Vector calculus

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Vector_calculus

Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, ${\displaystyle {\mathbb{R}}^3.}$  The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields, and fluid flow.

Vector calculus was developed from the theory of quaternions by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, Vector Analysis. In its standard form using the cross product, vector calculus does not generalize to higher dimensions, but the alternative approach of geometric algebra, which uses the exterior product, does (see § Generalizations below for more).

Basic objects

Scalar fields

Main article: Scalar field

A scalar field associates a scalar value to every point in a space. The scalar is a mathematical number representing a physical quantity. Examples of scalar fields in applications include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields (known as scalar bosons), such as the Higgs field. These fields are the subject of scalar field theory.

Vector fields

Main article: Vector field

A vector field is an assignment of a vector to each point in a space. A vector field in the plane, for instance, can be visualized as a collection of arrows with a given magnitude and direction each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point. This can be used, for example, to calculate work done over a line.

Vectors and pseudovectors

In more advanced treatments, one further distinguishes pseudovector fields and pseudoscalar fields, which are identical to vector fields and scalar fields, except that they change sign under an orientation-reversing map: for example, the curl of a vector field is a pseudovector field, and if one reflects a vector field, the curl points in the opposite direction. This distinction is clarified and elaborated in geometric algebra, as described below.

Vector algebra

Main article: Euclidean vector § Basic properties

The algebraic (non-differential) operations in vector calculus are referred to as vector algebra, being defined for a vector space and then applied pointwise to a vector field. The basic algebraic operations consist of:

Notations in vector calculus

Operation	Notation	Description
Vector addition	${\displaystyle {\mathbf{v}}_1+{\mathbf{v}}_2}$	Addition of two vectors, yielding a vector.
Scalar multiplication	${\displaystyle a\mathbf{v}}$	Multiplication of a scalar and a vector, yielding a vector.
Dot product	${\displaystyle {\mathbf{v}}_1\cdot {\mathbf{v}}_2}$	Multiplication of two vectors, yielding a scalar.
Cross product	${\displaystyle {\mathbf{v}}_1\times {\mathbf{v}}_2}$	Multiplication of two vectors in ${\displaystyle {\mathbb{R}}^3}$, yielding a (pseudo)vector.

Also commonly used are the two triple products:

Vector calculus triple products

Operation	Notation	Description
Scalar triple product	${\displaystyle {\mathbf{v}}_1\cdot \left({\mathbf{v}}_2\times {\mathbf{v}}_3\right)}$	The dot product of the cross product of two vectors.
Vector triple product	${\displaystyle {\mathbf{v}}_1\times \left({\mathbf{v}}_2\times {\mathbf{v}}_3\right)}$	The cross product of the cross product of two vectors.

Operators and theorems

Main article: Vector calculus identities

Differential operators

Main articles: Gradient, Divergence, Curl (mathematics), and Laplacian

Vector calculus studies various differential operators defined on scalar or vector fields, which are typically expressed in terms of the del operator (${\displaystyle \mathrm{\nabla }}$), also known as "nabla". The three basic vector operators are:

Differential operators in vector calculus

Operation	Notation	Description	Notational analogy	Domain/Range
Gradient	${\displaystyle \mathrm{grad}(f)=\mathrm{\nabla }f}$	Measures the rate and direction of change in a scalar field.	Scalar multiplication	Maps scalar fields to vector fields.
Divergence	${\displaystyle \mathrm{div}(\mathbf{F})=\mathrm{\nabla }\cdot \mathbf{F}}$	Measures the scalar of a source or sink at a given point in a vector field.	Dot product	Maps vector fields to scalar fields.
Curl	${\displaystyle \mathrm{curl}(\mathbf{F})=\mathrm{\nabla }\times \mathbf{F}}$	Measures the tendency to rotate about a point in a vector field in ${\displaystyle {\mathbb{R}}^3}$.	Cross product	Maps vector fields to (pseudo)vector fields.
f denotes a scalar field and F denotes a vector field

Also commonly used are the two Laplace operators:

