Binary-coded decimal

Last updated

A binary clock might use LEDs to express binary values. In this clock, each column of LEDs shows a binary-coded decimal numeral of the traditional sexagesimal time. Binary clock.svg
A binary clock might use LEDs to express binary values. In this clock, each column of LEDs shows a binary-coded decimal numeral of the traditional sexagesimal time.

In computing and electronic systems, binary-coded 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).

Contents

In byte-oriented 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 4-bit encoding, however, may vary for technical reasons (e.g. Excess-3).

The ten states representing a BCD digit are sometimes called tetrades [2] [3] (for the nibble typically needed to hold them is also known as a tetrade) while the unused, don't care-states are named pseudo-tetrad(e)s  [ de ], [4] [5] [6] [7] [8] pseudo-decimals [3] or pseudo-decimal 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 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 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 68000-series processors. BCD per se is not as widely used as in the past and it is no longer implemented in newer computers' instruction sets (e.g. ARM); x86 does not support its BCD instructions in long mode any more. However, decimal fixed-point and floating-point 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 floating point binary representations cannot be tolerated. [11]

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.

Decimal digitBCD
8421
00000
10001
20010
30011
40100
50101
60110
70111
81000
91001

This scheme can also be referred to as Simple Binary-Coded Decimal (SBCD) or BCD 8421, and is the most common encoding. [12] Others include the so-called "4221" and "7421" encoding – named after the weighting used for the bits – and "Excess-3". [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 Excess-3 it is 1001'b ().

4-bit BCD codes and pseudo-tetrades
BitWeight 0 1 2 3 4 5 6 7 8 9101112131415        Comment        
480000000011111111Binary
340000111100001111
220011001100110011
110101010101010101
Name0123456789101112131415Decimal
8421 (XS-0)0123456789101112131415 [14] [15] [16] [17] [nb 2]
74210123456 789      [18] [19] [20]
Aiken (2421)01234      56789 [14] [15] [16] [17] [nb 3]
Excess-3 (XS-3)-3-2-10123456789101112 [14] [15] [16] [17] [nb 2]
Excess-6 (XS-6)-6-5-4-3-2-10123456789 [18] [nb 2]
Jump-at-2 (2421)01      23456789 [16] [17]
Jump-at-8 (2421)01234567      89 [21] [22] [16] [17] [nb 4]
4221 (I)0123  45    6789 [16] [17]
4221 (II)0123  45  67  89 [21] [22]
542101234   56789    [18] [14] [16] [17]
52210123  4 5678  9  [14] [16] [17]
51210123   45678   9 [19]
531101 234  56 789   [16] [17]
White (5211)01 2 3 456 7 8 9 [23] [18] [14] [16] [17]
521101 2 3 45 6 7 89 [24]
 0123456789101112131415
Magnetic tape 1234567890      [15]
Paul 1326754 0  89   [25]
Gray 0132675415141213891110 [26] [14] [15] [16] [17] [nb 2]
Glixon 013267549   8    [27] [14] [15] [16] [17]
Ledley01327645    8 9  [28]
431101 23  54  67 89 [19]
LARC 01 2  4356 7  98 [29]
Klar01 2  4398 7  56 [2] [3]
Petherick (RAE) 132 04  867 95  [30] [31] [nb 5]
O'Brien I (Watts)0132  4 9867  5  [32] [14] [16] [17] [nb 6]
5-cyclic0132  4 5687  9  [28]
Tompkins I 0132  4  9  8756 [33] [14] [16] [17]
Lippel0123  4  9  8765 [34] [35] [14]
O'Brien II  0214 3  9785 6  [32] [14] [16] [17]
Tompkins II   0143 2 79856   [33] [14] [16] [17]
Excess-3 Gray -3-20-1431212119105687 [16] [17] [20] [nb 7] [nb 2]
63−2−1 (I)    3210 5489 76 [29] [36]
63−2−1 (II)0   321 654 987  [29] [36]
84−2−10   43218765   9 [29]
Lucal 0151411232138769411105 [37]
Kautz I0  2 513 79 86 4 [18]
Kautz II 94 1 328 67 05  [18] [14]
Susskind I 0 1 432 9 85 67 [35]
Susskind II 0 1 9 84 325 67 [35]
 0123456789101112131415

The following table represents decimal 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.

