Shortcut: WD:MATH

# Wikidata:WikiProject Mathematics

## Goals

### Long term goals

• Organize data about mathematics

### Short term goals

• Use the property has facet polytope (P1312) on all applicable items (7/??), and create missing items
• Gather information from infoboxes
• Create more properties that semantically describe items
• Create featured items that show how properties should be used
• Create "has part" property entries for formula components / identifiers -> discussion

## Useful lists and queries

Error detection: please make sure the query below returns zero results, and fix any item that appears.

## Properties

See also: Wikidata:List of properties/mathematics (bot-generated list)

### Properties used in Wikipedia

The following properties are available in Wikipedia, Wikiversity, and other Wikimedia wikis and are displayed when clicking on the annotated formulae. See w:Help:Displaying_a_formula#Semantics_and_links for a more detailed documentation. The assumption when implementing the semantics feature in mw:Extension Math was that relations or formulae like the Mass-Energy-Equivalence are annotated on Wikidata.

Title ID Data type Description Examples Inverse
defining formulaP2534Mathematical expressionformula: mathematical formula representing a theorem or law. Maximum length: 400 charactersideal gas law <defining formula> $PV=nRT$ -
has part(s)P527Itemhas part, consist of and meronymy: part of this subject; inverse property of "part of" (P361). See also "has parts of the class" (P2670).ideal gas law <has part(s)> pressurepart of
quantity symbol (string)P416Stringquantity symbol: symbol for a mathematical or physical quantityelectric charge <quantity symbol (string)> Q and q-

### General Properties

As opposed to the properties above that are used for semantic annotations in Wikipedia the following properties are used under the assumption that the corresponding wikidata item defines a mathematical/physical concept which has a definiendum that can be derived (calculated from) other symbols.

Title ID Data type Description Examples Inverse
defining formulaP2534Mathematical expressionformula: mathematical formula representing a theorem or law. Maximum length: 400 charactersPythagorean theorem <defining formula> $a^{2}+b^{2}=c^{2}$ -
in defining formulaP7235Mathematical expressionvariable, identifier and pronumeral: any symbol used in the defining formula (P2534)f-number <in defining formula> $N$ -
quantity symbol (LaTeX)P7973Mathematical expressionquantity symbol: symbol for a mathematical or physical quantity in LaTeXelectric charge <quantity symbol (LaTeX)> $Q$ -
calculated fromP4934Itemcalculation: value of the subject item cannot be measured directly or can be calculated from the following measurands or attributesbody mass index <calculated from> human body weight and human height-
relative toP2210Itemreference point: qualifier: what a statement value is relative toFTCS scheme <has part(s)> Euler method
<relative to> time
-
studied byP2579Itemacademic discipline: subject is studied by this science or domainnumerical method <studied by> numerical analysisstudies
has qualityP1552Itemquality: the entity has an inherent or distinguishing non-material characteristicmathematical proof <has quality> validity-
notationP913Itemnotation: mathematical notation or another symbolequality <notation> equals sign-
statement describesP2384Itemformalization of the statement contains a bound variable in this classPythagorean theorem <statement describes> right triangle-
admissible rule inP2577Itemadmissible rule: this logic inference rule is admissible in that logical systemmodus ponens <admissible rule in> classical logic-
generalization ofP7719Itemlogical implication: the subject statement is a particular case of this object statement, or the truth of this object statement is implied by the truth of the subject statementlaw of cosines <generalization of> Pythagorean theorem-

### Numbers

Title ID Data type Description Examples Inverse
numeric valueP1181Quantitynumeric value and quantity value: numerical value of a number, a mathematical constant, or a physical constantgolden ratio <numeric value> 1,6180339887±0,0000000001-
prime factorP5236Itemprime factor: one of the prime numbers that can be multiplied to give this number2334 <prime factor> 2, 389 and 3-
greater thanP5135Itemgreater than: instances of the item have a greater value than corresponding instances of the object, for the given measure100 <greater than> 99less than
less thanP5136Itemless than: instances of the item have a lesser value than corresponding instances of the object, for the given measure−1 <less than> zerogreater than
number of decimal digitsP7316Quantitynumerical digit: number of decimal digits of a natural number10^90 <number of decimal digits> 91-

