![Binary clock](https://upload.wikimedia.org/wikipedia/commons/2/27/Binary_clock.svg)
In
computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes, and development of both hardware and software. Computing has scientific, e ...
and
electronic
Electronic may refer to:
*Electronics, the science of how to control electric energy in semiconductor
* ''Electronics'' (magazine), a defunct American trade journal
*Electronic storage, the storage of data using an electronic device
*Electronic co ...
systems, binary-coded decimal (BCD) is a class of
binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that t ...
encodings of
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
numbers where each
digit is represented by a fixed number of
bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
s, usually four or eight. Sometimes, special bit patterns are used for a
sign
A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...
or other indications (e.g. error or overflow).
In
byte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit ...
-oriented systems (i.e. most modern computers), the term ''unpacked'' BCD
usually implies a full byte for each digit (often including a sign), whereas ''packed'' BCD typically encodes two digits within a single byte by taking advantage of the fact that four bits are enough to represent the range 0 to 9. The precise 4-bit encoding, however, may vary for technical reasons (e.g.
Excess-3
Excess-3, 3-excess or 10-excess-3 binary code (often abbreviated as XS-3, 3XS or X3), shifted binary or Stibitz code (after George Stibitz, who built a relay-based adding machine in 1937) is a self-complementary binary-coded decimal (BCD) cod ...
).
The ten states representing a BCD digit are sometimes called ''
tetrades''
(for the
nibble
In computing, a nibble (occasionally nybble, nyble, or nybl to match the spelling of byte) is a four-bit aggregation, or half an octet. It is also known as half-byte or tetrade. In a networking or telecommunication context, the nibble is oft ...
typically needed to hold them is also known as a tetrade) while the unused,
don't care
In digital logic, a don't-care term (abbreviated DC, historically also known as ''redundancies'', ''irrelevancies'', ''optional entries'', ''invalid combinations'', ''vacuous combinations'', ''forbidden combinations'', ''unused states'' or ''l ...
-states are named ,
''pseudo-decimals''
or ''pseudo-decimal digits''.
BCD's main virtue, in comparison to binary
positional system
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which th ...
s, is its more accurate representation and rounding of decimal quantities, as well as its ease of conversion into conventional human-readable representations. Its principal drawbacks are a slight increase in the complexity of the circuits needed to implement basic arithmetic as well as slightly less dense storage.
BCD was used in many early
decimal computer
Decimal computers are computers which can represent numbers and addresses in decimal as well as providing instructions to operate on those numbers and addresses directly in decimal, without conversion to a pure binary representation. Some also h ...
s, and is implemented in the instruction set of machines such as the
IBM System/360
The IBM System/360 (S/360) is a family of mainframe computer systems that was announced by IBM on April 7, 1964, and delivered between 1965 and 1978. It was the first family of computers designed to cover both commercial and scientific applica ...
series and its descendants,
Digital Equipment Corporation
Digital Equipment Corporation (DEC ), using the trademark Digital, was a major American company in the computer industry from the 1960s to the 1990s. The company was co-founded by Ken Olsen and Harlan Anderson in 1957. Olsen was president unt ...
's
VAX
VAX (an acronym for Virtual Address eXtension) is a series of computers featuring a 32-bit instruction set architecture (ISA) and virtual memory that was developed and sold by Digital Equipment Corporation (DEC) in the late 20th century. The V ...
, the
Burroughs B1700
The Burroughs B1000 Series was a series of mainframe computers, built by the Burroughs Corporation, and originally introduced in the 1970s with continued software development until 1987. The series consisted of three major generations which were t ...
, and the Motorola
68000
The Motorola 68000 (sometimes shortened to Motorola 68k or m68k and usually pronounced "sixty-eight-thousand") is a 16/32-bit complex instruction set computer (CISC) microprocessor, introduced in 1979 by Motorola Semiconductor Products Sector ...
-series processors. BCD ''per se'' is not as widely used as in the past, and is unavailable or limited in newer instruction sets (e.g.,
ARM
In human anatomy, the arm refers to the upper limb in common usage, although academically the term specifically means the upper arm between the glenohumeral joint (shoulder joint) and the elbow joint. The distal part of the upper limb between th ...
;
x86
x86 (also known as 80x86 or the 8086 family) is a family of complex instruction set computer (CISC) instruction set architectures initially developed by Intel based on the Intel 8086 microprocessor and its 8088 variant. The 8086 was introd ...
in
long mode
In the x86-64 computer architecture, long mode is the mode where a 64-bit operating system can access 64-bit instructions and registers. 64-bit programs are run in a sub-mode called 64-bit mode, while 32-bit programs and 16-bit protected mode pr ...
). However, decimal
fixed-point and decimal
floating-point
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can b ...
formats are still important and continue to be used in financial, commercial, and industrial computing, where the subtle conversion and fractional
rounding errors that are inherent in binary floating point formats cannot be tolerated.
Background
BCD takes advantage of the fact that any one decimal numeral can be represented by a four-bit pattern. The most obvious way of encoding digits is ''Natural BCD'' (NBCD), where each decimal digit is represented by its corresponding four-bit binary value, as shown in the following table. This is also called "8421" encoding.
This scheme can also be referred to as ''Simple Binary-Coded Decimal'' (''SBCD'') or ''BCD 8421'', and is the most common encoding.
