ISO 80000-2

ISO 80000-2:2009 is a standard describing mathematical signs and symbols developed by the International Organization for Standardization (ISO), superseding ISO 31-11.[1] The Standard, whose full name is Quantities and units — Part 2: Mathematical signs and symbols to be used in the natural sciences and technology, is a part of the group of standards called ISO/IEC 80000.

Contents list

The Standard is divided into the following chapters:

  • Foreword
  • Introduction
  1. Scope
  2. Normative references
  3. Variables, functions, and operators
  4. Mathematical logic
  5. Sets
  6. Standard number sets and intervals
  7. Miscellaneous signs and symbols
  8. Elementary geometry
  9. Operations
  10. Combinatorics
  11. Functions
  12. Exponential and logarithmic functions
  13. Circular and hyperbolic functions
  14. Complex numbers
  15. Matrices
  16. Coordinate systems
  17. Scalars, vectors, and tensors
  18. Transforms
  19. Special functions
  • Annex A (normative) - Clarification of the symbols used
  • Bibliography

Symbols for variables and constants

Clause 3 specifies that variables such as x and y, and functions in general (e.g., ƒ(x)) are printed in italic type, while mathematical constants and functions that do not depend on the context (e.g., sin(x + π)) are in roman (upright) type. Examples given of mathematical (upright) constants are e, π and i. The numbers 1, 2, 3, etc. are also upright.

Additions to ISO 31-11

Examples of additions at an elementary level are the inclusions of int a for the integer part of a real number, frac a for the fractional part of a real number and P (often typeset as blackboard bold ℙ) for the set of primes.

Function symbols and definitions

Clause 13 defines trigonometric and hyperbolic functions such as sin and tanh and their respective inverses arcsin and artanh. The popular way of writing these inverses as sin−1 and tanh−1 is not included in ISO 80000-2.

Clause 19 defines numerous special functions, including the gamma function, Riemann zeta function, beta function, exponential integral, logarithmic integral, sine integral, Fresnel integrals, error function, incomplete elliptic integrals, hypergeometric functions, Legendre polynomials, spherical harmonics, Hermite polynomials, Laguerre polynomials, Chebyshev polynomials, Bessel functions, Neumann functions, Hankel functions and Airy functions.

References

  1. ^ "ISO 80000-2:2009". International Organization for Standardization. Retrieved 1 July 2010.
Backslash

The backslash (\) is a typographical mark used mainly in computing and is the mirror image of the common slash (/). It is sometimes called a hack, whack, escape (from C/UNIX), reverse slash, slosh, downwhack, backslant, backwhack, bash, reverse slant, and reversed virgule. In Unicode, it is encoded at U+005C \ REVERSE SOLIDUS (HTML \).

Binary logarithm

In mathematics, the binary logarithm (log2n) is the power to which the number 2 must be raised to obtain the value n. That is, for any real number x,

For example, the binary logarithm of 1 is 0, the binary logarithm of 2 is 1, the binary logarithm of 4 is 2, and the binary logarithm of 32 is 5.

The binary logarithm is the logarithm to the base 2. The binary logarithm function is the inverse function of the power of two function. As well as log2, alternative notations for the binary logarithm include lg, ld, lb (the notation preferred by ISO 31-11 and ISO 80000-2), and (with a prior statement that the default base is 2) log.

Historically, the first application of binary logarithms was in music theory, by Leonhard Euler: the binary logarithm of a frequency ratio of two musical tones gives the number of octaves by which the tones differ. Binary logarithms can be used to calculate the length of the representation of a number in the binary numeral system, or the number of bits needed to encode a message in information theory. In computer science, they count the number of steps needed for binary search and related algorithms. Other areas in which the binary logarithm is frequently used include combinatorics, bioinformatics, the design of sports tournaments, and photography.

Binary logarithms are included in the standard C mathematical functions and other mathematical software packages. The integer part of a binary logarithm can be found using the find first set operation on an integer value, or by looking up the exponent of a floating point value. The fractional part of the logarithm can be calculated efficiently.

Decimal separator

A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form.

Different countries officially designate different symbols for the decimal separator. The choice of symbol for the decimal separator also affects the choice of symbol for the thousands separator used in digit grouping, so the latter is also treated in this article.

