Math wiki


This is a graphical construction of the various trigonometric functions from a unit circle centered at the origin, O, and two points, A and D, on the circle separated by a central angle θ. The triangle AOC has side lengths cos θ (OC, the side adjacent to the angle θ) and sin θ (AC, the side opposite the angle), and a hypotenuse of length 1 (because the circle has unitradius). When the tangent line AE to the circle at point A is drawn to meet the extension of OD beyond the limits of the circle, the triangle formed, AOE, contains sides of length tan θ (AE) and sec θ (OE). When the tangent line is extended in the other direction to meet the line OF drawn perpendicular to OC, the triangle formed, AOF, has sides of length cot θ (AF) and csc θ (OF). In addition to these common trigonometric functions, the diagram also includes some functions that have fallen into disuse: the chord (AD), versine (CD), exsecant (DE), coversine (GH), and excosecant (FH). First used in the early Middle Ages by Indian and Islamic mathematicians to solve simple geometrical problems (e.g., solving triangles), the trigonometric functions today are used in sophisticated two- and three-dimensional computer modeling (especially when rotating modeled objects), as well as in the study of sound and other mechanical waves, light (electromagnetic waves), and electrical networks.

This extension is enabled by default on Fandom.

Math formulas can be displayed on articles using the tag.

For the technically inclined, MediaWiki (which is used by Fandom wikis) uses a subset of AMS-LaTeX markup, a superset of LaTeX markup which is in turn a superset of TeX markup, for mathematical formulas.

Step by step

  • Open source editing mode for an article.
  • Write your math code in the following format: .
  • For example, the equation '3 x 2 = 6' can be displayed using:
  • Which generates this:
    {\displaystyle {3~\times ~2~=~6}}
  • In-line formulas require attribute:
  • Which can be used to generate this:
    The volume of a sphere is {\textstyle V={\frac {4}{3}}\pi R^{3}}.
  • Regular text can be added to a math formula like so:
  • Which will render like this:
    {\textstyle \rho _{\text{Air}}=1.225~{\text{kg/m}}^{3}}
  • For details on how to write the math formulas themselves, see Help:Displaying a formula, on Wikipedia.


  • Large formulas may become wider than the maximum page width. Consider breaking them into multiple lines when possible.
    • Another solution is to place the formula inside a simple div with "overflow-x:scroll" set, so that a scrollbar show up. For example:
<div style="overflow-x:scroll;"> <math> ... </math> </div>
  • In-line formulas can have poor vertical alignment relative to the rest of the text. A work-around is described on Wikipedia.

See also

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Mathematics Jobs Wiki

Welcome to the Mathematics Jobs Wiki 2020-2021 research positions page. This page collects information about the academic mathematics job market: positions, short lists, offers, acceptances, etc. It lists positions at PhD-granting departments (including stat and applied math), and at departments that are research-oriented by other reasonable criteria. See the teaching positions page for more teaching-oriented academic math jobs.

To post news or corrections anonymously, please edit this wiki page yourself by clicking "edit" at an appropriate place. It is better to first register, but you can edit by IP number as well. You can contact the moderators through [email protected] to report evil postings, (e.g. inaccurate information about yourself or other applicants), and other issues. We are interested in information provided in good faith. Do not post wild hunches, and please respect the trust of your friends and colleagues.

Have you accepted a position? If so, it's a great courtesy to other job applicants to post that information here.

This site is currently supported by Matthias Koeppe, Greg Kuperberg, and Tao Mei. Tao reads the e-mail and he will keep the correspondence confidential as appropriate. Disclaimer: There is no guarantee that any information listed here is accurate; no warranty is expressed or implied. UC Davis and the UC Davis Math Department do not endorse this project and disavow any responsibility for the information on this wiki.

[edit] Key

Departments   (a) applied (b) biomath/biostat (c) computational
(o) optimization/OR (p) physics/QCQI (s) statistics
* MathJobs position
Positions (p) merely preferred (t) tenured position (u) tenure-track
(o) open rank (n) n positions
Apply by (∞) open indefinitely (*) created position
Namesitalic offer bold accepted withdrawn/declined
... and others search completed

"Apply by" can mean a strict deadline, or full consideration, or something else. Expired deadlines may be replaced by the last deadline listed. See the ads for details.

Note that the links should be directed to the job announcements rather than the department homepage.

[edit] United States

[edit] Long-term positions

[edit] A-K

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[edit] UA-UL

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[edit] Temporary positions

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[edit] Germany

Key:   (W1) assistant professor   (W2) associate professor   (W3) professor

A shortlist reported on this wiki is the list of candidates who were invited to give a talk ("Einladung zum Vorstellungsvortrag"); the order of the names has no significance. Official rankings of candidates ("Listenplätze"), if known, are indicated by a number in parentheses after a name, or they are implied by the order in which candidates who received offers ("Rufe") are listed.

Preferred sorting order of names:Offers and acceptances in chronological order, followed by all other invited candidates in alphabetical order.

If possible, include a hyperlink or HTML comment after a listed name, indicating the source of the information posted. For example, many university publications provide official information on offers and filled positions.

[edit] Long-term/temp W positions

Faculty searches in Germany do not follow the academic year schedule and sometimes take very long. We continue to collect information about these faculty positions (for jobs with deadlines before July 2019) in the Mathematics Jobs Wiki 2018-2019 or pages corresponding to earlier years (use the navigation sidebar or Main Page).

[edit] A-C

InstitutionAreasApply byShort lists/offers
RWTH Aachen applied math (W3) Jan 31 M. Bachmayr, C. Lehrenfeld, M. Melenk, O. Mula Herandez, J. Pietschmann, B. Stamm (3)
TU Berlin algebraic and geometric methods in data analysis (W1) Mar 18 C. Améndola Cerón (1), E. Duarte (2), A.-L. Sattelberger (3)
U Bochum numerics (W2) Oct 11 K. Kormann, R. Altmann (2), M. Jensen (3), M. Weimar (4), L. Einkemmer, B. Verfuerth, A. Rademacher

[edit] D-F

InstitutionAreasApply byShort lists/offers
KU Eichstätt-Ingolstadt Applied Math (W3) Jan 20 M. Friedrich, S. Grad, Y. Kolomoitsew, W. Düll, M. Kronbichler, M. Oliver, R. Zhang
KU Eichstätt-Ingolstadt Geomatik und Geomathematik (W3) Jan 20 S. Keller, M. Oliver, N. Ray, M. Scholze
KU Eichstätt-Ingolstadt reliable machine learning (W3) Jan 20 F. Voigtlaender, M. Golbabaee

[edit] G-L

InstitutionAreasApply byShort lists/offers
Uni Leipzig PDE (W2) Mar 19 H. Dietert (Paris), J. Sbierski (Oxford), S. Czimek (Brown), J. Lührmann (Texas), P. Bella (Dortmund), K. Widmayer (EPFL), C. Yu (Florida), C. Mooney (UC Irvine)
Uni Halle optimization (W2) Sturm, Kroener, Schmidt, Heiland, Voigt, Grundel, Mehlitz

[edit] M-P

InstitutionAreasApply byShort lists/offers
Uni Magdeburg Numerical Analysis (W2) B. Verfurth, J. Heiland, S. Grundel, R. Altmann, F. Bertrand, G. Visconti, K.Papafitsoros, C. Moosmüller

[edit] R-Z

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[edit] Other temporary positions

Please do not list doctoral positions.

InstitutionAreasTypeApply byShort lists/offers
University of Göttingen Analytic number theory, additive combinatorics, growth in groups Research Assistant (five-year) Oct 20, 2021 Short lists/offers

[edit] Rest of the World

[edit] Temporary positions

Please do not list doctoral positions.

Austria [1] University Assistant with doctorate Sept 29th, 2021
Austria [2] University Assistant with doctorate Sept 29th, 2021

[edit] Mathematics jobs sites

This section is for web sites that are specific to mathematics employment. Do not list sites with general employment assistance or solicitations.

  • — Our favorite!
  • Academic Jobs Europe Academic job board for European countries. Academic science and research jobs including Mathematics section in each country of the Europe.

