In computing and electronics systems, binarycoded decimal ( BCD) is a class of binary encodings of decimal numbers where each decimal numerical digit is represented by a fixed number of , usually four or eight. Special bit patterns are sometimes used for a sign or for other indications (e.g., error or overflow).
In byteoriented 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 decimal 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 4bit encoding may vary however, for technical reasons, see Excess3 for instance. The ten states representing a BCD decimal digit are sometimes called tetrades (for the nibble typically needed to hold them also known as tetrade) with those don't carestates unused named or pseudodecimal digit).
BCD's main virtue is its more accurate representation and rounding of decimal quantities as well as an ease of conversion into humanreadable representations, in comparison to binary positional systems. BCD's principal drawbacks are a small increase in the complexity of the circuits needed to implement basic arithmetics and a slightly less dense storage.
BCD was used in many early , and is implemented in the instruction set of machines such as the IBM System/360 series and its descendants, Digital Equipment Corporation's VAX and the Motorola 68000series processors. Although BCD per se is not as widely used as in the past and is no longer implemented in newer computers' instruction sets (such as ARM architecture; x86 does not support its BCD instructions in long mode any more), decimal fixedpoint and floatingpoint formats are still important and continue to be used in financial, commercial, and industrial computing, where subtle conversion and fractional rounding roundoff error that are inherent in floating point binary representations cannot be tolerated.
0 
1 
0 
1 
0 
1 
0 
1 
0 
1 
Other encodings are also used, including socalled "4221" and "7421"—named after the weighting used for the bits—and "Excess3".
For example, the BCD digit 6, '0110'b in 8421 notation, is '1100'b in 4221 (two encodings are possible), '0110'b in 7421, and '1001'b (6+3=9) in excess3.As most computers deal with data in 8bit , it is possible to use one of the following methods to encode a BCD number:
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 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: 1 2 3 4 5 Binary : 0000 0001 0010 0011 0100 0101
Note that the most significant nibble of the most significant byte is zero, implying that the number is in actuality 012345. Also note how packed BCD is more efficient in storage usage as compared to unpacked BCD; encoding the same number (with the leading zero) in unpacked format would consume twice the storage.
logical shift and masking operations are used to pack or unpack a packed BCD digit. Other logical operations are used to convert a numeral to its equivalent bit pattern or reverse the process.
The same argument applies when hardware of this type uses an embedded microcontroller or other small processor. Often, smaller code results when representing numbers internally in BCD format, since a conversion from or to binary representation can be expensive on such limited processors. For these applications, some small processors feature BCD arithmetic modes, which assist when writing routines that manipulate BCD quantities.
Standard sign values are 1100 (hexadecimal 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). 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 is, there are 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 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 −''
(Note that, like character strings, the first byte of the packed decimal – 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 (about onesixth of the memory used is wasted), conversion to ASCII, EBCDIC, or the various encodings of Unicode is still trivial, as no arithmetic operations are required. The extra storage requirements are usually offset by the need for the accuracy 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". Besides 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 Corporation 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, a 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.)
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.
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''
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.
F1 F2 F7 F9 F5 C0 '' 1 2 7 9. 5 +0''
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. IBM BM 1401/1440/1460/1410/7010 Character Code Chart in BCD Order
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 processors and databases, such as IBM DB2, mainframes, and Power6. In these products, the BCD is usually zoned BCD (as in EBCDIC or ASCII), Packed BCD (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.http://publib.boulder.ibm.com/infocenter/zos/v1r12/index.jsp?topic=%2Fcom.ibm.zos.r12.iceg200%2Fenf.htm
The Intel x86 architecture supports a unique 18digit (tenbyte) BCD format that can be loaded into and stored from the floating point registers, and computations can be performed there.
The Motorola 68000 series had BCD instructions.url=http://www.tigernt.com/onlineDoc/68000.pdf
In more recent computers such capabilities are almost always implemented in software rather than the CPU's instruction set, but BCD numeric data is 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. For example, the following code (written in C) computes an unsigned 8digit packed BCD add using 32bit binary operations:
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}
1001 + 1000 = 10001 9 + 8 = 17
Note that 10001 is the binary, not decimal, representation of the desired result. Also note that it cannot fit in a 4bit 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 that sum and then the result is treated as two :
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.
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. 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). 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 check the answer, note that the first digit is 9, which means negative. This seems to be correct, since 357 − 432 should result in a negative number. To check the rest of the digits, represent them in decimal. 1001 0010 0101 is 925. The ten's complement of 925 is 1000 − 925 = 999 − 925 + 1 = 074 + 1 = 75, so the calculated answer is −75. To check, perform standard subtraction to verify that 357 − 432 is −75.
Note that in the event that there are a different number of nibbles being added together (such as 1053 − 122), the number with the fewest number of digits must first be padded with zeros before taking the ten's complement or subtracting. So, with 1053 − 122, 122 would have to first be represented as 0122, and the ten's complement of 0122 would have to be calculated.
In the headers to the table, the '', indicates the weight of each bit shown; note that in the fifth column, "BCD 8 4 −2 −1", two of the weights are negative. Both ASCII and EBCDIC character codes for the digits are examples of zoned BCD, and are also shown in the table.
The following table represents decimal digits from 0 to 9 in various BCD systems:
The Atari 8bit family of computers used BCD to implement floatingpoint algorithms. The MOS 6502 processor has a BCD mode that affects the addition and subtraction instructions. The Psion Organiser handheld computer’s manufacturersupplied software also used entirely BCD to implement floating point; later Psion models used 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.
Unsigned  No sign nibble  F1 F2 <u>F</u>3 
Signed trailing (canonical format)  Sign nibble in the last (least significant) byte  F1 F2 <u>C</u>3 
Signed leading (overpunch)  Sign nibble in the first (most significant) byte  <u>C</u>1 F2 F3 
Signed trailing separate  Separate sign character byte ('+' or '−') following the digit bytes  F1 F2 F3 <u>2B</u> 
Signed leading separate  Separate sign character byte ('+' or '−') preceding the digit bytes  <u>2B</u> F1 F2 F3 
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.
Chen–Ho encoding provides a boolean transformation for converting groups of three BCDencoded digits to and from 10bit values that can be efficiently encoded in hardware with only 2 or 3 gate delays. Densely packed decimal is a similar scheme that is used for most of the significand, except the lead digit, for one of the two alternative decimal encodings specified in the IEEE 7542008 standard.

