Binarycoded decimal
In computing and electronic systems, binarycoded decimal (BCD) is a class of binary encodings of decimal numbers where each digit is represented by a fixed number of bits, usually four or eight. Sometimes, special bit patterns are used for a sign or other indications (e.g. error or overflow).
In byteoriented systems (i.e. most modern computers), the term unpacked BCD^{[1]} 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 fourbit encoding, however, may vary for technical reasons (e.g. Excess3).
The ten states representing a BCD digit are sometimes called tetrades^{[2]}^{[3]} (the nibble typically needed to hold them is also known as a tetrade) while the unused, don't carestates are named pseudotetrad(e)s ,^{[4]}^{[5]}^{[6]}^{[7]}^{[8]} pseudodecimals^{[3]} or pseudodecimal digits.^{[9]}^{[10]}^{[nb 1]}
BCD's main virtue, in comparison to binary positional systems, is its more accurate representation and rounding of decimal quantities, as well as its ease of conversion into conventional humanreadable 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 computers, and is implemented in the instruction set of machines such as the IBM System/360 series and its descendants, Digital Equipment Corporation's VAX, the Burroughs B1700, and the Motorola 68000series 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; x86 in long mode). However, decimal fixedpoint and decimal floatingpoint 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.^{[11]}
Background
[edit]BCD takes advantage of the fact that any one decimal numeral can be represented by a fourbit pattern. An obvious way of encoding digits is Natural BCD (NBCD), where each decimal digit is represented by its corresponding fourbit binary value, as shown in the following table. This is also called "8421" encoding.
Decimal digit  BCD  

8  4  2  1  
0  0  0  0  0 
1  0  0  0  1 
2  0  0  1  0 
3  0  0  1  1 
4  0  1  0  0 
5  0  1  0  1 
6  0  1  1  0 
7  0  1  1  1 
8  1  0  0  0 
9  1  0  0  1 
This scheme can also be referred to as Simple BinaryCoded Decimal (SBCD) or BCD 8421, and is the most common encoding.^{[12]} Others include the socalled "4221" and "7421" encoding – named after the weighting used for the bits – and "Excess3".^{[13]} For example, the BCD digit 6, 0110'b
in 8421 notation, is 1100'b
in 4221 (two encodings are possible), 0110'b
in 7421, while in Excess3 it is 1001'b
().
Bit  Weight  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  Comment 

4  8  0  0  0  0  0  0  0  0  1  1  1  1  1  1  1  1  Binary 
3  4  0  0  0  0  1  1  1  1  0  0  0  0  1  1  1  1  
2  2  0  0  1  1  0  0  1  1  0  0  1  1  0  0  1  1  
1  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1  
Name  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  Decimal  
8 4 2 1 (XS0)  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  ^{[14]}^{[15]}^{[16]}^{[17]}^{[nb 2]}  
7 4 2 1  0  1  2  3  4  5  6  7  8  9  ^{[18]}^{[19]}^{[20]}  
Aiken (2 4 2 1)  0  1  2  3  4  5  6  7  8  9  ^{[14]}^{[15]}^{[16]}^{[17]}^{[nb 3]}  
Excess3 (XS3)  3  2  1  0  1  2  3  4  5  6  7  8  9  10  11  12  ^{[14]}^{[15]}^{[16]}^{[17]}^{[nb 2]}  
Excess6 (XS6)  6  5  4  3  2  1  0  1  2  3  4  5  6  7  8  9  ^{[18]}^{[nb 2]}  
Jumpat2 (2 4 2 1)  0  1  2  3  4  5  6  7  8  9  ^{[16]}^{[17]}  
Jumpat8 (2 4 2 1)  0  1  2  3  4  5  6  7  8  9  ^{[21]}^{[22]}^{[16]}^{[17]}^{[nb 4]}  
4 2 2 1 (I)  0  1  2  3  4  5  6  7  8  9  ^{[16]}^{[17]}  
4 2 2 1 (II)  0  1  2  3  4  5  6  7  8  9  ^{[21]}^{[22]}  
5 4 2 1  0  1  2  3  4  5  6  7  8  9  ^{[18]}^{[14]}^{[16]}^{[17]}  
5 2 2 1  0  1  2  3  4  5  6  7  8  9  ^{[14]}^{[16]}^{[17]}  
5 1 2 1  0  1  2  3  4  5  6  7  8  9  ^{[19]}  
5 3 1 1  0  1  2  3  4  5  6  7  8  9  ^{[16]}^{[17]}  
White (5 2 1 1)  0  1  2  3  4  5  6  7  8  9  ^{[23]}^{[18]}^{[14]}^{[16]}^{[17]}  
5 2 1 1  0  1  2  3  4  5  6  7  8  9  ^{[24]}  
0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  
Magnetic tape  1  2  3  4  5  6  7  8  9  0  ^{[15]}  
Paul  1  3  2  6  7  5  4  0  8  9  ^{[25]}  
Gray  0  1  3  2  6  7  5  4  15  14  12  13  8  9  11  10  ^{[26]}^{[14]}^{[15]}^{[16]}^{[17]}^{[nb 2]}  
Glixon  0  1  3  2  6  7  5  4  9  8  ^{[27]}^{[14]}^{[15]}^{[16]}^{[17]}  
Ledley  0  1  3  2  7  6  4  5  8  9  ^{[28]}  
4 3 1 1  0  1  2  3  5  4  6  7  8  9  ^{[19]}  
LARC  0  1  2  4  3  5  6  7  9  8  ^{[29]}  
Klar  0  1  2  4  3  9  8  7  5  6  ^{[2]}^{[3]}  
Petherick (RAE)  1  3  2  0  4  8  6  7  9  5  ^{[30]}^{[31]}^{[nb 5]}  
O'Brien I (Watts)  0  1  3  2  4  9  8  6  7  5  ^{[32]}^{[14]}^{[16]}^{[17]}^{[nb 6]}  
5cyclic  0  1  3  2  4  5  6  8  7  9  ^{[28]}  
Tompkins I  0  1  3  2  4  9  8  7  5  6  ^{[33]}^{[14]}^{[16]}^{[17]}  
Lippel  0  1  2  3  4  9  8  7  6  5  ^{[34]}^{[35]}^{[14]}  
O'Brien II  0  2  1  4  3  9  7  8  5  6  ^{[32]}^{[14]}^{[16]}^{[17]}  
Tompkins II  0  1  4  3  2  7  9  8  5  6  ^{[33]}^{[14]}^{[16]}^{[17]}  
Excess3 Gray  3  2  0  1  4  3  1  2  12  11  9  10  5  6  8  7  ^{[16]}^{[17]}^{[20]}^{[nb 7]}^{[nb 2]}  
6 3 −2 −1 (I)  3  2  1  0  5  4  8  9  7  6  ^{[29]}^{[36]}  
6 3 −2 −1 (II)  0  3  2  1  6  5  4  9  8  7  ^{[29]}^{[36]}  
8 4 −2 −1  0  4  3  2  1  8  7  6  5  9  ^{[29]}  
Lucal  0  15  14  1  12  3  2  13  8  7  6  9  4  11  10  5  ^{[37]}  
Kautz I  0  2  5  1  3  7  9  8  6  4  ^{[18]}  
Kautz II  9  4  1  3  2  8  6  7  0  5  ^{[18]}^{[14]}  
Susskind I  0  1  4  3  2  9  8  5  6  7  ^{[35]}  
Susskind II  0  1  9  8  4  3  2  5  6  7  ^{[35]}  
0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15 
The following table represents decimal digits from 0 to 9 in various BCD encoding systems. In the headers, the "8 4 2 1
" indicates the weight of each bit. In the fifth column ("BCD 8 4 −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.
Digit 
BCD 8 4 2 1 
Stibitz code or Excess3  AikenCode or BCD 2 4 2 1 
BCD 8 4 −2 −1 
IBM 702, IBM 705, IBM 7080, IBM 1401 8 4 2 1 
ASCII 0000 8421 
EBCDIC 0000 8421 

