75-471: Unpacked may refer to: Unpacked BCD , a specific binary encoding of decimals Unpacked by Flybuys , a song Unpacked , a news site focused on Jewish and Israeli issues See also [ edit ] Unpacking Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Unpacked . If an internal link led you here, you may wish to change
150-417: 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
225-524: 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. The Intel 8080 , the Zilog Z80 and the CPUs of the x86 family provide the opcode DAA (Decimal Adjust Accumulator). Subtraction is done by adding the ten's complement of the subtrahend to the minuend . To represent
300-461: A computer system specially designed to carry out operations on floating-point numbers. A number representation specifies some way of encoding a number, usually as a string of digits. There are several mechanisms by which strings of digits can represent numbers. In standard mathematical notation, the digit string can be of any length, and the location of the radix point is indicated by placing an explicit "point" character (dot or comma) there. If
375-497: A few details. Five of these formats are called basic formats , and others are termed extended precision formats and extendable precision format . Three formats are especially widely used in computer hardware and languages: Increasing the precision of the floating-point representation generally reduces the amount of accumulated round-off error caused by intermediate calculations. Other IEEE formats include: Any integer with absolute value less than 2 can be exactly represented in
450-475: 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
525-408: A number, the base (10) need not be stored, since it will be the same for the entire range of supported numbers, and can thus be inferred. Symbolically, this final value is: s b p − 1 × b e , {\displaystyle {\frac {s}{b^{\,p-1}}}\times b^{e},} where s is the significand (ignoring any implied decimal point), p
600-409: A numeral to its equivalent bit pattern or reverse the process. In packed BCD (or packed decimal ), each nibble represents 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
675-558: A single byte by taking advantage of the fact that four bits are enough to represent the range 0 to 9. The precise four-bit encoding, however, may vary for technical reasons (e.g. Excess-3 ). The ten states representing a BCD digit are sometimes called tetrades (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 ] , pseudo-decimals or pseudo-decimal digits . BCD's main virtue, in comparison to binary positional systems ,
750-1115: A syntax to be used that could be entered through a typewriter , as was the case of his Electromechanical Arithmometer in 1920. In 1938, Konrad Zuse of Berlin completed the Z1 , the first binary, programmable mechanical computer ; it uses a 24-bit binary floating-point number representation with a 7-bit signed exponent, a 17-bit significand (including one implicit bit), and a sign bit. The more reliable relay -based Z3 , completed in 1941, has representations for both positive and negative infinities; in particular, it implements defined operations with infinity, such as 1 / ∞ = 0 {\displaystyle ^{1}/_{\infty }=0} , and it stops on undefined operations, such as 0 × ∞ {\displaystyle 0\times \infty } . Zuse also proposed, but did not complete, carefully rounded floating-point arithmetic that includes ± ∞ {\displaystyle \pm \infty } and NaN representations, anticipating features of
825-453: 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: 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
SECTION 10
#1732858574299900-428: 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
975-423: A word of n bytes can contain up to (2 n )−1 decimal digits, which is always an odd number of digits. A decimal number with d digits requires 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: Like character strings,
1050-420: Is 1001'b ( 6 + 3 = 9 {\displaystyle 6+3=9} ). The following table represents decimal digits from 0 to 9 in various BCD encoding systems. In the headers, the " 8 4 2 1 " indicates the weight of each bit. In the fifth column ("BCD 8 4 −2 −1"), two of the weights are negative. Both ASCII and EBCDIC character codes for
1125-401: 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. IBM used
1200-500: Is a rational number , because it can be represented as one integer divided by another; for example 1.45 × 10 is (145/100)×1000 or 145,000 /100. The base determines the fractions that can be represented; for instance, 1/5 cannot be represented exactly as a floating-point number using a binary base, but 1/5 can be represented exactly using a decimal base ( 0.2 , or 2 × 10 ). However, 1/3 cannot be represented exactly by either binary (0.010101...) or decimal (0.333...), but in base 3 , it
