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{{Short description|Origin and evolution of the symbols used to write equations and formulas}}{{Use dmy dates|date=June 2021}}
{{for|the book by Florian Cajori|A History of Mathematical Notations}}
The '''history of mathematical notation'''<ref>[[Florian Cajori]]. A History of Mathematical Notations: Two Volumes in One. Cosimo, Inc., 1 Dec 2011</ref> covers the introduction, development, and [[cultural diffusion]] of [[mathematical symbol]]s and the conflicts between notational methods that arise during a notation's move to popularity or obsolescence. [[Mathematical notation]]<ref>A Dictionary of Science, Literature, & Art, Volume 2. Edited by [[William Thomas Brande]], [[George William Cox]]. Pg [https://books.google.com/books?id=J5tPAAAAMAAJ&pg=PA683 683]</ref> comprises the symbols used to write mathematical [[equation]]s and [[formula]]s. Notation generally implies a set of [[well-defined]] representations of quantities and symbols operators.<ref>{{cite web|url=http://mathworld.wolfram.com/Notation.html |title=Notation – from Wolfram MathWorld |publisher=Mathworld.wolfram.com |access-date=2014-06-24}}</ref> The history includes [[Hindu–Arabic numerals]], letters from the [[Roman alphabet|Roman]], [[Greek alphabet|Greek]], [[Hebrew alphabet|Hebrew]], and [[German alphabet|German]] [[alphabet]]s, and a variety of symbols invented by mathematicians over the past several centuries.

The historical development of mathematical notation can be divided into three stages:<ref>Diophantos of Alexandria: A Study in the History of Greek Algebra. By Sir Thomas Little Heath. Pg [https://archive.org/details/diophantosalexa00heatgoog/page/n91 77].</ref><ref>Mathematics: Its Power and Utility. By Karl J. Smith. Pg [https://books.google.com/books?id=-0x2JszrkooC&pg=PA86 86].</ref>

* ''[[Rhetorical]] stage''—where calculations are performed by words and tallies, and no symbols are used.<ref>The Commercial Revolution and the Beginnings of Western Mathematics in Renaissance Florence, 1300–1500. Warren Van Egmond. 1976. Page 233.</ref>
* ''[[Syncopated]] stage''—where frequently used operations and quantities are represented by symbolic [[syntax|syntactical]] abbreviations, such as letters or numerals. During antiquity and the medieval periods, bursts of mathematical creativity were often followed by centuries of stagnation. As the [[early modern age]] opened and the worldwide spread of knowledge began, written examples of mathematical developments came to light.
* ''Symbolic stage''—where comprehensive systems of notation supersede rhetoric. The increasing pace of new mathematical developments, interacting with new scientific discoveries, led to a robust and complete usage of symbols. This began with mathematicians of medieval India and mid-16th century Europe,<ref>[[Solomon Gandz]]. "The Sources of al-Khowarizmi's Algebra"</ref> and continues through the present day.

The more general area of study known as the [[history of mathematics]] primarily investigates the origins of discoveries in mathematics. The specific focus of this article is the investigation of mathematical methods and notations of the past.

==Rhetorical stage==
{{See also|Measurement}}
Many areas of mathematics began with the study of [[Applied mathematics|real world problems]], before the underlying rules and concepts were identified and defined as [[abstract structure]]s. For example, geometry has its origins in the [[distance|calculation of distances]] and [[area]]s in the real world; algebra started with methods of solving problems in [[arithmetic]]. The earliest mathematical notations emerged from these problems.

There can be no doubt that most early peoples who left records knew something of [[numeration]] and [[mechanics]] and that a few were also acquainted with the elements of [[Surveying|land-surveying]]. In particular, the [[ancient Egyptian]]s paid attention to geometry and numbers, and the [[Phoenicia|ancient Phoenicians]] performed practical arithmetic, [[book-keeping]], [[navigation]], and land-surveying. The results attained by these people seem to have been accessible (under certain conditions) to travelers, facilitating [[Knowledge transfer|dispersal of the methods]]. It is probable that the knowledge of the Egyptians and Phoenicians was largely the result of [[observation]] and [[measurement]], and represented the accumulated experience of many ages. Subsequent studies of mathematics by the Greeks were largely indebted to these previous investigations.