Others include the so-called "4221" and "7421" encoding – named after the weighting used for the bits – and "
Excess-3
Excess-3, 3-excess or 10-excess-3 binary code (often abbreviated as XS-3, 3XS or X3), shifted binary or Stibitz code (after George Stibitz, who built a relay-based adding machine in 1937) is a self-complementary binary-coded decimal (BCD) cod ...
".
For example, the BCD digit 6, in 8421 notation, is in 4221 (two encodings are possible), in 7421, while in Excess-3 it is (
).
The following table represents
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
digits from 0 to 9 in various BCD encoding systems. In the headers, the "
8421
" indicates the weight of each bit. In the fifth column ("BCD 84−2−1"), two of the weights are negative. Both ASCII and EBCDIC character codes for the digits, which are examples of zoned BCD, are also shown.
As most computers deal with data in 8-bit
byte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit ...
s, it is possible to use one of the following methods to encode a BCD number:
* Unpacked: Each decimal digit is encoded into one byte, with four bits representing the number and the remaining bits having no significance.
* Packed: Two decimal digits are encoded into a single byte, with one digit in the least significant
nibble
In computing, a nibble (occasionally nybble, nyble, or nybl to match the spelling of byte) is a four-bit aggregation, or half an octet. It is also known as half-byte or tetrade. In a networking or telecommunication context, the nibble is oft ...
(
bits 0 through 3) and the other numeral in the most significant nibble (bits 4 through 7).
As an example, encoding the decimal number
91
using unpacked BCD results in the following binary pattern of two bytes:
Decimal: 9 1
Binary : 0000 1001 0000 0001
In packed BCD, the same number would fit into a single byte:
Decimal: 9 1
Binary: 1001 0001
Hence the numerical range for one unpacked BCD byte is zero through nine inclusive, whereas the range for one packed BCD byte is zero through ninety-nine inclusive.
To represent numbers larger than the range of a single byte any number of contiguous bytes may be used. For example, to represent the decimal number
12345
in packed BCD, using
big-endian
In computing, endianness, also known as byte sex, is the order or sequence of bytes of a word of digital data in computer memory. Endianness is primarily expressed as big-endian (BE) or little-endian (LE). A big-endian system stores the most sig ...
format, a program would encode as follows:
Decimal: 0 1 2 3 4 5
Binary : 0000 0001 0010 0011 0100 0101
Here, the most significant nibble of the most significant byte has been encoded as zero, so the number is stored as
012345
(but formatting routines might replace or remove leading zeros). Packed BCD is more efficient in storage usage than unpacked BCD; encoding the same number (with the leading zero) in unpacked format would consume twice the storage.
Shifting and
masking operations are used to pack or unpack a packed BCD digit. Other
bitwise operation
In computer programming, a bitwise operation operates on a bit string, a bit array or a binary numeral (considered as a bit string) at the level of its individual bits. It is a fast and simple action, basic to the higher-level arithmetic operati ...
s are used to convert a numeral to its equivalent bit pattern or reverse the process.
Packed BCD
In packed BCD (or simply packed decimal
), each of the two
nibble
In computing, a nibble (occasionally nybble, nyble, or nybl to match the spelling of byte) is a four-bit aggregation, or half an octet. It is also known as half-byte or tetrade. In a networking or telecommunication context, the nibble is oft ...
s of each byte represent a decimal digit.
Packed BCD has been in use since at least the 1960s and is implemented in all IBM mainframe hardware since then. Most implementations are
big endian
In computing, endianness, also known as byte sex, is the order or sequence of bytes of a word of digital data in computer memory. Endianness is primarily expressed as big-endian (BE) or little-endian (LE). A big-endian system stores the most sig ...
, i.e. with the more significant digit in the upper half of each byte, and with the leftmost byte (residing at the lowest memory address) containing the most significant digits of the packed decimal value. The lower nibble of the rightmost byte is usually used as the sign flag, although some unsigned representations lack a sign flag. As an example, a 4-byte value consists of 8 nibbles, wherein the upper 7 nibbles store the digits of a 7-digit decimal value, and the lowest nibble indicates the sign of the decimal integer value.
Standard sign values are 1100 (
hex C) for positive (+) and 1101 (D) for negative (−). This convention comes from the zone field for
EBCDIC
Extended Binary Coded Decimal Interchange Code (EBCDIC; ) is an eight-bit character encoding used mainly on IBM mainframe and IBM midrange computer operating systems. It descended from the code used with punched cards and the corresponding six- ...
characters and the
signed overpunch
In computing, a signed overpunch is a coding scheme which stores the sign of a number by changing (usually) the last digit. It is used in character data on IBM mainframes by languages such as COBOL, PL/I, and RPG. Its purpose is to save a charac ...
representation. Other allowed signs are 1010 (A) and 1110 (E) for positive and 1011 (B) for negative. IBM System/360 processors will use the 1010 (A) and 1011 (B) signs if the A bit is set in the PSW, for the ASCII-8 standard that never passed. Most implementations also provide unsigned BCD values with a sign nibble of 1111 (F).
ILE RPG uses 1111 (F) for positive and 1101 (D) for negative.
These match the EBCDIC zone for digits without a sign overpunch. In packed BCD, the number 127 is represented by 0001 0010 0111 1100 (127C) and −127 is represented by 0001 0010 0111 1101 (127D). Burroughs systems used 1101 (D) for negative, and any other value is considered a positive sign value (the processors will normalize a positive sign to 1100 (C)).