[edit] Other disciplines (wikis and job rumor sites)


Areas of mathematics

Grouping by subject of mathematics

Mathematics encompasses a growing variety and depth of subjects over its history, and comprehension of it requires a system to categorize and organize these various subjects into a more general areas of mathematics. A number of different classification schemes have arisen, and though they share some similarities, there are differences due in part to the different purposes they serve.

A traditional division of mathematics is into pure mathematics; mathematics studied for its intrinsic interest, and applied mathematics; the mathematics that can be directly applied to real-world problems.[note 1] This division is not always clear and many subjects have been developed as pure mathematics to find unexpected applications later on. Broad divisions, such as discrete mathematics, computational mathematics and so on have emerged more recently.

An ideal system of classification permits adding new areas into the organization of previous knowledge, and fitting surprising discoveries and unexpected interactions into the outline. For example, the Langlands program has found unexpected connections between areas previously thought unconnected, at least Galois groups, Riemann surfaces and number theory.

Classification systems[edit]

  • Wikipedia uses a Category: Mathematics system on its articles, and also has a list of mathematics lists.
  • The Mathematics Subject Classification (MSC) is produced by the staff of the review databases Mathematical Reviews and Zentralblatt MATH. Many mathematics journals ask authors to label their papers with MSC subject codes. The MSC divides mathematics into over 60 areas, with further subdivisions within each area.
  • In the Library of Congress Classification, mathematics is assigned the many subclass QA within the class Q (Science). The LCC defines broad divisions, and individual subjects are assigned specific numerical values.
  • The Dewey Decimal Classification assigns mathematics to division 510, with subdivisions for Algebra & Number theory, Arithmetic, Topology, Analysis, Geometry, Numerical analysis, and Probabilities & Applied mathematics.
  • The Categories within Mathematics list is used by the arXiv for categorizing preprints. It differs from MSC; for example, it includes things like Quantum algebra.
  • The IMU uses its programme structure for organizing the lectures at its ICM every four years. One top-level section that MSC doesn't have is Lie theory.
  • The ACM Computing Classification System includes a couple of mathematical categories F. Theory of Computation and G. Mathematics of Computing.
  • MathOverflow has a tag system.
  • Mathematics book publishers such as Springer (subdisciplines), Cambridge (Browse Mathematics and statistics) and the AMS (subject area) use their own subject lists on their websites to enable customers to browse books or filter searches by subdiscipline, including topics such as mathematical biology and mathematical finance as top-level headings.
  • Schools and other educational bodies have syllabuses.
  • SIAM divides the areas of applied mathematics in activity groups.

Major divisions of mathematics[edit]

Pure mathematics[edit]


Mathematicians have always worked with logic and symbols, but for centuries the underlying laws of logic were taken for granted, and never expressed symbolically. Mathematical logic, also known as symbolic logic, was developed when people finally realized that the tools of mathematics can be used to study the structure of logic itself. Areas of research in this field have expanded rapidly, and are usually subdivided into several distinct subfields.

  • Proof theory and constructive mathematics : Proof theory grew out of David Hilbert's ambitious program to formalize all the proofs in mathematics. The most famous result in the field is encapsulated in Gödel's incompleteness theorems. A closely related and now quite popular concept is the idea of Turing machines. Constructivism is the outgrowth of Brouwer's unorthodox view of the nature of logic itself; constructively speaking, mathematicians cannot assert "Either a circle is round, or it is not" until they have actually exhibited a circle and measured its roundness.
  • Set theory : A set can be thought of as a collection of distinct things united by some common feature. Set theory is subdivided into three main areas. Naive set theory is the original set theory developed by mathematicians at the end of the 19th century. Axiomatic set theory is a rigorous axiomatic theory developed in response to the discovery of serious flaws (such as Russell's paradox) in naive set theory. It treats sets as "whatever satisfies the axioms", and the notion of collections of things serves only as motivation for the axioms. Internal set theory is an axiomatic extension of set theory that supports a logically consistent identification of illimited (enormously large) and infinitesimal (unimaginably small) elements within the real numbers. See also List of set theory topics.
History and biography[edit]

The history of mathematics is inextricably intertwined with the subject itself. This is perfectly natural: mathematics has an internal organic structure, deriving new theorems from those that have come before. As each new generation of mathematicians builds upon the achievements of their ancestors, the subject itself expands and grows new layers, like an onion.

Recreational mathematics[edit]

From magic squares to the Mandelbrot set, numbers have been a source of amusement and delight for millions of people throughout the ages. Many important branches of "serious" mathematics have their roots in what was once a mere puzzle and/or game.

Number theory[edit]

Number theory is the study of numbers and the properties of operations between them. Number theory is traditionally concerned with the properties of integers, but more recently, it has come to be concerned with wider classes of problems that have arisen naturally from the study of integers.

  • Arithmetic : An elementary part of number theory that primarily focuses upon the study of natural numbers, integers, fractions, and decimals, as well as the properties of the traditional operations on them: addition, subtraction, multiplication and division. Up until the 19th century, arithmetic and number theory were synonyms, but the evolution and growth of the field has resulted in arithmetic referring only to the elementary branch of number theory.
  • Elementary number theory: The study of integers at a higher level than arithmetic, where the term 'elementary' here refers to the fact that no techniques from other mathematical fields are used.


The study of structure begins with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra. The deeper properties of these numbers are studied in number theory. The investigation of methods to solve equations leads to the field of abstract algebra, which, among other things, studies rings and fields, structures that generalize the properties possessed by everyday numbers. Long standing questions about compass and straightedge constructions were finally settled by Galois theory. The physically important concept of vectors, generalized to vector spaces, is studied in linear algebra. Themes common to all kinds of algebraic structures are studied in universal algebra.

  • General algebraic systems : Given a set, different ways of combining or relating members of that set can be defined. If these obey certain rules, then a particular algebraic structure is formed. Universal algebra is the more formal study of these structures and systems.
  • Field theory and polynomials : Field theory studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined. A polynomial is an expression in which constants and variables are combined using only addition, subtraction, and multiplication.
  • Commutative rings and algebras : In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. This means that if a and b are any elements of the ring, then a×b = b×a. Commutative algebra is the field of study of commutative rings and their ideals, modules and algebras. It is foundational both for algebraic geometry and for algebraic number theory. The most prominent examples of commutative rings are rings of polynomials.


Combinatorics is the study of finite or discrete collections of objects that satisfy specified criteria. In particular, it is concerned with "counting" the objects in those collections (enumerative combinatorics) and with deciding whether certain "optimal" objects exist (extremal combinatorics). It includes graph theory, used to describe interconnected objects (a graph in this sense is a network, or collection of connected points). See also the list of combinatorics topics, list of graph theory topics and glossary of graph theory. A combinatorial flavour is present in many parts of problem solving.


Geometry deals with spatial relationships, using fundamental qualities or axioms. Such axioms can be used in conjunction with mathematical definitions for points, straight lines, curves, surfaces, and solids to draw logical conclusions. See also List of geometry topics.


Deals with the properties of a figure that do not change when the figure is continuously deformed. The main areas are point set topology (or general topology), algebraic topology, and the topology of manifolds, defined below.


Within mathematics, analysis is the branch that focuses on rates of change (derivatives), integrals, and multiple things changing relative to (or independently of) one another.

Modern analysis is a vast and rapidly expanding branch of mathematics that touches almost every other subdivision of the discipline, finding direct and indirect applications in topics as diverse as number theory, cryptography, and abstract algebra. It is also the language of science itself and is used across chemistry, biology, and physics, from astrophysics to X-ray crystallography.

Applied mathematics[edit]

Probability and statistics[edit]

See also: Glossary of probability and statistics

Computational sciences[edit]

  • Computer algebra: This area is also called symbolic computation or algebraic computation. It deals with exact computation, for example with integers of arbitrary size, polynomials or elements of finite fields. It includes also the computation with non numeric mathematical objects like polynomial ideals or series.