===Beginning of notation===
{{See also|Ancient history|History of writing ancient numbers|History of science in early cultures}}
[[File:Ybc7289-bw.jpg|thumb|Babylonian tablet (c. 1800–1600 BCE), showing an approximation of {{radic|2}} (1 24 51 10 in [[sexagesimal]]) in the context of the [[Pythagorean theorem]] for an [[isosceles triangle]].]]
Written mathematics began with numbers expressed as [[tally marks]], with each tally representing a single unit. Numerical symbols consisted probably of strokes or notches cut in wood or stone, which were intelligible across cultures. For example, one notch in a bone represented one animal, person, or object. Numerical notation's distinctive feature—symbols having both local and intrinsic values—implies a state of [[civilization]] at the period of its invention.

The earliest evidence of written mathematics dates back to the ancient [[Sumer]]ians and the system of [[metrology]] from 3000 BC. From around 2500 BC onwards, the Sumerians wrote [[multiplication table]]s on clay tablets and dealt with [[geometry|geometrical]] exercises and [[Division (mathematics)|division]] problems. The earliest traces of Babylonian numerals also date back to this period.<ref>{{Cite web |last=Melville |first=Duncan J. |date=28 August 2003 |title=Third Millennium Chronology |url=http://it.stlawu.edu/~dmelvill/mesomath/3Mill/chronology.html |url-status=dead |archive-url=http://web.archive.org/web/20200115175552/http://it.stlawu.edu:80/~dmelvill/mesomath/3Mill/chronology.html |archive-date=2020-01-15 |access-date=2025-01-02 |website=stlawu.edu}}</ref> Babylonian mathematics has been reconstructed from more than 400 clay tablets unearthed since the 1850s.<ref>Boyer, C. B. ''A History of Mathematics'', 2nd ed. rev. by [[Uta Merzbach|Uta C. Merzbach]]. New York: Wiley, 1989 {{ISBN|0-471-09763-2}} (1991 pbk ed. {{ISBN|0-471-54397-7}}). "Mesopotamia" p. 25.</ref> Written in [[cuneiform]], these tablets were inscribed whilst the clay was soft and then baked hard in an oven or by the heat of the sun. Some of these appear to be graded homework.{{Source needed|date=January 2025}}

The majority of Mesopotamian clay tablets date from 1800 to 1600 BC, and cover topics which include fractions, algebra, quadratic and cubic equations, and the calculation of [[regular number]]s, [[Multiplicative inverse|reciprocal]]s, and [[Twin prime|pairs]].<ref>{{cite book |author-link=Aaboe |last=Aaboe |first=Asger |title=Episodes from the Early History of Mathematics |year=1998 |publisher=Random House |location=New York |pages=30–31}}</ref> The tablets also include multiplication tables and methods for solving [[linear equation|linear]] and [[quadratic equation]]s. The Babylonian tablet [[YBC 7289]] gives an approximation of {{radic|2}} that is accurate to an equivalent of six decimal places.

Babylonian mathematics were written using a [[sexagesimal]] (base-60) [[numeral system]]. From this derives the modern-day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 × 6) degrees in a circle, as well as the use of [[minutes and seconds of arc]] to denote fractions of a degree. Babylonian advances in mathematics were facilitated by the fact that 60 has many divisors: the reciprocal of any integer which is a multiple of divisors of 60 has a finite expansion in base 60. (In decimal arithmetic, only reciprocals of multiples of 2 and 5 have finite decimal expansions.) Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a true place-value system, where digits written in the left column represented larger values, much as in the [[decimal]] system. They lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context.

Initially, the [[Mesopotamians]] had symbols for each power of ten.<ref>{{Cite web |title=Mathematics in Egypt and Mesopotamia |url=http://www.metu.edu.tr/~beyaz/303/presentations/egyptMesopotamia.pdf |url-status=dead |archive-url=https://web.archive.org/web/20221228002317/http://www.metu.edu.tr/~beyaz/303/presentations/egyptMesopotamia.pdf |archive-date=28 December 2022 |access-date=25 July 2013}}</ref> Later, they wrote numbers in almost exactly the same way as in modern times. Instead of using unique symbols for each power of ten, they wrote only the [[coefficient]]s of each power of ten, with each digit separated by only a space. By the time of [[Alexander the Great]], they had created a symbol that represented zero and was a placeholder.

