text:
Leon High School
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Leon High School
Address
550 East Tennessee Street
Tallahassee, Florida 32308
United States
Information
Type
Public coeducational secondary
Established
1831
School district
Leon County Schools
Principal
Billy Epting
Grades
9–12
Enrollment
2,051 (2014-15)[1]
Color(s)
Red and White
Mascot
Leo the Lion
Newspaper
The Leon High Life
Yearbook
The Lion's Tale
Website
School website

Leon High School in 2008
Leon High School
U.S. National Register of Historic Places
Show map of Florida Show map of the US Show all
Coordinates
30°26′43″N 84°16′32″W / 30.44528°N 84.27556°W / 30.44528; -84.27556Coordinates: 30°26′43″N 84°16′32″W / 30.44528°N 84.27556°W / 30.44528; -84.27556
Architect
T.A. Monk, M. Leo Elliot
Architectural style
Mission/Spanish Revival
NRHP reference #
93000982[2]
Added to NRHP
September 21, 1993
Leon High School is a public high school in Tallahassee, Florida, United States. It is the oldest public high school in the state, and is a part of the Leon County Schools System.
Contents
[hide]
1 History
2 Athletics
3 Notable alumni
4 References
5 External links
History[edit]
Leon High School 1910
Leon High School is one of the oldest high schools in the country. It was founded 14 years before Florida entered into statehood. It was founded as Leon Academies in 1831 in Tallahassee, Leon County, and helped shape education in the capital city. In 1903 it became the Leon County Graded and High School, which was originally set as the first public high school for whitesin Leon County? on the corner of Duval, Tennessee and Bronough Streets.
In 1911, a need for more room for the growing student population led to the construction of a second Leon High School where the LeRoy Collins Public Library now stands.
The current building at 550 East Tennessee Street was built in controversy. Built by the Works Progress Administration (WPA) for $500,000, citizens thought the school was too expensive, too far out of town and much too big with its three stories.
The architect was M. Leo Elliott, designer of many public buildings of merit, including the Old Jail on East Gaines Street in Tallahassee.[3] The builder was Thomas Monk, who built dozens of high schools and commercial buildings around Florida, including the Monk Building in downtown Bradenton, renovated in 2000. In Tallahassee, Monk also built the Mayo Building in the Capital area, and a house in Los Robles for his daughter and son-in-law.
Leon High School was segregated (as the Florida Constitution of 1885 mandated), until the closure of the black high school, Old Lincoln High School, in 1967.
Athletics[edit]
Leon High School's mascot is the Lion, and their colors are red and white. They were one of the 29 original member schools of the FHSAA. They have been a member of the Association since it formed in 1920. Leon has teams in the following FHSAA sports:[4]
Basketball (boys and girls)
Baseball (boys)
State champions - 1940, 1944, 1951, 1998[5]
Competitive cheer (girls) - 2016 2A State Runner-up; 2017 Regional Champion, 2017 Nationals- 7th place
Cross Country (boys and girls)
Boys state champions - 1969, 2007, 2008, 2009[6]
Girls state champions - 1977, 1978, 1986, 1987, 1988[7]
Flag football (girls)
Football (boys)
State champions - 1969, 1974[8]
Golf (boys and girls)
Lacrosse (boys)
Soccer (boys and girls)
Boys state champions - 1988[9]
Softball (girls)
Swimming and diving (boys and girls)
Track and field (boys and girls)
Tennis (boys and girls)
Volleyball (girls)
State champions - 2011[10]
Weightlifting (boys and girls)
Notable alumni[edit]
Loranne Ausley, State Representative (Class of 1981)
Ricky Carmichael, motocross national champion; three-time X-Games gold medalist