 
Digit
BCD
8421
Stibitz  code or Excess-3 Aiken-Code or BCD
2421
BCD
84−2−1
IBM 702, IBM 705, IBM 7080, IBM 1401
8421
ASCII
0000 8421
EBCDIC
0000 8421
0000000110000000010100011 00001111 0000
1000101000001011100010011 00011111 0001
2001001010010011000100011 00101111 0010
3001101100011010100110011 00111111 0011
4010001110100010001000011 01001111 0100
5010110001011101101010011 01011111 0101
6011010011100101001100011 01101111 0110
7011110101101100101110011 01111111 0111
8100010111110100010000011 10001111 1000
9100111001111111110010011 10011111 1001

As most computers deal with data in 8-bit bytes, 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 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 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

In packed BCD (or simply packed decimal [38] ), each of the two nibbles of each byte represent 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 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 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 ASCII-8 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
SignNotes
A1 0 1 0+ 
B1 0 1 1 
C1 1 0 0+Preferred
D1 1 0 1Preferred
E1 1 1 0+ 
F1 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 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 of the machine.

In contrast, a 4-byte 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 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 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 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 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 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, PL/I and Ada). 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 4th and 5th 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 210 (1,024) is greater than 103 (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

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

BCD digitHexadecimalEBCDIC character
0+C0A0E0F0{ (*) \ (*)0
1+C1A1E1F1A~ (*) 1
2+C2A2E2F2BsS2
3+C3A3E3F3CtT3
4+C4A4E4F4DuU4
5+C5A5E5F5EvV5
6+C6A6E6F6FwW6
7+C7A7E7F7GxX7
8+C8A8E8F8HyY8
9+C9A9E9F9IzZ9
0−D0B0  }  (*)^  (*)  
1−D1B1  J   
2−D2B2  K   
3−D3B3  L   
4−D4B4  M   
5−D5B5  N   
6−D6B6  O   
7−D7B7  P   
8−D8B8  Q   
9−D9B9  R   

(*) Note: These characters vary depending on the local character code page 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 (BCDIC, sometimes just called BCD), for 6-bit alphanumeric 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 (introduced in 1959), IBM 1400 series, and non-Decimal Architecture members of the IBM 700/7000 series.

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 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 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 7072, and IBM 7074 alphamerics are encoded using digit pairs (using two-out-of-five code 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, 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 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. [44]

Other computers

The Digital Equipment Corporation VAX-11 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 system-supplied 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.

The Intel x86 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. [45]

The Motorola 68000 series had BCD instructions. [46]

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. [47] For example, the following code (written in C) computes an unsigned 8-digit packed BCD addition using 32-bit binary operations:

uint32_tBCDadd(uint32_ta,uint32_tb){uint32_tt1,t2;// unsigned 32-bit intermediate valuest1=a+0x06666666;t2=t1^b;// sum without carry propagationt1=t1+b;// provisional sumt2=t1^t2;// all the binary carry bitst2=~t2&0x11111110;// just the BCD carry bitst2=(t2>>2)|(t2>>3);// correctionreturnt1-t2;// corrected BCD sum}

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 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. [48] [49]

Operations with BCD

Addition

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 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 24 = 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 to facilitate BCD arithmetic adjustments following binary addition and subtraction operations.

Subtraction

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.

Comparison with pure binary

Advantages

Disadvantages

Representational variations

Various BCD implementations exist that employ other representations for numbers. Programmable calculators manufactured by Texas Instruments, Hewlett-Packard, and others typically employ a floating-point 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

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:

TypeDescriptionExample
Unsigned No sign nibble F1 F2 F3
Signed trailing (canonical format)Sign nibble in the last (least significant) byteF1 F2 C3
Signed leading (overpunch)Sign nibble in the first (most significant) byteC1 F2 F3
Signed trailing separateSeparate sign character byte ('+' or '−') following the digit bytesF1 F2 F3 2B
Signed leading separateSeparate sign character byte ('+' or '−') preceding the digit bytes2B F1 F2 F3

Telephony binary-coded decimal (TBCD)

3GPP developed TBCD, [51] an expansion to BCD where the remaining (unused) bit combinations are used to add specific telephony characters, [52] [53] 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
a1 1 0 0
b1 1 0 1
c1 1 1 0
Used as filler when there is an odd number of digits1 1 1 1

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 2n

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 2×101.