Rhetorical algebra was first developed by the ancient [[Babylonians]] and remained dominant up to the 16th century. In this system, equations are written in full sentences. For example, the rhetorical form of <math>x + 1 = 2</math> is "The thing plus one equals two" or possibly "The thing plus 1 equals 2".{{Citation needed|date=January 2025}}

The ancient Egyptians [[numeration by Hieroglyphics|numerated by hieroglyphics]].<ref>Encyclopædia Americana. By Thomas Gamaliel Bradford. Pg [https://books.google.com/books?id=hrRPAAAAMAAJ&pg=PA314 314]</ref><ref>Mathematical Excursion, Enhanced Edition: Enhanced Webassign Edition By Richard N. Aufmann, Joanne Lockwood, Richard D. Nation, Daniel K. Cleg. Pg [https://books.google.com/books?id=GTgTnSGMukgC&pg=PA186 186]</ref> Egyptian mathematics had symbols for one, ten, one hundred, one thousand, ten thousand, one hundred thousand, and one million. Smaller digits were placed on the left of the number, as they are in Hindu–Arabic numerals. Later, the Egyptians used [[hieratic]] instead of [[Egyptian hieroglyphs|hieroglyphic]] script to show numbers. Hieratic was more like cursive and replaced several groups of symbols with individual ones. For example, the four vertical lines used to represent the number 'four' were replaced by a single horizontal line. This is found in the [[Rhind Mathematical Papyrus]] (c. 2000–1800 BC) and the [[Moscow Mathematical Papyrus]] (c. 1890 BC). The system the Egyptians used was discovered and modified by many other civilizations in the Mediterranean. The Egyptians also had symbols for basic operations: legs going forward represented addition, and legs walking backward to represent subtraction.

The peoples with whom the Greeks of Asia Minor (amongst whom notation in western history begins) were likely to have come into frequent contact were those inhabiting the eastern [[littoral]] of the Mediterranean; Greek tradition uniformly assigned the special development of geometry to the Egyptians, and the [[number theory|science of numbers]] to either the Egyptians or the Phoenicians.

==Syncopated stage==
[[File:Death of Archimedes (1815) by Thomas Degeorge.png|thumb|''Death of Archimedes'' (1815) by [[Thomas Degeorge]]. The last words attributed to [[Archimedes]] are "[[Noli turbare circulos meos|Do not disturb my circles]]", a reference to the circles in the mathematical drawing that he was studying when disturbed by the Roman soldier.]]

{{See also|Fundamental theorem of arithmetic|Naive set theory}}

The history of mathematics cannot with certainty be traced back to any school or period before that of the Ionian Greeks. Still, the subsequent history may be divided into periods, the distinctions between which are tolerably well-marked. [[Greek mathematics]], which originated with the study of geometry, tended to be deductive and scientific from its commencement. Since the fourth century AD, [[Pythagoras]] has commonly been given credit for discovering the [[Pythagorean theorem]], a theorem in geometry that states that in a right-angled triangle the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares of the other two sides.<ref name="group">That is, <math>a^2 + b^2 = c^2</math>.</ref> However, this geometric relationship appears in a few earlier ancient mathematical texts (albeit not as a formalized theorem), notably [[Plimpton 322]], a Babylonian tablet of mathematics from around 1900 BC. The study of mathematics as a subject in its own right began in the 6th century BC with the [[Pythagoreans]], who coined the term "mathematics" from the ancient Greek ''mathema'' (μάθημα), meaning "subject of instruction".<ref>{{cite journal|author=Heath|title=A Manual of Greek Mathematics|journal=Nature|year=1931|volume=128|issue=3235|page=5|doi=10.1038/128739a0|bibcode=1931Natur.128..739T|s2cid=3994109}}</ref>

[[Plato]]'s influence was especially strong in mathematics and the sciences. He helped to distinguish between [[pure mathematics|pure]] and [[applied mathematics]] by widening the gap between "arithmetic" (now called [[number theory]]) and "logistic" (now called [[arithmetic]]). Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning and [[mathematical rigor]] in [[mathematical proof|proofs]]) and expanded the subject matter of mathematics.<ref>Sir Thomas L. Heath, ''A Manual of Greek Mathematics'', Dover, 1963, p. 1: "In the case of mathematics, it is the Greek contribution which it is most essential to know, for it was the Greeks who made mathematics a science."</ref> [[Aristotle]] is credited with what later would be called the [[law of excluded middle]].

