Binarycoded decimal

In computing and electronic systems, binarycoded decimal (BCD) is a digital encoding method for numbers using decimal notation, with each decimal digit represented by its own binary sequence. In BCD, a numeral is usually represented by four bits which, in general, represent the decimal range 0 through 9. Other bit patterns are sometimes used for a sign or for other indications (e.g., error or overflow). Uncompressed (or zoned) BCD consumes a byte for each represented numeral, whereas compressed (or packed) BCD typically carries two numerals in a single byte by taking advantage of the fact that four bits will represent the full numeral range.
BCD's main virtue is ease of conversion between machine and humanreadable formats, as well as a more precise machineformat representation of decimal quantities. As compared to typical binary formats, BCD's principal drawbacks are a small increase in the complexity of the circuits needed to implement basic mathematical operations and less efficient usage of storage facilities.
BCD was used in many early decimal computers. Although BCD is not as widely used as in the past, decimal fixedpoint and floatingpoint formats are still important and continue to be used in financial, commercial, and industrial computing, where subtle conversion and rounding errors that are inherent to floating point binary representations cannot be tolerated.^{[1]}
Basics
As described in the introduction, BCD takes advantage of the fact that any one decimal numeral can be represented by a four bit pattern:
Decimal: 0 1 2 3 4 5 6 7 8 9 Binary : 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001
As most computers store data in 8bit bytes, it is possible to use one of the following methods to encode a BCD number:
 Uncompressed: each numeral is encoded into one byte, with four bits representing the numeral and the remaining bits having no significance.
 Packed: two numerals are encoded into a single byte, with one numeral in the least significant nibble (bits 03) and the other numeral in the most significant nibble (bits 47).
As an example, encoding the decimal number 91 using uncompressed 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 uncompressed 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 uncompressed BCD; encoding the same number in uncompressed format would consume 67 percent more storage.
Shifting 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.
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 utilizing 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 to such a display would require complex circuitry. Therefore, in cases where the calculations are relatively simple working throughout with BCD can lead to a simpler overall system than converting to binary.
The same argument applies when hardware of this type uses an embedded microcontroller or other small processor. Often, smaller code results when representing numbers internally in BCD format, since a conversion from or to binary representation can be expensive on such limited processors. For these applications, some small processors feature BCD arithmetic modes, which assist when writing routines that manipulate BCD quantities.
Packed BCD
A common variation of the twodigitsperbyte encoding is called packed BCD (or simply packed decimal), which has been in use since the 1960s or earlier and implemented in all IBM mainframe hardware since then. In most representations, one or more bytes hold a decimal integer, where each of the two nibbles of each byte represent a decimal digit, with the more significant digit in the upper half of each byte, and with 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 in some representations this nibble may be used as the least significant digit if the packed decimal value does not have a sign at all, i.e., is purely unsigned). 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 was derived from abbreviations for accounting terms (Credit and Debit), as packed decimal coding was widely used in accounting systems.^{[citation needed]} Other allowed signs are 1010 (A) and 1110 (E) for positive and 1011 (B) for negative. Some implementations also provide unsigned BCD values with a sign nibble of 1111 (F)^{[citation needed]}. ILE RPG uses 1111 (F) for positive and 1101 (D) for negative.^{[2]} 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 was considered a positive sign value (the processors would normalize a positive sign to 1100 (C)).
Sign
DigitBCD
8 4 2 1Sign 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 are 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 ½(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 ^{1}/_{6} 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. Besides the IBM System/360 and later compatible mainframes, packed BCD was implemented in the native instruction set of the original VAX processors from Digital Equipment Corporation and was the native format for the Burroughs Corporation Medium Systems line of mainframes (descended from the 1950's Electrodata 200 series).
Fixedpoint packed decimal
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 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.
