001 
0111 
0001 
2 
0010 
0101 
0010 
0110 
0010 
3 
0011 
0110 
0011 
0101 
0011 
4 
0100 
0111 
0100 
0100 
0100 
5 
0101 
1000 
1011 
1011 
0101 
6 
0110 
1001 
1100 
1010 
0110 
7 
0111 
1010 
1101 
1001 
0111 
8 
1000 
1011 
1110 
1000 
1000 
9 
1001 
1100 
1111 
1111 
1001 
Legal history
In 1972, the U.S. Supreme Court overturned a lower court decision
which had allowed a patent for converting BCD encoded numbers to binary
on a computer. This was an important case in determining the patentability of software and algorithms.
Comparison with pure binary
Advantages
 Scaling by a factor of 10 (or a power of 10) is simple; this is
useful when a decimal scaling factor is needed to represent a
noninteger quantity (e.g., in financial calculations)
 Rounding at a decimal digit boundary is simpler.
 Alignment of two decimal numbers (for example 1.3 + 27.08) is a simple, exact, shift
 Conversion to a character form or for display (e.g., to a textbased format such as XML, or to drive signals for a sevensegment display) is a simple perdigit mapping, and can be done in linear (O(n)) time. Conversion from pure binary involves relatively complex logic that spans digits, and for large numbers no lineartime conversion algorithm is known (see Binary numeral system#Conversion_to_and_from_other_numeral_systems).
 Some nonintegral values, such as 0.2, have a finite placevalue
representation in decimal but not in binary; consequently a system
based on binary placevalue representations would introduce a small
error representing such a value, which may be compounded by further
computation if careful numerical considerations are not made. Note that
if computation is not performed on the value this is not an issue,
since it suffices to represent it using enough bits that when rounded
to the original number of decimal digits the original value is
correctly recovered.
Disadvantages
 Some operations are more complex to implement. Adders
require extra logic to cause them to wrap and generate a carry early.
15%–20% more circuitry is needed for BCD add compared to pure binary.
Multiplication requires the use of algorithms that are somewhat more
complex than shiftmaskadd (a binary multiplication, requiring binary shifts and adds or the equivalent, perdigit or group of digits is required)
 Standard BCD requires four bits per digit, roughly 20% more space
than a binary encoding. When packed so that three digits are encoded in
ten bits, the storage overhead is reduced to about 0.34%, at the
expense of an encoding that is unaligned with the 8bit byte boundaries
common on existing hardware, resulting in slower implementations on
these systems.
 Practical existing implementations of BCD are typically slower than
operations on binary representations, especially on embedded systems,
due to limited processor support for native BCD operations.
Applications
The BIOS
in many PCs keeps the date and time in BCD format, probably for
historical reasons (it avoided the need for binary to ASCII conversion).
Representational variations
Various BCD implementations exist that employ other representations for numbers. Programmable calculators manufactured by Texas Instruments, HewlettPackard,
and others typically employ a floatingpoint BCD format, typically with
two or three digits for the (decimal) exponent. The extra bits of the
sign digit may be used to indicate special numeric values, such as infinity, underflow/overflow, and error (a blinking display).
Alternative encodings
If error in representation and computation is the primary concern,
rather than efficiency of conversion to and from display form, a scaled
binary representation may be used, which stores a decimal number as a
binaryencoded integer and a binaryencoded signed decimal exponent.
For example, 0.2 can be represented as 2 × 10^{1}. This
representation allows rapid multiplication and division, but may
require multiplication by a power of 10 during addition and subtraction
to align the decimals. It is particularly appropriate for applications
with a fixed number of decimal places, which do not require adjustment
during addition and subtraction and need not store the exponent
explicitly.
ChenHo encoding
provides a boolean transformation for converting groups of three
BCDencoded digits to and from 10bit values that can be efficiently
encoded in hardware with only 2 or 3 gate delays. Densely Packed Decimal
is a similar scheme that deals more efficiently and conveniently with
the case where the number of digits is not a multiple of 3. Source: Wikipedia (All text is available under the terms of the GNU Free Documentation License and Creative Commons AttributionShareAlike License.)