Laplace operators in vector calculus

Operation	Notation	Description	Domain/Range
Laplacian	${\displaystyle \mathrm{\Delta }f={\mathrm{\nabla }}^{2}f=\mathrm{\nabla }\cdot \mathrm{\nabla }f}$	Measures the difference between the value of the scalar field with its average on infinitesimal balls.	Maps between scalar fields.
Vector Laplacian	${\displaystyle {\mathrm{\nabla }}^2\mathbf{F}=\mathrm{\nabla }(\mathrm{\nabla }\cdot \mathbf{F})-\mathrm{\nabla }\times (\mathrm{\nabla }\times \mathbf{F})}$	Measures the difference between the value of the vector field with its average on infinitesimal balls.	Maps between vector fields.
f denotes a scalar field and F denotes a vector field

A quantity called the Jacobian matrix is useful for studying functions when both the domain and range of the function are multivariable, such as a change of variables during integration.

Integral theorems

The three basic vector operators have corresponding theorems which generalize the fundamental theorem of calculus to higher dimensions:

Integral theorems of vector calculus

Theorem	Statement	Description
Gradient theorem	${\displaystyle {\int }_{L\subset {\mathbb{R}}^n}\mathrm{\nabla }\phi \cdot d\mathbf{r}=\phi \left(\mathbf{q}\right)-\phi \left(\mathbf{p}\right)\text{ for }L=L[p\to q]}$	The line integral of the gradient of a scalar field over a curve L is equal to the change in the scalar field between the endpoints p and q of the curve.
Divergence theorem	${\displaystyle \underset{n}{\underbrace{\int \cdots {\int }_{V\subset {\mathbb{R}}^n}}}(\mathrm{\nabla }\cdot \mathbf{F})dV=\underset{n-1}{\underbrace{\oint \cdots {\oint }_{\mathrm{\partial }V}}}\mathbf{F}\cdot d\mathbf{S}}$	The integral of the divergence of a vector field over an n-dimensional solid V is equal to the flux of the vector field through the (n−1)-dimensional closed boundary surface of the solid.
Curl (Kelvin–Stokes) theorem	${\displaystyle {\iint }_{\mathrm{\Sigma }\subset {\mathbb{R}}^3}(\mathrm{\nabla }\times \mathbf{F})\cdot d\mathbf{\Sigma }={\oint }_{\mathrm{\partial }\mathrm{\Sigma }}\mathbf{F}\cdot d\mathbf{r}}$	The integral of the curl of a vector field over a surface Σ in ${\displaystyle {\mathbb{R}}^3}$ is equal to the circulation of the vector field around the closed curve bounding the surface.
${\displaystyle \phi }$ denotes a scalar field and F denotes a vector field

In two dimensions, the divergence and curl theorems reduce to the Green's theorem:

Green's theorem of vector calculus

Theorem	Statement	Description
Green's theorem	${\displaystyle {\iint }_{A\subset {\mathbb{R}}^2}\left(\frac{\mathrm{\partial }M}{\mathrm{\partial }x}-\frac{\mathrm{\partial }L}{\mathrm{\partial }y}\right)dA={\oint }_{\mathrm{\partial }A}\left(Ldx+Mdy\right)}$	The integral of the divergence (or curl) of a vector field over some region A in ${\displaystyle {\mathbb{R}}^2}$ equals the flux (or circulation) of the vector field over the closed curve bounding the region.
For divergence, F = (M, −L). For curl, F = (L, M, 0). L and M are functions of (x, y).

Applications

Linear approximations

Main article: Linear approximation

Linear approximations are used to replace complicated functions with linear functions that are almost the same. Given a differentiable function f(x, y) with real values, one can approximate f(x, y) for (x, y) close to (a, b) by the formula

${\displaystyle f(x,y)\approx f(a,b)+\frac{\mathrm{\partial }f}{\mathrm{\partial }x}(a,b)(x-a)+\frac{\mathrm{\partial }f}{\mathrm{\partial }y}(a,b)(y-b).}$

The right-hand side is the equation of the plane tangent to the graph of z = f(x, y) at (a, b).

Optimization

Main article: Mathematical optimization

For a continuously differentiable function of several real variables, a point P (that is, a set of values for the input variables, which is viewed as a point in Rⁿ) is critical if all of the partial derivatives of the function are zero at P, or, equivalently, if its gradient is zero. The critical values are the values of the function at the critical points.