LeRoy Collins, Governor of Florida (Class of 1927)
Mary Call Darby Collins, Florida's First Lady
Brad Culpepper, former Tampa Bay Buccaneer; competitor on Survivor (Class of 1987)
Maddy Curley, actress (Class of 2000)
Tommy Curtis, 1969 Florida Prep/High School Basketball Player of the Year (Class of 1969)
Faye Dunaway, actress (Class of 1958)
Gwen Graham, current member of U.S. House of Representatives, Florida's 2nd district (Class of 1980)
Tony Hale, actor (Class of 1988)
Cheryl Hines, actress (Class of 1983)
Doug Jennings, former MLB player (Oakland Athletics, Chicago Cubs) (Class of 1982)
Allison Miller, actress
Fondren Mitchell, American football player
Michel Oksenberg, China scholar, member of National Security Council (Class of 1956)
Tony Robinson, former NFL quarterback (Class of 1982)
Wyatt Sexton, former Florida State University quarterback (Class of 2002)
Jeff Shaara, New York Times bestselling author (Class of 1970)
Sonny Shroyer, actor
E. Lee Spence, underwater archaeologist (Class of 1966)
George Tabb, punk rock guitarist and songwriter, author, television host
Tamarick Vanover, football player (Florida State Seminoles, NFL)
Bert Yancey, professional golfer (Class of 1956)
Carrie Englert Zimmerman, 1976 United States Gymnastics Champion in floor exercise and balance beam, member of 1976 U.S. Olympic team
References[edit]
Jump up ^ "LEON HIGH SCHOOL". National Center for Education Statistics. Retrieved November 25, 2017.
Jump up ^ National Park Service (2010-07-09). "National Register Information System". National Register of Historic Places. National Park Service.
Jump up ^ Lindsay Brown; Vivian Young; Barbara E. Mattick (August 2003). "National Register of Historic Places Registration Form: Leon High School" (PDF). National Park Service. Retrieved 2014-12-12. Photos
Jump up ^ "Leon Lions". c2cschools.com. C2C Schools LLC. Retrieved 21 January 2015.
Jump up ^ "BASEBALL 2014-15 CHAMPIONSHIP RECORDS" (pdf). fhsaa.org. FHSAA. pp. 2–5. Retrieved 21 January 2015.
Jump up ^ "BOYS CROSS COUNTRY 2014-15 CHAMPIONSHIP RECORDS" (pdf). fhsaa.org. FHSAA. pp. 2–4. Retrieved 21 January 2015.
Jump up ^ "GIRLS CROSS COUNTRY 2014-15 CHAMPIONSHIP RECORDS" (pdf). fhsaa.org. FHSAA. p. 5. Retrieved 21 January 2015.
Jump up ^ "FOOTBALL 2013-14 CHAMPIONSHIP RECORDS" (pdf). fhsaa.org. FHSAA. p. 2. Retrieved 21 January 2015.
Jump up ^ "BOYS SOCCER 2014-15 CHAMPIONSHIP RECORDS" (pdf). fhsaa.org. FHSAA. p. 2. Retrieved 21 January 2015.
Jump up ^ "GIRLS VOLLEYBALL 2015-16 CHAMPIONSHIP RECORDS" (pdf). fhsaa.org. FHSAA. p. 4. Retrieved 21 January 2015.
External links[edit]
Wikimedia Commons has media related to Leon High School.
Leon High School website
Leon County School Board
Florida portal
Schools portal
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Leon County, Florida Schools
Public schools
Leon County Schools
Chaires
Chiles
Godby
Leon
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Lively
Raa
James S. Rickards
SAIL
Other schools
Florida State University School
Florida A&M University Developmental Research School
Former schools
Old Lincoln High School
Station One School
Independent schools
Private schools
John Paul II Catholic
Maclay School
North Florida Christian
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text:
Binary-coded decimal
From Wikipedia, the free encyclopedia
Jump to: navigation, search
"BCD code" redirects here. For BCD character sets, see BCD (character encoding).
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 decimal digit is represented by a fixed number of bits, usually four or eight. Special bit patterns are sometimes used for a sign or for other indications (e.g., error or overflow).