0  0000  0011  0000  0000  1010  0011 0000  1111 0000 
1  0001  0100  0001  0111  0001  0011 0001  1111 0001 
2  0010  0101  0010  0110  0010  0011 0010  1111 0010 
3  0011  0110  0011  0101  0011  0011 0011  1111 0011 
4  0100  0111  0100  0100  0100  0011 0100  1111 0100 
5  0101  1000  1011  1011  0101  0011 0101  1111 0101 
6  0110  1001  1100  1010  0110  0011 0110  1111 0110 
7  0111  1010  1101  1001  0111  0011 0111  1111 0111 
8  1000  1011  1110  1000  1000  0011 1000  1111 1000 
9  1001  1100  1111  1111  1001  0011 1001  1111 1001 
As most computers deal with data in 8bit bytes, 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 (bits 0 through 3) and the other numeral in the most significant nibble (bits 4 through 7).^{[nb 8]}
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 ninetynine 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 bigendian 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 operations are used to convert a numeral to its equivalent bit pattern or reverse the process.
Packed BCD
[edit]In packed BCD (or packed decimal^{[38]}), each nibble represents a decimal digit.^{[nb 8]} 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, 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 4byte value consists of 8 nibbles, wherein the upper 7 nibbles store the digits of a 7digit 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 characters and the signed overpunch 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 ASCII8 standard that never passed. Most implementations also provide unsigned BCD values with a sign nibble of 1111 (F).^{[39]}^{[40]}^{[41]} ILE RPG uses 1111 (F) for positive and 1101 (D) for negative.^{[42]} 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)).
Sign digit 
BCD 8 4 2 1 
Sign  Notes 

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 
No matter how many bytes wide a word 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 (2n)−1 decimal digits, which is always an odd number of digits. A decimal number with d digits requires 1/2(d+1) bytes of storage space.
For example, a 4byte (32bit) 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 of the machine.
In contrast, a 4byte 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 (using about 20% more memory than binary notation to store the same numbers), conversion to ASCII, EBCDIC, or the various encodings of Unicode 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 fixedpoint 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 programming language as the "COMPUTATIONAL3" (an IBM extension adopted by many other compiler vendors) or "PACKEDDECIMAL" (part of the 1985 COBOL standard) data type. It is supported in PL/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 processors from Digital Equipment Corporation and some models of the SDS Sigma series mainframes, and is the native format for the Burroughs Medium Systems line of mainframes (descended from the 1950s Electrodata 200 series).
Ten's complement 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 32bit 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.)
Fixedpoint packed decimal
[edit]Fixedpoint 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 fixedpoint 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.
Higherdensity encodings
[edit]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 and densely packed decimal (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
[edit]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
[edit]BCD digit  Hexadecimal  EBCDIC character  

