# ISO 31-11

ISO 31-11:1992 was the part of international standard ISO 31 that defines mathematical signs and symbols for use in physical sciences and technology. It was superseded in 2009 by ISO 80000-2.[1]

Its definitions include the following:[2]

## Mathematical logic

Sign Example Name Meaning and verbal equivalent Remarks
pq conjunction sign p and q
pq disjunction sign p or q (or both)
¬ ¬ p negation sign negation of p; not p; non p
pq implication sign if p then q; p implies q Can also be written as qp. Sometimes → is used.
xA p(x)
(∀xA) p(x)
universal quantifier for every x belonging to A, the proposition p(x) is true The "∈A" can be dropped where A is clear from context.
xA p(x)
(∃xA) p(x)
existential quantifier there exists an x belonging to A for which the proposition p(x) is true The "∈A" can be dropped where A is clear from context.
∃! is used where exactly one x exists for which p(x) is true.

## Sets

Sign Example Meaning and verbal equivalent Remarks
xA x belongs to A; x is an element of the set A
xA x does not belong to A; x is not an element of the set A The negation stroke can also be vertical.
Ax the set A contains x (as an element) same meaning as xA
Ax the set A does not contain x (as an element) same meaning as xA
{ } {x1, x2, ..., xn} set with elements x1, x2, ..., xn also {xiiI}, where I denotes a set of indices
{ ∣ } {xAp(x)} set of those elements of A for which the proposition p(x) is true Example: {x ∈ ℝ ∣ x > 5}
The ∈A can be dropped where this set is clear from the context.
card card(A) number of elements in A; cardinal of A
AB difference between A and B; A minus B The set of elements which belong to A but not to B.
AB = { xxAxB }
AB should not be used.
the empty set
the set of natural numbers; the set of positive integers and zero ℕ = {0, 1, 2, 3, ...}
Exclusion of zero is denoted by an asterisk:
* = {1, 2, 3, ...}
k = {0, 1, 2, 3, ..., k − 1}
the set of integers ℤ = {..., −3, −2, −1, 0, 1, 2, 3, ...}