Mathematical physics[edit]

Further information: Continuum mechanics, Elasticity (physics), and Plasticity (physics)

Further information: Energy principles in structural mechanics, Flexibility method, Direct stiffness method, and Finite element method

Further information: Solid state physics, Materials science, Mechanics of materials, Mechanics of solids, Fracture mechanics, Deformation (mechanics), and Deformable bodies

Further information: Fluid dynamics, Mechanics of fluids, Rheology, Electrodynamics, Plasma Physics, Gas dynamics, and Aerodynamics

  • Classical Mechanics: Addresses and describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.
  • Mechanics of structures: Mechanics of structures is a field of study within applied mechanics that investigates the behavior of structures under mechanical loads, such as bending of a beam, buckling of a column, torsion of a shaft, deflection of a thin shell, and vibration of a bridge.
  • Particle mechanics: In mathematics, a particle is a point-like, perfectly rigid, solid object. Particle mechanics deals with the results of subjecting particles to forces. It includes celestial mechanics—the study of the motion of celestial objects.

Other applied mathematics[edit]

  • Mathematical programming: Mathematical programming (or mathematical optimization) minimizes (or maximizes) a real-valued function over a domain that is often specified by constraints on the variables. Mathematical programming studies these problems and develops iterative methods and algorithms for their solution.

See also[edit]


External links[edit]


Wiki math


Field of study

This article is about the field of study. For other uses, see Mathematics (disambiguation) and Math (disambiguation).

Mathematics (from Greek: μάθημα, máthēma, 'knowledge, study, learning') includes the study of such topics as quantity (number theory),[1]structure (algebra),[2]space (geometry),[1] and change (analysis).[3][4][5] It has no generally accepted definition.[6][7]

Mathematicians seek and use patterns[8][9] to formulate new conjectures; they resolve the truth or falsity of such by mathematical proof. When mathematical structures are good models of real phenomena, mathematical reasoning can be used to provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements.[10] Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorousdeduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.

Mathematics is essential in many fields, including natural science, engineering, medicine, finance, and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics (mathematics for its own sake) without having any application in mind, but practical applications for what began as pure mathematics are often discovered later.[13]


Main article: History of mathematics

The history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, which is shared by many animals,[14] was probably that of numbers: the realization that a collection of two apples and a collection of two oranges (for example) have something in common, namely the quantity of their members.

As evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have also recognized how to count abstract quantities, like time—days, seasons, or years.[15][16]

The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.

Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC.[18] Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.[19] It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication, and division) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system [19] which is still in use today for measuring angles and time.

Beginning in the 6th century BC with the Pythagoreans, with Greek mathematics the Ancient Greeks began a systematic study of mathematics as a subject in its own right.[21] Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, Elements, is widely considered the most successful and influential textbook of all time.[22] The greatest mathematician of antiquity is often held to be Archimedes (c. 287–212 BC) of Syracuse. He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are conic sections (Apollonius of Perga, 3rd century BC),trigonometry (Hipparchus of Nicaea, 2nd century BC),[26] and the beginnings of algebra (Diophantus, 3rd century AD).[27]

The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.

The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics.[28] Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine,[28] and an early form of infinite series.

A page from al-Khwārizmī's Algebra

During the Golden Age of Islam, especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra. Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system.[29][30] Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī.

During the early modern period, mathematics began to develop at an accelerating pace in Western Europe. The development of calculus by Isaac Newton and Gottfried Leibniz in the 17th century revolutionized mathematics.[31]Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries.[32] Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss,[33] who made numerous contributions to fields such as algebra, analysis, differential geometry, matrix theory, number theory, and statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved.[34]

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs."


The word mathematics comes from Ancient Greekmáthēma (μάθημα), meaning "that which is learnt,"[36] "what one gets to know," hence also "study" and "science". The word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times.[37] Its adjective is mathēmatikós (μαθηματικός), meaning "related to learning" or "studious," which likewise further came to mean "mathematical." In particular, mathēmatikḗ tékhnē (μαθηματικὴ τέχνη; Latin: ars mathematica) meant "the mathematical art."

Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense.[38]

In Latin, and in English until around 1700, the term mathematics more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This has resulted in several mistranslations. For example, Saint Augustine's warning that Christians should beware of mathematici, meaning astrologers, is sometimes mistranslated as a condemnation of mathematicians.[39]

The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivativela mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural ta mathēmatiká (τὰ μαθηματικά), used by Aristotle (384–322 BC), and meaning roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from Greek.[40] In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math.[41]

Definitions of mathematics

Main article: Definitions of mathematics

Mathematics has no generally accepted definition.[6][7]Aristotle defined mathematics as "the science of quantity" and this definition prevailed until the 18th century. However, Aristotle also noted a focus on quantity alone may not distinguish mathematics from sciences like physics; in his view, abstraction and studying quantity as a property "separable in thought" from real instances set mathematics apart.[42]

In the 19th century, when the study of mathematics increased in rigor and began to address abstract topics such as group theory and projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions.[43]

A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable.[6] There is not even consensus on whether mathematics is an art or a science.[7] Some just say, "Mathematics is what mathematicians do."[6]

Three leading types

Three leading types of definition of mathematics today are called logicist, intuitionist, and formalist, each reflecting a different philosophical school of thought.[44] All have severe flaws, none has widespread acceptance, and no reconciliation seems possible.[44]

Logicist definitions

An early definition of mathematics in terms of logic was that of Benjamin Peirce (1870): "the science that draws necessary conclusions."[45] In the Principia Mathematica, Bertrand Russell and Alfred North Whitehead advanced the philosophical program known as logicism, and attempted to prove that all mathematical concepts, statements, and principles can be defined and proved entirely in terms of symbolic logic. A logicist definition of mathematics is Russell's (1903) "All Mathematics is Symbolic Logic."[46]

Intuitionist definitions

Intuitionist definitions, developing from the philosophy of mathematician L. E. J. Brouwer, identify mathematics with certain mental phenomena. An example of an intuitionist definition is "Mathematics is the mental activity which consists in carrying out constructs one after the other."[44] A peculiarity of intuitionism is that it rejects some mathematical ideas considered valid according to other definitions. In particular, while other philosophies of mathematics allow objects that can be proved to exist even though they cannot be constructed, intuitionism allows only mathematical objects that one can actually construct. Intuitionists also reject the law of excluded middle (i.e., {\displaystyle P\vee \neg P}). While this stance does force them to reject one common version of proof by contradiction as a viable proof method, namely the inference of P from {\displaystyle \neg P\to \bot }, they are still able to infer \neg P from P\to \bot . For them, {\displaystyle \neg (\neg P)} is a strictly weaker statement than P.[47]

Formalist definitions

Formalist definitions identify mathematics with its symbols and the rules for operating on them. Haskell Curry defined mathematics simply as "the science of formal systems".[48] A formal system is a set of symbols, or tokens, and some rules on how the tokens are to be combined into formulas. In formal systems, the word axiom has a special meaning different from the ordinary meaning of "a self-evident truth", and is used to refer to a combination of tokens that is included in a given formal system without needing to be derived using the rules of the system.

Mathematics as science

The German mathematician Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences". More recently, Marcus du Sautoy has called mathematics "the Queen of Science ... the main driving force behind scientific discovery".[50] The philosopher Karl Popper observed that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently." Popper also noted that "I shall certainly admit a system as empirical or scientific only if it is capable of being tested by experience."[52]

Several authors consider that mathematics is not a science because it does not rely on empirical evidence.[53][54][55][56]

Mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences. Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics.

The opinions of mathematicians on this matter are varied. Many mathematicians[57] feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts; others feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics.[58] One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created (as in art) or discovered (as in science). In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the philosophy of mathematics.[59]

Inspiration, pure and applied mathematics, and aesthetics

Main article: Mathematical beauty

Mathematics arises from many different kinds of problems. At first these were found in commerce, land measurement, architecture and later astronomy; today, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. For example, the physicistRichard Feynman invented the path integral formulation of quantum mechanics using a combination of mathematical reasoning and physical insight, and today's string theory, a still-developing scientific theory which attempts to unify the four fundamental forces of nature, continues to inspire new mathematics.[60]

Some mathematics is relevant only in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. A distinction is often made between pure mathematics and applied mathematics. However pure mathematics topics often turn out to have applications, e.g. number theory in cryptography.