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 BCD-encoded digits to and from 10-bit 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 754-2008 floating-point standard.

Application

The BIOS in many personal computers stores the date and time in BCD because the MC6818 real-time 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. [54] [55]

The Atari 8-bit family of computers used BCD to implement floating-point algorithms. The MOS 6502 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 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.

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 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 itself". [56] This was a landmark judgement that determining the patentability of software and algorithms.

See also

Notes

  1. 1 2 3 In a standard packed 4-bit representation, there are 16 states (four bits for each digit) with 10 tetrades and 6 pseudo-tetrades, whereas in more densely packed schemes such as Hertz, Chen–Ho or DPD encodings there are fewere.g., only 24 unused states in 1024 states (10 bits for three digits).
  2. 1 2 3 4 5 Code states (shown in black) outside the decimal range 0–9 indicate additional states of the non-BCD variant of the code. In the BCD code variant discussed here, they are pseudo-tetrades.
  3. The Aiken code is one of several 2421 codes. It is also known as 2*421 code.
  4. The Jump-at-8 code is also known as unsymmetrical 2421 code.
  5. The Petherick code is also known as Royal Aircraft Establishment (RAE) code.
  6. The O'Brien code type I is also known as Watts code or Watts reflected decimal (WRD) code.
  7. The Excess-3 Gray code is also known as GrayStibitz code.
  8. 1 2 In a similar fashion, multiple characters were often packed into machine words on minicomputers, see IBM SQUOZE and DEC RADIX 50.

Related Research Articles

Floating-point arithmetic Computer format for representing real numbers

In computing, floating-point arithmetic (FP) is arithmetic using formulaic representation of real numbers as an approximation to support a trade-off between range and precision. For this reason, floating-point computation is often used in systems with very small and very large real numbers that require fast processing times. In general, a floating-point number is represented approximately with a fixed number of significant digits and scaled using an exponent in some fixed base; the base for the scaling is normally two, ten, or sixteen. A number that can be represented exactly is of the following form:

In mathematics and computing, the hexadecimal numeral system is a positional numeral system that represents numbers using a radix (base) of 16. Unlike the common way of representing numbers using 10 symbols, hexadecimal uses 16 distinct symbols, most often the symbols "0"–"9" to represent values 0 to 9, and "A"–"F" to represent values 10 to 15.

In computer science, an integer is a datum of integral data type, a data type that represents some range of mathematical integers. Integral data types may be of different sizes and may or may not be allowed to contain negative values. Integers are commonly represented in a computer as a group of binary digits (bits). The size of the grouping varies so the set of integer sizes available varies between different types of computers. Computer hardware nearly always provide a way to represent a processor register or memory address as an integer.

Nibble group of four bits (half a "byte"); unit of information

In computing, a nibble (occasionally nybble or nyble 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 often called a semi-octet, quadbit, or quartet. A nibble has sixteen (24) possible values. A nibble can be represented by a single hexadecimal digit and called a hex digit.

The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Octal numerals can be made from binary numerals by grouping consecutive binary digits into groups of three. For example, the binary representation for decimal 74 is 1001010. Two zeroes can be added at the left: (00)1 001 010, corresponding the octal digits 1 1 2, yielding the octal representation 112.

A computer number format is the internal representation of numeric values in digital device hardware and software, such as in programmable computers and calculators. Numerical values are stored as groupings of bits, such as bytes and words. The encoding between numerical values and bit patterns is chosen for convenience of the operation of the computer; the encoding used by the computer's instruction set generally requires conversion for external use, such as for printing and display. Different types of processors may have different internal representations of numerical values and different conventions are used for integer and real numbers. Most calculations are carried out with number formats that fit into a processor register, but some software systems allow representation of arbitrarily large numbers using multiple words of memory.