* = ℤ ∖ {0} = {..., −3, −2, −1, 1, 2, 3, ...}

the set of rational numbers * = ℚ ∖ {0}
the set of real numbers * = ℝ ∖ {0}
the set of complex numbers * = ℂ ∖ {0}
[,] [a,b] closed interval in ℝ from a (included) to b (included) [a,b] = {x ∈ ℝ ∣ axb}
],]
(,]
]a,b]
(a,b]
left half-open interval in ℝ from a (excluded) to b (included) ]a,b] = {x ∈ ℝ ∣ a < xb}
[,[
[,)
[a,b[
[a,b)
right half-open interval in ℝ from a (included) to b (excluded) [a,b[ = {x ∈ ℝ ∣ ax < b}
],[
(,)
]a,b[
(a,b)
open interval in ℝ from a (excluded) to b (excluded) ]a,b[ = {x ∈ ℝ ∣ a < x < b}
BA B is included in A; B is a subset of A Every element of B belongs to A. ⊂ is also used.
BA B is properly included in A; B is a proper subset of A Every element of B belongs to A, but B is not equal to A. If ⊂ is used for "included", then ⊊ should be used for "properly included".
CA C is not included in A; C is not a subset of A ⊄ is also used.
AB A includes B (as subset) A contains every element of B. ⊃ is also used. BA means the same as AB.
AB. A includes B properly. A contains every element of B, but A is not equal to B. If ⊃ is used for "includes", then ⊋ should be used for "includes properly".
AC A does not include C (as subset) ⊅ is also used. AC means the same as CA.
AB union of A and B The set of elements which belong to A or to B or to both A and B.
AB = { xxAxB }
${\displaystyle \bigcup _{i=1}^{n}A_{i}}$ union of a collection of sets ${\displaystyle \bigcup _{i=1}^{n}A_{i}=A_{1}\cup A_{2}\cup \ldots \cup A_{n}}$, the set of elements belonging to at least one of the sets A1, …, An. ${\displaystyle \bigcup {}_{i=1}^{n}}$ and ${\displaystyle \bigcup _{i\in I}}$, ${\displaystyle \bigcup {}_{i\in I}}$ are also used, where I denotes a set of indices.
AB intersection of A and B The set of elements which belong to both A and B.
AB = { xxAxB }
${\displaystyle \bigcap _{i=1}^{n}A_{i}}$ intersection of a collection of sets ${\displaystyle \bigcap _{i=1}^{n}A_{i}=A_{1}\cap A_{2}\cap \ldots \cap A_{n}}$, the set of elements belonging to all sets A1, …, An. ${\displaystyle \bigcap {}_{i=1}^{n}}$ and ${\displaystyle \bigcap _{i\in I}}$, ${\displaystyle \bigcap {}_{i\in I}}$ are also used, where I denotes a set of indices.
AB complement of subset B of A The set of those elements of A which do not belong to the subset B. The symbol A is often omitted if the set A is clear from context. Also ∁AB = AB.
(,) (a, b) ordered pair a, b; couple a, b (a, b) = (c, d) if and only if a = c and b = d.
a, b⟩ is also used.
(,…,) (a1a2, …, an) ordered n-tuple a1, a2, …, an⟩ is also used.
× A × B cartesian product of A and B The set of ordered pairs (a, b) such that aA and bB.
A × B = { (a, b) ∣ aAbB }
A × A × ⋯ × A is denoted by An, where n is the number of factors in the product.
Δ ΔA set of pairs (a, a) ∈ A × A where aA; diagonal of the set A × A ΔA = { (a, a) ∣ aA }
idA is also used.

## Miscellaneous signs and symbols

Sign Example Meaning and verbal equivalent Remarks

${\displaystyle \ {\stackrel {\mathrm {def} }{=}}}$
ab a is by definition equal to b [2] := is also used
= a = b a equals b ≡ may be used to emphasize that a particular equality is an identity.
ab a is not equal to b ${\displaystyle a\not \equiv b}$ may be used to emphasize that a is not identically equal to b.
ab a corresponds to b On a 1:106 map: 1 cm ≙ 10 km.
ab a is approximately equal to b The symbol ≃ is reserved for "is asymptotically equal to".

ab
ab
a is proportional to b
< a < b a is less than b
> a > b a is greater than b
ab a is less than or equal to b The symbol ≦ is also used.
ab a is greater than or equal to b The symbol ≧ is also used.
ab a is much less than b
ab a is much greater than b
infinity
()
[]
{}
${\displaystyle \langle \rangle }$
(a+b)c
[a+b]c
{a+b}c
${\displaystyle \langle }$a+b${\displaystyle \rangle }$c
ac+bc, parentheses
ac+bc, square brackets
ac+bc, braces
ac+bc, angle brackets
In ordinary algebra, the sequence of (), [], {}, ${\displaystyle \langle \rangle }$ in order of nesting is not standardized. Special uses are made of (), [], {}, ${\displaystyle \langle \rangle }$ in particular fields.[3]
AB ∥ CD the line AB is parallel to the line CD
${\displaystyle \perp }$ AB${\displaystyle \perp }$CD the line AB is perpendicular to the line CD[4]

## Operations

Sign Example Meaning and verbal equivalent Remarks
+ a + b a plus b
ab a minus b
± a ± b a plus or minus b
ab a minus or plus b −(a ± b) = −ab
... ... ... ...

## Functions

Example Meaning and verbal equivalent Remarks
${\displaystyle f:D\rightarrow C}$ function f has domain D and codomain C Used to explicitly define the domain and codomain of a function.
${\displaystyle f\left(S\right)}$ ${\displaystyle \left\{f\left(x\right)\mid x\in S\right\}}$ Set of all possible outputs in the codomain when given inputs from S, a subset of the domain of f.