No matter how many bytes wide a
word
A word is a basic element of language that carries an semantics, objective or pragmatics, practical semantics, meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of w ...
is, there is always an even number of nibbles because each byte has two of them. Therefore, a word of ''n'' bytes can contain up to (2''n'')−1 decimal digits, which is always an odd number of digits. A decimal number with ''d'' digits requires (''d''+1) bytes of storage space.
For example, a 4-byte (32-bit) word can hold seven decimal digits plus a sign and can represent values ranging from ±9,999,999. Thus the number −1,234,567 is 7 digits wide and is encoded as:
0001 0010 0011 0100 0101 0110 0111 1101
1 2 3 4 5 6 7 −
Like character strings, the first byte of the packed decimal that with the most significant two digits is usually stored in the lowest address in memory, independent of the
endianness
In computing, endianness, also known as byte sex, is the order or sequence of bytes of a word of digital data in computer memory. Endianness is primarily expressed as big-endian (BE) or little-endian (LE). A big-endian system stores the most sig ...
of the machine.
In contrast, a 4-byte binary
two's complement
Two's complement is a mathematical operation to reversibly convert a positive binary number into a negative binary number with equivalent (but negative) value, using the binary digit with the greatest place value (the leftmost bit in big- endian ...
integer can represent values from −2,147,483,648 to +2,147,483,647.
While packed BCD does not make optimal use of storage (using about 20% more memory than
binary notation
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one).
The base-2 numeral system is a positional notatio ...
to store the same numbers), conversion to
ASCII
ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because of ...
, EBCDIC, or the various encodings of
Unicode
Unicode, formally The Unicode Standard,The formal version reference is is an information technology Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expre ...
is made trivial, as no arithmetic operations are required. The extra storage requirements are usually offset by the need for the accuracy and compatibility with calculator or hand calculation that fixed-point decimal arithmetic provides. Denser packings of
BCD exist which avoid the storage penalty and also need no arithmetic operations for common conversions.
Packed BCD is supported in the
COBOL
COBOL (; an acronym for "common business-oriented language") is a compiled English-like computer programming language designed for business use. It is an imperative, procedural and, since 2002, object-oriented language. COBOL is primarily us ...
programming language as the "COMPUTATIONAL-3" (an IBM extension adopted by many other compiler vendors) or "PACKED-DECIMAL" (part of the 1985 COBOL standard) data type. It is supported in
PL/I
PL/I (Programming Language One, pronounced and sometimes written PL/1) is a procedural, imperative computer programming language developed and published by IBM. It is designed for scientific, engineering, business and system programming. I ...
as "FIXED DECIMAL". Beside the IBM System/360 and later compatible mainframes, packed BCD is implemented in the native instruction set of the original
VAX
VAX (an acronym for Virtual Address eXtension) is a series of computers featuring a 32-bit instruction set architecture (ISA) and virtual memory that was developed and sold by Digital Equipment Corporation (DEC) in the late 20th century. The V ...
processors from
Digital Equipment Corporation
Digital Equipment Corporation (DEC ), using the trademark Digital, was a major American company in the computer industry from the 1960s to the 1990s. The company was co-founded by Ken Olsen and Harlan Anderson in 1957. Olsen was president unt ...
and some models of the
SDS Sigma series
The SDS Sigma series is a series of third generation computers that were introduced by Scientific Data Systems of the United States in 1966.
The first machines in the series are the 16-bit Sigma 2 and the 32-bit Sigma 7; the Sigma 7 was the first ...
mainframes, and is the native format for the
Burroughs Corporation
The Burroughs Corporation was a major American manufacturer of business equipment. The company was founded in 1886 as the American Arithmometer Company. In 1986, it merged with Sperry UNIVAC to form Unisys. The company's history paralleled many ...
Medium Systems line of mainframes (descended from the 1950s
Electrodata 200 series).
Ten's complement
In mathematics and computing, the method of complements is a technique to encode a symmetric range of positive and negative integers in a way that they can use the same algorithm (hardware) for addition throughout the whole range. For a given n ...
representations for negative numbers offer an alternative approach to encoding the sign of packed (and other) BCD numbers. In this case, positive numbers always have a most significant digit between 0 and 4 (inclusive), while negative numbers are represented by the 10's complement of the corresponding positive number. As a result, this system allows for 32-bit packed BCD numbers to range from −50,000,000 to +49,999,999, and −1 is represented as 99999999. (As with two's complement binary numbers, the range is not symmetric about zero.)
Fixed-point packed decimal
Fixed-point decimal numbers are supported by some programming languages (such as COBOL and PL/I). These languages allow the programmer to specify 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 fixed-point value +1,234.567 when the implied decimal point is located between the fourth and fifth digits:
12 34 56 7C
''12 34.56 7+''
The decimal point is not actually stored in memory, as the packed BCD storage format does not provide for it. Its location is simply known to the compiler, and the generated code acts accordingly for the various arithmetic operations.
Higher-density encodings
If a decimal digit requires four bits, then three decimal digits require 12 bits. However, since 2
10 (1,024) is greater than 10
3 (1,000), if three decimal digits are encoded together, only 10 bits are needed. Two such encodings are ''
Chen–Ho encoding
Chen–Ho encoding is a memory-efficient alternate system of binary encoding for decimal digits.
The traditional system of binary encoding for decimal digits, known as binary-coded decimal (BCD), uses four bits to encode each digit, resulting in ...
'' and ''
densely packed decimal
Densely packed decimal (DPD) is an efficient method for binary encoding decimal digits.
The traditional system of binary encoding for decimal digits, known as binary-coded decimal (BCD), uses four bits to encode each digit, resulting in signifi ...