0+  C0  A0  E0  F0  { (*)  \ (*)  0  
1+  C1  A1  E1  F1  A  ~ (*)  1  
2+  C2  A2  E2  F2  B  s  S  2 
3+  C3  A3  E3  F3  C  t  T  3 
4+  C4  A4  E4  F4  D  u  U  4 
5+  C5  A5  E5  F5  E  v  V  5 
6+  C6  A6  E6  F6  F  w  W  6 
7+  C7  A7  E7  F7  G  x  X  7 
8+  C8  A8  E8  F8  H  y  Y  8 
9+  C9  A9  E9  F9  I  z  Z  9 
0−  D0  B0  } (*)  ^ (*)  
1−  D1  B1  J  
2−  D2  B2  K  
3−  D3  B3  L  
4−  D4  B4  M  
5−  D5  B5  N  
6−  D6  B6  O  
7−  D7  B7  P  
8−  D8  B8  Q  
9−  D9  B9  R 
(*) Note: These characters vary depending on the local character code page setting.
Fixedpoint zoned decimal
[edit]Some languages (such as COBOL and PL/I) directly support fixedpoint zoned decimal values, assigning 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 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
Operations with BCD
[edit]Addition
[edit]It is possible to perform addition 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 fivebit result of adding a pair of digits has a value greater than 9. The reason for adding 6 is that there are 16 possible 4bit 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 4bit 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 5bit result of each digitpair sum to 9. Some CPUs provide a halfcarry flag to facilitate BCD arithmetic adjustments following binary addition and subtraction operations. The Intel 8080, the Zilog Z80 and the CPUs of the x86 family provide the opcode DAA (Decimal Adjust Accumulator).
Subtraction
[edit]Subtraction is done by adding the ten's complement of the subtrahend to the minuend. To represent the sign of a number in BCD, the number 0000 is used to represent a positive number, and 1001 is used to represent a negative number. 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 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.
BCD in computers
[edit]IBM
[edit]IBM used the terms BinaryCoded Decimal Interchange Code (BCDIC, sometimes just called BCD), for 6bit alphanumeric codes that represented numbers, uppercase letters and special characters. Some variation of BCDIC alphamerics is used in most early IBM computers, including the IBM 1620 (introduced in 1959), IBM 1400 series, and nondecimal architecture members of the IBM 700/7000 series.
The IBM 1400 series are characteraddressable 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 4bit 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 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.^{[43]}
The memory of the IBM 1620 is organized into 6bit 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 evenaddressed digit and the "digit" in the oddaddressed 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 6bit BCD codes.
In the decimal architecture IBM 7070, IBM 7072, and IBM 7074 alphamerics are 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 6bit BCD codes.
With the introduction of System/360, IBM expanded 6bit BCD alphamerics to 8bit 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 and 1800, packed BCD is supported in software by IBM's Commercial Subroutine Package.
Today, BCD data is still heavily used in IBM databases such as IBM Db2 and processors such as z/Architecture and POWER6 and later Power ISA processors. 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.
Other computers
[edit]The Digital Equipment Corporation VAX series includes instructions that can perform arithmetic directly on packed BCD data and convert between packed BCD data and other integer representations.^{[41]} 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 systemsupplied software library. This is invoked automatically via exception handling when the defunct instructions are encountered, so that programs using them can execute without modification on the newer machines.
Many processors have hardware support for BCDencoded integer arithmetic. For example the 6502,^{[44]}^{[45]} the Motorola 68000 series,^{[46]} and the x86 series.^{[47]} The Intel x86 architecture supports a unique 18digit (tenbyte) BCD format that can be loaded into and stored from the floating point registers, from where computations can be performed.^{[48]}
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 wordparallel logic and binary arithmetic operations.^{[49]} For example, the following code (written in C) computes an unsigned 8digit packed BCD addition using 32bit binary operations:
uint32_t BCDadd(uint32_t a, uint32_t b)
{
uint32_t t1, t2; // unsigned 32bit intermediate values
t1 = a + 0x06666666;
t2 = t1 ^ b; // sum without carry propagation
t1 = t1 + b; // provisional sum
t2 = t1 ^ t2; // all the binary carry bits
t2 = ~t2 & 0x11111110; // just the BCD carry bits
t2 = (t2 >> 2)  (t2 >> 3); // correction
return t1  t2; // corrected BCD sum
}
BCD in electronics
[edit]This section has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages)

BCD is 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 subcircuit.
This matches much more closely the physical reality of display hardware—a designer might choose to use a series of separate identical sevensegment displays 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.^{[50]}^{[51]}
Comparison with pure binary
[edit]Advantages
[edit] Scaling by a power of 10 is simple.
 Rounding at a decimal digit boundary is simpler. Addition and subtraction in decimal do not require rounding.^{[dubious – discuss]}
 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 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 number § Conversion to and from other numeral systems).
 Many nonintegral values, such as decimal 0.2, have an infinite placevalue representation in binary (.001100110011...) but have a finite placevalue in binarycoded decimal (0.0010). Consequently, a system based on binarycoded decimal representations of decimal fractions avoids errors representing and calculating such values. This is useful in financial calculations.
Disadvantages
[edit] 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.^{[52]}
 Some operations are more complex to implement. Adders require extra logic to cause them to wrap and generate a carry early. Also, 15 to 20 per cent more circuitry is needed for BCD add compared to pure binary.^{[citation needed]} 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 per cent more space than a binary encoding (the ratio of 4 bits to log_{2}10 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 8bit byte boundaries common on existing hardware, resulting in slower implementations on these systems.
Representational variations
[edit]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).
Signed variations
[edit]Signed decimal values may be represented in several ways. The COBOL programming language, for example, supports five zoned decimal formats, with each one encoding the numeric sign in a different way:
Type  Description  Example 

Unsigned  No sign nibble  F1 F2 F3

Signed trailing (canonical format)  Sign nibble in the last (least significant) byte  F1 F2 C3

Signed leading (overpunch)  Sign nibble in the first (most significant) byte  C1 F2 F3

Signed trailing separate  Separate sign character byte ('+' or '−' ) following the digit bytes

F1 F2 F3 2B

Signed leading separate  Separate sign character byte ('+' or '−' ) preceding the digit bytes

2B F1 F2 F3

Telephony binarycoded decimal (TBCD)
[edit]3GPP developed TBCD,^{[53]} an expansion to BCD where the remaining (unused) bit combinations are used to add specific telephony characters,^{[54]}^{[55]} with digits similar to those found in telephone keypads original design.
Decimal digit 
TBCD 8 4 2 1 