If the function is smooth, or, at least twice continuously differentiable, a critical point may be either a local maximum, a local minimum or a saddle point. The different cases may be distinguished by considering the eigenvalues of the Hessian matrix of second derivatives.

By Fermat's theorem, all local maxima and minima of a differentiable function occur at critical points. Therefore, to find the local maxima and minima, it suffices, theoretically, to compute the zeros of the gradient and the eigenvalues of the Hessian matrix at these zeros.

Generalizations

Vector calculus can also be generalized to other 3-manifolds and higher-dimensional spaces.

Different 3-manifolds

Vector calculus is initially defined for Euclidean 3-space, ${\displaystyle {\mathbb{R}}^3,}$ which has additional structure beyond simply being a 3-dimensional real vector space, namely: a norm (giving a notion of length) defined via an inner product (the dot product), which in turn gives a notion of angle, and an orientation, which gives a notion of left-handed and right-handed. These structures give rise to a volume form, and also the cross product, which is used pervasively in vector calculus.

The gradient and divergence require only the inner product, while the curl and the cross product also requires the handedness of the coordinate system to be taken into account (see Cross product § Handedness for more detail).

Vector calculus can be defined on other 3-dimensional real vector spaces if they have an inner product (or more generally a symmetric nondegenerate form) and an orientation; this is less data than an isomorphism to Euclidean space, as it does not require a set of coordinates (a frame of reference), which reflects the fact that vector calculus is invariant under rotations (the special orthogonal group SO(3)).

More generally, vector calculus can be defined on any 3-dimensional oriented Riemannian manifold, or more generally pseudo-Riemannian manifold. This structure simply means that the tangent space at each point has an inner product (more generally, a symmetric nondegenerate form) and an orientation, or more globally that there is a symmetric nondegenerate metric tensor and an orientation, and works because vector calculus is defined in terms of tangent vectors at each point.

Other dimensions

Most of the analytic results are easily understood, in a more general form, using the machinery of differential geometry, of which vector calculus forms a subset. Grad and div generalize immediately to other dimensions, as do the gradient theorem, divergence theorem, and Laplacian (yielding harmonic analysis), while curl and cross product do not generalize as directly.

From a general point of view, the various fields in (3-dimensional) vector calculus are uniformly seen as being k-vector fields: scalar fields are 0-vector fields, vector fields are 1-vector fields, pseudovector fields are 2-vector fields, and pseudoscalar fields are 3-vector fields. In higher dimensions there are additional types of fields (scalar, vector, pseudovector or pseudoscalar corresponding to 0, 1, n − 1 or n dimensions, which is exhaustive in dimension 3), so one cannot only work with (pseudo)scalars and (pseudo)vectors.

In any dimension, assuming a nondegenerate form, grad of a scalar function is a vector field, and div of a vector field is a scalar function, but only in dimension 3 or 7 (and, trivially, in dimension 0 or 1) is the curl of a vector field a vector field, and only in 3 or 7 dimensions can a cross product be defined (generalizations in other dimensionalities either require ${\displaystyle n-1}$ vectors to yield 1 vector, or are alternative Lie algebras, which are more general antisymmetric bilinear products). The generalization of grad and div, and how curl may be generalized is elaborated at Curl § Generalizations; in brief, the curl of a vector field is a bivector field, which may be interpreted as the special orthogonal Lie algebra of infinitesimal rotations; however, this cannot be identified with a vector field because the dimensions differ – there are 3 dimensions of rotations in 3 dimensions, but 6 dimensions of rotations in 4 dimensions (and more generally ${\displaystyle (\genfrac{}{}{0ex}{}{n}{2})=\frac{1}{2}n(n-1)}$ dimensions of rotations in n dimensions).

There are two important alternative generalizations of vector calculus. The first, geometric algebra, uses k-vector fields instead of vector fields (in 3 or fewer dimensions, every k-vector field can be identified with a scalar function or vector field, but this is not true in higher dimensions). This replaces the cross product, which is specific to 3 dimensions, taking in two vector fields and giving as output a vector field, with the exterior product, which exists in all dimensions and takes in two vector fields, giving as output a bivector (2-vector) field. This product yields Clifford algebras as the algebraic structure on vector spaces (with an orientation and nondegenerate form). Geometric algebra is mostly used in generalizations of physics and other applied fields to higher dimensions.