'' (DPD). The latter has the advantage that subsets of the encoding encode two digits in the optimal seven bits and one digit in four bits, as in regular BCD.
Zoned decimal
Some implementations, for example
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:
F1 F2 D3
1 2 −3
EBCDIC zoned decimal conversion table
(*) ''Note: These characters vary depending on the local character
code page
In computing, a code page is a character encoding and as such it is a specific association of a set of printable characters and control characters with unique numbers. Typically each number represents the binary value in a single byte. (In some co ...
setting.''
Fixed-point zoned decimal
Some languages (such as COBOL and PL/I) directly support fixed-point zoned decimal values, assigning an implicit decimal point at some location between the decimal digits of a number. For example, given a six-byte signed zoned decimal value with an implied decimal point to the right of the fourth digit, the hex bytes F1 F2 F7 F9 F5 C0 represent the value +1,279.50:
F1 F2 F7 F9 F5 C0
1 2 7 9. 5 +0
BCD in computers
IBM
IBM used the terms ''
Binary-Coded Decimal Interchange Code
BCD (''binary-coded decimal''), also called alphanumeric BCD, alphameric BCD, BCD Interchange Code, or BCDIC, is a family of representations of numerals, uppercase Latin letters, and some special and control characters as six-bit character code ...
'' (BCDIC, sometimes just called BCD), for 6-bit ''
alphanumeric
Alphanumericals or alphanumeric characters are a combination of alphabetical and numerical characters. More specifically, they are the collection of Latin letters and Arabic digits. An alphanumeric code is an identifier made of alphanumeric ch ...
'' codes that represented numbers, upper-case letters and special characters. Some variation of BCDIC ''alphamerics'' is used in most early IBM computers, including the
IBM 1620
The IBM 1620 was announced by IBM on October 21, 1959, and marketed as an inexpensive scientific computer. After a total production of about two thousand machines, it was withdrawn on November 19, 1970. Modified versions of the 1620 were used as ...
(introduced in 1959),
IBM 1400 series
The IBM 1400 series were second-generation (transistor) mid-range business decimal computers that IBM marketed in the early 1960s. The computers were offered to replace tabulating machines like the IBM 407. The 1400-series machines stored infor ...
, and non-
Decimal Architecture
Decimal computers are computers which can represent numbers and addresses in decimal as well as providing instructions to operate on those numbers and addresses directly in decimal, without conversion to a pure binary representation. Some also ha ...
members of the
IBM 700/7000 series
The IBM 700/7000 series is a series of large-scale (Mainframe computer, mainframe) computer systems that were made by IBM through the 1950s and early 1960s. The series includes several different, incompatible processor architectures. The 700s ...
.
The IBM 1400 series are character-addressable machines, each location being six bits labeled ''B, A, 8, 4, 2'' and ''1,'' plus an odd parity check bit (''C'') and a word mark bit (''M''). For encoding digits ''1'' through ''9'', ''B'' and ''A'' are zero and the digit value represented by standard 4-bit BCD in bits ''8'' through ''1''. For most other characters bits ''B'' and ''A'' are derived simply from the "12", "11", and "0" "zone punches" in the
punched card
A punched card (also punch card or punched-card) is a piece of stiff paper that holds digital data represented by the presence or absence of holes in predefined positions. Punched cards were once common in data processing applications or to di ...
character code, and bits ''8'' through ''1'' from the ''1'' through ''9'' punches. A "12 zone" punch set both ''B'' and ''A'', an "11 zone" set ''B'', and a "0 zone" (a 0 punch combined with any others) set ''A''. Thus the letter A, which is ''(12,1)'' in the punched card format, is encoded ''(B,A,1)''. The currency symbol $, ''(11,8,3)'' in the punched card, was encoded in memory as ''(B,8,2,1)''. This allows the circuitry to convert between the punched card format and the internal storage format to be very simple with only a few special cases. One important special case is digit ''0'', represented by a lone ''0'' punch in the card, and ''(8,2)'' in core memory.
The memory of the IBM 1620 is organized into 6-bit addressable digits, the usual ''8, 4, 2, 1'' plus ''F'', used as a flag bit and ''C'', an odd parity check bit. BCD ''alphamerics'' are encoded using digit pairs, with the "zone" in the even-addressed digit and the "digit" in the odd-addressed digit, the "zone" being related to the ''12'', ''11'', and ''0'' "zone punches" as in the 1400 series. Input/Output translation hardware converted between the internal digit pairs and the external standard 6-bit BCD codes.
In the Decimal Architecture
IBM 7070
IBM 7070 was a decimal-architecture intermediate data-processing system that was introduced by IBM in 1958. It was part of the IBM 700/7000 series, and was based on discrete transistors rather than the vacuum tubes of the 1950s. It was the compan ...
,
IBM 7072, and
IBM 7074 ''alphamerics'' are encoded using digit pairs (using
two-out-of-five code
A two-out-of-five code is a constant-weight code that provides exactly ten possible combinations of two bits, and is thus used for representing the decimal digits using five bits. Each bit is assigned a weight, such that the set bits sum to the ...
in the digits, not BCD) of the 10-digit 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 6-bit BCD codes.
With the introduction of
System/360
The IBM System/360 (S/360) is a family of mainframe computer systems that was announced by IBM on April 7, 1964, and delivered between 1965 and 1978. It was the first family of computers designed to cover both commercial and scientific applica ...