## Exponential and logarithmic functions

Example Meaning and verbal equivalent Remarks
e base of natural logarithms e = 2.718 28...
ex exponential function to the base e of x
logax logarithm to the base a of x
lb x binary logarithm (to the base 2) of x lb x = log2x
ln x natural logarithm (to the base e) of x ln x = logex
lg x common logarithm (to the base 10) of x lg x = log10x
... ... ...

## Circular and hyperbolic functions

Example Meaning and verbal equivalent Remarks
π ratio of the circumference of a circle to its diameter π = 3.141 59...
... ... ...

## Complex numbers

Example Meaning and verbal equivalent Remarks
i   j imaginary unit; i² = −1 In electrotechnology, j is generally used.
Re z real part of z z = x + iy, where x = Re z and y = Im z
Im z imaginary part of z
z absolute value of z; modulus of z mod z is also used
arg z argument of z; phase of z z = reiφ, where r = ∣z∣ and φ = arg z, i.e. Re z = r cos φ and Im z = r sin φ
z* (complex) conjugate of z sometimes a bar above z is used instead of z*
sgn z signum z sgn z = z / ∣z∣ = exp(i arg z) for z ≠ 0, sgn 0 = 0

## Matrices

Example Meaning and verbal equivalent Remarks
A matrix A ...
... ... ...

## Coordinate systems

Coordinates Position vector and its differential Name of coordinate system Remarks
x, y, z [x y z] = [x y z]; [dx dy dz]; cartesian x1, x2, x3 for the coordinates and e1, e2, e3 for the base vectors are also used. This notation easily generalizes to n-mensional space. ex, ey, ez form an orthonormal right-handed system. For the base vectors, i, j, k are also used.
ρ, φ, z [x, y, z] = [ρ cos(φ), ρ sin(φ), z] cylindrical eρ(φ), eφ(φ), ez form an orthonormal right-handed system. lf z= 0, then ρ and φ are the polar coordinates.
r, θ, φ [x, y, z] = r [sin(θ)cos(φ), sin(θ)sin(φ), cos(θ)] spherical er(θ,φ), eθ(θ,φ),eφ(φ) form an orthonormal right-handed system.

## Vectors and tensors

Example Meaning and verbal equivalent Remarks
a
${\displaystyle {\vec {a}}}$
vector a Instead of italic boldface, vectors can also be indicated by an arrow above the letter symbol. Any vector a can be multiplied by a scalar k, i.e. ka.
... ... ...

## Special functions

Example Meaning and verbal equivalent Remarks
Jl(x) cylindrical Bessel functions (of the first kind) ...
... ... ...

## References and notes

1. ^ "ISO 80000-2:2009". International Organization for Standardization. Retrieved 1 July 2010.
2. ^ a b Thompson, Ambler; Taylor, Barry M (March 2008). Guide for the Use of the International System of Units (SI) — NIST Special Publication 811, 2008 Edition — Second Printing (PDF). Gaithersburg, MD, USA: NIST.
3. ^ These brace or fence characters are upper level unicode characters, fairly recently established and so may not display correctly in every browser. A close approximation of the appearance is found in the standard Latin characters: ( ), [ ], { }, < >. A more accurate glyph depiction of the mathematical angle bracket characters are found in the Chinese-Japanese-Korean (CJK) punctuation category: &#x3008h; &#x3009h;.
4. ^ If the perpendicular symbol, ⟂, does not display correctly, it is similar to ⊥ (up tack: sometimes meaning orthogonal to) and it also appears similar to ⏊ (the dentistry: symbol light up and horizontal)
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,

${\displaystyle x=\log _{2}n\quad \Longleftrightarrow \quad 2^{x}=n.}$

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.