*  1 0 1 0 
#  1 0 1 1 
a  1 1 0 0 
b  1 1 0 1 
c  1 1 1 0 
Used as filler when there is an odd number of digits  1 1 1 1 
The mentioned 3GPP document defines TBCDSTRING 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 2n
bits 4321 of octet n encoding digit 2(n – 1) + 1
Meaning number 1234
, would become 21 43
in TBCD.
This format is used in modern mobile telephony to send dialed numbers, as well as operator ID, IMEI, IMSI (SUPI), et.c.^{[56]}^{[57]}
Alternative encodings
[edit]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 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 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 is implied.
The Hertz and Chen–Ho encodings provide Boolean transformations for converting groups of three BCDencoded digits to and from 10bit values^{[nb 1]} that can be efficiently encoded in hardware with only 2 or 3 gate delays. Densely packed decimal (DPD) is a similar scheme^{[nb 1]} that is used for most of the significand, except the lead digit, for one of the two alternative decimal encodings specified in the IEEE 7542008 floatingpoint standard.
Application
[edit]The BIOS in many personal computers stores the date and time in BCD because the MC6818 realtime clock chip used in the original IBM PC AT motherboard provided the time encoded in BCD. This form is easily converted into ASCII for display.^{[58]}^{[59]}
The Atari 8bit computers use a BCD format for floating point numbers. The MOS Technology 6502 processor has a BCD mode for the addition and subtraction instructions. The Psion Organiser 1 handheld computer's manufacturersupplied software also uses BCD to implement floating point; later Psion models use binary exclusively.
Early models of the PlayStation 3 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.
Legal history
[edit]In the 1972 case Gottschalk v. Benson, the U.S. Supreme Court overturned a lower court's decision that had allowed a patent for converting BCDencoded numbers to binary on a computer.
The decision noted that a patent "would wholly preempt the mathematical formula and in practical effect would be a patent on the algorithm itself".^{[60]} This was a landmark judgement that determined the patentability of software and algorithms.
See also
[edit] Biquinary coded decimal
 Binarycoded ternary (BCT)
 Binary integer decimal (BID)
 Bitmask
 Chen–Ho encoding
 Decimal computer
 Densely packed decimal (DPD)
 Double dabble, an algorithm for converting binary numbers to BCD
 Year 2000 problem
Notes
[edit] ^ ^{a} ^{b} ^{c} In a standard packed 4bit representation, there are 16 states (four bits for each digit) with 10 tetrades and 6 pseudotetrades, whereas in more densely packed schemes such as Hertz, Chen–Ho or DPD encodings there are fewer—e.g., only 24 unused states in 1024 states (10 bits for three digits).
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} Code states (shown in black) outside the decimal range 0–9 indicate additional states of the nonBCD variant of the code. In the BCD code variant discussed here, they are pseudotetrades.
 ^ The Aiken code is one of several 2 4 2 1 codes. It is also known as 2* 4 2 1 code.
 ^ The Jumpat8 code is also known as unsymmetrical 2 4 2 1 code.
 ^ The Petherick code is also known as Royal Aircraft Establishment (RAE) code.
 ^ The O'Brien code type I is also known as Watts code or Watts reflected decimal (WRD) code.
 ^ The Excess3 Gray code is also known as Gray–Stibitz code.
 ^ ^{a} ^{b} In a similar fashion, multiple characters were often packed into machine words on minicomputers, see IBM SQUOZE and DEC RADIX 50.
References
[edit] ^ Intel. "ia32 architecture manual" (PDF). Intel. Archived (PDF) from the original on 20221009. Retrieved 20150701.
 ^ ^{a} ^{b} Klar, Rainer (19700201). "1.5.3 Konvertierung binär verschlüsselter Dezimalzahlen" [1.5.3 Conversion of binary coded decimal numbers]. Digitale Rechenautomaten – Eine Einführung [Digital Computers – An Introduction]. Sammlung Göschen (in German). Vol. 1241/1241a (1 ed.). Berlin, Germany: Walter de Gruyter & Co. / G. J. Göschen'sche Verlagsbuchhandlung . pp. 17, 21. ISBN 3110831600. . ArchivNr. 7990709. Archived from the original on 20200418. Retrieved 20200413. (205 pages) (NB. A 2019 reprint of the first edition is available under ISBN 3110027933, 9783110027938. A reworked and expanded 4th edition exists as well.)
 ^ ^{a} ^{b} ^{c} Klar, Rainer (1989) [19881001]. "1.4 Codes: Binär verschlüsselte Dezimalzahlen" [1.4 Codes: Binary coded decimal numbers]. Digitale Rechenautomaten – Eine Einführung in die Struktur von Computerhardware [Digital Computers – An Introduction into the structure of computer hardware]. Sammlung Göschen (in German). Vol. 2050 (4th reworked ed.). Berlin, Germany: Walter de Gruyter & Co. pp. 25, 28, 38–39. ISBN 3110117002. p. 25:
[…] Die nicht erlaubten 0/1Muster nennt man auch Pseudodezimalen. […]
(320 pages)  ^ Schneider, HansJochen (1986). Lexikon der Informatik und Datenverarbeitung (in German) (2 ed.). R. Oldenbourg Verlag München Wien. ISBN 3486226622.
 ^ Tafel, Hans Jörg (1971). Einführung in die digitale Datenverarbeitung [Introduction to digital information processing] (in German). Munich: Carl Hanser Verlag. ISBN 3446105697.