This remarkable fact, that even the "purest" mathematics often turns out to have practical applications, is what the physicist Eugene Wigner has named "the unreasonable effectiveness of mathematics".[13] The philosopher of mathematicsMark Steiner has written extensively on this matter and acknowledges that the applicability of mathematics constitutes “a challenge to naturalism.”[61] For the philosopher of mathematics Mary Leng, the fact that the physical world acts in accordance with the dictates of non-causal mathematical entities existing beyond the universe is "a happy coincidence".[62] On the other hand, for some anti-realists, connections, which are acquired among mathematical things, just mirror the connections acquiring among objects in the universe, so that there is no "happy coincidence".[62]

As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: there are now hundreds of specialized areas in mathematics and the latest Mathematics Subject Classification runs to 46 pages.[63] Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science.

For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty in a simple and elegant proof, such as Euclid's proof that there are infinitely many prime numbers, and in an elegant numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in A Mathematician's Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic.[64] Mathematical research often seeks critical features of a mathematical object. A theorem expressed as a characterization of the object by these features is the prize. Examples of particularly succinct and revelatory mathematical arguments have been published in Proofs from THE BOOK.

The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions. And at the other social extreme, philosophers continue to find problems in philosophy of mathematics, such as the nature of mathematical proof.[65]

Notation, language, and rigor

Main article: Mathematical notation

Leonhard Eulercreated and popularized much of the mathematical notation used today.

Most of the mathematical notation in use today was not invented until the 16th century.[66] Before that, mathematics was written out in words, limiting mathematical discovery.Euler (1707–1783) was responsible for many of the notations in use today. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. According to Barbara Oakley, this can be attributed to the fact that mathematical ideas are both more abstract and more encrypted than those of natural language.[68] Unlike natural language, where people can often equate a word (such as cow) with the physical object it corresponds to, mathematical symbols are abstract, lacking any physical analog.[69] Mathematical symbols are also more highly encrypted than regular words, meaning a single symbol can encode a number of different operations or ideas.[70]

Mathematical language can be difficult to understand for beginners because even common terms, such as or and only, have a more precise meaning than they have in everyday speech, and other terms such as open and field refer to specific mathematical ideas, not covered by their laymen's meanings. Mathematical language also includes many technical terms such as homeomorphism and integrable that have no meaning outside of mathematics. Additionally, shorthand phrases such as iff for "if and only if" belong to mathematical jargon. There is a reason for special notation and technical vocabulary: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".

Mathematical proof is fundamentally a matter of rigor. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "theorems", based on fallible intuitions, of which many instances have occurred in the history of the subject.[b] The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics. Today, mathematicians continue to argue among themselves about computer-assisted proofs. Since large computations are hard to verify, such proofs may be erroneous if the used computer program is erroneous.[c][71] On the other hand, proof assistants allow verifying all details that cannot be given in a hand-written proof, and provide certainty of the correctness of long proofs such as that of the Feit–Thompson theorem.[d]

Axioms in traditional thought were "self-evident truths", but that conception is problematic.[72] At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable formulas; and so a final axiomatization of mathematics is impossible. Nonetheless, mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.[73]

Fields of mathematics

See also: Areas of mathematics and Glossary of areas of mathematics

The abacusis a simple calculating tool used since ancient times.

Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry, and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty. While some areas might seem unrelated, the Langlands program has found connections between areas previously thought unconnected, such as Galois groups, Riemann surfaces and number theory.

Discrete mathematics conventionally groups together the fields of mathematics which study mathematical structures that are fundamentally discrete rather than continuous.

Foundations and philosophy

In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic includes the mathematical study of logic and the applications of formal logic to other areas of mathematics; set theory is the branch of mathematics that studies sets or collections of objects. The phrase "crisis of foundations" describes the search for a rigorous foundation for mathematics that took place from approximately 1900 to 1930.[74] Some disagreement about the foundations of mathematics continues to the present day. The crisis of foundations was stimulated by a number of controversies at the time, including the controversy over Cantor's set theory and the Brouwer–Hilbert controversy.

Mathematical logic is concerned with setting mathematics within a rigorous axiomatic framework, and studying the implications of such a framework. As such, it is home to Gödel's incompleteness theorems which (informally) imply that any effective formal system that contains basic arithmetic, if sound (meaning that all theorems that can be proved are true), is necessarily incomplete (meaning that there are true theorems which cannot be proved in that system). Whatever finite collection of number-theoretical axioms is taken as a foundation, Gödel showed how to construct a formal statement that is a true number-theoretical fact, but which does not follow from those axioms. Therefore, no formal system is a complete axiomatization of full number theory. Modern logic is divided into recursion theory, model theory, and proof theory, and is closely linked to theoretical computer science,[75] as well as to category theory. In the context of recursion theory, the impossibility of a full axiomatization of number theory can also be formally demonstrated as a consequence of the MRDP theorem.

Theoretical computer science includes computability theory, computational complexity theory, and information theory. Computability theory examines the limitations of various theoretical models of the computer, including the most well-known model—the Turing machine. Complexity theory is the study of tractability by computer; some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with the rapid advancement of computer hardware. A famous problem is the "P = NP?" problem, one of the Millennium Prize Problems.[76] Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence deals with concepts such as compression and entropy.

Pure mathematics

Main article: Pure mathematics

Number systems and number theory

Main articles: Arithmetic, Number system, and Number theory

The study of quantity starts with numbers, first the familiar natural numbers\mathbb {N} and integers\mathbb {Z} ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory, from which come such popular results as Fermat's Last Theorem. The twin prime conjecture and Goldbach's conjecture are two unsolved problems in number theory.

As the number system is further developed, the integers are recognized as a subset of the rational numbers\mathbb {Q} ("fractions"). These, in turn, are contained within the real numbers, \mathbb {R} which are used to represent limits of sequences of rational numbers and continuous quantities. Real numbers are generalized to the complex numbers\mathbb {C} . According to the fundamental theorem of algebra, all polynomial equations in one unknown with complex coefficients have a solution in the complex numbers, regardless of degree of the polynomial. {\displaystyle \mathbb {N} ,\ \mathbb {Z} ,\ \mathbb {Q} ,\ \mathbb {R} } and \mathbb {C} are the first steps of a hierarchy of numbers that goes on to include quaternions and octonions. Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of "infinity". Another area of study is the size of sets, which is described with the cardinal numbers. These include the aleph numbers, which allow meaningful comparison of the size of infinitely large sets.

{\displaystyle (0),1,2,3,\ldots }{\displaystyle \ldots ,-2,-1,0,1,2\,\ldots }{\displaystyle -2,{\frac {2}{3}},1.21}{\displaystyle -e,{\sqrt {2}},3,\pi }{\displaystyle 2,i,-2+3i,2e^{i{\frac {4\pi }{3}}}}{\displaystyle \aleph _{0},\aleph _{1},\aleph _{2},\ldots ,\aleph _{\alpha },\ldots .\ }
Natural numbersIntegersRational numbersReal numbersComplex numbersInfinite cardinals


Main article: Algebra

Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations or relations that are defined on the set. Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance number theory studies properties of the set of integers that can be expressed in terms of arithmetic operations. Moreover, it frequently happens that different such structured sets (or structures) exhibit similar properties, which makes it possible, by a further step of abstraction, to state axioms for a class of structures, and then study at once the whole class of structures satisfying these axioms. Thus one can study groups, rings, fields and other abstract systems; together such studies (for structures defined by algebraic operations) constitute the domain of abstract algebra.

By its great generality, abstract algebra can often be applied to seemingly unrelated problems; for instance a number of ancient problems concerning compass and straightedge constructions were finally solved using Galois theory, which involves field theory and group theory. Another example of an algebraic theory is linear algebra, which is the general study of vector spaces, whose elements called vectors have both quantity and direction, and can be used to model (relations between) points in space. This is one example of the phenomenon that the originally unrelated areas of geometry and algebra have very strong interactions in modern mathematics. Combinatorics studies ways of enumerating the number of objects that fit a given structure.