Any such symbol can be called a decimal mark, decimal marker or decimal sign. But symbol-specific names are also used; decimal point and decimal comma refer to an (either baseline or middle) dot and comma respectively, when it is used as a decimal separator; these are the usual terms used in English, with the aforementioned generic terms reserved for abstract usage.In many contexts, when a number is spoken, the function of the separator is assumed by the spoken name of the symbol: comma or point in most cases. In some specialized contexts, the word decimal is instead used for this purpose (such as in ICAO-regulated air traffic control communications).

In mathematics the decimal separator is a type of radix point, a term that also applies to number systems with bases other than ten.

Del in cylindrical and spherical coordinates

This is a list of some vector calculus formulae for working with common curvilinear coordinate systems.

Division (mathematics)

Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication. The mathematical symbols used for the division operator are the obelus (÷) and the slash (/).

At an elementary level the division of two natural numbers is – among other possible interpretations – the process of calculating the number of times one number is contained within another one. This number of times is not always an integer, and this led to two different concepts.

The division with remainder or Euclidean division of two natural numbers provides a quotient, which is the number of times the second one is contained in the first one, and a remainder, which is the part of the first number that remains, when in the course of computing the quotient, no further full chunk of the size of the second number can be allocated.

For a modification of this division to yield only one single result, the natural numbers must be extended to rational numbers or real numbers. In these enlarged number systems, division is the inverse operation to multiplication, that is a = c ÷ b means a × b = c, as long as b is not zero—if b = 0, then this is a division by zero, which is not defined.Both forms of divisions appear in various algebraic structures. Those in which a Euclidean division (with remainder) is defined are called Euclidean domains and include polynomial rings in one indeterminate. Those in which a division (with a single result) by all nonzero elements is defined are called fields and division rings. In a ring the elements by which division is always possible are called the units; e.g., within the ring of integers the units are 1 and –1.

E (mathematical constant)

The number e is a mathematical constant that is the base of the natural logarithm: the unique number whose natural logarithm is equal to one. It is approximately equal to 2.71828, and is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series

The constant can be characterized in many different ways. For example, e can be defined as the unique positive number a such that the graph of the function y = ax has unit slope at x = 0. The function f(x) = ex is called the (natural) exponential function, and is the unique exponential function equal to its own derivative. The natural logarithm, or logarithm to base e, is the inverse function to the natural exponential function. The natural logarithm of a number k > 1 can be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case e is the value of k for which this area equals one (see image). There are alternative characterizations.

Sometimes called Euler's number after the Swiss mathematician Leonhard Euler, e is not to be confused with γ, the Euler–Mascheroni constant, sometimes called simply Euler's constant. The number e is also known as Napier's constant, but Euler's choice of the symbol e is said to have been retained in his honor. The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest.

The number e is of eminent importance in mathematics, alongside 0, 1, π and i. All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity. Like the constant π, e is irrational: it is not a ratio of integers. Also like π, e is transcendental: it is not a root of any non-zero polynomial with rational coefficients. The numerical value of e truncated to 50 decimal places is

Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.

A function is a process or a relation that associates each element x of a set X, the domain of the function, to a single element y of another set Y (possibly the same set), the codomain of the function. If the function is called f, this relation is denoted y = f (x) (read f of x), the element x is the argument or input of the function, and y is the value of the function, the output, or the image of x by f. The symbol that is used for representing the input is the variable of the function (one often says that f is a function of the variable x).

A function is uniquely represented by its graph which is the set of all pairs (x, f (x)). When the domain and the codomain are sets of numbers, each such pair may be considered as the Cartesian coordinates of a point in the plane. In general, these points form a curve, which is also called the graph of the function. This is a useful representation of the function, which is commonly used everywhere. For example, graphs of functions are commonly used in newspapers for representing the evolution of price indexes and stock market indexes

Functions are widely used in science, and in most fields of mathematics. Their role is so important that it has been said that they are "the central objects of investigation" in most fields of mathematics.

ISO/IEC 80000

ISO 80000 or IEC 80000 is an international standard promulgated jointly by the International Organization for Standardization (ISO) and the International Electrotechnical Commission (IEC).

The standard introduces the International System of Quantities (ISQ). It is a style guide for the use of physical quantities and units of measurement, formulas involving them, and their corresponding units, in scientific and educational documents for worldwide use. In most countries, the notations used in mathematics and science textbooks at schools and universities follow closely the guidelines in this standard.The ISO/IEC 80000 family of standards was completed with the publication of Part 1 in November 2009.

ISO 31

ISO 31 (Quantities and units, International Organization for Standardization, 1992) is a deprecated international standard for the use of physical quantities and units of measurement, and formulas involving them, in scientific and educational documents. It is superseded by ISO/IEC 80000.