 ^ Steinbuch, Karl W.; Weber, Wolfgang; Heinemann, Traute, eds. (1974) [1967]. Taschenbuch der Informatik  Band II  Struktur und Programmierung von EDVSystemen. Taschenbuch der Nachrichtenverarbeitung (in German). Vol. 2 (3 ed.). Berlin, Germany: SpringerVerlag. ISBN 3540062416. LCCN 7380607.
 ^ Tietze, Ulrich; Schenk, Christoph (20121206). Advanced Electronic Circuits. Springer Science & Business Media. ISBN 9783642812415. 9783642812415. Retrieved 20150805.
 ^ Kowalski, Emil (20130308) [1970]. Nuclear Electronics. SpringerVerlag. doi:10.1007/9783642876639. ISBN 9783642876639. 9783642876639, 9783642876646. Retrieved 20150805.
 ^ Ferretti, Vittorio (20130313). Wörterbuch der Elektronik, Datentechnik und Telekommunikation / Dictionary of Electronics, Computing and Telecommunications: Teil 1: DeutschEnglisch / Part 1: GermanEnglish. Vol. 1 (2 ed.). SpringerVerlag. ISBN 9783642980886. 9783642980886. Retrieved 20150805.
 ^ Speiser, Ambrosius Paul (1965) [1961]. Digitale Rechenanlagen  Grundlagen / Schaltungstechnik / Arbeitsweise / Betriebssicherheit [Digital computers  Basics / Circuits / Operation / Reliability] (in German) (2 ed.). ETH Zürich, Zürich, Switzerland: SpringerVerlag / IBM. p. 209. LCCN 6514624. 0978.
 ^ Cowlishaw, Mike F. (2015) [1981, 2008]. "General Decimal Arithmetic". Retrieved 20160102.
 ^ Evans, David Silvester (March 1961). "Chapter Four: Ancillary Equipment: Outputdrive and paritycheck relays for digitizers". Digital Data: Their derivation and reduction for analysis and process control (1 ed.). London, UK: Hilger & Watts Ltd / Interscience Publishers. pp. 46–64 [56–57]. Retrieved 20200524. (8+82 pages) (NB. The 4bit 8421 BCD code with an extra parity bit applied as least significant bit to achieve odd parity of the resulting 5bit code is also known as Ferranti code.)
 ^ Lala, Parag K. (2007). Principles of Modern Digital Design. John Wiley & Sons. pp. 20–25. ISBN 9780470072967.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} ^{i} ^{j} ^{k} ^{l} ^{m} ^{n} Berger, Erich R. (1962). "1.3.3. Die Codierung von Zahlen". Written at Karlsruhe, Germany. In Steinbuch, Karl W. (ed.). Taschenbuch der Nachrichtenverarbeitung (in German) (1 ed.). Berlin / Göttingen / New York: SpringerVerlag OHG. pp. 68–75. LCCN 6214511. (NB. The shown Kautz code (II), containing all eight available binary states with an odd count of 1s, is a slight modification of the original Kautz code (I), containing all eight states with an even count of 1s, so that inversion of the mostsignificant bits will create a 9s complement.)
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} Kämmerer, Wilhelm [in German] (May 1969). "II.15. Struktur: Informationsdarstellung im Automaten". Written at Jena, Germany. In Frühauf, Hans [in German]; Kämmerer, Wilhelm; Schröder, Kurz; Winkler, Helmut (eds.). Digitale Automaten – Theorie, Struktur, Technik, Programmieren. Elektronisches Rechnen und Regeln (in German). Vol. 5 (1 ed.). Berlin, Germany: AkademieVerlag GmbH. p. 161. License no. 202100/416/69. Order no. 4666 ES 20 K 3. (NB. A second edition 1973 exists as well.)
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} ^{i} ^{j} ^{k} ^{l} ^{m} ^{n} ^{o} ^{p} ^{q} Dokter, Folkert; Steinhauer, Jürgen (19730618). Digital Electronics. Philips Technical Library (PTL) / Macmillan Education (Reprint of 1st English ed.). Eindhoven, Netherlands: The Macmillan Press Ltd. / N. V. Philips' Gloeilampenfabrieken. doi:10.1007/9781349014170. ISBN 9781349014194. SBN 333133609. Archived from the original on 20200716. Retrieved 20200511. (270 pages) (NB. This is based on a translation of volume I of the twovolume German edition.)
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} ^{i} ^{j} ^{k} ^{l} ^{m} ^{n} ^{o} ^{p} ^{q} Dokter, Folkert; Steinhauer, Jürgen (1975) [1969]. Digitale Elektronik in der Meßtechnik und Datenverarbeitung: Theoretische Grundlagen und Schaltungstechnik. Philips Fachbücher (in German). Vol. I (improved and extended 5th ed.). Hamburg, Germany: Deutsche Philips GmbH. p. 50. ISBN 3871452726. (xii+327+3 pages) (NB. The German edition of volume I was published in 1969, 1971, two editions in 1972, and 1975. Volume II was published in 1970, 1972, 1973, and 1975.)
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} Kautz, William H. (June 1954). "IV. Examples A. Binary Codes for Decimals, n = 4". Optimized Data Encoding for Digital Computers. Convention Record of the I.R.E., 1954 National Convention, Part 4  Electronic Computers and Information Theory. Session 19: Information Theory III  Speed and Computation. Stanford Research Institute, Stanford, California, USA: I.R.E. pp. 47–57 [49, 51–52, 57]. Archived from the original on 20200703. Retrieved 20200703. p. 52:
[…] The last column [of Table II], labeled "Best," gives the maximum fraction possible with any code—namely 0.60—half again better than any conventional code. This extremal is reached with the ten [heavilymarked vertices of the graph of Fig. 4 for n = 4, or, in fact, with any set of ten code combinations which include all eight with an even (or all eight with an odd) number of "1's." The second and third rows of Table II list the average and peak decimal change per undetected single binary error, and have been derived using the equations of Sec. II for Δ_{1} and δ_{1}. The confusion index for decimals using the criterion of "decimal change," is taken to be c_{ij} = i − j i,j = 0, 1, … 9. Again, the "Best" arrangement possible (the same for average and peak), one of which is shown in Fig. 4, is substantially better than the conventional codes. […] Fig. 4 Minimumconfusion code for decimals. […] δ_{1}=2 Δ_{1}=15 […]
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] (11 pages) (NB. Besides the combinatorial set of 4bit BCD "minimumconfusion codes for decimals", of which the author illustrates only one explicitly (here reproduced as code I) in form of a 4bit graph, the author also shows a 16state 4bit "binary code for analog data" in form of a code table, which, however, is not discussed here. The code II shown here is a modification of code I discussed by Berger.)  ^ ^{a} ^{b} ^{c} Chinal, Jean P. (January 1973). "Codes". Written at Paris, France. Design Methods for Digital Systems. Translated by Preston, Alan; Summer, Arthur (1st English ed.). Berlin, Germany: AkademieVerlag / SpringerVerlag. p. 46. doi:10.1007/9783642861871_3. ISBN 9780387058719. License No. 202100/542/73. Order No. 7617470(6047) ES 19 B 1 / 20 K 3. Retrieved 20200621. (xviii+506 pages) (NB. The French 1967 original book was named "Techniques Booléennes et Calculateurs Arithmétiques", published by Éditions Dunod .)
 ^ ^{a} ^{b} Military Handbook: Encoders  Shaft Angle To Digital (PDF). United States Department of Defense. 19910930. MILHDBK231A. Archived (PDF) from the original on 20200725. Retrieved 20200725. (NB. Supersedes MILHDBK231(AS) (19700701).)
 ^ ^{a} ^{b} Stopper, Herbert (March 1960). Written at Litzelstetten, Germany. Runge, Wilhelm Tolmé (ed.). "Ermittlung des Codes und der logischen Schaltung einer Zähldekade". TelefunkenZeitung (TZ)  TechnischWissenschaftliche Mitteilungen der Telefunken GMBH (in German). 33 (127). Berlin, Germany: Telefunken: 13–19. (7 pages)
 ^ ^{a} ^{b} Borucki, Lorenz; Dittmann, Joachim (1971) [July 1970, 1966, Autumn 1965]. "2.3 Gebräuchliche Codes in der digitalen Meßtechnik". Written at Krefeld / Karlsruhe, Germany. Digitale Meßtechnik: Eine Einführung (in German) (2 ed.). Berlin / Heidelberg, Germany: SpringerVerlag. pp. 10–23 [12–14]. doi:10.1007/9783642805608. ISBN 3540050582. LCCN 75131547. ISBN 9783642805615. (viii+252 pages) 1st edition
 ^ White, Garland S. (October 1953). "Coded Decimal Number Systems for Digital Computers". Proceedings of the Institute of Radio Engineers. 41 (10). Institute of Radio Engineers (IRE): 1450–1452. doi:10.1109/JRPROC.1953.274330. eISSN 21626634. ISSN 00968390. S2CID 51674710. (3 pages)
 ^ "Different Types of Binary Codes". Electronic Hub. 20190501 [20150128]. Section 2.4 5211 Code. Archived from the original on 20200518. Retrieved 20200804.
 ^ Paul, Matthias R. (19950810) [1994]. "Unterbrechungsfreier Schleifencode" [Continuous loop code]. 1.02 (in German). Retrieved 20080211. (NB. The author called this code Schleifencode (English: "loop code"). It differs from Gray BCD code only in the encoding of state 0 to make it a cyclic unitdistance code for fullcircle rotatory slip ring applications. Avoiding the allzero code pattern allows for loop selftesting and to use the data lines for uninterrupted power distribution.)
 ^ Gray, Frank (19530317) [19471113]. Pulse Code Communication (PDF). New York, USA: Bell Telephone Laboratories, Incorporated. U.S. patent 2,632,058. Serial No. 785697. Archived (PDF) from the original on 20200805. Retrieved 20200805. (13 pages)
 ^ Glixon, Harry Robert (March 1957). "Can You Take Advantage of the Cyclic BinaryDecimal Code?". Control Engineering. 4 (3). Technical Publishing Company, a division of DunDonnelley Publishing Corporation, Dun & Bradstreet Corp.: 87–91. ISSN 00108049. (5 pages)
 ^ ^{a} ^{b} Ledley, Robert Steven; Rotolo, Louis S.; Wilson, James Bruce (1960). "Part 4. Logical Design of DigitalComputer Circuitry; Chapter 15. Serial Arithmetic Operations; Chapter 157. Additional Topics". Digital Computer and Control Engineering (PDF). McGrawHill Electrical and Electronic Engineering Series (1 ed.). New York, USA: McGrawHill Book Company, Inc. (printer: The Maple Press Company, York, Pennsylvania, USA). pp. 517–518. ISBN 007036981X. ISSN 25747916. LCCN 59015055. OCLC 1033638267. OL 5776493M. SBN 07036981X. . ark:/13960/t72v3b312. Archived (PDF) from the original on 20210219. Retrieved 20210219. p. 517:
[…] The cyclic code is advantageous mainly in the use of relay circuits, for then a sticky relay will not give a false state as it is delayed in going from one cyclic number to the next. There are many other cyclic codes that have this property. […]
[12] (xxiv+835+1 pages) (NB. Ledley classified the described cyclic code as a cyclic decimalcoded binary code.)  ^ ^{a} ^{b} ^{c} ^{d} Savard, John J. G. (2018) [2006]. "Decimal Representations". quadibloc. Archived from the original on 20180716. Retrieved 20180716.