### Mathematicans and Mathematical History

Title ID Data type Description Examples Inverse
discoverer or inventorP61Iteminventor, innovator, discoverer, discovery and invention: subject who discovered, first described, invented, or developed this discovery or inventionPoincaré conjecture <discoverer or inventor> Henri Poincaré-
time of discovery or inventionP575Point in timeinvention, discovery and archaeological find: date or point in time when the item was discovered or inventedEuler method <time of discovery or invention> 1768-
named afterP138Itemeponym, memorial society and namesakes: entity or event that inspired the subject's name, or namesake (in at least one language). Qualifier "applies to name" (P5168) can be used to indicate which oneEuler method <named after> Leonhard Euler-
proved byP1318Itemperson who proved somethingPoincaré conjecture <proved by> Grigori Perelman-
solved byP1136Itemperson that solved a scientific questionFermat's Last Theorem <solved by> Andrew Wiles-
based onP144Itembased on: the work(s) used as the basis for subject itemPythagorean trigonometric identity <based on> Pythagorean theoremderivative work
Erdős numberP2021QuantityErdős number: the "collaborative distance" between mathematician Paul Erdős and another person. Use point in time (P585) as qualifier and should be used with a source.Paul Erdős <Erdős number> 0-
title in LaTeXP6835Mathematical expressionLaTeX: (qualifier) for title property (P1476), to write title in LaTeX/math notation, if the string can't render it normally. Enclose normal text with "\text{" and "}"On the Integers of the Form xk +yk <title in LaTeX> ${\text{On the Integers of the Form }}x^{k}+y^{k}$ -

### Sets, Algebras, and Topology

Title ID Data type Description Examples Inverse
cardinality of this setP2820Itemcardinality: measure of number of elements of a setrational number <cardinality of this set> aleph null-
group cardinalityP1164Quantitygroup order: number of elements in a finite groupdihedral group of order 6 <group cardinality> 6-
identity elementP8864Itemidentity element: value of the identity element of the mathematical operationaddition <identity element> zero-
has operatorP8866Itemoperator: mathematical operator associated with this algebraic structureadditive group <has operator> addition-
mathematical inverseP8865Itemmathematical inverse: inverse element with respect to binary operation given as a qualifier2 <mathematical inverse> ½mathematical inverse
containsP4330Iteminsertion: item or substance located within this item but not part of itsphere <contains> open ball-
has part(s) of the classP2670Itemhas part of class: the subject instance (the subject is not a class) has one or more parts of the object classset of real numbers <has part(s) of the class> real number-
graph radiusP7391Quantitygraph radius: the minimum eccentricity of any vertex of a graphclaw graph <graph radius> 1-

To model whether a set is open or closed, use

For a class item that is a union of other items, use

### Knots and Links

Title ID Data type Description Examples Inverse
crossing numberP11035Quantitycrossing number: crossing number for mathematical knots; the crossing number of a knot is the smallest number of crossings of any diagram of the knot. It is a knot invariant.unknot <crossing number> 0-
Alexander polynomialP5350Mathematical expressionAlexander polynomial: invariant of a knot or link. Use 't' as variable and list monomials in decreasing order.unknot <Alexander polynomial> $1$ -
Conway polynomialP5351Mathematical expressionConway–Alexander polynomial: invariant of a knot. Use z as variable and list monomials in decreasing order.unknot <Conway polynomial> $1$ -
Jones polynomialP5352Mathematical expressionJones polynomial: invariant of a knot or link. Use q as variable and list monomials in decreasing order.unknot <Jones polynomial> $1$ -
Alexander–Briggs notationP6432Mathematical expressionAlexander-Briggs notation: common notation of abstract knots and linksunknot <Alexander–Briggs notation> $0_{1}$ -
Dowker-Thistlethwaite notationP8378Mathematical expressionDowker–Thistlethwaite notation: descriptive property of mathematical knots, also known as Dowker notationtrefoil knot <Dowker-Thistlethwaite notation> $4,6,2$ -
Dowker-Thistlethwaite nameP8416Mathematical expressionDowker–Thistlethwaite name: unambiguous systematic identifier scheme for mathematical knotsunknot <Dowker-Thistlethwaite name> $0_{1}$ -
Gauss notationP11034Mathematical expressionGauss notation: Gauss notation for knotsunknot <Gauss notation> not applicable-