Colon (punctuation)

The colon ( : ) is a punctuation mark consisting of two equally sized dots centered on the same vertical line. A colon precedes an explanation or an enumeration, or list. A colon is also used with ratios, titles and subtitles of books, city and publisher in bibliographies, biblical citations between chapter and verse, and for salutations in business letters and other formal letter writing, and often to separate hours and minutes.The use of a colon-like character as an alphabetic letter rather than as punctuation is covered at colon (letter).

Complement (set theory)

In set theory, the complement of a set A refers to elements not in A.

When all sets under consideration are considered to be subsets of a given set U, the absolute complement of A is the set of elements in U but not in A.

The relative complement of A with respect to a set B, also termed the difference of sets A and B, written B ∖ A, is the set of elements in B but not in A.

Countable set

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set. Whether finite or infinite, the elements of a countable set can always be counted one at a time and, although the counting may never finish, every element of the set is associated with a unique natural number.

Some authors use countable set to mean countably infinite alone. To avoid this ambiguity, the term at most countable may be used when finite sets are included and countably infinite, enumerable, or denumerable otherwise.

Georg Cantor introduced the term countable set, contrasting sets that are countable with those that are uncountable (i.e., nonenumerable or nondenumerable). Today, countable sets form the foundation of a branch of mathematics called discrete mathematics.

Del in cylindrical and spherical coordinates

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

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.

ISO 31-0

ISO 31-0 is the introductory part of international standard ISO 31 on quantities and units. It provides guidelines for using physical quantities, quantity and unit symbols, and coherent unit systems, especially the SI. It is intended for use in all fields of science and technology and is augmented by more specialized conventions defined in other parts of the ISO 31 standard. ISO 31-0 was withdrawn on 17 November 2009. It is superseded by ISO 80000-1. Other parts of ISO 31 have also been withdrawn and replaced by parts of ISO 80000.

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. 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.

Interval (mathematics)

In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers x satisfying 0 ≤ x ≤ 1 is an interval which contains 0 and 1, as well as all numbers between them. Other examples of intervals are the set of all real numbers ${\displaystyle \mathbb {R} }$, the set of all negative real numbers, and the empty set.

Real intervals play an important role in the theory of integration, because they are the simplest sets whose "size" or "measure" or "length" is easy to define. The concept of measure can then be extended to more complicated sets of real numbers, leading to the Borel measure and eventually to the Lebesgue measure.

Intervals are central to interval arithmetic, a general numerical computing technique that automatically provides guaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematical approximations, and arithmetic roundoff.

Intervals are likewise defined on an arbitrary totally ordered set, such as integers or rational numbers. The notation of integer intervals is considered in the special section below.

List of mathematical abbreviations

This article is a listing of abbreviated names of mathematical functions, function-like operators and other mathematical terminology.

This list is limited to abbreviations of two or more letters. The capitalization of some of these abbreviations is not standardized – different authors use different capitalizations.
List of mathematical symbols

This is a list of mathematical symbols used in all branches of mathematics to express a formula or to represent a constant.

A mathematical concept is independent of the symbol chosen to represent it. For many of the symbols below, the symbol is usually synonymous with the corresponding concept (ultimately an arbitrary choice made as a result of the cumulative history of mathematics), but in some situations, a different convention may be used. For example, depending on context, the triple bar "≡" may represent congruence or a definition. However, in mathematical logic, numerical equality is sometimes represented by "≡" instead of "=", with the latter representing equality of well-formed formulas. In short, convention dictates the meaning.

Each symbol is shown both in HTML, whose display depends on the browser's access to an appropriate font installed on the particular device, and typeset as an image using TeX.

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.