The second generalization uses differential forms (k-covector fields) instead of vector fields or k-vector fields, and is widely used in mathematics, particularly in differential geometry, geometric topology, and harmonic analysis, in particular yielding Hodge theory on oriented pseudo-Riemannian manifolds. From this point of view, grad, curl, and div correspond to the exterior derivative of 0-forms, 1-forms, and 2-forms, respectively, and the key theorems of vector calculus are all special cases of the general form of Stokes' theorem.

From the point of view of both of these generalizations, vector calculus implicitly identifies mathematically distinct objects, which makes the presentation simpler but the underlying mathematical structure and generalizations less clear. From the point of view of geometric algebra, vector calculus implicitly identifies k-vector fields with vector fields or scalar functions: 0-vectors and 3-vectors with scalars, 1-vectors and 2-vectors with vectors. From the point of view of differential forms, vector calculus implicitly identifies k-forms with scalar fields or vector fields: 0-forms and 3-forms with scalar fields, 1-forms and 2-forms with vector fields. Thus for example the curl naturally takes as input a vector field or 1-form, but naturally has as output a 2-vector field or 2-form (hence pseudovector field), which is then interpreted as a vector field, rather than directly taking a vector field to a vector field; this is reflected in the curl of a vector field in higher dimensions not having as output a vector field.

Byte

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Byte

 

byte
Unit system	unit derived from bit
Unit of	digital information, data size
Symbol	B, o (when 8 bits)

The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit of memory in many computer architectures. To disambiguate arbitrarily sized bytes from the common 8-bit definition, network protocol documents such as the Internet Protocol (RFC 791) refer to an 8-bit byte as an octet. Those bits in an octet are usually counted with numbering from 0 to 7 or 7 to 0 depending on the bit endianness.

The size of the byte has historically been hardware-dependent and no definitive standards existed that mandated the size. Sizes from 1 to 48 bits have been used. The six-bit character code was an often-used implementation in early encoding systems, and computers using six-bit and nine-bit bytes were common in the 1960s. These systems often had memory words of 12, 18, 24, 30, 36, 48, or 60 bits, corresponding to 2, 3, 4, 5, 6, 8, or 10 six-bit bytes, and persisted, in legacy systems, into the twenty-first century. In this era, bit groupings in the instruction stream were often referred to as syllables or slab, before the term byte became common.

The modern de facto standard of eight bits, as documented in ISO/IEC 2382-1:1993, is a convenient power of two permitting the binary-encoded values 0 through 255 for one byte, as 2 to the power of 8 is 256. The international standard IEC 80000-13 codified this common meaning. Many types of applications use information representable in eight or fewer bits and processor designers commonly optimize for this usage. The popularity of major commercial computing architectures has aided in the ubiquitous acceptance of the 8-bit byte. Modern architectures typically use 32- or 64-bit words, built of four or eight bytes, respectively.

The unit symbol for the byte was designated as the upper-case letter B by the International Electrotechnical Commission (IEC) and Institute of Electrical and Electronics Engineers (IEEE). Internationally, the unit octet explicitly defines a sequence of eight bits, eliminating the potential ambiguity of the term "byte". The symbol for octet, 'o', also conveniently eliminates the ambiguity in the symbol 'B' between byte and bel.

Etymology and history

The term byte was coined by Werner Buchholz in June 1956, during the early design phase for the IBM Stretch computer, which had addressing to the bit and variable field length (VFL) instructions with a byte size encoded in the instruction. It is a deliberate respelling of bite to avoid accidental mutation to bit.

Another origin of byte for bit groups smaller than a computer's word size, and in particular groups of four bits, is on record by Louis G. Dooley, who claimed he coined the term while working with Jules Schwartz and Dick Beeler on an air defense system called SAGE at MIT Lincoln Laboratory in 1956 or 1957, which was jointly developed by Rand, MIT, and IBM. Later on, Schwartz's language JOVIAL actually used the term, but the author recalled vaguely that it was derived from AN/FSQ-31.