, IBM expanded 6-bit BCD ''alphamerics'' to 8-bit EBCDIC, allowing the addition of many more characters (e.g., lowercase letters). A variable length Packed BCD ''numeric'' data type is also implemented, providing machine instructions that perform arithmetic directly on packed decimal data.
On the
IBM 1130
The IBM 1130 Computing System, introduced in 1965, was IBM's least expensive computer at that time. A binary 16-bit machine, it was marketed to price-sensitive, computing-intensive technical markets, like education and engineering, succeeding th ...
and
1800, packed BCD is supported in software by IBM's Commercial Subroutine Package.
Today, BCD data is still heavily used in IBM processors and databases, such as
IBM Db2
Db2 is a family of data management products, including database servers, developed by IBM. It initially supported the relational model, but was extended to support object–relational features and non-relational structures like JSON a ...
, mainframes, and
Power6
The POWER6 is a microprocessor developed by IBM that implemented the Power ISA v.2.03. When it became available in systems in 2007, it succeeded the POWER5+ as IBM's flagship Power microprocessor. It is claimed to be part of the eCLipz projec ...
. In these products, the BCD is usually zoned BCD (as in EBCDIC or ASCII), Packed BCD (two decimal digits per byte), or "pure" BCD encoding (one decimal digit stored as BCD in the low four bits of each byte). All of these are used within hardware registers and processing units, and in software. To convert packed decimals in EBCDIC table unloads to readable numbers, you can use the OUTREC FIELDS mask of the JCL utility DFSORT.
Other computers
The Digital Equipment Corporation VAX-11 series includes
instructions
Instruction or instructions may refer to:
Computing
* Instruction, one operation of a processor within a computer architecture instruction set
* Computer program, a collection of instructions
Music
* Instruction (band), a 2002 rock band from Ne ...
that can perform arithmetic directly on packed BCD data and convert between packed BCD data and other integer representations.
The VAX's packed BCD format is compatible with that on IBM System/360 and IBM's later compatible processors. The MicroVAX and later VAX implementations dropped this ability from the CPU but retained code compatibility with earlier machines by implementing the missing instructions in an operating system-supplied software library. This is invoked automatically via
exception handling
In computing and computer programming, exception handling is the process of responding to the occurrence of ''exceptions'' – anomalous or exceptional conditions requiring special processing – during the execution of a program. In general, an ...
when the defunct instructions are encountered, so that programs using them can execute without modification on the newer machines.
The
Intel
Intel Corporation is an American multinational corporation and technology company headquartered in Santa Clara, California. It is the world's largest semiconductor chip manufacturer by revenue, and is one of the developers of the x86 seri ...
x86
x86 (also known as 80x86 or the 8086 family) is a family of complex instruction set computer (CISC) instruction set architectures initially developed by Intel based on the Intel 8086 microprocessor and its 8088 variant. The 8086 was introd ...
architecture supports a
unique 18-digit (ten-byte) BCD format that can be loaded into and stored from the floating point registers, from where computations can be performed.
The
Motorola 68000 series
The Motorola 68000 series (also known as 680x0, m68000, m68k, or 68k) is a family of 32-bit complex instruction set computer (CISC) microprocessors. During the 1980s and early 1990s, they were popular in personal computers and workstations and w ...
had BCD instructions.
In more recent computers such capabilities are almost always implemented in software rather than the CPU's instruction set, but BCD numeric data are still extremely common in commercial and financial applications. There are tricks for implementing packed BCD and zoned decimal add–or–subtract operations using short but difficult to understand sequences of word-parallel logic and binary arithmetic operations.
For example, the following code (written in
C) computes an unsigned 8-digit packed BCD addition using 32-bit binary operations:
uint32_t BCDadd(uint32_t a, uint32_t b)
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 employing BCD, the manipulation of numerical data for display can be greatly simplified by treating each digit as a separate single sub-circuit. This matches much more closely the physical reality of display hardware—a designer might choose to use a series of separate identical
seven-segment display
A seven-segment display is a form of electronic display device for displaying decimal numerals that is an alternative to the more complex dot matrix displays.
Seven-segment displays are widely used in digital clocks, electronic meters, basic ...
s to build a metering circuit, for example. If the numeric quantity were stored and manipulated as pure binary, interfacing with such a display would require complex circuitry. Therefore, in cases where the calculations are relatively simple, working throughout with BCD can lead to an overall simpler system than converting to and from binary. Most pocket calculators do all their calculations in BCD.
The same argument applies when hardware of this type uses an embedded microcontroller or other small processor. Often, representing numbers internally in BCD format results in smaller code, since a conversion from or to binary representation can be expensive on such limited processors. For these applications, some small processors feature dedicated arithmetic modes, which assist when writing routines that manipulate BCD quantities.
Operations with BCD
Addition
It is possible to perform
addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
by first adding in binary, and then converting to BCD afterwards. Conversion of the simple sum of two digits can be done by adding 6 (that is, 16 − 10) when the five-bit result of adding a pair of digits has a value greater than 9. The reason for adding 6 is that there are 16 possible 4-bit BCD values (since 2
4 = 16), but only 10 values are valid (0000 through 1001). For example:
1001 + 1000 = 10001
9 + 8 = 17
10001 is the binary, not decimal, representation of the desired result, but the most significant 1 (the "carry") cannot fit in a 4-bit binary number. In BCD as in decimal, there cannot exist a value greater than 9 (1001) per digit. To correct this, 6 (0110) is added to the total, and then the result is treated as two nibbles:
10001 + 0110 = 00010111 => 0001 0111
17 + 6 = 23 1 7
The two nibbles of the result, 0001 and 0111, correspond to the digits "1" and "7". This yields "17" in BCD, which is the correct result.