{\begin{matrix}(1,2,3)&(1,3,2)\\(2,1,3)&(2,3,1)\\(3,1,2)&(3,2,1)\end{matrix}}Elliptic curve simple.svgRubik's cube.svgGroup diagdram D6.svgLattice of the divisibility of 60.svgBraid-modular-group-cover.svg
CombinatoricsNumber theoryGroup theoryGraph theoryOrder theoryAlgebra


Main articles: Space (mathematics) and Geometry

The study of space originates with geometry—in particular, Euclidean geometry, which combines space and numbers, and encompasses the well-known Pythagorean theorem. Trigonometry is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Convex and discrete geometry were developed to solve problems in number theory and functional analysis but now are pursued with an eye on applications in optimization and computer science. Within differential geometry are the concepts of fiber bundles and calculus on manifolds, in particular, vector and tensor calculus. Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may have been the greatest growth area in 20th-century mathematics; it includes point-set topology, set-theoretic topology, algebraic topology and differential topology. In particular, instances of modern-day topology are metrizability theory, axiomatic set theory, homotopy theory, and Morse theory. Topology also includes the now solved Poincaré conjecture, and the still unsolved areas of the Hodge conjecture. Other results in geometry and topology, including the four color theorem and Kepler conjecture, have been proven only with the help of computers.


Main article: Calculus

Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a tool to investigate it. Functions arise here as a central concept describing a changing quantity. The rigorous study of real numbers and functions of a real variable is known as real analysis, with complex analysis the equivalent field for the complex numbers. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.

Applied mathematics

Main article: Applied mathematics

Applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge. The term applied mathematics also describes the professional specialty in which mathematicians work on practical problems; as a profession focused on practical problems, applied mathematics focuses on the "formulation, study, and use of mathematical models" in science, engineering, and other areas of mathematical practice.

In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Thus, the activity of applied mathematics is vitally connected with research in pure mathematics.

Statistics and other decision sciences

Main article: Statistics

Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with probability theory. Statisticians (working as part of a research project) "create data that makes sense" with random sampling and with randomized experiments;[77] the design of a statistical sample or experiment specifies the analysis of the data (before the data becomes available). When reconsidering data from experiments and samples or when analyzing data from observational studies, statisticians "make sense of the data" using the art of modelling and the theory of inference—with model selection and estimation; the estimated models and consequential predictions should be tested on new data.[e]

Statistical theory studies decision problems such as minimizing the risk (expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best. In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost, under specific constraints: For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence.[78] Because of its use of optimization, the mathematical theory of statistics shares concerns with other decision sciences, such as operations research, control theory, and mathematical economics.[79]

Computational mathematics

Computational mathematics proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis includes the study of approximation and discretisation broadly with special concern for rounding errors. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmicmatrix and graph theory. Other areas of computational mathematics include computer algebra and symbolic computation.

Mathematical awards

Arguably the most prestigious award in mathematics is the Fields Medal, established in 1936 and awarded every four years (except around World War II) to as many as four individuals. The Fields Medal is often considered a mathematical equivalent to the Nobel Prize.

The Wolf Prize in Mathematics, instituted in 1978, recognizes lifetime achievement, and another major international award, the Abel Prize, was instituted in 2003. The Chern Medal was introduced in 2010 to recognize lifetime achievement. These accolades are awarded in recognition of a particular body of work, which may be innovational, or provide a solution to an outstanding problem in an established field.

A famous list of 23 open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. This list achieved great celebrity among mathematicians, and at least nine of the problems have now been solved. A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Only one of them, the Riemann hypothesis, duplicates one of Hilbert's problems. A solution to any of these problems carries a 1 million dollar reward. Currently, only one of these problems, the Poincaré conjecture, has been solved.

See also


  1. ^No likeness or description of Euclid's physical appearance made during his lifetime survived antiquity. Therefore, Euclid's depiction in works of art depends on the artist's imagination (see Euclid).
  2. ^See false proof for simple examples of what can go wrong in a formal proof.
  3. ^For considering as reliable a large computation occurring in a proof, one generally requires two computations using independent software
  4. ^The book containing the complete proof has more than 1,000 pages.
  5. ^Like other mathematical sciences such as physics and computer science, statistics is an autonomous discipline rather than a branch of applied mathematics. Like research physicists and computer scientists, research statisticians are mathematical scientists. Many statisticians have a degree in mathematics, and some statisticians are also mathematicians.


  1. ^ ab"mathematics, n.". Oxford English Dictionary. Oxford University Press. 2012. Archived from the original on November 16, 2019. Retrieved June 16, 2012.
  2. ^Kneebone, G.T. (1963). Mathematical Logic and the Foundations of Mathematics: An Introductory Survey. Dover. p. 4. ISBN .
  3. ^LaTorre, Donald R.; Kenelly, John W.; Biggers, Sherry S.; Carpenter, Laurel R.; Reed, Iris B.; Harris, Cynthia R. (2011). Calculus Concepts: An Informal Approach to the Mathematics of Change. Cengage Learning. p. 2. ISBN .
  4. ^Ramana (2007). Applied Mathematics. Tata McGraw–Hill Education. p. 2.10. ISBN .
  5. ^Ziegler, Günter M. (2011). "What Is Mathematics?". An Invitation to Mathematics: From Competitions to Research. Springer. p. vii. ISBN .
  6. ^ abcdMura, Roberta (December 1993). "Images of Mathematics Held by University Teachers of Mathematical Sciences". Educational Studies in Mathematics. 25 (4): 375–85. doi:10.1007/BF01273907. JSTOR 3482762. S2CID 122351146.
  7. ^ abcTobies, Renate & Helmut Neunzert (2012). Iris Runge: A Life at the Crossroads of Mathematics, Science, and Industry. Springer. p. 9. ISBN .
  8. ^Steen, L.A. (April 29, 1988). The Science of PatternsScience, 240: 611–16. And summarized at Association for Supervision and Curriculum DevelopmentArchived October 28, 2010, at the Wayback Machine,
  9. ^Devlin, Keith, Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe (Scientific American Paperback Library) 1996, ISBN 978-0-7167-5047-5
  10. ^Wise, David. "Eudoxus' Influence on Euclid's Elements with a close look at The Method of Exhaustion". Archived from the original on June 1, 2019. Retrieved October 26, 2019.
  11. ^ abWigner, Eugene (1960). "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". Communications on Pure and Applied Mathematics. 13 (1): 1–14. Bibcode:1960CPAM...13....1W. doi:10.1002/cpa.3160130102. Archived from the original on February 28, 2011.
  12. ^Dehaene, Stanislas; Dehaene-Lambertz, Ghislaine; Cohen, Laurent (August 1998). "Abstract representations of numbers in the animal and human brain". Trends in Neurosciences. 21 (8): 355–61. doi:10.1016/S0166-2236(98)01263-6. PMID 9720604. S2CID 17414557.
  13. ^See, for example, Raymond L. Wilder, Evolution of Mathematical Concepts; an Elementary Study, passim
  14. ^Zaslavsky, Claudia. (1999). Africa Counts : Number and Pattern in African Culture. Chicago Review Press. ISBN . OCLC 843204342. Archived from the original on March 31, 2021. Retrieved May 29, 2020.
  15. ^"Egyptian Mathematics". The Story of Mathematics. Archived from the original on September 16, 2018. Retrieved October 27, 2019.
  16. ^ ab"Sumerian/Babylonian Mathematics". The Story of Mathematics. Archived from the original on September 7, 2019. Retrieved October 27, 2019.
  17. ^Heath, Thomas Little (1981) [1921]. A History of Greek Mathematics: From Thales to Euclid. New York: Dover Publications. p. 1. ISBN .
  18. ^Boyer 1991, "Euclid of Alexandria" p. 119.
  19. ^Boyer 1991, "Greek Trigonometry and Mensuration" p. 162.
  20. ^Boyer 1991, "Revival and Decline of Greek Mathematics" p. 180.
  21. ^ ab"Indian Mathematics – The Story of Mathematics". Archived from the original on April 13, 2019. Retrieved October 27, 2019.
  22. ^"Islamic Mathematics – The Story of Mathematics". Archived from the original on October 17, 2019. Retrieved October 27, 2019.
  23. ^Saliba, George. (1994). A history of Arabic astronomy : planetary theories during the golden age of Islam. New York University Press. ISBN . OCLC 28723059. Archived from the original on March 31, 2021. Retrieved May 29, 2020.
  24. ^"17th Century Mathematics – The Story of Mathematics". Archived from the original on September 16, 2018. Retrieved October 27, 2019.
  25. ^"Euler – 18th Century Mathematics – The Story of Mathematics". Archived from the original on May 2, 2019. Retrieved October 27, 2019.
  26. ^"Gauss – 19th Century Mathematics – The Story of Mathematics". Archived from the original on July 25, 2019. Retrieved October 27, 2019.
  27. ^"20th Century Mathematics – Gödel". The Story of Mathematics. Archived from the original on September 16, 2018. Retrieved October 27, 2019.
  28. ^"mathematic (n.)". Online Etymology Dictionary. Archived from the original on March 7, 2013.
  29. ^Both meanings can be found in Plato, the narrower in Republic510cArchived February 24, 2021, at the Wayback Machine, but Plato did not use a math- word; Aristotle did, commenting on it. μαθηματική. Liddell, Henry George; Scott, Robert; A Greek–English Lexicon at the Perseus Project. OED Online, "Mathematics".
  30. ^"Pythagoras – Greek Mathematics – The Story of Mathematics". Archived from the original on September 17, 2018. Retrieved October 27, 2019.
  31. ^Boas, Ralph (1995) [1991]. "What Augustine Didn't Say About Mathematicians". Lion Hunting and Other Mathematical Pursuits: A Collection of Mathematics, Verse, and Stories by the Late Ralph P. Boas, Jr. Cambridge University Press. p. 257. ISBN . Archived from the original on May 20, 2020. Retrieved January 17, 2018.
  32. ^The Oxford Dictionary of English Etymology, Oxford English Dictionary, sub "mathematics", "mathematic", "mathematics"
  33. ^"maths, n." and "math, n.3"Archived April 4, 2020, at the Wayback Machine. Oxford English Dictionary, on-line version (2012).
  34. ^Franklin, James (July 8, 2009). Philosophy of Mathematics. pp. 104–106. ISBN . Archived from the original on September 6, 2015. Retrieved July 1, 2020.
  35. ^Cajori, Florian (1893). A History of Mathematics. American Mathematical Society (1991 reprint). pp. 285–86. ISBN .
  36. ^ abcSnapper, Ernst (September 1979). "The Three Crises in Mathematics: Logicism, Intuitionism, and Formalism". Mathematics Magazine. 52 (4): 207–16. doi:10.2307/2689412. JSTOR 2689412.
  37. ^Peirce, Benjamin (1882). Linear Associative Algebra. Van Nostrand. p. 1.
  38. ^Russell, Bertrand (1903). The Principles of Mathematics