Early computers used a variety of four-bit binary-coded decimal (BCD) representations and the six-bit codes for printable graphic patterns common in the U.S. Army (FIELDATA) and Navy. These representations included alphanumeric characters and special graphical symbols. These sets were expanded in 1963 to seven bits of coding, called the American Standard Code for Information Interchange (ASCII) as the Federal Information Processing Standard, which replaced the incompatible teleprinter codes in use by different branches of the U.S. government and universities during the 1960s. ASCII included the distinction of upper- and lowercase alphabets and a set of control characters to facilitate the transmission of written language as well as printing device functions, such as page advance and line feed, and the physical or logical control of data flow over the transmission media. During the early 1960s, while also active in ASCII standardization, IBM simultaneously introduced in its product line of System/360 the eight-bit Extended Binary Coded Decimal Interchange Code (EBCDIC), an expansion of their six-bit binary-coded decimal (BCDIC) representations used in earlier card punches. The prominence of the System/360 led to the ubiquitous adoption of the eight-bit storage size, while in detail the EBCDIC and ASCII encoding schemes are different.

In the early 1960s, AT&T introduced digital telephony on long-distance trunk lines. These used the eight-bit μ-law encoding. This large investment promised to reduce transmission costs for eight-bit data.

In Volume 1 of The Art of Computer Programming (first published in 1968), Donald Knuth uses byte in his hypothetical MIX computer to denote a unit which "contains an unspecified amount of information ... capable of holding at least 64 distinct values ... at most 100 distinct values. On a binary computer a byte must therefore be composed of six bits". He notes that "Since 1975 or so, the word byte has come to mean a sequence of precisely eight binary digits...When we speak of bytes in connection with MIX we shall confine ourselves to the former sense of the word, harking back to the days when bytes were not yet standardized."

The development of eight-bit microprocessors in the 1970s popularized this storage size. Microprocessors such as the Intel 8080, the direct predecessor of the 8086, could also perform a small number of operations on the four-bit pairs in a byte, such as the decimal-add-adjust (DAA) instruction. A four-bit quantity is often called a nibble, also nybble, which is conveniently represented by a single hexadecimal digit.

The term octet unambiguously specifies a size of eight bits. It is used extensively in protocol definitions.

Historically, the term octad or octade was used to denote eight bits as well at least in Western Europe; however, this usage is no longer common. The exact origin of the term is unclear, but it can be found in British, Dutch, and German sources of the 1960s and 1970s, and throughout the documentation of Philips mainframe computers.

Unit symbol

The unit symbol for the byte is specified in IEC 80000-13, IEEE 1541 and the Metric Interchange Format as the upper-case character B.

In the International System of Quantities (ISQ), B is also the symbol of the bel, a unit of logarithmic power ratio named after Alexander Graham Bell, creating a conflict with the IEC specification. However, little danger of confusion exists, because the bel is a rarely used unit. It is used primarily in its decadic fraction, the decibel (dB), for signal strength and sound pressure level measurements, while a unit for one-tenth of a byte, the decibyte, and other fractions, are only used in derived units, such as transmission rates.

The lowercase letter o for octet is defined as the symbol for octet in IEC 80000-13 and is commonly used in languages such as French and Romanian, and is also combined with metric prefixes for multiples, for example ko and Mo.

Multiple-byte units

Several terms redirect here. For other uses, see Megabyte (disambiguation), Gigabyte (disambiguation), Yottabyte (disambiguation), KIB (disambiguation), MIB (disambiguation), GIB (disambiguation), TIB (disambiguation), PIB (disambiguation), EIB (disambiguation), and Zib (disambiguation).

For the company, see Exabyte (company).

Multiple-byte units
Decimal Value Metric 1000 kB kilobyte 10002 MB megabyte 10003 GB gigabyte 10004 TB terabyte 10005 PB petabyte 10006 EB exabyte 10007 ZB zettabyte 10008 YB yottabyte 10009 RB ronnabyte 100010 QB quettabyte	Binary Value IEC Memory 1024 KiB kibibyte KB kilobyte 10242 MiB mebibyte MB megabyte 10243 GiB gibibyte GB gigabyte 10244 TiB tebibyte TB terabyte 10245 PiB pebibyte — 10246 EiB exbibyte — 10247 ZiB zebibyte — 10248 YiB yobibyte — 10249 RiB robibyte — 102410 QiB quebibyte —
Orders of magnitude of data

More than one system exists to define unit multiples based on the byte. Some systems are based on powers of 10, following the International System of Units (SI), which defines for example the prefix kilo as 1000 (10³); other systems are based on powers of two. Nomenclature for these systems has led to confusion. Systems based on powers of 10 use standard SI prefixes (kilo, mega, giga, ...) and their corresponding symbols (k, M, G, ...). Systems based on powers of 2, however, might use binary prefixes (kibi, mebi, gibi, ...) and their corresponding symbols (Ki, Mi, Gi, ...) or they might use the prefixes K, M, and G, creating ambiguity when the prefixes M or G are used.