This technique can be extended to adding multiple digits by adding in groups from right to left, propagating the second digit as a carry, always comparing the 5-bit result of each digit-pair sum to 9. Some CPUs provide a
half-carry flag
A half-carry flag (also known as an auxiliary flag or decimal adjust flag) is a condition flag bit in the status register of many CPU families, such as the Intel 8080, Zilog Z80, the x86, and the Atmel AVR series, among others. It indicates when a ...
to facilitate BCD arithmetic adjustments following binary addition and subtraction operations. The
Intel 8080
The Intel 8080 (''"eighty-eighty"'') is the second 8-bit microprocessor designed and manufactured by Intel. It first appeared in April 1974 and is an extended and enhanced variant of the earlier 8008 design, although without binary compatibil ...
, the
Zilog Z80
The Z80 is an 8-bit microprocessor introduced by Zilog as the startup company's first product. The Z80 was conceived by Federico Faggin in late 1974 and developed by him and his 11 employees starting in early 1975. The first working samples wer ...
and the CPUs of the x86 family
provide the opcode DAA (Decimal Adjust Accumulator).
Subtraction
Subtraction is done by adding the ten's complement of the
subtrahend
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
to the
minuend
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
. To represent the sign of a number in BCD, the number 0000 is used to represent a
positive number
In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
, and 1001 is used to represent a
negative number
In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed m ...
. The remaining 14 combinations are invalid signs. To illustrate signed BCD subtraction, consider the following problem: 357 − 432.
In signed BCD, 357 is 0000 0011 0101 0111. The ten's complement of 432 can be obtained by taking the
nine's complement
In mathematics and computing, the method of complements is a technique to encode a symmetric range of positive and negative integers in a way that they can use the same algorithm (hardware) for addition throughout the whole range. For a given num ...
of 432, and then adding one. So, 999 − 432 = 567, and 567 + 1 = 568. By preceding 568 in BCD by the negative sign code, the number −432 can be represented. So, −432 in signed BCD is 1001 0101 0110 1000.
Now that both numbers are represented in signed BCD, they can be added together:
0000 0011 0101 0111
0 3 5 7
+ 1001 0101 0110 1000
9 5 6 8
= 1001 1000 1011 1111
9 8 11 15
Since BCD is a form of decimal representation, several of the digit sums above are invalid. In the event that an invalid entry (any BCD digit greater than 1001) exists, 6 is added to generate a carry bit and cause the sum to become a valid entry. So, adding 6 to the invalid entries results in the following:
1001 1000 1011 1111
9 8 11 15
+ 0000 0000 0110 0110
0 0 6 6
= 1001 1001 0010 0101
9 9 2 5
Thus the result of the subtraction is 1001 1001 0010 0101 (−925). To confirm the result, note that the first digit is 9, which means negative. This seems to be correct since 357 − 432 should result in a negative number. The remaining nibbles are BCD, so 1001 0010 0101 is 925. The ten's complement of 925 is 1000 − 925 = 75, so the calculated answer is −75.
If there are a different number of nibbles being added together (such as 1053 − 2), the number with the fewer digits must first be prefixed with zeros before taking the ten's complement or subtracting. So, with 1053 − 2, 2 would have to first be represented as 0002 in BCD, and the ten's complement of 0002 would have to be calculated.
Comparison with pure binary
Advantages
* Many non-integral values, such as decimal 0.2, have an infinite place-value representation in binary (.001100110011...) but have a finite place-value in binary-coded decimal (0.0010). Consequently, a system based on binary-coded decimal representations of decimal fractions avoids errors representing and calculating such values. This is useful in financial calculations.
* Scaling by a power of 10 is simple.
*
Rounding
Rounding means replacing a number with an approximate value that has a shorter, simpler, or more explicit representation. For example, replacing $ with $, the fraction 312/937 with 1/3, or the expression with .
Rounding is often done to obta ...
at a decimal digit boundary is simpler. Addition and subtraction in decimal do not require rounding.
* The 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 text-based format such as
XML
Extensible Markup Language (XML) is a markup language and file format for storing, transmitting, and reconstructing arbitrary data. It defines a set of rules for encoding documents in a format that is both human-readable and machine-readable. T ...
, or to drive signals for a
seven-segment display
A seven-segment display is a form of electronic display device for displaying decimal numerals that is an alternative to the more complex dot matrix displays.
Seven-segment displays are widely used in digital clocks, electronic meters, basic ...
) is a simple per-digit mapping, and can be done in linear (
O(''n'')) time. Conversion from pure
binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that t ...
involves relatively complex logic that spans digits, and for large numbers, no linear-time conversion algorithm is known (see ).
Disadvantages
* Some operations are more complex to implement.
Adders require extra logic to cause them to wrap and generate a carry early. 15 to 20 per cent more circuitry is needed for BCD add compared to pure binary. Multiplication requires the use of algorithms that are somewhat more complex than shift-mask-add (a
binary multiplication
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one).
The base-2 numeral system is a positional notatio ...
, requiring binary shifts and adds or the equivalent, per-digit or group of digits is required).
* Standard BCD requires four bits per digit, roughly 20 per cent more space than a binary encoding (the ratio of 4 bits to log
210 bits is 1.204). When packed so that three digits are encoded in ten bits, the storage overhead is greatly reduced, at the expense of an encoding that is unaligned with the 8-bit 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.