 ^ Petherick, Edward John (October 1953). A Cyclic Progressive Binarycodeddecimal System of Representing Numbers (Technical Note MS15). Farnborough, UK: Royal Aircraft Establishment (RAE). (4 pages) (NB. Sometimes referred to as A CyclicCoded BinaryCodedDecimal System of Representing Numbers.)
 ^ Petherick, Edward John; Hopkins, A. J. (1958). Some Recently Developed Digital Devices for Encoding the Rotations of Shafts (Technical Note MS21). Farnborough, UK: Royal Aircraft Establishment (RAE).
 ^ ^{a} ^{b} O'Brien, Joseph A. (May 1956) [19551115, 19550623]. "Cyclic Decimal Codes for Analogue to Digital Converters". Transactions of the American Institute of Electrical Engineers, Part I: Communication and Electronics. 75 (2). Bell Telephone Laboratories, Whippany, New Jersey, USA: 120–122. doi:10.1109/TCE.1956.6372498. ISSN 00972452. S2CID 51657314. Paper 5621. Archived from the original on 20200518. Retrieved 20200518. (3 pages) (NB. This paper was prepared for presentation at the AIEE Winter General Meeting, New York, USA, 19560130 to 19560203.)
 ^ ^{a} ^{b} Tompkins, Howard E. (September 1956) [19560716]. "UnitDistance BinaryDecimal Codes for TwoTrack Commutation". IRE Transactions on Electronic Computers. Correspondence. EC5 (3). Moore School of Electrical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania, USA: 139. doi:10.1109/TEC.1956.5219934. ISSN 03679950. Archived from the original on 20200518. Retrieved 20200518. (1 page)
 ^ Lippel, Bernhard (December 1955). "A Decimal Code for AnalogtoDigital Conversion". IRE Transactions on Electronic Computers. EC4 (4): 158–159. doi:10.1109/TEC.1955.5219487. ISSN 03679950. (2 pages)
 ^ ^{a} ^{b} ^{c} Susskind, Alfred Kriss; Ward, John Erwin (19580328) [1957, 1956]. "III.F. UnitDistance Codes / VI.E.2. Reflected Binary Codes". Written at Cambridge, Massachusetts, USA. In Susskind, Alfred Kriss (ed.). Notes on AnalogDigital Conversion Techniques. Technology Books in Science and Engineering. Vol. 1 (3 ed.). New York, USA: Technology Press of the Massachusetts Institute of Technology / John Wiley & Sons, Inc. / Chapman & Hall, Ltd. pp. 37–38 [37], 310–316 [313–316], 665–660 [660]. (x+416+2 pages) (NB. The contents of the book was originally prepared by staff members of the Servomechanisms Laboraratory, Department of Electrical Engineering, MIT, for Special Summer Programs held in 1956 and 1957. The code Susskind actually presented in his work as "readingtype code" is shown as code type II here, whereas the type I code is a minor derivation with the two most significant bit columns swapped to better illustrate symmetries.)
 ^ ^{a} ^{b} Yuen, ChunKwong (December 1977). "A New Representation for Decimal Numbers". IEEE Transactions on Computers. C26 (12): 1286–1288. doi:10.1109/TC.1977.1674792. S2CID 40879271. Archived from the original on 20200808. Retrieved 20200808.
 ^ Lucal, Harold M. (December 1959). "Arithmetic Operations for Digital Computers Using a Modified Reflected Binary". IRE Transactions on Electronic Computers. EC8 (4): 449–458. doi:10.1109/TEC.1959.5222057. ISSN 03679950. S2CID 206673385. (10 pages)
 ^ Dewar, Robert Berriedale Keith; Smosna, Matthew (1990). Microprocessors  A Programmer's View (1 ed.). Courant Institute, New York University, New York, USA: McGrawHill Publishing Company. p. 14. ISBN 0070166382. LCCN 8977320. (xviii+462 pages)
 ^ "Chapter 8: Decimal Instructions". IBM System/370 Principles of Operation. IBM. March 1980.
 ^ "Chapter 3: Data Representation". PDP11 Architecture Handbook. Digital Equipment Corporation. 1983.
 ^ ^{a} ^{b} VAX11 Architecture Handbook. Digital Equipment Corporation. 1985.
 ^ "ILE RPG Reference".
 ^ IBM BM 1401/1440/1460/1410/7010 Character Code Chart in BCD Order^{[permanent dead link]}
 ^ 6502 Instruction Set Archived 20180508 at the Wayback Machine.
 ^ NMOS 6502 Opcodes. Archived 20160114 at the Wayback Machine.
 ^ "The 68000's Instruction Set" (PDF). Archived (PDF) from the original on 20231120. Retrieved 20231121. (58 pages)
 ^ Intel 64 and IA32 Architectures Software Developer's Manual Volume 1: Basic Architecture (PDF). Intel. March 2013. Section 4.7. Archived (PDF) from the original on 20130402. Retrieved 20130423.
 ^ "4.7 BCD and packed BCD integers". Intel 64 and IA32 Architectures Software Developer's Manual, Volume 1: Basic Architecture (PDF). Version 072. Vol. 1. Intel Corporation. 20200527 [1997]. pp. 3–2, 49–411 [410]. 253665072US. Archived (PDF) from the original on 20200806. Retrieved 20200806. p. 410:
[…] When operating on BCD integers in generalpurpose registers, the BCD values can be unpacked (one BCD digit per byte) or packed (two BCD digits per byte). The value of an unpacked BCD integer is the binary value of the low halfbyte (bits 0 through 3). The high halfbyte (bits 4 through 7) can be any value during addition and subtraction, but must be zero during multiplication and division. Packed BCD integers allow two BCD digits to be contained in one byte. Here, the digit in the high halfbyte is more significant than the digit in the low halfbyte. […] When operating on BCD integers in x87 FPU data registers, BCD values are packed in an 80bit format and referred to as decimal integers. In this format, the first 9 bytes hold 18 BCD digits, 2 digits per byte. The leastsignificant digit is contained in the lower halfbyte of byte 0 and the mostsignificant digit is contained in the upper halfbyte of byte 9. The most significant bit of byte 10 contains the sign bit (0 = positive and 1 = negative; bits 0 through 6 of byte 10 are don't care bits). Negative decimal integers are not stored in two's complement form; they are distinguished from positive decimal integers only by the sign bit. The range of decimal integers that can be encoded in this format is −10^{18} + 1 to 10^{18} − 1. The decimal integer format exists in memory only. When a decimal integer is loaded in an x87 FPU data register, it is automatically converted to the doubleextendedprecision floatingpoint format. All decimal integers are exactly representable in double extendedprecision format. […]