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 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 4-bit encoding may vary however, for technical reasons, see Excess-3 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 care-states unused named pseudo-tetrad(e)s (de)[2][3][4][5][6] or pseudo-decimal digit[7][8]).[nb 1]
BCD's main virtue is its more accurate representation and rounding of decimal quantities as well as an ease of conversion into human-readable 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 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 and the Motorola 68000-series 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; x86 does not support BCD instructions in long mode any more) decimal fixed-point and floating-point formats are still important and continue to be used in financial, commercial, and industrial computing, where subtle conversion and fractional rounding errors that are inherent in floating point binary representations cannot be tolerated.[9]
Contents
[hide]
1 Basics
2 BCD in electronics
3 Packed BCD
3.1 Fixed-point packed decimal
3.2 Higher-density encodings
4 Zoned decimal
4.1 EBCDIC zoned decimal conversion table
4.2 Fixed-point zoned decimal
5 IBM and BCD
6 Other computers and BCD
7 Addition with BCD
8 Subtraction with BCD
9 Background
10 Legal history
11 Comparison with pure binary
11.1 Advantages
11.2 Disadvantages
12 Application
13 Representational variations
13.1 Signed variations
13.2 Telephony Binary Coded Decimal (TBCD)
14 Alternative encodings
15 See also
16 Notes
17 References
18 Further reading
19 External links
Basics[edit]
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 digit
BCD
8
4
2
1
0 0 0 0 0
1 0 0 0 1
2 0 0 1 0
3 0 0 1 1
4 0 1 0 0
5 0 1 0 1
6 0 1 1 0
7 0 1 1 1
8 1 0 0 0
9 1 0 0 1
Other encodings are also used, including so-called "4221" and "7421" — named after the weighting used for the bits — and "excess-3".[10] 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 excess-3.
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:
Unpacked: 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 0 through 3) and the other numeral in the most significant nibble (bits 4 through 7).
As an example, encoding the decimal number 91 using unpacked BCD results in the following binary pattern of two bytes:
 Decimal:          9          1
 Binary :  0000 1001  0000 0001

In packed BCD, the same number would fit into a single byte:
 Decimal:     9    1
 Binary :  1001 0001

Hence the numerical range for one unpacked BCD byte is zero through nine inclusive, whereas the range for one packed BCD 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:          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.
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[edit]
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 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 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, 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.[11][12]
Packed BCD[edit]
In Packed BCD (or simply packed decimal), each of the two nibbles of each byte represent a decimal digit. Packed BCD has been in use since at least the 1960s and is implemented in all IBM mainframe hardware since then. Most implementations are big endian, 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).[13][14][15] ILE RPG uses 1111 (F) for positive and 1101 (D) for negative.[16] These match the EBCDIC zone for digits without a sign overpunch. In packed BCD, the number 127 is represented by 0001 0010 0111 1100 (127C) and −127 is represented by 0001 0010 0111 1101 (127D). Burroughs systems used 1101 (D) for negative, and any other value is considered a positive sign value (the processors will normalize a positive sign to 1100 (C)).
Sign
Digit
BCD
8 4 2 1
Sign
Notes
A
1 0 1 0 +
B
1 0 1 1 −
C
1 1 0 0 + Preferred
D
1 1 0 1 − Preferred
E
1 1 1 0 +
F
1 1 1 1 + Unsigned
No matter how many bytes wide a word is, there 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 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 −

(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 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 (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 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". 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 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[edit]
Fixed-point decimal numbers are supported by some programming languages (such as COBOL and PL/I). These languages allow the programmer to specify an implicit decimal point in front of one of the digits. For example, a packed decimal value encoded with the bytes 12 34 56 7C represents the fixed-point value +1,234.567 when the implied decimal point is located between the 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[edit]
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[edit]
Some implementations, for example IBM mainframe systems, support zoned decimal numeric representations. Each decimal digit is stored in one byte, with the lower four bits encoding the digit in BCD form. The upper four bits, called the "zone" bits, are usually set to a fixed value so that the byte holds a character value corresponding to the digit. EBCDIC systems use a zone value of 1111 (hex F); this yields bytes in the range F0 to F9 (hex), which are the EBCDIC codes for the characters "0" through "9". Similarly, ASCII systems use a zone value of 0011 (hex 3), giving character codes 30 to 39 (hex).