A binary code represents text, computer processor instructions, or any other data using a two-symbol system. The two-symbol system used is often "0" and "1" from the binary number system. The binary code assigns a pattern of binary digits, also known as bits, to each character, instruction, etc. For example, a binary string of eight bits can represent any of 256 possible values and can, therefore, represent a wide variety of different items.

Excess-3, 3-excess or 10-excess-3 binary code, shifted binary or Stibitz code is a self-complementary binary-coded decimal (BCD) code and numeral system. It is a biased representation. Excess-3 code was used on some older computers as well as in cash registers and hand-held portable electronic calculators of the 1970s, among other uses.

The Intel BCD opcodes are a set of six x86 instructions that operate with binary-coded decimal numbers. The radix used for the representation of numbers in the x86 processors is 2. This is called a binary numeral system. However, the x86 processors do have limited support for the decimal numeral system.

Chen–Ho encoding is a memory-efficient alternate system of binary encoding for decimal digits.

Densely packed decimal (DPD) is an efficient method for binary encoding decimal digits.

Decimal floating-point (DFP) arithmetic refers to both a representation and operations on decimal floating-point numbers. Working directly with decimal (base-10) fractions can avoid the rounding errors that otherwise typically occur when converting between decimal fractions and binary (base-2) fractions.

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 had a variable wordlength, which enabled operations on numbers with a large number of digits.

The IEEE 754-2008 standard includes decimal floating-point number formats in which the significand and the exponent can be encoded in two ways, referred to as binary encoding and decimal encoding.

Offset binary, also referred to as excess-K, excess-N, excess-e, excess code or biased representation, is a method for signed number representation where a signed number n is represented by the bit pattern corresponding to the unsigned numbern+K, K being the biasing value or offset. There is no standard for offset binary, but most often the K for an n-bit binary word is K = 2n−1 (for example, the offset for a four-digit binary number would be 23=8). This has the consequence that the minimal negative value is represented by all-zeros, the "zero" value is represented by a 1 in the most significant bit and zero in all other bits, and the maximal positive value is represented by all-ones (conveniently, this is the same as using two's complement but with the most significant bit is inverted). It also has the consequence that in a logical comparison operation, one gets the same result as with a true form numerical comparison operation, whereas, in two's complement notation a logical comparison will agree with true form numerical comparison operation if and only if the numbers being compared have the same sign. Otherwise the sense of the comparison will be inverted, with all negative values being taken as being larger than all positive values.

In computing, decimal32 is a decimal floating-point computer numbering format that occupies 4 bytes (32 bits) in computer memory. It is intended for applications where it is necessary to emulate decimal rounding exactly, such as financial and tax computations. Like the binary16 format, it is intended for memory saving storage.

In computing, decimal64 is a decimal floating-point computer numbering format that occupies 8 bytes in computer memory. It is intended for applications where it is necessary to emulate decimal rounding exactly, such as financial and tax computations.

In computing, decimal128 is a decimal floating-point computer numbering format that occupies 16 bytes (128 bits) in computer memory. It is intended for applications where it is necessary to emulate decimal rounding exactly, such as financial and tax computations.

BCD, 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 codes.

A half-carry 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 carry or borrow has been generated out of the least significant four bits of the accumulator register following the execution of an arithmetic instruction. It is primarily used in decimal (BCD) arithmetic instructions.