While the difference between the decimal and binary interpretations is relatively small for the kilobyte (about 2% smaller than the kibibyte), the systems deviate increasingly as units grow larger (the relative deviation grows by 2.4% for each three orders of magnitude). For example, a power-of-10-based terabyte is about 9% smaller than power-of-2-based tebibyte.

Units based on powers of 10

Definition of prefixes using powers of 10—in which 1 kilobyte (symbol kB) is defined to equal 1,000 bytes—is recommended by the International Electrotechnical Commission (IEC). The IEC standard defines eight such multiples, up to 1 yottabyte (YB), equal to 1000⁸ bytes. The additional prefixes ronna- for 1000⁹ and quetta- for 1000¹⁰ were adopted by the International Bureau of Weights and Measures (BIPM) in 2022.

This definition is most commonly used for data-rate units in computer networks, internal bus, hard drive and flash media transfer speeds, and for the capacities of most storage media, particularly hard drives, flash-based storage, and DVDs. Operating systems that use this definition include macOS, iOS, Ubuntu, and Debian. It is also consistent with the other uses of the SI prefixes in computing, such as CPU clock speeds or measures of performance.

Prior art, the IBM System 360 and the related tape systems set the byte at 8 bits. Early 5.25" disks used decimal even though they used 128 byte and 256 byte sectors. Hard disks used mostly 256 byte and then 512 byte before 4096 byte blocks became standard. RAM was always sold in powers of 2.

Units based on powers of 2

A system of units based on powers of 2 in which 1 kibibyte (KiB) is equal to 1,024 (i.e., 2¹⁰) bytes is defined by international standard IEC 80000-13 and is supported by national and international standards bodies (BIPM, IEC, NIST). The IEC standard defines eight such multiples, up to 1 yobibyte (YiB), equal to 1024⁸ bytes. The natural binary counterparts to ronna- and quetta- were given in a consultation paper of the International Committee for Weights and Measures' Consultative Committee for Units (CCU) as robi- (Ri, 1024⁹) and quebi- (Qi, 1024¹⁰), but have not yet been adopted by the IEC or ISO.

An alternative system of nomenclature for the same units (referred to here as the customary convention), in which 1 kilobyte (KB) is equal to 1,024 bytes, 1 megabyte (MB) is equal to 1024² bytes and 1 gigabyte (GB) is equal to 1024³ bytes is mentioned by a 1990s JEDEC standard. Only the first three multiples (up to GB) are mentioned by the JEDEC standard, which makes no mention of TB and larger. While confusing and incorrect, the customary convention is used by the Microsoft Windows operating system and random-access memory capacity, such as main memory and CPU cache size, and in marketing and billing by telecommunication companies, such as Vodafone, AT&T, Orange and Telstra.

For storage capacity, the customary convention was used by macOS and iOS through Mac OS X 10.5 Leopard and iOS 10, after which they switched to units based on powers of 10.

Parochial units

Various computer vendors have coined terms for data of various sizes, sometimes with different sizes for the same term even within a single vendor. These terms include double word, half word, long word, quad word, slab, superword and syllable. There are also informal terms. e.g., half byte and nybble for 4 bits, octal K for 1000₈.