Abstract or [[pure mathematics]]<ref name="Encyclopaedia Perthensi" /> deals with concepts like [[Magnitude (mathematics)|magnitude]] and [[quantity]] without regard to any practical application or situation, and includes [[arithmetic]] and [[geometry]]. In contrast, in mixed or [[applied mathematics]], mathematical properties and relationships are applied to real-world objects to model laws of physics, for example in [[hydrostatics]], [[optics]], and [[navigation]].<ref name="Encyclopaedia Perthensi">The new encyclopædia; or, Universal dictionary of arts and sciences. By Encyclopaedia Perthensi. Pg 49</ref>

[[Archimedes]] is generally considered to be the greatest mathematician of antiquity and one of the greatest of all time.<ref>{{cite book |last=Calinger |first=Ronald |title=A Contextual History of Mathematics |year=1999 |publisher=Prentice-Hall |isbn=0-02-318285-7 |page=150 |quote=Shortly after Euclid, compiler of the definitive textbook, came Archimedes of Syracuse (ca. 287&nbsp;212 BC), the most original and profound mathematician of antiquity.}}</ref><ref>{{cite web |url=http://www-history.mcs.st-and.ac.uk/Biographies/Archimedes.html |title=Archimedes of Syracuse |access-date=2008-06-09 |publisher=The MacTutor History of Mathematics archive |date=January 1999}}</ref> He used the [[method of exhaustion]] to calculate the area under the arc of a [[parabola]] with the [[Series (mathematics)|summation of an infinite series]], and gave a remarkably accurate approximation of [[pi]].<ref>{{cite web|title = A history of calculus |author1=O'Connor, J.J.  |author2=Robertson, E.F.|publisher = [[University of St Andrews]]| url = http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html |date=February 1996|access-date= 2007-08-07| archive-url= https://web.archive.org/web/20070715191704/http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html| archive-date= 15 July 2007 | url-status= live}}</ref> He also defined the [[Archimedes spiral|spiral]] bearing his name, formulae for the volumes of [[surface of revolution|surfaces of revolution]], and an ingenious system for expressing very large numbers.

[[File:Euclid Vat ms no 190 XI prop 31.jpg|thumbnail|Propositions 31, 32, and 33 in the ninth book of [[Euclid's Elements|Euclid's ''Elements'']] (volume 2 of the manuscript, sheets 207–208 recto.)]]
The ancient Greeks made steps in the abstraction of geometry. [[Euclid's Elements|Euclid's ''Elements'']] (c. 300 BC) is the earliest extant documentation of the axioms of plane geometry—though Proclus tells of an earlier [[Axiomatic system|axiomatisation]] by [[Hippocrates of Chios]]<ref>{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/Extras/Proclus_history_geometry.html |title=Proclus' Summary |publisher=Gap.dcs.st-and.ac.uk |access-date=2014-06-24 |archive-url=https://web.archive.org/web/20150923114020/http://www-gap.dcs.st-and.ac.uk/~history/Extras/Proclus_history_geometry.html |archive-date=23 September 2015  }}</ref> —and is one of the oldest extant Greek mathematical treatises. Consisting of thirteen books, it collects theorems proven by other mathematicians, supplemented by some original work. The document is a successful collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions, and covers topics such as Euclidean geometry, geometric algebra, elementary number theory, and the ancient Greek version of algebraic systems. The first theorem given in the text, [[Euclid's lemma]], captures a fundamental property of [[prime number]]s. The text was ubiquitous in the ''[[quadrivium]]'' and was instrumental in the development of logic, mathematics, and science. [[Autolycus]]' ''[[On the Moving Sphere]]'' is another ancient mathematical manuscript of the time.{{fact|date=August 2024}}

The next phase of notation for algebra was syncopated algebra, in which some symbolism is used, but which does not contain all of the characteristics of symbolic algebra. For instance, there may be a restriction that subtraction may be used only once within one side of an equation, which is not the case with symbolic algebra. Syncopated algebraic expression [[Arithmetica#Syncopated algebra|first appeared]] in a serious of books called ''[[Arithmetica]]'', by [[Diophantus of Alexandria]] (3rd century AD; many lost), followed by [[Brahmagupta]]'s ''[[Brahma Sphuta Siddhanta]]'' (7th century).