References

  1. Intel. "ia32 architecture manual" (PDF). Intel . Retrieved 2015-07-01.
  2. 1 2 Klar, Rainer (1970-02-01). "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). 1241/1241a (1 ed.). Berlin, Germany: Walter de Gruyter & Co. / G. J. Göschen'sche Verlagsbuchhandlung  [ de ]. pp. 17, 21. ISBN   3-11-083160-0. . Archiv-Nr. 7990709. Archived from the original on 2020-04-18. Retrieved 2020-04-13. (205 pages) (NB. A 2019 reprint of the first edition is available under ISBN   3-11002793-3, 978-3-11002793-8. A reworked and expanded 4th edition exists as well.)
  3. 1 2 3 Klar, Rainer (1989) [1988-10-01]. "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). 2050 (4th reworked ed.). Berlin, Germany: Walter de Gruyter & Co. pp. 25, 28, 38–39. ISBN   3-11011700-2. p. 25: […] Die nicht erlaubten 0/1-Muster nennt man auch Pseudodezimalen. […] (320 pages)
  4. Schneider, Hans-Jochen (1986). Lexikon der Informatik und Datenverarbeitung (in German) (2 ed.). R. Oldenbourg Verlag München Wien. ISBN   3-486-22662-2.
  5. Tafel, Hans Jörg (1971). Einführung in die digitale Datenverarbeitung[Introduction to digital information processing] (in German). Munich: Carl Hanser Verlag. ISBN   3-446-10569-7.
  6. Steinbuch, Karl W.; Weber, Wolfgang; Heinemann, Traute, eds. (1974) [1967]. Taschenbuch der Informatik - Band II - Struktur und Programmierung von EDV-Systemen. Taschenbuch der Nachrichtenverarbeitung (in German). 2 (3 ed.). Berlin, Germany: Springer-Verlag. ISBN   3-540-06241-6. LCCN   73-80607.
  7. Tietze, Ulrich; Schenk, Christoph (2012-12-06). Advanced Electronic Circuits. Springer Science & Business Media. ISBN   978-3642812415. 9783642812415. Retrieved 2015-08-05.
  8. Kowalski, Emil (2013-03-08) [1970]. Nuclear Electronics. Springer-Verlag. doi:10.1007/978-3-642-87663-9. ISBN   978-3642876639. 9783642876639, 978-3-642-87664-6. Retrieved 2015-08-05.
  9. Ferretti, Vittorio (2013-03-13). Wörterbuch der Elektronik, Datentechnik und Telekommunikation / Dictionary of Electronics, Computing and Telecommunications: Teil 1: Deutsch-Englisch / Part 1: German-English. 1 (2 ed.). Springer-Verlag. ISBN   978-3642980886. 9783642980886. Retrieved 2015-08-05.
  10. 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: Springer-Verlag / IBM. p. 209. LCCN   65-14624. 0978.
  11. Cowlishaw, Mike F. (2015) [1981, 2008]. "General Decimal Arithmetic" . Retrieved 2016-01-02.
  12. Evans, David Silvester (March 1961). "Chapter Four: Ancillary Equipment: Output-drive and parity-check 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 2020-05-24. (8+82 pages) (NB. The 4-bit 8421 BCD code with an extra parity bit applied as least significant bit to achieve odd parity of the resulting 5-bit code is also known as Ferranti code.)
  13. Lala, Parag K. (2007). Principles of Modern Digital Design. John Wiley & Sons. pp. 20–25. ISBN   978-0-470-07296-7.
  14. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 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: Springer-Verlag OHG. pp. 68–75. LCCN   62-14511. (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 most-significant bits will create a 9s complement.)
  15. 1 2 3 4 5 6 Kämmerer, Wilhelm (May 1969). "II.15. Struktur: Informationsdarstellung im Automaten". Written at Jena, Germany. In Frühauf, Hans; Kämmerer, Wilhelm; Schröder, Kurz; Winkler, Helmut (eds.). Digitale Automaten – Theorie, Struktur, Technik, Programmieren. Elektronisches Rechnen und Regeln (in German). 5 (1 ed.). Berlin, Germany: Akademie-Verlag GmbH. p. 161. License no. 202-100/416/69. Order no. 4666 ES 20 K 3. (NB. A second edition 1973 exists as well.)