### Methods and Algorithms

Title ID Data type Description Examples Inverse
computes solution toP2159Itemproblem that this algorithm or method solvesDPLL algorithm <computes solution to> boolean satisfiability problem-
solution toP9030Itema mathematical object that satisfies the criteria for a mathematical problemeigenvalue <solution to> characteristic equation-
usesP2283Itemuse: item or concept used by the subject or in the operation (see also instrument [P1303] and armament [P520])method of lines <uses> discretizationused by
approximation algorithmP1171Itemapproximation algorithm: method used to approximate a numberpi <approximation algorithm> Gauss–Legendre algorithm-
computational complexityP10374Itemcomputational complexity: the most specific complexity class this computational problem has been proved to belong to; if available, provide the tight complexity, otherwise both hardness and membership can be specifiedst-connectivity <computational complexity> NL-complete-
best-case time complexityP3753Mathematical expressiontime complexity: time complexity of an algorithm at leastquicksort <best-case time complexity> $O(n\log n)$ -
average time complexityP3754Mathematical expressiontime complexity: time complexity of an algorithm on averagequicksort <average time complexity> $O(n\log n)$ -
worst-case time complexityP3752Mathematical expressiontime complexity: time complexity of an algorithm at mostquicksort <worst-case time complexity> $O(n^{2})$ -
best-case space complexityP3756Mathematical expressionspace complexity: space complexity of an algorithm at leastmerge sort <best-case space complexity> $O(1)$ -
average space complexityP3757Mathematical expressionspace complexity: space complexity of an algorithm on averagequicksort <average space complexity> $O(\log n)$ -
worst-case space complexityP3755Mathematical expressionspace complexity: space complexity of an algorithm at mostquicksort <worst-case space complexity> $O(n)$ -
Butcher tableauP8558Mathematical expressionButcher tableau: table containing the coefficients of a Runge-Kutta methodbackward Euler method <Butcher tableau> ${\begin{array}{l|l}1&1\\\hline &1\end{array}}$ -

The following are also useful for iterative numerical methods

### Functions

Title ID Data type Description Examples Inverse
definition domainP1568Itemdomain of a function: set of "input" or argument values for which a mathematical function is definedsquare function <definition domain> real number-
codomainP1571Itemcodomain: codomain of a functionsquare function <codomain> real number-
image of functionP2396Itemimage: set of values that a mathematical function actually takessquare function <image of function> set of non-negative real numbers-
input setP1851Iteminput set: a superset of the domain of a function or relation that may include some inputs for which the function is not defined; to specify the set of only those inputs for which the function is defined use domain (P1568)tangent <input set> set of real numbers-
support of a functionP10731Mathematical expressionsupport: subset of the domain containing those elements which are not mapped to zeronormal distribution <support of a function> $\displaystyle \mathbb {R}$ -

Several properties of functions are modeled in Wikdata using the has quality (P1552) and does not have quality (P6477) properties.

To specify symmetry, use

### Geometry

Title ID Data type Description Examples Inverse
has facet polytopeP1312Itemfacet: facet of a polytope, in the next-lower dimensiondodecahedron <has facet polytope> pentagon-
has vertex figureP1678Itemvertex figure: the figure exposed when a corner of a polytope is sliced offcuboctahedron <has vertex figure> rectangle-
baseP3263Itembase and base face: configuration of a polytop vertices around the symmetry axispentagonal prism <base> pentagon-
Euler characteristicP6438QuantityEuler characteristic: topological invariant of a space; the alternating sum of the dimensions of the (co)homology of a spacesphere <Euler characteristic> 2-
dual toP1322Itemduality, dual polytope and dual graph: dual of a polytope, graph or curvecube <dual to> octahedron-
Schläfli symbolP3228StringSchläfli symbol: notation that defines regular polytopes and tessellationsregular polygon <Schläfli symbol> {n}-