===Acrophonic and Milesian numeration===
The ancient Greeks employed [[Attic numeration]],<ref name=":0">Mathematics and Measurement By Oswald Ashton Wentworth Dilk. Pg [https://books.google.com/books?id=AKJZvXOS7n4C&pg=PA14 14]</ref> which was based on the system of the Egyptians and was later adapted and used by the [[Ancient Rome|Romans]]. Greek numerals one through four were written as vertical lines, as in the hieroglyphics. The symbol for five was the Greek letter Π (pi), representing the Greek word for 'five' (''pente''). Numbers six through nine were written as a Π with vertical lines beside it. Ten was represented by the letter Δ (delta), from word for 'ten' (''deka''), one hundred by the letter from the word for hundred, and so on. This system was 'acrophonic' since it was based on the first sound of the numeral.<ref name=":0" />

[[Ionian numeration|Milesian (Ionian) numeration]] was another Greek numeral system. It was constructed by partitioning the twenty-four letters of the Greek alphabet, plus three archaic letters, into three classes of nine letters each, and using them to represent the units, tens, and hundreds.<ref name=":0" /> ([[Jean Baptiste Joseph Delambre]]'s Astronomie Ancienne, t. ii.)

{| class=wikitable
 |-
 |[[Α]] (α)
 |[[Β]] (β)
 |[[Gamma|Г]] (γ)
 |[[Δ]] (δ)
 |[[Ε]] (ε)
 |[[Ϝ]] (ϝ)
 |[[Ζ]] (ζ)
 |[[Η]] (η)
 |[[θ]] (θ)
 |[[Ι]] (ι)
 |[[Κ]] (κ)
 |[[Λ]] (λ)
 |[[Μ]] (μ)
 |[[Ν]] (ν)
 |[[Ξ]] (ξ)
 |[[Ο]] (ο)
 |[[Pi (letter)|Π]] (π)
 |[[Ϟ]] (ϟ)
 |[[Ρ]] (ρ)
 |[[Sigma|Σ]] (σ)
 |[[Τ]] (τ)
 |[[Υ]] (υ)
 |[[Φ]] (φ)
 |[[Χ]] (χ)
 |[[Ψ]] (ψ)
 |[[Ω]] (ω)
 |[[Ϡ]] (ϡ)
 |-
 |1
 |2
 |3
 |4
 |5
 |6
 |7
 |8
 |9
 |10
 |20
 |30
 |40
 |50
 |60
 |70
 |80
 |90
 |100
 |200
 |300
 |400
 |500
 |600
 |700
 |800
 |900
|}
This system appeared in the third century BC, before the letters digamma (Ϝ), koppa (Ϟ), and sampi (Ϡ) became obsolete. When lowercase letters became differentiated from uppercase letters, the lowercase letters were used as the symbols for notation. Multiples of one thousand were written as the nine numbers with a stroke in front of them: thus, one thousand was ",α", two thousand was ",β", etc. The letter M (for ''μύριοι'', as in "myriad") was used to multiply numbers by ten thousand. For example, the number 88,888,888 would be written as M,ηωπη*ηωπη.<ref name=Boyer>Boyer, Carl B. ''A History of Mathematics'', 2nd edition, John Wiley & Sons, Inc., 1991.</ref>

Milesian numeration, though far less convenient than modern numerals, was formed on a perfectly regular and scientific plan,<ref name="Brande">A dictionary of science, literature and art, ed. by W.T. Brande. Pg [https://books.google.com/books?id=yo4DAAAAQAAJ&pg=PA683 683]</ref> and could be used with tolerable effect as an instrument of calculation, to which purpose the [[Roman numeral|Roman system]] was totally inapplicable.

Greek mathematical reasoning was almost entirely geometric (albeit often used to reason about non-geometric subjects such as [[number theory]]), and hence the Greeks had no interest in algebraic symbols. An exception was the great algebraist [[Diophantus]] of [[Alexandria]].<ref>[http://www.ms.uky.edu/~carl/ma330/projects/diophanfin1.html Diophantine Equations]. Submitted by: Aaron Zerhusen, Chris Rakes, & Shasta Meece. MA 330-002. Dr. Carl Eberhart. 16 February 1999.</ref> His ''[[Arithmetica]]'' was one of the texts to use symbols in equations. It was not completely symbolic, but was much more so than previous books. In it, an unknown number was called ''s''; the square of ''s'' was <math>\Delta^y</math>; the cube was <math>K^y</math>; the fourth power was <math>\Delta^y\Delta</math>; and the fifth power was <math>\Delta K^y</math>.<ref>{{Cite book |last=Heath |first=Sir Thomas Little |url=https://archive.org/details/bub_gb_7DDQAAAAMAAJ/page/n472 |title=A History of Greek Mathematics |date=1921 |publisher=Oxford : Clarendon Press |pages=456, 458}}</ref> So for example, the expression:
:<math>2x^4+3x^3-4x^2+5x-6</math>

would be written as:{{citation needed|date=July 2013}}
:{{math|SS2 C3 x5 M S4 u6}}

