Binary digits crossword
In computing and electronic systems, binary-coded decimal BCD is a binary digits crossword of binary encodings of decimal numbers binary digits crossword each decimal digit is represented by a fixed number of bitsusually four or eight. Special bit patterns are sometimes used for a sign or for other indications e.
In byte-oriented systems i. The precise 4-bit encoding may vary however, for technical reasons, see Excess-3 for instance. 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.
Although BCD binary digits crossword 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 moredecimal 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.
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" NBCDwhere each decimal digit is represented by its corresponding four-bit binary value, as shown in the following table. This is also called "" encoding. Other encodings are also used, including so-called "" and ""—named binary digits crossword the weighting used for the bits—and " Excess-3 ".
As most computers deal with data in 8-bit bytesit is possible to use one of the following methods to encode a BCD number:. As an example, encoding the binary digits crossword number 91 using unpacked BCD results in the following binary binary digits crossword of two bytes:. 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 binary digits crossword than the range of a single byte any number of contiguous bytes may be used. For example, to represent the decimal number in packed BCD, using big-endian format, a program would encode as follows:. Note that the most significant nibble of the most significant byte is zero, implying that the number is in actuality 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 binary digits crossword a packed BCD digit. Other logical operations are used to convert a numeral to its equivalent bit pattern or reverse the process.
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 binary digits crossword a microprocessor. By employing BCD, the manipulation of numerical data for display can be greatly simplified by treating each digit as binary digits crossword 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, binary digits crossword 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 binary digits crossword 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 binary digits crossword 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.
In packed BCD or simply packed decimaleach of the two nibbles of each byte represent a decimal digit. Most implementations are big endiani. 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. Other allowed signs are A and E for positive and B for negative. Most implementations also provide unsigned BCD values with a sign nibble of F. Burroughs systems used D for negative, and any other value is considered a positive sign value the processors binary digits crossword normalize a positive sign to C.
No matter how many binary digits crossword wide a word is, there are always an even number of nibbles because each byte has two of them. Note that, like character binary digits crossword, 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.
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. 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 inclusivewhile negative numbers are represented by the 10's complement of the corresponding positive number. As a result, this system allows for, a bit packed BCD numbers to range from , to 49,, and -1 is represented as As with two's complement binary numbers, the range is not symmetric about zero.
These languages allow the programmer to specify an implicit decimal point in front of one of the digits. The decimal binary digits crossword 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. If a decimal digit requires four bits, then three decimal digits require 12 bits.
However, since 2 10 1, binary digits crossword greater than 10 3 1,if three decimal digits are encoded together, only 10 bits are needed.
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. 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 binary digits crossword a fixed value so that the byte holds a character value corresponding to the digit.
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.
These characters vary depending on the local character binary digits crossword page setting. The IBM 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 9B and Binary digits crossword 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 Aan "11 zone" set Band a "0 zone" a 0 punch combined with any others set A.
Thus the letter Awhich is 12,1 in the punched card format, is encoded B,A,1. This allows the circuitry to convert between the punched card format and the internal storage format to be very binary digits crossword with only a few special cases.
One important special case is digit 0represented by a lone 0 punch in the card, and 8,2 in core memory. The memory of the IBM is organized into 6-bit addressable digits, the usual 8, 4, 2, 1 plus Fused as a flag bit and Can 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 1211and 0 "zone punches" as in the series.
A variable length Binary digits crossword BCD numeric data type is also implemented, providing machine instructions that perform arithmetic directly on packed decimal data. All of these are used binary digits crossword hardware registers and processing units, and in software. 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 binary digits crossword them can execute without modification on the newer machines. The Intel x86 architecture supports a unique digit ten-byte BCD format that can be loaded into and stored from the floating point registers, and computations can be performed there.
The Motorola series had BCD instructions. 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 Binary digits crossword and zoned decimal add or subtract operations using short but difficult to understand sequences of word-parallel binary digits crossword and binary arithmetic operations.
Conversion of the simple sum of two binary digits crossword 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. Note that 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 per digit. To correct this, 6 is added to that sum and then the result is treated as two nibbles:. The two binary digits crossword of the result, andcorrespond 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, binary digits crossword 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 binary digits crossword BCD arithmetic adjustments following binary addition and subtraction operations.
Subtraction is done by adding the ten's binary digits crossword of the subtrahend. To represent binary digits crossword sign of a number in BCD, the number is used to represent a positive numberand is used to represent a negative number. The binary digits crossword 14 combinations are invalid signs.
To illustrate signed BCD subtraction, consider the following problem: In signed BCD, is The ten's complement of can be obtained by taking the nine's complement ofand then adding one.
Since BCD is a form of decimal representation, several of the digit sums above binary digits crossword invalid. In the event that an invalid entry any BCD digit greater than exists, 6 is added to generate a carry bit and cause the sum to become a valid entry. So adding 6 to the invalid entries results in the following:. To check binary digits crossword answer, note that the first digit is 9, which means negative.
To check the rest of the digits, represent them in decimal. The binary-coded decimal scheme described in this article is the most common encoding, but there are many others. The following table represents decimal digits from 0 to 9 in various BCD systems:. In binary digits crossword case Gottschalk v. Bensonthe U. 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. The Atari 8-bit family of computers used BCD to implement floating-point algorithms.