### Probability

invalid 'id' parameter should be a number or 'new', none provided

Title ID Data type Description Examples Inverse
probability mass functionP10732Mathematical expressionprobability mass function: function that gives the probability that a discrete random variable is exactly equal to some valuePoisson distribution <probability mass function> ${\frac {\lambda ^{k}e^{-\lambda }}{k!}}$ -
cumulative distribution functionP10736Mathematical expressiondistribution function: probability that X will take a value less than or equal to xnormal distribution <cumulative distribution function> ${\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]$ -
quantile functionP10737Mathematical expressionquantile function: value at which the probability of the random variable is less than or equal to the given probabilitynormal distribution <quantile function> $\mu +\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(2F-1)$ -
mean of a probability distributionP10738Mathematical expressionmean: long-run average value of repetitions of the experiment it representsnormal distribution <mean of a probability distribution> $\mu$ -
median of a probability distributionP10739Mathematical expressionmedian: value separating the higher half of a data sample, a population, or a probability distribution, from the lower halfnormal distribution <median of a probability distribution> $\mu$ -
mode of a probability distributionP10740Mathematical expressionmode: value that appears most often in a set of datanormal distribution <mode of a probability distribution> $\mu$ -
variance of a probability distributionP10743Mathematical expressionvariance: expectation of the squared deviation of a random variable from its meannormal distribution <variance of a probability distribution> $\sigma ^{2}$ -
skewnessP10744Mathematical expressioncoefficient of skewness: measure of the asymmetry of the probability distribution of a real-valued random variable about its meannormal distribution <skewness> $0$ -
excess kurtosisP10745Mathematical expressionexcess kurtosis: measure of the "tailedness" of the probability distribution of a real-valued random variablenormal distribution <excess kurtosis> $0$ -
information entropyP10746Mathematical expressioninformation entropy: expected value (average) of the information contained in each messagenormal distribution <information entropy> ${\tfrac {1}{2}}\ln(2\sigma ^{2}\pi \,e\,)$ -
moment-generating functionP10747Mathematical expressionmoment-generating function: alternative specification of its probability distributionnormal distribution <moment-generating function> $\exp\{\mu t+{\frac {1}{2}}\sigma ^{2}t^{2}\}$ -
characteristic functionP10735Mathematical expressioncharacteristic function: Fourier transform of the probability density functionnormal distribution <characteristic function> $\mathrm {e} ^{\mathrm {i} \mu t-{\frac {\sigma ^{2}t^{2}}{2}}}$ -
Fisher informationP10734Mathematical expressionFisher information: way of measuring the amount of information that an observable random variable X carries about an unknown parameter θ of a distribution that models Xnormal distribution <Fisher information> ${\begin{pmatrix}1/\sigma ^{2}&0\\0&1/(2\sigma ^{4})\end{pmatrix}}$ -
probability generating functionP10733Mathematical expressionprobability-generating function: power series representation (the generating function) of the probability mass function of the random variablePoisson distribution <probability generating function> $\exp(\lambda (z-1))$ -
support of a functionP10731Mathematical expressionsupport: subset of the domain containing those elements which are not mapped to zeronormal distribution <support of a function> $\displaystyle \mathbb {R}$ -

### External Identifiers

Title ID Data type Description Examples Inverse
OEIS IDP829External identifierOn-Line Encyclopedia of Integer Sequences: identifer on the On-Line Encyclopedia of Integer SequencesFermat number <OEIS ID> A000215-
Mathematics Genealogy Project IDP549External identifierMathematics Genealogy Project: identifier for mathematicians and computer scientists at the Mathematics Genealogy ProjectPatrick Ion <Mathematics Genealogy Project ID> 8691-
zbMATH author IDP1556External identifierzbMATH Open: identifier of a person in the Zentralblatt MATH databasePatrick Ion <zbMATH author ID> ion.patrick-d-f-
zbMATH work IDP894External identifierzbMATH Open: identifier in the zbMath databaseGame Theory: A Multi-Leveled Approach <zbMATH work ID> 1147.91001-
MathWorld identifierP2812External identifierMathWorld: identifier for entries in MathWorld, online mathematics reference workaleph null <MathWorld identifier> Aleph-0-
ProofWiki IDP6781External identifierProofWiki: identifier for a ProofWiki articleEuclid's theorem <ProofWiki ID> Euclid's Theorem-

## Polytopes

Root item Query Query description Count date Count quantity Count link
polytope (Q747980) tree[] subclasses 2014-10-07 813 autolist2
vertex (Q26401) web polytopes with property "has facet polytope" (direct, not inherited) 2014-10-07 534 autolist2

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## Related WikiProjects

### Wikidata

• Shared symmetry properties (crystallography):