===Chinese mathematical notation===
{{Main|Suzhou numerals}}
{{See also|Chinese numerals}}
[[File:Huama numerals.svg|thumb|The numbers 0–9 in Chinese ''huama'' ({{lang|zh|花碼}}) numerals]]
The ancient Chinese used numerals that look much like the tally system.<ref>The American Mathematical Monthly, Volume 16. Pg [https://books.google.com/books?id=JggPAAAAIAAJ&pg=PA131 131]</ref> Numbers one through four were horizontal lines. Five was an X between two horizontal lines; it looked almost exactly the same as the [[Roman numeral]] for ten. Nowadays, this [[Numerals in Unicode#CJK Suzhou (huāmǎ) numerals|''huama'' numeral system]] is only used for displaying prices in Chinese markets or on traditional handwritten invoices.

[[Mathematics in China]] emerged independently by the 11th century BC,<ref>{{cite web|url=http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Chinese_overview.html |title=Overview of Chinese mathematics |publisher=Groups.dcs.st-and.ac.uk |access-date=2014-06-24}}</ref> but has much older roots. The ancient Chinese were acquainted with astronomical cycles, geometrical implements like the [[Ruler|rule]], [[compass]], and [[plumb-bob]], and machines like the [[wheel]] and [[axle]]. The Chinese independently developed very large and [[negative number]]s, [[decimal]]s, a place value decimal system, a [[Binary numeral system|binary system]], algebra, geometry, and trigonometry. As in other early societies, the purpose of astronomy was to perfect the agricultural calendar and other practical tasks, not to establish a [[formal systems|formal system]]; thus, the duties of the [[Chinese Board of Mathematics]] were confined to the annual preparation of the dates and predictions of the almanac.

[[File:Chounumerals.svg|thumb|[[Counting rods|Counting rod]] numerals]]
Early Chinese mathematical inventions include a [[place value system]] known as [[counting rods]]<ref>George Gheverghese Joseph, ''The Crest of the Peacock: Non-European Roots of Mathematics'', Penguin Books, London, 1991, pp.&nbsp;140—148</ref><ref>Georges Ifrah, ''Universalgeschichte der Zahlen'', Campus, Frankfurt/New York, 1986, pp.&nbsp;428—437</ref> (which emerged during the [[Warring States period]]), certain geometrical theorems (such as the [[Pythagorean theorem|ratio of sides]]), and the ''[[suanpan]]'' (abacus) for performing arithmetic calculations. Mathematical results were expressed in writing. Ancient Chinese mathematicians did not develop an axiomatic approach, but made advances in algorithm development and algebra. Chinese algebra reached its zenith in the 13th century, when [[Zhu Shijie]] invented the method of four unknowns.{{Clarify|date=January 2025|reason=What is the "method of four unknowns"?}} Early China exemplifies how a civilization may possess considerable skill in the applied arts with only scarce understanding of the formal mathematics on which those arts are founded.

Due to linguistic and geographic barriers, as well as content, the mathematics of ancient China and the mathematics of the ancient Mediterranean world are presumed to have developed more or less independently. The final form of ''[[The Nine Chapters on the Mathematical Art]]'' and the ''[[Book on Numbers and Computation]]'' and ''[[Huainanzi]]'' are roughly contemporary with classical Greek mathematics. Some exchange of ideas across Asia through known cultural exchanges from at least Roman times is likely. Frequently, elements of the mathematics of early societies correspond to rudimentary results found later in branches of modern mathematics such as geometry or [[number theory]]. For example, the Pythagorean theorem was attested in the ''[[Zhoubi Suanjing]]'', and knowledge of [[Pascal's triangle]] has also been shown to have existed in China centuries before [[Blaise Pascal]],<ref>{{cite web|url=http://www.psupress.psu.edu/books/titles/0-271-01238-2.html |title=Frank J. Swetz and T. I. Kao: Was Pythagoras Chinese? |publisher=Psupress.psu.edu |access-date=2014-06-24}}</ref> articulated by mathematicians like the polymath [[Shen Kuo]].