Higherdensity encodings
If a decimal digit requires four bits, then three decimal digits require 12 bits. However, since 2^{10} (1,024) is greater than 10^{3} (1,000), if three decimal digits are encoded together, only 10 bits are needed. Two such encodings are ChenHo encoding and Densely Packed Decimal. 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 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
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
IBM and BCD
Main article: BCD (6bit)IBM used the terms binarycoded decimal and BCD for 6bit alphameric codes that represented numbers, uppercase letters and special characters. Some variation of BCD alphamerics was used in most early IBM computers, including the IBM 1620, IBM 1400 series, and nonDecimal Architecture members of the IBM 700/7000 series.
The IBM 1400 series were 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 were zero and the digit value represented by standard 4bit BCD in bits 8 through 1. For most other characters bits B and A were 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, (12,1) in the punched card format, was encoded (B,A,1) and the currency symbol $, (11,8,3) in the punched card, as (B,8,3). This allowed 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 was digit 0, represented by a lone 0 punch in the card, and (8,2) in core memory. ^{[3]}
The memory of the IBM 1620 was 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 were 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 were encoded using digit pairs (using twooutoffive code in the digits, not BCD) of the 10digit word, with the "zone" in the left digit and the "digit" in the right digit. Input/Output translation hardware converted between the internal digit pairs and the external standard 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 was also implemented, providing machine instructions that performed arithmetic directly on packed decimal data.
On the IBM 1130 and 1800, packed BCD was 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.
Other computers and BCD
The Digital Equipment Corporation VAX11 series included instructions that could perform arithmetic directly on packed BCD data and convert between packed BCD data and other integer representations. The VAX's packed BCD format was 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 was invoked automatically via exception handling when the no longer implemented instructions were encountered, so that programs using them could execute without modification on the newer machines.
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.
Addition with BCD
It is possible to perform addition in BCD 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 result has a value greater than 9. For example:
1001 + 1000 = 10001 = 0001 0001 9 + 8 = 17 = 1 1
In BCD, there cannot exist a value greater than 9 (1001) per nibble. To correct this, 6 (0110) is added to that sum to get the correct first two digits:
0001 0001 + 0000 0110 = 0001 0111 1 1 + 0 6 = 1 7
which gives two nibbles, 0001 and 0111, which 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.
Subtraction with BCD
Subtraction is done by adding the ten's complement of the subtrahend. 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, −568 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 + 1001 0101 0110 1000 = 1001 1000 1011 1111 0 3 5 7 + 9 5 6 8 = 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 + 0000 0000 0110 0110 = 1001 1001 0010 0101 9 8 11 15 + 0 0 6 6 = 9 9 2 5
Thus the result of the subtraction is 1001 1001 0010 0101 (925). To check the answer, note that the first bit is the sign bit, which is 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.
Background
The binarycoded decimal scheme described in this article is the most common encoding, but there are many others. The method here can be referred to as Simple BinaryCoded Decimal (SBCD) or BCD 8421. In the headers to the table, the '8 4 2 1', etc., indicates the weight of each bit shown; note that in the fifth column 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:
DigitBCD
8 4 2 1Excess3
or Stibitz CodeBCD 2 4 2 1
or Aiken CodeBCD
8 4 −2 −1IBM 702 IBM 705
IBM 7080 IBM 1401
8 4 2 1ASCII
0000 8421EBCDIC
0000 84210 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 Legal history
In the 1972 case Gottschalk v. Benson, the U.S. Supreme Court overturned a lower court decision which had allowed a patent for converting BCD encoded numbers to binary on a computer. This was an important case in determining the patentability of software and algorithms.
Comparison with pure binary
Advantages
 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.
 Scaling by a factor of 10 (or a power of 10) is simple; this is useful when a decimal scaling factor is needed to represent a noninteger quantity (e.g., in financial calculations)
 Rounding at a decimal digit boundary is simpler. Addition and subtraction in decimal does not require rounding.
 Alignment of two decimal numbers (for example 1.3 + 27.08) is a simple, exact, shift.