  16. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Dokter, Folkert; Steinhauer, Jürgen (1973-06-18). Digital Electronics. Philips Technical Library (PTL) / Macmillan Education (Reprint of 1st English ed.). Eindhoven, Netherlands: The Macmillan Press Ltd. / N. V. Philips' Gloeilampenfabrieken. doi:10.1007/978-1-349-01417-0. ISBN   978-1-349-01419-4. SBN   333-13360-9 . Retrieved 2020-05-11. (270 pages) (NB. This is based on a translation of volume I of the two-volume German edition.)
  17. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Dokter, Folkert; Steinhauer, Jürgen (1975) [1969]. Digitale Elektronik in der Meßtechnik und Datenverarbeitung: Theoretische Grundlagen und Schaltungstechnik. Philips Fachbücher (in German). I (improved and extended 5th ed.). Hamburg, Germany: Deutsche Philips GmbH. p. 50. ISBN   3-87145-272-6. (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.)
  18. 1 2 3 4 5 6 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 2020-07-03. Retrieved 2020-07-03. p. 52: […] The last column [of Table II], labeled "Best," gives the maximum fraction possible with any codenamely 0.60half again better than any conventional code. This extremal is reached with the ten [heavily-marked 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 Δ1 and δ1. The confusion index for decimals using the criterion of "decimal change," is taken to be cij = |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 Minimum-confusion code for decimals. […] δ1=2   Δ1=15 […] (11 pages) (NB. Besides the combinatorial set of 4-bit BCD "minimum-confusion codes for decimals", of which the author illustrates only one explicitly (here reproduced as code I) in form of a 4-bit graph, the author also shows a 16-state 4-bit "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.)
  19. 1 2 3 Chinal, Jean P. (January 1973). "3.3. Unit Distance Codes". Written at Paris, France. Design Methods for Digital Systems. Translated by Preston, Alan; Summer, Arthur (1st English ed.). Berlin, Germany: Akademie-Verlag / Springer-Verlag. p. 46. doi:10.1007/978-3-642-86187-1_3. ISBN   978-0-387-05871-9. License No. 202-100/542/73. Order No. 7617470(6047) ES 19 B 1 / 20 K 3. Retrieved 2020-06-21. (xviii+506 pages) (NB. The French 1967 original book was named "Techniques Booléennes et Calculateurs Arithmétiques", published by Éditions Dunod  [ fr ].)
  20. 1 2 Military Handbook: Encoders - Shaft Angle To Digital (PDF). United States Department of Defense. 1991-09-30. MIL-HDBK-231A. Archived (PDF) from the original on 2020-07-25. Retrieved 2020-07-25. (NB. Supersedes MIL-HDBK-231(AS) (1970-07-01).)
  21. 1 2 Stopper, Herbert (March 1960). Written at Litzelstetten, Germany. Runge, Wilhelm Tolmé (ed.). "Ermittlung des Codes und der logischen Schaltung einer Zähldekade". Telefunken-Zeitung (TZ) - Technisch-Wissenschaftliche Mitteilungen der Telefunken GMBH (in German). Berlin, Germany: Telefunken. 33 (127): 13–19. (7 pages)
  22. 1 2 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: Springer-Verlag. pp. 10–23 [12–14]. doi:10.1007/978-3-642-80560-8. ISBN   3-540-05058-2. LCCN   75-131547. ISBN   978-3-642-80561-5. (viii+252 pages) 1st edition
  23. White, Garland S. (October 1953). "Coded Decimal Number Systems for Digital Computers". Proceedings of the Institute of Radio Engineers . Institute of Radio Engineers (IRE). 41 (10): 1450–1452. doi:10.1109/JRPROC.1953.274330. eISSN   2162-6634. ISSN   0096-8390. S2CID   51674710. (3 pages)
  24. "Different Types of Binary Codes". Electronic Hub. 2019-05-01 [2015-01-28]. Section 2.4 5211 Code. Archived from the original on 2017-11-14. Retrieved 2020-08-04.