1275-450: Is a class of binary encodings of decimal numbers where each digit is represented by a fixed number of bits , usually four or eight. Sometimes, special bit patterns are used for a sign or other indications (e.g. error or overflow). In byte -oriented systems (i.e. most modern computers), the term unpacked BCD usually implies a full byte for each digit (often including a sign), whereas packed BCD typically encodes two digits within
1350-524: Is a smallest positive normal floating-point number, which has a 1 as the leading digit and 0 for the remaining digits of the significand, and the smallest possible value for the exponent. There is a largest floating-point number, which has B − 1 as the value for each digit of the significand and the largest possible value for the exponent. In addition, there are representable values strictly between −UFL and UFL. Namely, positive and negative zeros , as well as subnormal numbers . The IEEE standardized
1425-503: Is also called "8421" encoding. This scheme can also be referred to as Simple Binary-Coded Decimal ( SBCD ) or BCD 8421 , and is the most common encoding. Others include the so-called "4221" and "7421" encoding – named after the weighting used for the bits – and " Excess-3 ". 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
1500-457: Is digit 0 , represented by a lone 0 punch in the card, and (8,2) in core memory. 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
1575-432: Is from 2 ≈ 2 × 10 to approximately 2 ≈ 2 × 10 . The number of normal floating-point numbers in a system ( B , P , L , U ) where is 2 ( B − 1 ) ( B P − 1 ) ( U − L + 1 ) {\displaystyle 2\left(B-1\right)\left(B^{P-1}\right)\left(U-L+1\right)} . There
SECTION 20
#17328585742991650-424: 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
1725-447: Is not a floating-point number in base ten with five digits—it needs six digits. The nearest floating-point number with only five digits is 12.346. And 1/3 = 0.3333… is not a floating-point number in base ten with any finite number of digits. In practice, most floating-point systems use base two , though base ten ( decimal floating point ) is also common. Floating-point arithmetic operations, such as addition and division, approximate
1800-492: Is often used to allow very small and very large real numbers that require fast processing times. The result of this dynamic range is that the numbers that can be represented are not uniformly spaced; the difference between two consecutive representable numbers varies with their exponent. Over the years, a variety of floating-point representations have been used in computers. In 1985, the IEEE 754 Standard for Floating-Point Arithmetic
1875-399: Is similar in concept to scientific notation. Logically, a floating-point number consists of: To derive the value of the floating-point number, the significand is multiplied by the base raised to the power of the exponent , equivalent to shifting the radix point from its implied position by a number of places equal to the value of the exponent—to the right if the exponent is positive or to
1950-418: Is simply known to the compiler, and the generated code acts accordingly for the various arithmetic operations. If a decimal digit requires four bits, then three decimal digits require 12 bits. However, since 2 (1,024) is greater than 10 (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
2025-1354: Is stored in memory using the IEEE 754 encoding, this becomes the significand s . The significand is assumed to have a binary point to the right of the leftmost bit. So, the binary representation of π is calculated from left-to-right as follows: ( ∑ n = 0 p − 1 bit n × 2 − n ) × 2 e = ( 1 × 2 − 0 + 1 × 2 − 1 + 0 × 2 − 2 + 0 × 2 − 3 + 1 × 2 − 4 + ⋯ + 1 × 2 − 23 ) × 2 1 ≈ 1.57079637 × 2 ≈ 3.1415927 {\displaystyle {\begin{aligned}&\left(\sum _{n=0}^{p-1}{\text{bit}}_{n}\times 2^{-n}\right)\times 2^{e}\\={}&\left(1\times 2^{-0}+1\times 2^{-1}+0\times 2^{-2}+0\times 2^{-3}+1\times 2^{-4}+\cdots +1\times 2^{-23}\right)\times 2^{1}\\\approx {}&1.57079637\times 2\\\approx {}&3.1415927\end{aligned}}} where p
2100-596: 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
2175-458: Is the precision ( 24 in this example), n is the position of the bit of the significand from the left (starting at 0 and finishing at 23 here) and e is the exponent ( 1 in this example). It can be required that the most significant digit of the significand of a non-zero number be non-zero (except when the corresponding exponent would be smaller than the minimum one). This process is called normalization . For binary formats (which uses only
2250-621: Is the precision (the number of digits in the significand), b is the base (in our example, this is the number ten ), and e is the exponent. Historically, several number bases have been used for representing floating-point numbers, with base two ( binary ) being the most common, followed by base ten ( decimal floating point ), and other less common varieties, such as base sixteen ( hexadecimal floating point ), base eight (octal floating point ), base four (quaternary floating point ), base three ( balanced ternary floating point ) and even base 256 and base 65,536 . A floating-point number