History of the conflicting definitions

Percentage difference between decimal and binary interpretations of the unit prefixes grows with increasing storage size

Contemporary computer memory has a binary architecture making a definition of memory units based on powers of 2 most practical. The use of the metric prefix kilo for binary multiples arose as a convenience, because 1024 is approximately 1000. This definition was popular in early decades of personal computing, with products like the Tandon 51⁄4-inch DD floppy format (holding 368640 bytes) being advertised as "360 KB", following the 1024-byte convention. It was not universal, however. The Shugart SA-400 51⁄4-inch floppy disk held 109,375 bytes unformatted, and was advertised as "110 Kbyte", using the 1000 convention. Likewise, the 8-inch DEC RX01 floppy (1975) held 256256 bytes formatted, and was advertised as "256k". Some devices were advertised using a mixture of the two definitions: most notably, floppy disks advertised as "1.44 MB" have an actual capacity of 1440 KiB, the equivalent of 1.47 MB or 1.41 MiB.

In 1995, the International Union of Pure and Applied Chemistry's (IUPAC) Interdivisional Committee on Nomenclature and Symbols attempted to resolve this ambiguity by proposing a set of binary prefixes for the powers of 1024, including kibi (kilobinary), mebi (megabinary), and gibi (gigabinary).

In December 1998, the IEC addressed such multiple usages and definitions by adopting the IUPAC's proposed prefixes (kibi, mebi, gibi, etc.) to unambiguously denote powers of 1024. Thus one kibibyte (1 KiB) is 1024¹ bytes = 1024 bytes, one mebibyte (1 MiB) is 1024² bytes = 1048576 bytes, and so on.

In 1999, Donald Knuth suggested calling the kibibyte a "large kilobyte" (KKB).

Modern standard definitions

The IEC adopted the IUPAC proposal and published the standard in January 1999.^[54][55] The IEC prefixes are part of the International System of Quantities. The IEC further specified that the kilobyte should only be used to refer to 1000 bytes.

Lawsuits over definition

Lawsuits arising from alleged consumer confusion over the binary and decimal definitions of multiples of the byte have generally ended in favor of the manufacturers, with courts holding that the legal definition of gigabyte or GB is 1 GB = 1000000000 (10⁹) bytes (the decimal definition), rather than the binary definition (2³⁰, i.e., 1073741824). Specifically, the United States District Court for the Northern District of California held that "the U.S. Congress has deemed the decimal definition of gigabyte to be the 'preferred' one for the purposes of 'U.S. trade and commerce' [...] The California Legislature has likewise adopted the decimal system for all 'transactions in this state.'"

Earlier lawsuits had ended in settlement with no court ruling on the question, such as a lawsuit against drive manufacturer Western Digital. Western Digital settled the challenge and added explicit disclaimers to products that the usable capacity may differ from the advertised capacity. Seagate was sued on similar grounds and also settled.

Practical examples

Unit	Approximate equivalent
bit	a Boolean variable indicating true (1) or false (0).
byte	a basic Latin character.
kilobyte	text of "Jabberwocky"
	a typical favicon
megabyte	text of Harry Potter and the Goblet of Fire
gigabyte	about half an hour of DVD video
	CD-quality uncompressed audio of The Lamb Lies Down on Broadway
terabyte	the largest consumer hard drive in 2007
	75 hours of video, encoded at 30 Mbit/second
petabyte	2000 years of MP3-encoded music
exabyte	global monthly Internet traffic in 2004
zettabyte	global yearly Internet traffic in 2016 (known as the Zettabyte Era)

Common uses

Many programming languages define the data type byte.

The C and C++ programming languages define byte as an "addressable unit of data storage large enough to hold any member of the basic character set of the execution environment" (clause 3.6 of the C standard). The C standard requires that the integral data type unsigned char must hold at least 256 different values, and is represented by at least eight bits (clause 5.2.4.2.1). Various implementations of C and C++ reserve 8, 9, 16, 32, or 36 bits for the storage of a byte. In addition, the C and C++ standards require that there be no gaps between two bytes. This means every bit in memory is part of a byte.

Java's primitive data type byte is defined as eight bits. It is a signed data type, holding values from −128 to 127.

.NET programming languages, such as C#, define byte as an unsigned type, and the sbyte as a signed data type, holding values from 0 to 255, and −128 to 127, respectively.

In data transmission systems, the byte is used as a contiguous sequence of bits in a serial data stream, representing the smallest distinguished unit of data. For asynchronous communication a full transmission unit usually additionally includes a start bit, 1 or 2 stop bits, and possibly a parity bit, and thus its size may vary from seven to twelve bits for five to eight bits of actual data. For synchronous communication the error checking usually uses bytes at the end of a frame.