Inverse hyperbolic functions

In mathematics, the inverse hyperbolic functions are the inverse functions of the hyperbolic functions.

For a given value of a hyperbolic function, the corresponding inverse hyperbolic function provides the corresponding hyperbolic angle. The size of the hyperbolic angle is equal to the area of the corresponding hyperbolic sector of the hyperbola xy = 1, or twice the area of the corresponding sector of the unit hyperbola x2 − y2 = 1, just as a circular angle is twice the area of the circular sector of the unit circle. Some authors have called inverse hyperbolic functions "area functions" to realize the hyperbolic angles.Hyperbolic functions and their inverses occur in many linear differential equations, for example the equation defining a catenary, of some cubic equations, in calculations of angles and distances in hyperbolic geometry and of Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.

Inverse trigonometric functions

In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

List of mathematical symbols by subject

This list of mathematical symbols by subject shows a selection of the most common symbols that are used in modern mathematical notation within formulas, grouped by mathematical topic. As it is virtually impossible to list all the symbols ever used in mathematics, only those symbols which occur often in mathematics or mathematics education are included. Many of the characters are standardized, for example in DIN 1302 General mathematical symbols or DIN EN ISO 80000-2 Quantities and units – Part 2: Mathematical signs for science and technology.

The following list is largely limited to non-alphanumeric characters. It is divided by areas of mathematics and grouped within sub-regions. Some symbols have a different meaning depending on the context and appear accordingly several times in the list. Further information on the symbols and their meaning can be found in the respective linked articles.

Mathematical notation

Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics. Mathematical notations include relatively simple symbolic representations, such as the numbers 0, 1 and 2; function symbols such as sin; operator symbols such as "+"; conceptual symbols such as lim and dy/dx; equations and variables; and complex diagrammatic notations such as Penrose graphical notation and Coxeter–Dynkin diagrams.

Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). In common mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers".

Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers (including negative integers).The natural numbers are a basis from which many other number sets may be built by extension: the integers (Grothendieck group), by including (if not yet in) the neutral element 0 and an additive inverse (−n) for each nonzero natural number n; the rational numbers, by including a multiplicative inverse (1/n) for each nonzero integer n (and also the product of these inverses by integers); the real numbers by including with the rationals the limits of (converging) Cauchy sequences of rationals; the complex numbers, by including with the real numbers the unresolved square root of minus one (and also the sums and products thereof); and so on. These chains of extensions make the natural numbers canonically embedded (identified) in the other number systems.

Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics.

In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, and also to contrast the discreteness of counting to the continuity of measurement, established by the real numbers.

The natural numbers can, at times, appear as a convenient set of names (labels), that is, as what linguists call nominal numbers, foregoing many or all of the properties of being a number in a mathematical sense.

Obelus

An obelus (symbol: ÷ or †, plural: obeluses or obeli) is a symbol consisting of a short horizontal line with a dot above and another dot below, and in other uses it is a symbol resembling a small dagger. In mathematics it is mainly used to represent the mathematical operation of division. It is therefore commonly called the division sign. In editing texts an obelus takes the form of a dagger mark (†) and is used as a reference mark, or to indicate that a person is dead, and often used to indicate a footnote.The word "obelus" comes from ὀβελός, the Ancient Greek word for a sharpened stick, spit, or pointed pillar. This is the same root as that of the word "obelisk".

Vector notation

Vector notation is a commonly used mathematical notation for working with mathematical vectors, which may be geometric vectors or members of vector spaces.

For representing a vector, the common typographic convention is lower case, upright boldface type, as in for a vector named ‘v’. The International Organization for Standardization (ISO) recommends either bold italic serif, as in v or a, or non-bold italic serif accented by a right arrow, as in v → {\displaystyle {\vec {v}}} or a → {\displaystyle {\vec {a}}} . This arrow notation for vectors is commonly used in handwriting, where boldface is impractical. The arrow represents right-pointing arrow notation or harpoons. Shorthand notations include tildes and straight lines placed respectively, below or above the name of a vector.

Between 1880 and 1887, Oliver Heaviside developed operational calculus, a method of solving differential equations by transforming them into ordinary algebraic equations which caused much controversy when introduced because of the lack of rigour in its derivation. After the turn of the 20th century, Josiah Willard Gibbs would in physical chemistry supply notation for the scalar product and vector products, which was introduced in Vector Analysis.

ISO standards by standard number
1–9999
10000–19999
20000+

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