For signed zoned decimal values, the rightmost (least significant) zone nibble holds the sign digit, which is the same set of values that are used for signed packed decimal numbers (see above). Thus a zoned decimal value encoded as the hex bytes F1 F2 D3 represents the signed decimal value −123:
F1 F2 D3
1 2 −3

EBCDIC zoned decimal conversion table[edit]
BCD Digit
Hexadecimal
EBCDIC Character
0+ C0 A0 E0 F0 { (*) \ (*) 0
1+ C1 A1 E1 F1 A ~ (*) 1
2+ C2 A2 E2 F2 B s S 2
3+ C3 A3 E3 F3 C t T 3
4+ C4 A4 E4 F4 D u U 4
5+ C5 A5 E5 F5 E v V 5
6+ C6 A6 E6 F6 F w W 6
7+ C7 A7 E7 F7 G x X 7
8+ C8 A8 E8 F8 H y Y 8
9+ C9 A9 E9 F9 I z Z 9
0− D0 B0 } (*) ^ (*)
1− D1 B1 J
2− D2 B2 K
3− D3 B3 L
4− D4 B4 M
5− D5 B5 N
6− D6 B6 O
7− D7 B7 P
8− D8 B8 Q
9− D9 B9 R
(*) Note: These characters vary depending on the local character code page setting.
Fixed-point zoned decimal[edit]
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

IBM and BCD[edit]
Main article: BCDIC
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, 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 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 is digit 0, represented by a lone 0 punch in the card, and (8,2) in core memory.[17]
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.[18]
Other computers and BCD[edit]
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.[15] 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 no longer implemented 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, and computations can be performed there.[19]
The Motorola 68000 series had BCD instructions.[20]
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 word-parallel logic and binary arithmetic operations.[21] For example, the following code (written in C) computes an unsigned 8-digit packed BCD add using 32-bit binary operations:
uint32_t BCDadd(uint32_t a,uint32_t b) { uint32_t t1, t2; // unsigned 32-bit 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 }
Addition with BCD[edit]
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 five-bit result of adding a pair of digits has a value greater than 9. For example:
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 4-bit 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 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 with BCD[edit]
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, −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 4-bit BCD values (since 24 = 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.
Background[edit]
The binary-coded 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 Binary-Coded Decimal (SBCD) or BCD 8421. In the headers to the table, the '8 4 2 1', 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:

Digit
BCD
8 4 2 1
Stibitz Code or Excess-3
Aiken-Code or BCD
2 4 2 1
BCD
8 4 −2 −1
IBM 702, IBM 705, IBM 7080, IBM 1401 8 4 2 1
ASCII 0000 8421
EBCDIC 0000 8421
0
0000 0011 0000 0000 1010 0011 0000 1111 0000
1
0001 0100 0001 0111 0001 0011 0001 1111 0001
2
0010 0101 0010 0110 0010 0011 0010 1111 0010
3
0011 0110 0011 0101 0011 0011 0011 1111 0011
4
0100 0111 0100 0100 0100 0011 0100 1111 0100
5
0101 1000 1011 1011 0101 0011 0101 1111 0101
6
0110 1001 1100 1010 0110 0011 0110 1111 0110
7
0111 1010 1101 1001 0111 0011 0111 1111 0111
8
1000 1011 1110 1000 1000 0011 1000 1111 1000
9
1001 1100 1111 1111 1001 0011 1001 1111 1001
Legal history[edit]
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[edit]
Advantages[edit]
Many non-integral values, such as decimal 0.2, have an infinite place-value representation in binary (.001100110011...) but have a finite place-value in binary-coded decimal (0.0010). Consequently, a system based on binary-coded decimal representations of decimal fractions avoids errors representing and calculating such values.
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 non-integer 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 text-based format such as XML, or to drive signals for a seven-segment display) is a simple per-digit 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 linear-time conversion algorithm is known (see Binary numeral system).
Disadvantages[edit]
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 shift-mask-add (a binary multiplication, requiring binary shifts and adds or the equivalent, per-digit or group of digits is required)
Standard BCD requires four bits per digit, roughly 20 percent more space than a binary encoding (the ratio of 4 bits to log210 bits is 1.204). When packed so that three digits are encoded in ten bits, the storage overhead is greatly reduced, at the expense of an encoding that is unaligned with the 8-bit byte boundaries common on existing hardware, resulting in slower implementations on these systems.