[13]  ^ Jones, Douglas W. (20151125) [1999]. "BCD Arithmetic, a tutorial". Arithmetic Tutorials. Iowa City, Iowa, USA: The University of Iowa, Department of Computer Science. Retrieved 20160103.
 ^ University of Alicante. "A Cordicbased Architecture for High Performance Decimal Calculations" (PDF). IEEE. Archived (PDF) from the original on 20100105. Retrieved 20150815.
 ^ "Decimal CORDIC Rotation based on Selection by Rounding: Algorithm and Architecture" (PDF). British Computer Society. Archived (PDF) from the original on 20221009. Retrieved 20150814.
 ^ Mathur, Aditya P. (1989). Introduction to Microprocessors (3 ed.). Tata McGrawHill Publishing Company Limited. ISBN 9780074602225.
 ^ 3GPP TS 29.002: Mobile Application Part (MAP) specification (Technical report). 2013. sec. 17.7.8 Common data types.
 ^ "Signalling Protocols and Switching (SPS) Guidelines for using Abstract Syntax Notation One (ASN.1) in telecommunication application protocols" (PDF). p. 15. Archived (PDF) from the original on 20131204.
 ^ "XOM Mobile Application Part (XMAP) Specification" (PDF). p. 93. Archived from the original (PDF) on 20150221. Retrieved 20130627.
 ^ "NonAccessStratum (NAS) protocol for 5G System (5GS); Stage 3. (3GPP TS 24.501 version 16.10.0 Release 16) TS 24.501 release 16.10.0" (PDF). ETSI and 3GPP. Archived (PDF) from the original on 20220217. Retrieved 20220226. (TS 24.501)
 ^ "Digital cellular telecommunications system (Phase 2+) (GSM); Universal Mobile Telecommunications System (UMTS); LTE; 5G; Numbering, addressing and identification (3GPP TS 23.003 version 16.8.0 Release 16)" (PDF). ETSI and 3GPP. Archived (PDF) from the original on 20220226. Retrieved 20220226. (TS 23.003)
 ^ "Timer Counter Circuits in an IBM PC" (PDF). www.se.ecu.edu.au. Archived from the original (PDF) on 20081010. Retrieved 20220522. (7 pages)
 ^ MC6818 datasheet
 ^ Gottschalk v. Benson, 409 U.S. 63, 72 (1972).
Further reading
[edit] Mackenzie, Charles E. (1980). Coded Character Sets, History and Development (PDF). The Systems Programming Series (1 ed.). AddisonWesley Publishing Company, Inc. ISBN 9780201144604. LCCN 7790165. Archived (PDF) from the original on May 26, 2016. Retrieved August 25, 2019.
 Richards, Richard Kohler (1955). Arithmetic Operations in Digital Computers. New York, USA: van Nostrand. pp. 397–.
 Schmid, Hermann (1974). Decimal Computation (1 ed.). Binghamton, New York, USA: John Wiley & Sons. ISBN 047176180X. and Schmid, Hermann (1983) [1974]. Decimal Computation (1 (reprint) ed.). Malabar, Florida, USA: Robert E. Krieger Publishing Company. ISBN 0898743184. (NB. At least some batches of the Krieger reprint edition were misprints with defective pages 115–146.)
 Massalin, Henry (October 1987). Katz, Randy (ed.). "Superoptimizer: A look at the smallest program" (PDF). ACM Sigops Operating Systems Review. 21 (4): 122–126. doi:10.1145/36204.36194. ISBN 0818608056. Archived (PDF) from the original on 20170704. Retrieved 20120425. (Also: ACM SIGPLAN Notices, Vol. 22 #10, IEEE Computer Society Press #87CH24406, October 1987)
 "GNU Superoptimizer". HPUX.
 Shirazi, Behrooz; Yun, David Y. Y.; Zhang, Chang N. (March 1988). VLSI designs for redundant binarycoded decimal addition. IEEE Seventh Annual International Phoenix Conference on Computers and Communications, 1988. IEEE. pp. 52–56.
 Brown; Vranesic (2003). Fundamentals of Digital Logic.
 Thapliyal, Himanshu; Arabnia, Hamid R. (November 2006). Modified Carry Look Ahead BCD Adder With CMOS and Reversible Logic Implementation. Proceedings of the 2006 International Conference on Computer Design (CDES'06). CSREA Press. pp. 64–69. ISBN 1601320094.
 Kaivani, A.; Alhosseini, A. Zaker; Gorgin, S.; Fazlali, M. (December 2006). Reversible Implementation of DenselyPackedDecimal Converter to and from BinaryCodedDecimal Format Using in IEEE754R. 9th International Conference on Information Technology (ICIT'06). IEEE. pp. 273–276.
 Cowlishaw, Mike F. (2009) [2002, 2008]. "Bibliography of material on Decimal Arithmetic – by category". General Decimal Arithmetic. IBM. Retrieved 20160102.
External links
[edit] Cowlishaw, Mike F. (2014) [2000]. "A Summary of ChenHo Decimal Data encoding". General Decimal Arithmetic. IBM. Retrieved 20160102.
 Cowlishaw, Mike F. (2007) [2000]. "A Summary of Densely Packed Decimal encoding". General Decimal Arithmetic. IBM. Retrieved 20160102.
 Convert BCD to decimal, binary and hexadecimal and vice versa
 BCD for Java