One of the greatest motivating forces for Donald Knuth when he began developing the original TeX system was to create something that allowed simple construction of mathematical formulae, while looking professional when printed. The fact that he succeeded was most probably why TeX (and later on, LaTeX) became so popular within the scientific community. Typesetting mathematics is one of LaTeX's greatest strengths. It is also a large topic due to the existence of so much mathematical notation.

If your document requires only a few simple mathematical formulas, plain LaTeX has most of the tools that you will ever need. If you are writing a scientific document that contains numerous complex formulas, the amsmath package[1] introduces several new commands that are more powerful and flexible than the ones provided by basic LaTeX. The mathtools package fixes some amsmath quirks and adds some useful settings, symbols, and environments to amsmath.[2] To use either package, include:


in the preamble of the document. The mathtools package loads the amsmath package and hence there is no need to in the preamble if mathtools is used.

Mathematics environments[edit | edit source]

LaTeX needs to know when the text is mathematical. This is because LaTeX typesets math notation differently from normal text. Therefore, special environments have been declared for this purpose. They can be distinguished into two categories depending on how they are presented:

  • text — text formulas are displayed inline, that is, within the body of text where it is declared, for example, I can say that {\displaystyle a+a=2a} within this sentence.
  • displayed — displayed formulas are on a line by themselves.

As math requires special environments, there are naturally the appropriate environment names you can use in the standard way. Unlike most other environments, however, there are some handy shorthands for declaring your formulas. The following table summarizes them:

Type Inline (within text) formulas Displayed equations Displayed and automatically numbered equations
LaTeX shorthand
TeX shorthand
Comment (starred version) suppresses numbering, but requires amsmath

Suggestion: Using the should be avoided, as it may cause problems, particularly with the AMS-LaTeX macros. Furthermore, should a problem occur, the error messages may not be helpful.

The and environments are functionally equivalent.

If you are typing text normally, you are said to be in text mode, but while you are typing within one of those mathematical environments, you are said to be in math mode, that has some differences compared to the text mode:

  1. Most spaces and line breaks do not have any significance, as all spaces are either derived logically from the mathematical expressions or have to be specified with special commands such as
  2. Empty lines are not allowed. Only one paragraph per formula.
  3. Each letter is considered to be the name of a variable and will be typeset as such. If you want to typeset normal text within a formula (normal upright font with normal spacing), then you have to enter the text using dedicated commands.

[edit | edit source]

In order for some operators, such as or , to be displayed correctly inside some math environments (read ), it might be convenient to write the class inside the environment. Doing so might cause the line to be taller, but will cause exponents and indices to be displayed correctly for some math operators. For example, the will print a smaller Σ and will print a bigger one \displaystyle \sum , like in equations (This only works with AMSMATH package). It is possible to force this behaviour for all math environments by declaring at the very beginning (i.e. before ).

Symbols[edit | edit source]

Mathematics has many symbols! The following is a set of symbols that can be accessed directly from the keyboard:

+ - = ! / ( ) [ ] < > | ' : *

Beyond those listed above, distinct commands must be issued in order to display the desired symbols. There are many examples such as Greek letters, set and relations symbols, arrows, binary operators, etc.

For example:

\forall x \in X, \quad\exists y \leq\epsilon

\forall x\in X,\quad \exists y\leq \epsilon \,

Fortunately, there's a tool that can greatly simplify the search for the command for a specific symbol. Look for "Detexify" in the external links section below. Another option would be to look in "The Comprehensive LaTeX Symbol List" in the external links section below.

Greek letters[edit | edit source]

Greek letters are commonly used in mathematics, and they are very easy to type in math mode. You just have to type the name of the letter after a backslash: if the first letter is lowercase, you will get a lowercase Greek letter, if the first letter is uppercase (and only the first letter), then you will get an uppercase letter. Note that some uppercase Greek letters look like Latin ones, so they are not provided by LaTeX (e.g. uppercase Alpha and Beta are just "A" and "B", respectively). Lowercase epsilon, theta, kappa, phi, pi, rho, and sigma are provided in two different versions. The alternate, or variant, version is created by adding "var" before the name of the letter:

\alpha, \Alpha, \beta, \Beta, \gamma, \Gamma, \pi, \Pi, \phi, \varphi, \mu, \Phi

{\displaystyle \alpha ,\mathrm {A} ,\beta ,\mathrm {B} ,\gamma ,\Gamma ,\pi ,\Pi ,\phi ,\varphi ,\mu ,\Phi }

Scroll down to #List of mathematical symbols for a complete list of Greek symbols.