Practical existing implementations of BCD are typically slower than operations on binary representations, especially on embedded systems,[citation needed] due to limited processor support for native BCD operations.
Application[edit]
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.[22]
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 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.
Representational variations[edit]
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[edit]
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 bytes F1 F2 F3 2B
Signed leading separate Separate sign character byte ('+' or '−') preceding the digit bytes 2B F1 F2 F3
Telephony Binary Coded Decimal (TBCD)[edit]
3GPP developed TBCD,[23] an expansion to BCD where the remaining (unused) bit combinations are used to add specific telephony characters,[24][25] with digits similar to those found in telephone keypads original design. It is backward compatible with BCD.
Decimal
Digit
TBCD
8 4 2 1
*
1 0 1 0
#
1 0 1 1
a
1 1 0 0
b
1 1 0 1
c
1 1 1 0
Used as filler when there is an odd number of digits
1 1 1 1
Alternative encodings[edit]
If errors in representation and computation are more important than the speed of conversion to and from display, a scaled binary representation may be used, which stores a decimal number as a binary-encoded integer and a binary-encoded 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.
Chen-Ho encoding provides a boolean transformation for converting groups of three BCD-encoded digits to and from 10-bit values that can be efficiently encoded in hardware with only 2 or 3 gate delays. Densely Packed Decimal 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 754-2008 standard.
See also[edit]
Bi-quinary coded decimal
Chen-Ho encoding
Densely packed decimal
Double dabble, an algorithm for converting binary numbers to BCD
Gray code
Year 2000 problem
Decimal computer
Notes[edit]
Jump up ^ That is, 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 Chen-Ho or DPD coding there are less, f.e. only 24 pseudo-tetrades in 1024 states (10 bits for three digits).
References[edit]
Jump up ^ Intel. "ia32 architecture manual" (PDF). Intel. Retrieved 2015-07-01.
Jump up ^ Schneider, Hans-Jochen (1986). Lexikon der Informatik und Datenverarbeitung (in German) (2 ed.). R. Oldenbourg Verlag München Wien. ISBN 3-486-22662-2.
Jump up ^ Tafel, Hans Jörg (1971). Written at RWTH, Aachen, Germany. Einführung in die digitale Datenverarbeitung [Introduction to digital information processing] (in German). Munich, Germany: Carl Hanser Verlag. ISBN 3-446-10569-7.
Jump up ^ 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.
Jump up ^ Tietze, Ulrich; Schenk, Christoph (2012-12-06). Advanced Electronic Circuits. Springer Science & Business Media. ISBN 3642812414. 9783642812415. Retrieved 2015-08-05.
Jump up ^ Kowalski, Emil (2013-03-08) [1970]. Nuclear Electronics. Springer-Verlag. doi:10.1007/978-3-642-87663-9. ISBN 3642876633. 9783642876639, 978-3-642-87664-6. Retrieved 2015-08-05.
Jump up ^ 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 3642980880. 9783642980886. Retrieved 2015-08-05.
Jump up ^ 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.
Jump up ^ Cowlishaw, Mike F. (2015) [1981,2008]. "General Decimal Arithmetic". Retrieved 2016-01-02.
Jump up ^ Parag K., Lala (2007). Principles of Modern Digital Design. John Wiley & Sons. pp. 20–25. ISBN 978-0-470-07296-7.
Jump up ^ University of Alicante. "A Cordic-based Architecture for High Performance Decimal Calculations" (PDF). IEEE. Retrieved 2015-08-15.
Jump up ^ "Decimal CORDIC Rotation based on Selection by Rounding: Algorithm and Architecture" (PDF). British Computer Society. Retrieved 2015-08-14.
