# Binary signals and coding in computing what is half a bytes

In most computer systems, a byte is a unit of data that is eight binary digits long. A byte is the unit most computers use to represent a character such as a letter, number or typographic symbol. Each byte can hold a string of bits that need to be used in a larger unit for binary signals and coding in computing what is half a bytes purposes.

For example, the stream of bits that constitute a visual image for a program that displays images or the string of bits that constitutes the machine code of a computer program. You forgot to provide an Email Address. This email address is already registered. You have exceeded the maximum character limit. Please provide a Corporate E-mail Address. By submitting your personal information, you agree that TechTarget and its partners may contact you regarding relevant content, products and special offers.

You also agree that your personal information may be transferred and processed in the United States, and that you have read and agree to the Terms of Use binary signals and coding in computing what is half a bytes the Privacy Policy. In some computer systems, four bytes constitute a worda unit that a computer processor can be designed to handle efficiently as it reads and processes each instruction.

Depending on the capabilities, some computer processors can handle two-byte or single-byte instructions. Language scripts can sometimes require two bytes to represent a character. These are called double-byte character sets. A bit is represented by a lowercase b. While a byte can hold a letter or symbol, a bit is the smallest unit of storage, storing just one binary digit. The standard number of bits in a byte is eight, but that number can vary from system to system, depending on the hardware.

Initially, there could be one to six bits in a byte because the equipment at the time used 6-bit pieces of information. Another unit of data representing eight bits is an octet. Unlike a byte, an octet **binary signals and coding in computing what is half a bytes** consists of eight bits, no matter the architecture. Octets, as a measurement, can be used to avoid possible ambiguity associated with bytes, particularly with legacy systems, but the terms are often used synonymously. While bytes are measured in bit multiples, computer storage is typically measured in byte multiples.

In many computer architectures, a byte is the smallest addressable unit of memory. For example, an megabyte MB hard drive holds a nominal million bytes of data. Due to massive increases in storage capacity over time, there are now eight additional units of measurement following the byte.

The eight different types of bytes currently used in computer architectures range from kilobytes 1, bytes to yottabytes 1, zettabytes. Byte multiples can be measured using two systems: A base-2, or binary, system is commonly expressed as a rounded off decimal number.

One megabyte 1 million bytes is actually made up of 1, bytes by the base-2 definition. A base system states that bytes for computer storage should be calculated as powers of In that system, a MB would actually be 1 million decimal bytes. This system is now most common among manufacturers and consumers. While the difference between the base-2 and base systems was once fairly insignificant, as capacity has increased, the discrepancy has widened considerably.

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In computing and electronic systems, binary-coded decimal BCD is a class of binary encodings of decimal numbers where each decimal digit is represented **binary signals and coding in computing what is half a bytes** a fixed number of bitsusually four or eight.

Special bit patterns are sometimes used for a sign or for other indications e. Binary signals and coding in computing what is half a bytes 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 binary signals and coding in computing what is half a bytes 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 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 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 after the weighting used for the bits—and " Excess-3 ". As most computers deal binary signals and coding in computing what is half a bytes data in 8-bit bytesit 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:. 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 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 unpack 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 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. 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 will normalize a positive sign to C. No binary signals and coding in computing what is half a bytes how many bytes wide a word is, there are always an even number of nibbles because each byte has two of them.

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. 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 binary signals and coding in computing what is half a bytes 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 binary signals and coding in computing what is half a bytes 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 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. If a decimal digit requires four bits, then three decimal digits require 12 bits.

However, since 2 10 1, is 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 to 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 code 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 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 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 simple 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 Packed BCD numeric data type is also implemented, providing machine instructions that perform arithmetic directly on packed decimal data. All of these are used within 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 using 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 BCD and zoned decimal add or subtract operations using short but difficult to understand sequences of word-parallel logic and binary arithmetic operations.

Conversion of the simple sum of two digits can be done by adding 6 that is, 16 — 10 when binary signals and coding in computing what is half a bytes 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 nibbles 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 binary signals and coding in computing what is half a bytes multiple digits by adding in groups from binary signals and coding in computing what is half a bytes 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 is done by adding the ten's complement of the subtrahend.

To represent the sign of a number in BCD, the number is used to represent a positive numberand is used to represent a negative number. The remaining 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 are 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 the answer, note that the first digit is 9, which means negative. To check the rest of the digits, represent them in **binary signals and coding in computing what is half a bytes.** 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 the 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.