Operators[edit | edit source]

An operator is a function that is written as a word: e.g. trigonometric functions (sin, cos, tan), logarithms and exponentials (log, exp), limits (lim), as well as trace and determinant (tr, det). LaTeX has many of these defined as commands:

\cos (2\theta) = \cos^2 \theta - \sin^2 \theta

\cos(2\theta )=\cos ^{2}\theta -\sin ^{2}\theta \,

For certain operators such as limits, the subscript is placed underneath the operator:

\lim\limits_{x \to\infty}\exp(-x) = 0

\lim _{x\to \infty }\exp(-x)=0

For the modular operator there are two commands: and :

a\,{\bmod {\,}}b\,

x\equiv a{\pmod {b}}\,

To use operators that are not pre-defined, such as argmax, see custom operators

Powers and indices[edit | edit source]

Powers and indices are equivalent to superscripts and subscripts in normal text mode. The caret (; also known as the circumflex accent) character is used to raise something, and the underscore () is for lowering. If an expression containing more than one character is raised or lowered, it should be grouped using curly braces ( and ).

k_{n+1} = n^2 + k_n^2 - k_{n-1}


For powers with more than one digit, surround the power with {}.


An underscore () can be used with a vertical bar (|) to denote evaluation using subscript notation in mathematics:

f(n) = n^5 + 4n^2 + 2 |_{n=17}


Fractions and Binomials[edit | edit source]

A fraction is created using the command (for those who need their memories refreshed, that's the top and bottom respectively!). Likewise, the binomial coefficient (a.k.a, the Choose function) may be written using the command[3]:

\frac{n!}{k!(n-k)!} = \binom{n}{k}

{\frac {n!}{k!(n-k)!}}={\binom {n}{k}}

You can embed fractions within fractions:


{\frac {{\frac {1}{x}}+{\frac {1}{y}}}{y-z}}

Note that when appearing inside another fraction, or in inline text {\tfrac {a}{b}}, a fraction is noticeably smaller than in displayed mathematics. The and commands[3] force the use of the respective styles, and . Similarly, the and commands typeset the binomial coefficient.

For relatively simple fractions, especially within the text, it may be more aesthetically pleasing to use powers and indices instead:


If this looks a little "loose" (i.e., overspaced), a tightened version can be defined by inserting some negative space

%running fraction with slash - requires math mode.\newcommand*\rfrac[2]{{}^{#1}\!/_{#2}}\rfrac{3}{7}


If you use them throughout the document, usage of xfrac package is recommended. This package provides command to create slanted fractions. Usage:

Take $\sfrac{1}{2}$ cup of sugar, \dots 3\times\sfrac{1}{2}=1\sfrac{1}{2} Take ${}^1/_2$ cup of sugar, \dots 3\times{}^1/_2=1{}^1/_2


If fractions are used as an exponent, curly braces have to be used around the command:

$x^\frac{1}{2}$ % no error $x^\sfrac{1}{2}$ % error $x^{\sfrac{1}{2}}$ % no error
$x^\frac{1}{2}$% no error

x^{\frac {1}{2}}

In some cases, using the package alone will result in errors about certain font shapes not being available. In that case, the lmodern and fix-cm packages need to be added as well.

Alternatively, the nicefrac package provides the command, whose usage is similar to .

Continued fractions[edit | edit source]

Continued fractions should be written using command[3]:

\begin{equation} x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{a_4}}}}\end{equation}

x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\cfrac {1}{a_{4}}}}}}}}}

Multiplication of two numbers[edit | edit source]

To make multiplication visually similar to a fraction, a nested array can be used. For example, multiplication of numbers written one below the other can be typeset as follows:

\begin{equation}\frac{\begin{array}[b]{r}\left( x_1 x_2 \right)\\\times\left( x'_1 x'_2 \right) \end{array}}{\left( y_1y_2y_3y_4 \right) }\end{equation}

{\frac {\begin{array}{r}\left(x_{1}x_{2}\right)\\\times \left(x'_{1}x'_{2}\right)\end{array}}{\left(y_{1}y_{2}y_{3}y_{4}\right)}}

Roots[edit | edit source]

The command creates a square root surrounding an expression. It accepts an optional argument specified in square brackets ( and ) to change magnitude:

{\sqrt {\frac {a}{b}}}


{\displaystyle {\sqrt[{n}]{1+x+x^{2}+x^{3}+\dots +x^{n}}}}

Some people prefer writing the square root "closing" it over its content. This method arguably makes it more clear what is in the scope of the root sign. This habit is not normally used while writing with the computer, but if you still want to change the output of the square root, LaTeX gives you this possibility. Just add the following code in the preamble of your document:

% New definition of square root:% it renames \sqrt as \oldsqrt\let\oldsqrt\sqrt% it defines the new \sqrt in terms of the old one\def\sqrt{\mathpalette\DHLhksqrt}\def\DHLhksqrt#1#2{%\setbox0=\hbox{$#1\oldsqrt{#2\,}$}\dimen0=\ht0 \advance\dimen0-0.2\ht0 \setbox2=\hbox{\vrule height\ht0 depth -\dimen0}%{\box0\lower0.4pt\box2}}
The new style is on left, the old one on right

This TeX code first renames the command as , then redefines in terms of the old one, adding something more. The new square root can be seen in the picture on the left, compared to the old one on the right. Unfortunately this code won't work if you want to use multiple roots: if you try to write {\sqrt[{b}]{a}} as after you used the code above, you'll just get a wrong output. In other words, you can redefine the square root this way only if you are not going to use multiple roots in the whole document.

An alternative piece of TeX code that does allow multiple roots is

\usepackage{letltxmacro}\makeatletter\let\[email protected]@t\[email protected]@t \def\[email protected]@t#1#2{%\setbox0=\hbox{$\[email protected]@t#1{#2\,}$}\dimen0=\ht0 \advance\dimen0-0.2\ht0 \setbox2=\hbox{\vrule height\ht0 depth -\dimen0}%{\box0\lower0.4pt\box2}}\LetLtxMacro{\oldsqrt}{\sqrt}\renewcommand*{\sqrt}[2][\ ]{\oldsqrt[#1]{#2}}\makeatother$\sqrt[a]{b} \quad\oldsqrt[a]{b}$

LaTeX example sqrt.png

However, this requires the package.

Sums and integrals[edit | edit source]

The and commands insert the sum and integral symbols respectively, with limits specified using the caret () and underscore (). The typical notation for sums is:

\textstyle \sum _{i=1}^{10}t_{i}\,


\displaystyle\sum_{i=1}^{10} t_i

\displaystyle \sum _{i=1}^{10}t_{i}\,

The limits for the integrals follow the same notation. It's also important to represent the integration variables with an upright d, which in math mode is obtained through the \mathrm{} command, and with a small space separating it from the integrand, which is attained with the \, command.


\int _{0}^{\infty }\mathrm {e} ^{-x}\,\mathrm {d} x\,

There are many other "big" commands which operate in a similar manner:

For more integral symbols, including those not included by default in the Computer Modern font, try the esint package.

The command[3] allows the use of to write the limits over multiple lines:

\sum_{\substack{ 0<i<m \\ 0<j<n }} P(i,j)

\sum _{\overset {\scriptstyle 0<i<m}{\scriptstyle 0<j<n}}P(i,j)\,

If you want the limits of an integral to be specified above and below the symbol (like the sum), use the command:

\int \limits _{a}^{b}\,

However, if you want this to apply to all integrals, it is preferable to specify the intlimits option when loading the amsmath package:


Subscripts and superscripts in other contexts, as well as other parameters to amsmath package related to them, are described in Advanced Mathematics chapter.

For bigger integrals, you may use personal declarations, or the bigints package [4].

Brackets, braces and delimiters[edit | edit source]

How to use braces in multi line equations is described in the Advanced Mathematics chapter.

The use of delimiters such as brackets soon becomes important when dealing with anything but the most trivial equations. Without them, formulas can become ambiguous. Also, special types of mathematical structures, such as matrices, typically rely on delimiters to enclose them.

There are a variety of delimiters available for use in LaTeX:

( a ), [ b ], \{ c \}, | d |, \| e \|, \langle f \rangle, \lfloor g \rfloor, \lceil h \rceil, \ulcorner i \urcorner, / j \backslash

{\displaystyle (a),[b],\{c\},|d|,\|e\|,\langle f\rangle ,\lfloor g\rfloor ,\lceil h\rceil ,\ulcorner i\urcorner ,/j\backslash }

where and may be used in place of [ and ].