Representational variations
Various BCD implementations exist that employ other representations for numbers.
Programmable calculator
Programmable calculators are calculators that can automatically carry out a sequence of operations under control of a stored computer programming, program. Most are Turing complete, and, as such, are theoretically general-purpose computers. Howe ...
s manufactured by
Texas Instruments
Texas Instruments Incorporated (TI) is an American technology company headquartered in Dallas, Texas, that designs and manufactures semiconductors and various integrated circuits, which it sells to electronics designers and manufacturers globall ...
,
Hewlett-Packard
The Hewlett-Packard Company, commonly shortened to Hewlett-Packard ( ) or HP, was an American multinational information technology company headquartered in Palo Alto, California. HP developed and provided a wide variety of hardware components ...
, and others typically employ a
floating-point
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can b ...
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
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
,
underflow/
overflow, and
error
An error (from the Latin ''error'', meaning "wandering") is an action which is inaccurate or incorrect. In some usages, an error is synonymous with a mistake. The etymology derives from the Latin term 'errare', meaning 'to stray'.
In statistics ...
(a blinking display).
Signed variations
Signed decimal values may be represented in several ways. The
COBOL
COBOL (; an acronym for "common business-oriented language") is a compiled English-like computer programming language designed for business use. It is an imperative, procedural and, since 2002, object-oriented language. COBOL is primarily us ...
programming language, for example, supports five zoned decimal formats, with each one encoding the numeric sign in a different way:
Telephony binary-coded decimal (TBCD)
3GPP developed TBCD,
an expansion to BCD where the remaining (unused) bit combinations are used to add specific
telephony
Telephony ( ) is the field of technology involving the development, application, and deployment of telecommunication services for the purpose of electronic transmission of voice, fax, or data, between distant parties. The history of telephony is i ...
characters,
with digits similar to those found in
telephone keypad
A telephone keypad is a keypad installed on a push-button telephone or similar telecommunication device for dialing a telephone number. It was standardized when the dual-tone multi-frequency signaling (DTMF) system was developed in the Bell Syste ...
s original design.
The mentioned 3GPP document defines TBCD-STRING with swapped nibbles in each byte. Bits, octets and digits indexed from 1, bits from the right, digits and octets from the left.
bits 8765 of octet ''n'' encoding digit 2''n''
bits 4321 of octet ''n'' encoding digit 2(''n'' – 1) + 1
Meaning number
1234
, would become
21 43
in TBCD.
Alternative encodings
If errors in representation and computation are more important than the speed of conversion to and from display, a scaled binary representation may be used, which stores a decimal number as a binary-encoded integer and a binary-encoded signed decimal exponent. For example, 0.2 can be represented as 2.
This representation allows rapid multiplication and division, but may require shifting by a power of 10 during addition and subtraction to align the decimal points. It is appropriate for applications with a fixed number of decimal places that do not then require this adjustment—particularly financial applications where 2 or 4 digits after the decimal point are usually enough. Indeed, this is almost a form of
fixed point arithmetic since the position of the
radix point
A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The choi ...
is implied.
The
Hertz
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that on ...
and
Chen–Ho encoding
Chen–Ho encoding is a memory-efficient alternate system of binary encoding for decimal digits.
The traditional system of binary encoding for decimal digits, known as binary-coded decimal (BCD), uses four bits to encode each digit, resulting in ...
s provide Boolean transformations for converting groups of three BCD-encoded digits to and from 10-bit values
that can be efficiently encoded in hardware with only 2 or 3 gate delays.
Densely packed decimal
Densely packed decimal (DPD) is an efficient method for binary encoding decimal digits.
The traditional system of binary encoding for decimal digits, known as binary-coded decimal (BCD), uses four bits to encode each digit, resulting in signifi ...
(DPD) is a similar scheme
that is used for most of the
significand
The significand (also mantissa or coefficient, sometimes also argument, or ambiguously fraction or characteristic) is part of a number in scientific notation or in floating-point representation, consisting of its significant digits. Depending on ...
, except the lead digit, for one of the two alternative decimal encodings specified in the
IEEE 754-2008
The Institute of Electrical and Electronics Engineers (IEEE) is a 501(c)(3) professional association for electronic engineering and electrical engineering (and associated disciplines) with its corporate office in New York City and its operation ...
floating-point standard.
Application
The
BIOS
In computing, BIOS (, ; Basic Input/Output System, also known as the System BIOS, ROM BIOS, BIOS ROM or PC BIOS) is firmware used to provide runtime services for operating systems and programs and to perform hardware initialization during the ...
in many
personal computer
A personal computer (PC) is a multi-purpose microcomputer whose size, capabilities, and price make it feasible for individual use. Personal computers are intended to be operated directly by an end user, rather than by a computer expert or tec ...
s stores the date and time in BCD because the
MC6818
The 6800 ("''sixty-eight hundred''") is an 8-bit microprocessor designed and first manufactured by Motorola in 1974. The MC6800 microprocessor was part of the M6800 Microcomputer System (latter dubbed ''68xx'') that also included serial and para ...
real-time clock chip used in the original
IBM PC AT
The IBM Personal Computer/AT (model 5170, abbreviated as IBM AT or PC/AT) was released in 1984 as the fourth model in the IBM Personal Computer line, following the IBM PC/XT and its IBM Portable PC variant. It was designed around the Intel 8028 ...
motherboard provided the time encoded in BCD. This form is easily converted into ASCII for display.