Jump up ^ "Chapter 8: Decimal Instructions", IBM System/370 Principles of Operation, IBM, March 1980
Jump up ^ "Chapter 3: Data Representation", PDP-11 Architecture Handbook, Digital Equipment Corporation, 1983
^ Jump up to: a b VAX-11 Architecture Handbook, Digital Equipment Corporation, 1985
Jump up ^ "ILE RPG Reference".
Jump up ^ IBM BM 1401/1440/1460/1410/7010 Character Code Chart in BCD Order[permanent dead link]
Jump up ^ http://publib.boulder.ibm.com/infocenter/zos/v1r12/index.jsp?topic=%2Fcom.ibm.zos.r12.iceg200%2Fenf.htm[permanent dead link]
Jump up ^ Intel® 64 and IA-32 Architectures Software Developer’s Manual, Volume 1: Basic Architecture. Intel Corporation. December 2016. p. 4-9.
Jump up ^ url=http://www.tigernt.com/onlineDoc/68000.pdf
Jump up ^ 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.
Jump up ^ http://www.se.ecu.edu.au/units/ens1242/lectures/ens_Notes_08.pdf[permanent dead link]
Jump up ^ 3GPP TS 29.002: Mobile Application Part (MAP) specification (Technical report). 2013. sec. 17.7.8 Common data types.
Jump up ^ "Signalling Protocols and Switching (SPS) Guidelines for using Abstract Syntax Notation One (ASN.1) in telecommunication application protocols" (PDF). p. 15.
Jump up ^ "XOM Mobile Application Part (XMAP) Specification" (PDF). p. 93.
Further reading[edit]
Mackenzie, Charles E. (1980). Coded Character Sets, History and Development. The Systems Programming Series (1 ed.). Addison-Wesley Publishing Company, Inc. p. xii. ISBN 0-201-14460-3. LCCN 77-90165. ISBN 978-0-201-14460-4. Retrieved 2016-05-22. [1]
Arithmetic Operations in Digital Computers, R. K. Richards, 397pp, D. Van Nostrand Co., NY, 1955
Schmid, Hermann (1974). Decimal Computation (1 ed.). Binghamton, New York, USA: John Wiley & Sons. ISBN 0-471-76180-X. and Schmid, Hermann (1983) [1974]. Decimal Computation (1 (reprint) ed.). Malabar, Florida, USA: Robert E. Krieger Publishing Company. ISBN 0-89874-318-4. (NB. At least some batches of the Krieger reprint edition were misprints with defective pages 115–146.)
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 #87CH2440-6, October 1987
VLSI designs for redundant binary-coded 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 1-60132-009-4, pp64–69, CSREA Press, November 2006
Reversible Implementation of Densely-Packed-Decimal Converter to and from Binary-Coded-Decimal Format Using in IEEE-754R, A. Kaivani, A. Zaker Alhosseini, S. Gorgin, and M. Fazlali, 9th International Conference on Information Technology (ICIT'06), pp273–276, IEEE, December 2006.
Cowlishaw, Mike F. (2009) [2002,2008]. "Bibliography of material on Decimal Arithmetic – by category". General Decimal Arithmetic. IBM. Retrieved 2016-01-02.
External links[edit]
Cowlishaw, Mike F. (2014) [2000]. "A Summary of Chen-Ho Decimal Data encoding". General Decimal Arithmetic. IBM. Retrieved 2016-01-02.
Cowlishaw, Mike F. (2007) [2000]. "A Summary of Densely Packed Decimal encoding". General Decimal Arithmetic. IBM. Retrieved 2016-01-02.
Convert BCD to decimal, binary and hexadecimal and vice versa
BCD for Java
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Open system (systems theory)
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Open system have input and output flows, representing exchanges of matter, energy or information with its surroundings.
An open system is a system that has external interactions. Such interactions can take the form of information, energy, or material transfers into or out of the system boundary, depending on the discipline which defines the concept. An open system is contrasted with the concept of an isolated system which exchanges neither energy, matter, nor information with its environment. An open system is also known as a constant volume system or a flow system.
The concept of an open system was formalized within a framework that enabled one to interrelate the theory of the organism, thermodynamics, and evolutionary theory.[1] This concept was expanded upon with the advent of information theory and subsequently systems theory. Today the concept has its applications in the natural and social sciences.