Automatic sizing[edit | edit source]

Very often, mathematical features will differ in size, in which case the delimiters surrounding the expression should vary accordingly. This can be done automatically using the , , and commands. Any of the previous delimiters may be used in combination with these:


\left({\frac {x^{2}}{y^{3}}}\right)\,


Latex example middle.png

Curly braces are defined differently by using and ,


\left\{{\frac {x^{2}}{y^{3}}}\right\}\,

If a delimiter on only one side of an expression is required, then an invisible delimiter on the other side may be denoted using a period ().


\left.{\frac {x^{3}}{3}}\right|_{0}^{1}\,

Manual sizing[edit | edit source]

In certain cases, the sizing produced by the and commands may not be desirable, or you may simply want finer control over the delimiter sizes. In this case, the , , and modifier commands may be used:

( \big( \Big( \bigg( \Bigg(

({\big (}{\Big (}{\bigg (}{\Bigg (}\,

These commands are primarily useful when dealing with nested delimiters. For example, when typesetting

\frac{\mathrm d}{\mathrm d x}\left( k g(x) \right)

{\frac {\mathrm {d} }{\mathrm {d} x}}\left(kg(x)\right)

we notice that the and commands produce the same size delimiters as those nested within it. This can be difficult to read. To fix this, we write

\frac{\mathrm d}{\mathrm d x}\big( k g(x) \big)

{\frac {\mathrm {d} }{\mathrm {d} x}}{\big (}kg(x){\big )}

Manual sizing can also be useful when an equation is too large, trails off the end of the page, and must be separated into two lines using an align command. Although the commands and can be used to balance the delimiters on each line, this may lead to wrong delimiter sizes. Furthermore, manual sizing can be used to avoid overly large delimiters — if an or a similar command appears between the delimiters.

Matrices and arrays[edit | edit source]

A basic matrix may be created using the matrix environment[3]: in common with other table-like structures, entries are specified by row, with columns separated using an ampersand () and new rows separated with a double backslash ()

\[\begin{matrix} a & b & c \\ d & e & f \\ g & h & i\end{matrix}\]


To specify alignment of columns in the table, use starred version[5]:

\begin{matrix} -1 & 3 \\ 2 & -4 \end{matrix} = \begin{matrix*}[r] -1 & 3 \\ 2 & -4 \end{matrix*}


The alignment by default is c, but it can be any column type valid in array environment.

However matrices are usually enclosed in delimiters of some kind, and while it is possible to use the and commands, there are various other predefined environments which automatically include delimiters:

Environment name Surrounding delimiter Notes
pmatrix[3](\,)centers columns by default
pmatrix*[5](\,)allows to specify alignment of columns in optional parameter
bmatrix[3][\,]centers columns by default
bmatrix*[5][\,]allows to specify alignment of columns in optional parameter
Bmatrix[3]\{\,\}centers columns by default
Bmatrix*[5]\{\,\}allows to specify alignment of columns in optional parameter
vmatrix[3]|\,|centers columns by default
vmatrix*[5]|\,|allows to specify alignment of columns in optional parameter
Vmatrix[3]\|\,\|centers columns by default
Vmatrix*[5]\|\,\|allows to specify alignment of columns in optional parameter

When writing down arbitrary sized matrices, it is common to use horizontal, vertical and diagonal triplets of dots (known as ellipses) to fill in certain columns and rows. These can be specified using the , and respectively:

A_{m,n} = \begin{pmatrix} a_{1,1}& a_{1,2}&\cdots& a_{1,n}\\ a_{2,1}& a_{2,2}&\cdots& a_{2,n}\\\vdots&\vdots&\ddots&\vdots\\ a_{m,1}& a_{m,2}&\cdots& a_{m,n}\end{pmatrix}

A_{m,n}={\begin{pmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,n}\\a_{2,1}&a_{2,2}&\cdots &a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\a_{m,1}&a_{m,2}&\cdots &a_{m,n}\end{pmatrix}}

In some cases, you may want to have finer control of the alignment within each column, or to insert lines between columns or rows. This can be achieved using the array environment, which is essentially a math-mode version of the environment, which requires that the columns be pre-specified:

\begin{array}{c|c} 1 & 2 \\\hline 3 & 4 \end{array}

{\begin{array}{c|c}1&2\\\hline 3&4\end{array}}

You may see that the AMS matrix class of environments doesn't leave enough space when used together with fractions resulting in output similar to this:

M={\begin{bmatrix}{\frac {5}{6}}&{\frac {1}{6}}&0\\{\frac {5}{6}}&0&{\frac {1}{6}}\\0&{\frac {5}{6}}&{\frac {1}{6}}\end{bmatrix}}

To counteract this problem, add additional leading space with the optional parameter to the command:

M = \begin{bmatrix}\frac{5}{6}&\frac{1}{6}& 0 \\[0.3em]\frac{5}{6}& 0 &\frac{1}{6}\\[0.3em] 0 &\frac{5}{6}&\frac{1}{6}\end{bmatrix}

M={\begin{bmatrix}{\frac {5}{6}}&{\frac {1}{6}}&0\\[0.3em]{\frac {5}{6}}&0&{\frac {1}{6}}\\[0.3em]0&{\frac {5}{6}}&{\frac {1}{6}}\end{bmatrix}}

If you need "border" or "indexes" on your matrix, plain TeX provides the macro

M = \bordermatrix{~ & x & y \cr A & 1 & 0 \cr B & 0 & 1 \cr}


Matrices in running text[edit | edit source]

To insert a small matrix without increasing leading in the line containing it, use smallmatrix environment:

A matrix in text must be set smaller: $\bigl(\begin{smallmatrix}a&b \\ c&d\end{smallmatrix} \bigr)$ to not increase leading in a portion of text.


Adding text to equations[edit | edit source]

The math environment differs from the text environment in the representation of text. Here is an example of trying to represent text within the math environment:

50 apples \times 100 apples = lots of apples^2

50apples\times 100apples=lotsofapples^{2}\,

There are two noticeable problems: there are no spaces between words or numbers, and the letters are italicized and more spaced out than normal. Both issues are simply artifacts of the maths mode, in that it treats it as a mathematical expression: spaces are ignored (LaTeX spaces mathematics according to its own rules), and each character is a separate element (so are not positioned as closely as normal text).

There are a number of ways that text can be added properly. The typical way is to wrap the text with the command [3] (a similar command is , though this causes problems with subscripts, and has a less descriptive name). Let's see what happens when the above equation code is adapted:

50 \text{apples}\times 100 \text{apples} = \text{lots of apples}^2

50{\text{apples}}\times 100{\text{apples}}={\text{lots of apples}}^{2}\,

The text looks better. However, there are no gaps between the numbers and the words. Unfortunately, you are required to explicitly add these. There are many ways to add spaces between math elements, but for the sake of simplicity we may simply insert space characters into the commands.

50 \text{ apples}\times 100 \text{ apples} = \text{lots of apples}^2

50{\text{ apples}}\times 100{\text{ apples}}={\text{lots of apples}}^{2}\,

Formatted text[edit | edit source]

Using the is fine and gets the basic result. Yet, there is an alternative that offers a little more flexibility. You may recall the introduction of font formatting commands, such as , , , etc. These commands format the argument accordingly, e.g., gives bold text. These commands are equally valid within a maths environment to include text. The added benefit here is that you can have better control over the font formatting, rather than the standard text achieved with .

50 \textrm{ apples}\times 100 \textbf{ apples} = \textit{lots of apples}^2

{\displaystyle 50\;{\textrm {apples}}\times 100\;{\textbf {apples}}={\textit {lotsofapples}}^{2}\,}

Formatting mathematics symbols[edit | edit source]

See also: w:Mathematical Alphanumeric Symbols, w:Help:Displaying a formula#Alphabets and typefaces and w:Wikipedia:LaTeX symbols#Fonts

We can now format text; what about formatting mathematical expressions? There are a set of formatting commands very similar to the font formatting ones just used, except that they are specifically aimed at text in math mode (requires amsfonts)


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