 Conversion to a character form or for display (e.g., to a textbased format such as XML, or to drive signals for a sevensegment display) is a simple perdigit mapping, and can be done in linear (O(n)) time. Conversion from pure binary involves relatively complex logic that spans digits, and for large numbers no lineartime conversion algorithm is known (see Binary numeral system).
Disadvantages
 Some operations are more complex to implement. Adders require extra logic to cause them to wrap and generate a carry early. 15–20 percent 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 percent 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.
 Practical existing implementations of BCD are typically slower than operations on binary representations, especially on embedded systems,^{[citation needed]} due to limited processor support for native BCD operations.
Application
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.^{[4]}
The Atari 8bit family of computers used BCD to implement floatingpoint algorithms. The MOS 6502 processor used has a BCD mode that affects the addition and subtraction instructions.
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 a paradox in the unit's date, rendering most functionalities inoperable.
Representational variations
Various BCD implementations exist that employ other representations for numbers. Programmable calculators manufactured by Texas Instruments, HewlettPackard, and others typically employ a floatingpoint BCD format, typically with two or three digits for the (decimal) exponent. The extra bits of the sign digit may be used to indicate special numeric values, such as infinity, underflow/overflow, and error (a blinking display).
Signed variations
Signed decimal values may be represented in several ways. The COBOL programming language, for example, supports a total of five zoned decimal formats, 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 bytesF1 F2 F3 2B
Signed leading separate Separate sign character byte ( '+'
or'−'
) preceding the digit bytes2B F1 F2 F3
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 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.
ChenHo encoding provides a boolean transformation for converting groups of three BCDencoded digits to and from 10bit values that can be efficiently encoded in hardware with only 2 or 3 gate delays. Densely Packed Decimal is a similar scheme that 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.
See also
 Biquinary coded decimal
 ChenHo encoding
 Densely packed decimal
 Double dabble, an algorithm for converting binary numbers to BCD
 Gray code
 Year 2000 problem
 Decimal computer
References
 ^ "General Decimal Arithmetic". http://speleotrove.com/decimal/.
 ^ "ILE RPG Reference". http://publib.boulder.ibm.com/iseries/v5r2/ic2924/books/c0925083170.htm.
 ^ IBM BM 1401/1440/1460/1410/7010 Character Code Chart in BCD Order
 ^ http://www.se.ecu.edu.au/units/ens1242/lectures/ens_Notes_08.pdf
Further reading
 Mackenzie, Charles E. (1980). Coded Character Sets, History and Development. AddisonWesley. ISBN 0201144601.
 Arithmetic Operations in Digital Computers, R. K. Richards, 397pp, D. Van Nostrand Co., NY, 1955
 Schmid, Hermann, Decimal computation, ISBN 047176180X, 266pp, Wiley, 1974
 Superoptimizer: A Look at the Smallest Program, Henry Massalin, ACM Sigplan Notices, Vol. 22 #10 (Proceedings of the Second International Conference on Architectural support for Programming Languages and Operating Systems), pp122–126, ACM, also IEEE Computer Society Press #87CH24406, October 1987
 VLSI designs for redundant binarycoded decimal addition, Behrooz Shirazi, David Y. Y. Yun, and Chang N. Zhang, IEEE Seventh Annual International Phoenix Conference on Computers and Communications, 1988, pp52–56, IEEE, March 1988
 Fundamentals of Digital Logic by Brown and Vranesic, 2003
 Modified Carry Look Ahead BCD Adder With CMOS and Reversible Logic Implementation, Himanshu Thapliyal and Hamid R. Arabnia, Proceedings of the 2006 International Conference on Computer Design (CDES'06), ISBN 1601320094, pp64–69, CSREA Press, November 2006
 Reversible Implementation of DenselyPackedDecimal Converter to and from BinaryCodedDecimal Format Using in IEEE754R, A. Kaivani, A. Zaker Alhosseini, S. Gorgin, and M. Fazlali, 9th International Conference on Information Technology (ICIT'06), pp273–276, IEEE, December 2006.
 Decimal Arithmetic Bibliography
External links
Categories: Computer arithmetic
 Numeration
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