2325-469: Is trivial (0.1 or 1×3 ) . The occasions on which infinite expansions occur depend on the base and its prime factors . The way in which the significand (including its sign) and exponent are stored in a computer is implementation-dependent. The common IEEE formats are described in detail later and elsewhere, but as an example, in the binary single-precision (32-bit) floating-point representation, p = 24 {\displaystyle p=24} , and so
Unpacked - Misplaced Pages Continue
2400-446: 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: Here, the most significant nibble of the most significant byte has been encoded as zero, so
2475-522: 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. Floating-point In computing , floating-point arithmetic ( FP ) is arithmetic on subsets of real numbers formed by a signed string of a fixed number of digits in some base , called a significand , scaled by an integer exponent of that base. Numbers of this form are called floating-point numbers . For example,
2550-678: The English Electric DEUCE . The arithmetic is actually implemented in software, but with a one megahertz clock rate, the speed of floating-point and fixed-point operations in this machine were initially faster than those of many competing computers. The mass-produced IBM 704 followed in 1954; it introduced the use of a biased exponent . For many decades after that, floating-point hardware was typically an optional feature, and computers that had it were said to be "scientific computers", or to have " scientific computation " (SC) capability (see also Extensions for Scientific Computation (XSC)). It
2625-636: 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 is unavailable or limited in newer instruction sets (e.g., ARM ; x86 in long mode ). However, decimal fixed-point and decimal floating-point formats are still important and continue to be used in financial, commercial, and industrial computing, where
2700-566: The SDS Sigma series mainframes, and is the native format for the Burroughs Medium Systems line of mainframes (descended from the 1950s Electrodata 200 series ). Ten's complement representations for negative numbers offer an alternative approach to encoding the sign of packed (and other) BCD numbers. In this case, positive numbers always have a most significant digit between 0 and 4 (inclusive), while negative numbers are represented by
2775-404: 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 decimal numbers are supported by some programming languages (such as COBOL and PL/I). These languages allow
2850-478: The EBCDIC zone for digits without a sign overpunch. In packed BCD, the number 127 is represented by 0001 0010 0111 1100 (127C) and −127 is represented by 0001 0010 0111 1101 (127D). Burroughs systems used 1101 (D) for negative, and any other value is considered a positive sign value (the processors will normalize a positive sign to 1100 (C)). No matter how many bytes wide a word is, there is always an even number of nibbles because each byte has two of them. Therefore,
2925-690: The IEEE Standard by four decades. In contrast, von Neumann recommended against floating-point numbers for the 1951 IAS machine , arguing that fixed-point arithmetic is preferable. The first commercial computer with floating-point hardware was Zuse's Z4 computer, designed in 1942–1945. In 1946, Bell Laboratories introduced the Model ;V , which implemented decimal floating-point numbers . The Pilot ACE has binary floating-point arithmetic, and it became operational in 1950 at National Physical Laboratory, UK . Thirty-three were later sold commercially as
3000-603: The Spanish engineer Leonardo Torres Quevedo published Essays on Automatics , where he designed a special-purpose electromechanical calculator based on Charles Babbage 's analytical engine and described a way to store floating-point numbers in a consistent manner. He stated that numbers will be stored in exponential format as n x 10 m {\displaystyle ^{m}} , and offered three rules by which consistent manipulation of floating-point numbers by machines could be implemented. For Torres, " n will always be
3075-406: 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. 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
Unpacked - Misplaced Pages Continue
3150-565: The computer representation for binary floating-point numbers in IEEE 754 (a.k.a. IEC 60559) in 1985. This first standard is followed by almost all modern machines. It was revised in 2008 . IBM mainframes support IBM's own hexadecimal floating point format and IEEE 754-2008 decimal floating point in addition to the IEEE 754 binary format. The Cray T90 series had an IEEE version, but the SV1 still uses Cray floating-point format. The standard provides for many closely related formats, differing in only
3225-420: The corresponding real number arithmetic operations by rounding any result that is not a floating-point number itself to a nearby floating-point number. For example, in a floating-point arithmetic with five base-ten digits, the sum 12.345 + 1.0001 = 13.3451 might be rounded to 13.345. The term floating point refers to the fact that the number's radix point can "float" anywhere to the left, right, or between