The
Atari 8-bit family
The Atari 8-bit family is a series of 8-bit home computers introduced by Atari, Inc. in 1979 as the Atari 400 and Atari 800. The series was successively upgraded to Atari 1200XL , Atari 600XL, Atari 800XL, Atari 65XE, Atari 130XE, Atari 800XE, ...
of computers used BCD to implement floating-point algorithms. The
MOS 6502
The MOS Technology 6502 (typically pronounced "sixty-five-oh-two" or "six-five-oh-two") William Mensch and the moderator both pronounce the 6502 microprocessor as ''"sixty-five-oh-two"''. is an 8-bit microprocessor that was designed by a small te ...
processor has a BCD mode that affects the addition and subtraction instructions. The
Psion Organiser 1 handheld computer's manufacturer-supplied software also entirely used BCD to implement floating point; later Psion models used binary exclusively.
Early models of the
PlayStation 3
The PlayStation 3 (PS3) is a home video game console developed by Sony Interactive Entertainment, Sony Computer Entertainment. The successor to the PlayStation 2, it is part of the PlayStation brand of consoles. It was first released on Novemb ...
store the date and time in BCD. This led to a worldwide outage of the console on 1 March 2010. The last two digits of the year stored as BCD
were misinterpreted as 16 causing an error in the unit's date, rendering most functions inoperable. This has been referred to as the
Year 2010 problem
The year 2000 problem, also known as the Y2K problem, Y2K scare, millennium bug, Y2K bug, Y2K glitch, Y2K error, or simply Y2K refers to potential computer errors related to the Time formatting and storage bugs, formatting and storage of cale ...
.
Legal history
In the 1972 case ''
Gottschalk v. Benson
''Gottschalk v. Benson'', 409 U.S. 63 (1972), was a Supreme Court of the United States, United States Supreme Court case in which the Court ruled that a process claim directed to a numerical algorithm, as such, was not patentable because "the pate ...
'', the
U.S. Supreme Court
The Supreme Court of the United States (SCOTUS) is the highest court in the federal judiciary of the United States. It has ultimate appellate jurisdiction over all U.S. federal court cases, and over state court cases that involve a point o ...
overturned a
lower court
A lower court or inferior court is a court from which an appeal may be taken, usually referring to courts other than supreme court. In relation to an appeal from one court to another, the lower court is the court whose decision is being reviewed ...
's decision that had allowed a
patent
A patent is a type of intellectual property that gives its owner the legal right to exclude others from making, using, or selling an invention for a limited period of time in exchange for publishing an enabling disclosure of the invention."A p ...
for converting BCD-encoded numbers to binary on a computer. The decision noted that a patent "would wholly pre-empt the mathematical formula and in practical effect would be a patent on the
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
itself".
This was a landmark judgement that determining the patentability of software and algorithms.
See also
*
Bi-quinary coded decimal
*
Binary-coded ternary (BCT)
*
Binary integer decimal
The IEEE 754-2008 standard includes decimal floating-point number formats in which the significand and the exponent (and the payloads of NaNs) can be encoded in two ways, referred to as binary encoding and ''decimal encoding''.
Both formats bre ...
(BID)
*
Bitmask
In computer science, a mask or bitmask is data that is used for bitwise operations, particularly in a bit field. Using a mask, multiple bits in a byte, nibble, word, etc. can be set either on or off, or inverted from on to off (or vice versa) in ...
*
Chen–Ho encoding
Chen–Ho encoding is a memory-efficient alternate system of binary encoding for decimal digits.
The traditional system of binary encoding for decimal digits, known as binary-coded decimal (BCD), uses four bits to encode each digit, resulting in ...
*
Decimal computer
Decimal computers are computers which can represent numbers and addresses in decimal as well as providing instructions to operate on those numbers and addresses directly in decimal, without conversion to a pure binary representation. Some also h ...
*
Densely packed decimal
Densely packed decimal (DPD) is an efficient method for binary encoding decimal digits.
The traditional system of binary encoding for decimal digits, known as binary-coded decimal (BCD), uses four bits to encode each digit, resulting in signifi ...
(DPD)
*
Double dabble
In computer science, the double dabble algorithm is used to convert binary numbers into binary-coded decimal (BCD) notation. It is also known as the shift-and-add-3 algorithm, and can be implemented using a small number of gates in computer hardwa ...
, an algorithm for converting binary numbers to BCD
*
Year 2000 problem
The year 2000 problem, also known as the Y2K problem, Y2K scare, millennium bug, Y2K bug, Y2K glitch, Y2K error, or simply Y2K refers to potential computer errors related to the Time formatting and storage bugs, formatting and storage of cale ...
Notes
References
Further reading
*
*
* and (NB. At least some batches of the Krieger reprint edition were
misprint
A typographical error (often shortened to typo), also called a misprint, is a mistake (such as a spelling mistake) made in the typing of printed (or electronic) material. Historically, this referred to mistakes in manual type-setting (typography). ...
s with defective pages 115–146.)
* (Also: ACM SIGPLAN Notices, Vol. 22 #10, IEEE Computer Society Press #87CH2440-6, October 1987)
**
*
*
*
*
*
External links
*
*
Convert BCD to decimal, binary and hexadecimal and vice versaBCD for Java
{{DEFAULTSORT:Binary-Coded Decimal
Computer arithmetic
Numeral systems
Non-standard positional numeral systems
Binary arithmetic
Articles with example C code