In the natural sciences an open system is one whose border is permeable to both energy and mass.[2] In thermodynamics a closed system, by contrast, is permeable to energy but not to matter.
Open systems have a number of consequences. A closed system contains limited energies. The definition of an open system assumes that there are supplies of energy that cannot be depleted; in practice, this energy is supplied from some source in the surrounding environment, which can be treated as infinite for the purposes of study. One type of open system is the radiant energy system, which receives its energy from solar radiation – an energy source that can be regarded as inexhaustible for all practical purposes.
Contents
[hide]
1 Social sciences
2 Thermodynamics
3 Systems engineering
4 See also
5 References
6 Further reading
7 External links
Social sciences[edit]
In the social sciences an open system is a process that exchanges material, energy, people, capital and information with its environment. French/Greek philosopher Kostas Axelos argued that seeing the "world system" as inherently open (though unified) would solve many of the problems in the social sciences, including that of praxis (the relation of knowledge to practice), so that various social scientific disciplines would work together rather than create a monopoly whereby the world appears only sociological, political, historical, or psychological. Axelos argues that theorizing a closed system contributes to making it closed, and is thus a conservative approach.[3] The Althusserian concept of overdetermination (drawing on Sigmund Freud) posits that there are always multiple causes in every event.[4] David Harvey uses this to argue that when systems such as capitalism enter a phase of crisis, it could happen through one of a number of elements, such as gender roles, the relation to nature/the environment, or crises in accumulation.[5] Looking at the crisis in accumulation, Harvey argues that phenomena such as foreign direct investment, privatization of state-owned resources, and accumulation by dispossession act as necessary outlets when capital has overaccumulated too much in private hands and cannot circulate effectively in the marketplace. He cites the forcible displacement of Mexican and Indian peasants since the 1970s and the Asian and South-East Asian financial crisis of 1997-8, involving "hedge fund raising" of national currencies, as examples of this.[6]
Structural functionalists such as Talcott Parsons and neofunctionalists such as Niklas Luhmann have incorporated system theory to describe society and its components.
Thermodynamics[edit]
See also: Thermodynamic system § Open system, and Thermodynamics § System models
Systems engineering[edit]
See also: Systems engineering, Signal processing, Control theory, and Black box
Scheme of a black box
See also[edit]
Business process
Complex system
Dynamical system
Glossary of systems theory
Maximum power principle
Non-equilibrium thermodynamics
Open system (computing)
Open System Environment Reference Model
Openness
Open and Closed Systems in Social Science
Phantom loop
Thermodynamic system
References[edit]
Jump up ^ Luhmann, Niklas. Social Systems. Stanford: Stanford University Press, 1995; pp. 6-7
Jump up ^ Glossary Maxwell Demon, 1998.
Jump up ^ Axelos, K. ([2006] 1984). "The World: Being Becoming Totality," from Systematique ouverte (Trans. Gerald Moore, Les Editions de Minuit: Paris). Environment and Planning D: Society and Space, Vol. 24, 643-651.
Jump up ^ Althusser, L. ([2005] 1969). For Marx. London: Verso Books, Ch. 3: "Contradiction and Overdetermination," [1].
Jump up ^ RSA Animate - David Harvey, The Crises of Capitalism: [2].
Jump up ^ Harvey, D. (2005). The New Imperialism. New York: Oxford University Press USA, Ch. 3: "Accumulation by Dispossession," 137-182.
Further reading[edit]
Khalil, E.L. (1995). Nonlinear thermodynamics and social science modeling: fad cycles, cultural development and identificational slips. The American Journal of Economics and Sociology, Vol. 54, Issue 4, pp. 423–438.
Weber, B.H. (1989). Ethical Implications Of The Interface Of Natural And Artificial Systems. Delicate Balance: Technics, Culture and Consequences: Conference Proceedings for the Institute of Electrical and Electronic Engineers.
External links[edit]
OPEN SYSTEM, Principia Cybernetica Web, 2007.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Open_system_(systems_theory)&oldid=823141637"
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