3300-477: The creation of the IEEE 754 standard once the 32-bit (or 64-bit) word had become commonplace. This standard was significantly based on a proposal from Intel, which was designing the i8087 numerical coprocessor; Motorola, which was designing the 68000 around the same time, gave significant input as well. In 1989, mathematician and computer scientist William Kahan was honored with the Turing Award for being
3375-413: 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: 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
3450-503: The digits 0 and 1 ), this non-zero digit is necessarily 1 . Therefore, it does not need to be represented in memory, allowing the format to have one more bit of precision. This rule is variously called the leading bit convention , the implicit bit convention , the hidden bit convention , or the assumed bit convention . The floating-point representation is by far the most common way of representing in computers an approximation to real numbers. However, there are alternatives: In 1914,
3525-432: The digits, which are examples of zoned BCD, are also shown. 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: In packed BCD, the same number would fit into a single byte: Hence the numerical range for one unpacked BCD byte
3600-448: 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
3675-418: 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 2 = 16), but only 10 values are valid (0000 through 1001). For example: 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
3750-492: The given number is scaled by a power of 10 , so that it lies within a specific range—typically between 1 and 10, with the radix point appearing immediately after the first digit. As a power of ten, the scaling factor is then indicated separately at the end of the number. For example, the orbital period of Jupiter 's moon Io is 152,853.5047 seconds, a value that would be represented in standard-form scientific notation as 1.528535047 × 10 seconds. Floating-point representation
3825-453: The left if the exponent is negative. Using base-10 (the familiar decimal notation) as an example, the number 152,853.5047 , which has ten decimal digits of precision, is represented as the significand 1,528,535,047 together with 5 as the exponent. To determine the actual value, a decimal point is placed after the first digit of the significand and the result is multiplied by 10 to give 1.528535047 × 10 , or 152,853.5047 . In storing such
SECTION 50
#17328585742993900-434: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Unpacked&oldid=1227544023 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Unpacked BCD In computing and electronic systems, binary-coded decimal ( BCD )
3975-538: 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
4050-411: The mainframe level was an ongoing problem by the early 1970s for those writing and maintaining higher-level source code; these manufacturer floating-point standards differed in the word sizes, the representations, and the rounding behavior and general accuracy of operations. Floating-point compatibility across multiple computing systems was in desperate need of standardization by the early 1980s, leading to
4125-405: 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: 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
4200-567: The number 2469/200 is a floating-point number in base ten with five digits: 2469 / 200 = 12.345 = 12345 ⏟ significand × 10 ⏟ base − 3 ⏞ exponent {\displaystyle 2469/200=12.345=\!\underbrace {12345} _{\text{significand}}\!\times \!\underbrace {10} _{\text{base}}\!\!\!\!\!\!\!\overbrace {{}^{-3}} ^{\text{exponent}}} However, unlike 2469/200 = 12.345, 7716/625 = 12.3456
4275-399: 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
4350-420: The primary architect behind this proposal; he was aided by his student Jerome Coonen and a visiting professor, Harold Stone . Among the x86 innovations are these: A floating-point number consists of two fixed-point components, whose range depends exclusively on the number of bits or digits in their representation. Whereas components linearly depend on their range, the floating-point range linearly depends on
4425-403: 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 fourth and fifth digits: The decimal point is not actually stored in memory, as the packed BCD storage format does not provide for it. Its location
4500-457: The radix point is not specified, then the string implicitly represents an integer and the unstated radix point would be off the right-hand end of the string, next to the least significant digit. In fixed-point systems, a position in the string is specified for the radix point. So a fixed-point scheme might use a string of 8 decimal digits with the decimal point in the middle, whereby "00012345" would represent 0001.2345. In scientific notation ,