  25. Paul, Matthias R. (1995-08-10) [1994]. "Unterbrechungsfreier Schleifencode" [Continuous loop code]. 1.02 (in German). Retrieved 2008-02-11. (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 unit-distance code for full-circle rotatory applications. Avoiding the all-zero code pattern allows for loop self-testing and to use the data lines for uninterrupted power distribution.)
  26. Gray, Frank (1953-03-17) [1947-11-13]. 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 2020-08-05. Retrieved 2020-08-05. (13 pages)
  27. Glixon, Harry Robert (March 1957). "Can You Take Advantage of the Cyclic Binary-Decimal Code?". Control Engineering . Technical Publishing Company, a division of Dun-Donnelley Publishing Corporation, Dun & Bradstreet Corp. 4 (3): 87–91. ISSN   0010-8049. (5 pages)
  28. 1 2 Ledley, Robert Steven; Rotolo, Louis S.; Wilson, James Bruce (1960). "Part 4. Logical Design of Digital-Computer Circuitry; Chapter 15. Serial Arithmetic Operations; Chapter 15-7. Additional Topics". Digital Computer and Control Engineering (PDF). McGraw-Hill Electrical and Electronic Engineering Series (1 ed.). New York, USA: McGraw-Hill Book Company, Inc. (printer: The Maple Press Company, York, Pennsylvania, USA). pp. 517–518. ISBN   0-07036981-X. ISSN   2574-7916. LCCN   59015055. OCLC   1033638267. OL   5776493M. SBN   07036981-X. . ark:/13960/t72v3b312. Archived (PDF) from the original on 2021-02-19. Retrieved 2021-02-19. 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. […] (xxiv+835+1 pages) (NB. Ledley classified the described cyclic code as a cyclic decimal-coded binary code.)
  29. 1 2 3 4 Savard, John J. G. (2018) [2006]. "Decimal Representations". quadibloc. Archived from the original on 2018-07-16. Retrieved 2018-07-16.
  30. Petherick, Edward John (October 1953). A Cyclic Progressive Binary-coded-decimal System of Representing Numbers (Technical Note MS15). Farnborough, UK: Royal Aircraft Establishment (RAE). (4 pages) (NB. Sometimes referred to as A Cyclic-Coded Binary-Coded-Decimal System of Representing Numbers.)
  31. 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).
  32. 1 2 O'Brien, Joseph A. (May 1956) [1955-11-15, 1955-06-23]. "Cyclic Decimal Codes for Analogue to Digital Converters". Transactions of the American Institute of Electrical Engineers, Part I: Communication and Electronics . Bell Telephone Laboratories, Whippany, New Jersey, USA. 75 (2): 120–122. doi:10.1109/TCE.1956.6372498. ISSN   0097-2452. S2CID   51657314. Paper 56-21. Retrieved 2020-05-18. (3 pages) (NB. This paper was prepared for presentation at the AIEE Winter General Meeting, New York, USA, 1956-01-30 to 1956-02-03.)
  33. 1 2 Tompkins, Howard E. (September 1956) [1956-07-16]. "Unit-Distance Binary-Decimal Codes for Two-Track Commutation". IRE Transactions on Electronic Computers . Correspondence. Moore School of Electrical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania, USA. EC-5 (3): 139. doi:10.1109/TEC.1956.5219934. ISSN   0367-9950 . Retrieved 2020-05-18. (1 page)
  34. Lippel, Bernhard (December 1955). "A Decimal Code for Analog-to-Digital Conversion". IRE Transactions on Electronic Computers . EC-4 (4): 158–159. doi:10.1109/TEC.1955.5219487. ISSN   0367-9950. (2 pages)
  35. 1 2 3 Susskind, Alfred Kriss; Ward, John Erwin (1958-03-28) [1957, 1956]. "III.F. Unit-Distance Codes / VI.E.2. Reflected Binary Codes". Written at Cambridge, Massachusetts, USA. In Susskind, Alfred Kriss (ed.). Notes on Analog-Digital Conversion Techniques. Technology Books in Science and Engineering. 1 (3 ed.). New York, USA: Technology Press of the Massachusetts Institute of Technology / John Wiley & Sons, Inc. / Chapman & Hall, Ltd. pp. 3-7–3-8 [3-7], 3-10–3-16 [3-13–3-16], 6-65–6-60 [6-60]. (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 "reading-type 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.)