The state of trigonometry advanced during the Song dynasty (960–1279), when Chinese mathematicians had greater need of spherical trigonometry in calendrical science and astronomical calculations.<ref name="needham volume 3 109 2" /> [[Shen Kuo]] used trigonometric functions to solve mathematical problems of chords and arcs.<ref name="needham volume 3 109 2">Needham, Joseph (1986). Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd.</ref> Shen's work on arc lengths provided the basis for [[spherical trigonometry]] developed in the 13th century by the mathematician and astronomer [[Guo Shoujing]].<ref name="restivo 32">[[Sal Restivo]]</ref> As the historians L. Gauchet and Joseph Needham state, Guo Shoujing used spherical trigonometry in his calculations to improve the [[Chinese calendar|calendar system]] and [[Chinese astronomy]].<ref name="needham volume 3 109 2" /><ref name="gauchet 151">[[Marcel Gauchet]], 151.</ref> Chinese mathematics later incorporated the work and teaching of Arab missionaries with knowledge of spherical trigonometry who had come to China during the 13th century.

===Indian and Arabic numerals and notation===
{{See also|Arabic numerals|Hindu–Arabic numeral system|History of the Hindu–Arabic numeral system|Mathematics in medieval Islam}}

The [[Hindu–Arabic numeral system]] and the rules for the use of its operations, in use throughout the world today, likely evolved over the course of the first millennium AD in [[Indian mathematics|India]] and was transmitted to the west via Islamic mathematics.<ref>Robert Kaplan, "The Nothing That Is: A Natural History of Zero", Allen Lane/The Penguin Press, London, 1999</ref><ref>{{cite web |last1=O'Connor |first1=J. J. |last2=Robertson |first2=E. F. |date=November 2000 |title=Indian numerals |url=http://www-history.mcs.st-and.ac.uk/HistTopics/Indian_numerals.html |url-status=dead |archive-url=https://web.archive.org/web/20191022210400/http://www-history.mcs.st-and.ac.uk/HistTopics/Indian_numerals.html |archive-date=22 October 2019 |access-date=2014-06-24 |publisher= |quote="The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius." – Pierre-Simon Laplace}}</ref> [[Islamic mathematics]] developed and expanded the mathematics known to [[Central Asia]]n civilizations,<ref>[[Adolf Yushkevich|A.P. Juschkewitsch]], "Geschichte der Mathematik im Mittelalter", Teubner, Leipzig, 1964</ref> including the addition of the [[decimal point]] notation to the [[Arabic numerals]].{{contradict-inline|Decimal separator#Hellenistic–Renaissance eras|date=August 2024}}

The algebraic notation of the Indian mathematician [[Brahmagupta]] was [[History of algebra#Diophantus|syncopated]] (that is, some operations and quantities had symbolic representations). Addition was indicated by placing the numbers side by side, subtraction by placing a dot over the [[subtrahend]] (the number to be subtracted), and division by placing the divisor below the dividend, similar to our notation but without the bar. Multiplication, evolution, and unknown quantities were represented by abbreviations of appropriate terms.<ref name="Boyer Brahmagupta Indeterminate equations">{{cite book |last=Boyer |first=C. B. |title=A History of Mathematics |edition=2nd |editor=Uta C. Merzbach |place=New York |publisher=Wiley |year=1989 |isbn=0-471-09763-2 |chapter=China and India |page=221 |quote="[...] he was the first one to give a ''general'' solution of the linear Diophantine equation ax + by = c, where a, b, and c are integers. [...] It is greatly to the credit of Brahmagupta that he gave ''all'' integral solutions of the linear Diophantine equation, whereas Diophantus himself had been satisfied to give one particular solution of an indeterminate equation. Inasmuch as Brahmagupta used some of the same examples as Diophantus, we see again the likelihood of Greek influence in India – or the possibility that they both made use of a common source, possibly from Babylonia. It is interesting to note also that the algebra of Brahmagupta, like that of Diophantus, was syncopated. Addition was indicated by juxtaposition, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, as in our fractional notation but without the bar. The operations of multiplication and evolution (the taking of roots), as well as unknown quantities, were represented by abbreviations of appropriate words."}}</ref>

[[File:Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala.jpg|thumb|A page from al-Khwārizmī's ''Algebra'']]