4575-437: The same number of digits (e.g. six), the first digit of n will be of order of tenths, the second of hundredths, etc, and one will write each quantity in the form: n ; m ." The format he proposed shows the need for a fixed-sized significand as is presently used for floating-point data, fixing the location of the decimal point in the significand so that each representation was unique, and how to format such numbers by specifying
SECTION 60
#17328585742994650-464: 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
4725-455: 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: (*) Note: These characters vary depending on the local character code page setting. Some languages (such as COBOL and PL/I) directly support fixed-point zoned decimal values, assigning an implicit decimal point at some location between
4800-486: 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
4875-496: The significand is a string of 24 bits . For instance, the number π 's first 33 bits are: 11001001 00001111 1101101 0 _ 10100010 0. {\displaystyle 11001001\ 00001111\ 1101101{\underline {0}}\ 10100010\ 0.} In this binary expansion, let us denote the positions from 0 (leftmost bit, or most significant bit) to 32 (rightmost bit). The 24-bit significand will stop at position 23, shown as
4950-417: The significand range and exponentially on the range of exponent component, which attaches outstandingly wider range to the number. On a typical computer system, a double-precision (64-bit) binary floating-point number has a coefficient of 53 bits (including 1 implied bit), an exponent of 11 bits, and 1 sign bit. Since 2 = 1024, the complete range of the positive normal floating-point numbers in this format
5025-408: The significant digits of the number. This position is indicated by the exponent, so floating point can be considered a form of scientific notation . A floating-point system can be used to represent, with a fixed number of digits, numbers of very different orders of magnitude — such as the number of meters between galaxies or between protons in an atom . For this reason, floating-point arithmetic
5100-500: The single-precision format, and any integer with absolute value less than 2 can be exactly represented in the double-precision format. Furthermore, a wide range of powers of 2 times such a number can be represented. These properties are sometimes used for purely integer data, to get 53-bit integers on platforms that have double-precision floats but only 32-bit integers. The standard specifies some special values, and their representation: positive infinity ( +∞ ), negative infinity ( −∞ ),
5175-413: The subtle conversion and fractional rounding errors that are inherent in binary floating point formats cannot be tolerated. BCD takes advantage of the fact that any one decimal numeral can be represented by a four-bit pattern. An 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
5250-401: The sum to become a valid entry. So, adding 6 to the invalid entries results in the following: 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
5325-603: 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
5400-547: The underlined bit 0 above. The next bit, at position 24, is called the round bit or rounding bit . It is used to round the 33-bit approximation to the nearest 24-bit number (there are specific rules for halfway values , which is not the case here). This bit, which is 1 in this example, is added to the integer formed by the leftmost 24 bits, yielding: 11001001 00001111 1101101 1 _ . {\displaystyle 11001001\ 00001111\ 1101101{\underline {1}}.} When this
5475-572: 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). ILE RPG uses 1111 (F) for positive and 1101 (D) for negative. These match
5550-408: Was established, and since the 1990s, the most commonly encountered representations are those defined by the IEEE. The speed of floating-point operations, commonly measured in terms of FLOPS , is an important characteristic of a computer system , especially for applications that involve intensive mathematical calculations. A floating-point unit (FPU, colloquially a math coprocessor ) is a part of
5625-1002: Was not until the launch of the Intel i486 in 1989 that general-purpose personal computers had floating-point capability in hardware as a standard feature. The UNIVAC 1100/2200 series , introduced in 1962, supported two floating-point representations: The IBM 7094 , also introduced in 1962, supported single-precision and double-precision representations, but with no relation to the UNIVAC's representations. Indeed, in 1964, IBM introduced hexadecimal floating-point representations in its System/360 mainframes; these same representations are still available for use in modern z/Architecture systems. In 1998, IBM implemented IEEE-compatible binary floating-point arithmetic in its mainframes; in 2005, IBM also added IEEE-compatible decimal floating-point arithmetic. Initially, computers used many different representations for floating-point numbers. The lack of standardization at
#298701