  36. 1 2 Yuen, Chun-Kwong (December 1977). "A New Representation for Decimal Numbers". IEEE Transactions on Computers . C-26 (12): 1286–1288. doi:10.1109/TC.1977.1674792. S2CID   40879271. Archived from the original on 2020-08-08. Retrieved 2020-08-08.
  37. Lucal, Harold M. (December 1959). "Arithmetic Operations for Digital Computers Using a Modified Reflected Binary". IRE Transactions on Electronic Computers . EC-8 (4): 449–458. doi:10.1109/TEC.1959.5222057. ISSN   0367-9950. S2CID   206673385. (10 pages)
  38. Dewar, Robert Berriedale Keith; Smosna, Matthew (1990). Microprocessors - A Programmer's View (1 ed.). Courant Institute, New York University, New York, USA: McGraw-Hill Publishing Company. p. 14. ISBN   0-07-016638-2. LCCN   89-77320. (xviii+462 pages)
  39. "Chapter 8: Decimal Instructions". IBM System/370 Principles of Operation. IBM. March 1980.
  40. "Chapter 3: Data Representation". PDP-11 Architecture Handbook. Digital Equipment Corporation. 1983.
  41. 1 2 VAX-11 Architecture Handbook. Digital Equipment Corporation. 1985.
  42. "ILE RPG Reference".
  43. IBM BM 1401/1440/1460/1410/7010 Character Code Chart in BCD Order [ permanent dead link ]
  44. http://publib.boulder.ibm.com/infocenter/zos/v1r12/index.jsp?topic=%2Fcom.ibm.zos.r12.iceg200%2Fenf.htm%5B%5D
  45. "4.7 BCD and packed BCD integers". Intel 64 and IA-32 Architectures Software Developer's Manual, Volume 1: Basic Architecture (PDF). Version 072. 1. Intel Corporation. 2020-05-27 [1997]. pp. 3–2, 4-9–4-11 [4-10]. 253665-072US. Archived (PDF) from the original on 2020-08-06. Retrieved 2020-08-06. p. 4-10: […] When operating on BCD integers in general-purpose 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 half-byte (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 half-byte is more significant than the digit in the low half-byte. […] When operating on BCD integers in x87 FPU data registers, BCD values are packed in an 80-bit format and referred to as decimal integers. In this format, the first 9 bytes hold 18 BCD digits, 2 digits per byte. The least-significant digit is contained in the lower half-byte of byte 0 and the most-significant digit is contained in the upper half-byte 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 1018 + 1 to 1018 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 double-extended-precision floating-point format. All decimal integers are exactly representable in double extended-precision format. […]
  46. url=http://www.tigernt.com/onlineDoc/68000.pdf
  47. Jones, Douglas W. (2015-11-25) [1999]. "BCD Arithmetic, a tutorial". Arithmetic Tutorials. Iowa City, Iowa, USA: The University of Iowa, Department of Computer Science. Retrieved 2016-01-03.
  48. University of Alicante. "A Cordic-based Architecture for High Performance Decimal Calculations" (PDF). IEEE . Retrieved 2015-08-15.
  49. "Decimal CORDIC Rotation based on Selection by Rounding: Algorithm and Architecture" (PDF). British Computer Society . Retrieved 2015-08-14.
  50. Mathur, Aditya P. (1989). Introduction to Microprocessors (3 ed.). Tata McGraw-Hill Publishing Company Limited. ISBN   978-0-07-460222-5.
  51. 3GPP TS 29.002: Mobile Application Part (MAP) specification (Technical report). 2013. sec. 17.7.8 Common data types.
  52. "Signalling Protocols and Switching (SPS) Guidelines for using Abstract Syntax Notation One (ASN.1) in telecommunication application protocols" (PDF). p. 15.
  53. "XOM Mobile Application Part (XMAP) Specification" (PDF). p. 93. Archived from the original (PDF) on 2015-02-21. Retrieved 2013-06-27.
  54. http://www.se.ecu.edu.au/units/ens1242/lectures/ens_Notes_08.pdf%5B%5D
  55. MC6818 datasheet
  56. Gottschalk v. Benson , 409 U.S. 63, 72 (1972).

Further reading