Despite their name, [[Arabic numerals]] have roots in India. The reason for this [[misnomer]] is Europeans saw the numerals used in an Arabic book, ''Concerning the Hindu Art of Reckoning'', by [[al-Khwarizmi|Muhammed ibn-Musa al-Khwarizmi]]. Al-Khwārizmī wrote several important books on the Hindu–Arabic numerals and on methods for solving equations. His book ''On the Calculation with Hindu Numerals'' (c. 825), along with the work of [[Al-Kindi]], were instrumental in spreading Indian mathematics and numerals to the West. Al-Khwarizmi did not claim the numerals as Arabic, but over several Latin translations, the fact that the numerals were Indian in origin was lost. The word ''[[algorithm]]'' is derived from the Latinization of Al-Khwārizmī's name, Algoritmi, and the word ''[[algebra]]'' from the title of one of his works, ''[[The Compendious Book on Calculation by Completion and Balancing|Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa'l-muqābala]]'' (''The Compendious Book on Calculation by Completion and Balancing'').

The modern [[Arabic numeral]] symbols used around the world first appeared in Islamic [[North Africa]] in the 10th century. A distinctive Western Arabic variant of the [[Eastern Arabic numerals]] began to emerge around the 10th century in the [[Maghreb]] and [[Al-Andalus]] (sometimes called ''ghubar'' numerals, though the term is not always accepted), which are the direct ancestor of the modern Arabic numerals used throughout the world.<ref>{{citation |first=Paul |last= Kunitzsch |chapter=The Transmission of Hindu-Arabic Numerals Reconsidered |editor1=J. P. Hogendijk |editor2=A. I. Sabra |title=The Enterprise of Science in Islam: New Perspectives |chapter-url=https://books.google.com/books?id=_AUtLNtg3nsC&pg=PA3 |year=2003 |publisher=MIT Press |isbn=978-0-262-19482-2 |pages=3–22 (12–13)}}</ref>

Many Greek and Arabic texts on mathematics were then [[Latin translations of the 12th century|translated into Latin]], which led to further development of mathematics in medieval Europe. In the 12th century, scholars traveled to Spain and Sicily seeking scientific Arabic texts, including al-Khwārizmī's (translated into Latin by [[Robert of Chester]]) and the complete text of [[Euclid's Elements|Euclid's ''Elements'']] (translated in various versions by [[Adelard of Bath]], [[Herman of Carinthia]], and [[Gerard of Cremona]]).<ref>[[Marie-Thérèse d'Alverny]], "Translations and Translators", pp. 421–62 in Robert L. Benson and Giles Constable, ''Renaissance and Renewal in the Twelfth Century'', (Cambridge: Harvard University Press, 1982).</ref><ref>Guy Beaujouan, "The Transformation of the Quadrivium", pp. 463–87 in Robert L. Benson and Giles Constable, ''Renaissance and Renewal in the Twelfth Century'', (Cambridge: Harvard University Press, 1982).</ref> One of the European books that advocated using the numerals was ''[[Liber Abaci]]'', by Leonardo of Pisa, better known as [[Fibonacci]]. ''Liber Abaci'' is better known for containing a mathematical problem in which the growth of a rabbit population ends up being the [[Fibonacci sequence]].

==Symbolic stage==
;Symbols by popular introduction date
<timeline>
ImageSize = width:678 height:285
PlotArea  = left:65 right:55 bottom:20 top:15
AlignBars = justify

Colors =
    id:time value:rgb(0.7,0.7,1) #
    id:period value:rgb(1,0.7,0.5) #
    id:age value:rgb(0.95,0.85,0.5) #
    id:era value:rgb(1,0.85,0.5) #
    id:eon value:rgb(1,0.85,0.7) #
    id:filler value:gray(0.8) # background bar
    id:black value:black

Period     = from:1360 till:1962
TimeAxis   = orientation:horizontal
ScaleMajor = unit:year increment:100 start:1360
ScaleMinor = unit:year increment:10 start:1360

PlotData =
  align:center textcolor:black fontsize:8 mark:(line,black) width:15

  bar:&nbsp; color:filler width:15
  from:1360 till:1962 shift:(0,10) text:[[Mathematical notation|Symbolic Notation]]
  from:1360 till:1704 shift:(-40,0) text:Early
  from:1704 till:1876 text:High
  from:1876 till:1962 text:Late

  bar:&nbsp; color:age width:5
  from:1360 till:1618  shift:(0,-10) text:Arithmetic
  from:1618 till:1718 shift:(0,-10) text:Multiplication
  from:1718 till:1846 shift:(0,-10) text:Division
  from:1846 till:1962 shift:(10,-10) text:