Logarithm

In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case, the logarithm counts repeated multiplication of the same factor; e.g., since 1000 = 10 × 10 × 10 = 103, the "logarithm to base 10" of 1000 is 3. The logarithm of x to base b is denoted as logb (x) (or, without parentheses, as logbx, or even without explicit base as log x, when no confusion is possible). More generally, exponentiation allows any positive real number to be raised to any real power, always producing a positive result, so the logarithm for any two positive real numbers b and x where b is not equal to 1, is always a unique real number y. More explicitly, the defining relation between exponentiation and logarithm is:

${\displaystyle \log _{b}(x)=y\quad }$ exactly if ${\displaystyle \quad b^{y}=x.}$

For example, log2 64 = 6, as 26 = 64.

The logarithm to base 10 (that is b = 10) is called the common logarithm and has many applications in science and engineering. The natural logarithm has the number e (that is b ≈ 2.718) as its base; its use is widespread in mathematics and physics, because of its simpler derivative. The binary logarithm uses base 2 (that is b = 2) and is commonly used in computer science.

Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations. They were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables. Tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition because of the fact—important in its own right—that the logarithm of a product is the sum of the logarithms of the factors:

${\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y,\,}$

provided that b, x and y are all positive and b ≠ 1. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century.

Logarithmic scales reduce wide-ranging quantities to tiny scopes. For example, the decibel (dB) is a unit used to express ratio as logarithms, mostly for signal power and amplitude (of which sound pressure is a common example). In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and of geometric objects called fractals. They help describing frequency ratios of musical intervals, appear in formulas counting prime numbers or approximating factorials, inform some models in psychophysics, and can aid in forensic accounting.

In the same way as the logarithm reverses exponentiation, the complex logarithm is the inverse function of the exponential function applied to complex numbers. The discrete logarithm is another variant; it has uses in public-key cryptography.

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.

Outline of mathematics

Mathematics is a field of study that investigates topics including number, space, structure, and change. For more on the relationship between mathematics and science, refer to the article on science.

Scientific notation

Scientific notation (also referred to as scientific form or standard index form, or standard form in the UK) is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. It is commonly used by scientists, mathematicians and engineers, in part because it can simplify certain arithmetic operations. On scientific calculators it is usually known as "SCI" display mode.

In scientific notation all numbers are written in the form

m × 10n(m times ten raised to the power of n), where the exponent n is an integer, and the coefficient m is any real number. The integer n is called the

order of magnitude and the real number m is called the significand or mantissa. However, the term "mantissa" may cause confusion because it is the name of the fractional part of the common logarithm. If the number is negative then a minus sign precedes m (as in ordinary decimal notation). In normalized notation, the exponent is chosen so that the absolute value of the coefficient is at least one but less than ten.

Decimal floating point is a computer arithmetic system closely related to scientific notation.

Spherical coordinate system

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

The radial distance is also called the radius or radial coordinate. The polar angle may be called colatitude, zenith angle, normal angle, or inclination angle.

The use of symbols and the order of the coordinates differs between sources. In one system frequently encountered in physics (r, θ, φ) gives the radial distance, polar angle, and azimuthal angle, whereas in another system used in many mathematics books (r, θ, φ) gives the radial distance, azimuthal angle, and polar angle. In both systems ρ is often used instead of r. Other conventions are also used, so great care needs to be taken to check which one is being used.

A number of different spherical coordinate systems following other conventions are used outside mathematics. In a geographical coordinate system positions are measured in latitude, longitude and height or altitude. There are a number of different celestial coordinate systems based on different fundamental planes and with different terms for the various coordinates. The spherical coordinate systems used in mathematics normally use radians rather than degrees and measure the azimuthal angle counterclockwise from the x-axis to the y-axis rather than clockwise from north (0°) to east (+90°) like the horizontal coordinate system. The polar angle is often replaced by the elevation angle measured from the reference plane. Elevation angle of zero is at the horizon.

The spherical coordinate system generalizes the two-dimensional polar coordinate system. It can also be extended to higher-dimensional spaces and is then referred to as a hyperspherical coordinate system.

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 ${\displaystyle \mathbf {v} }$ 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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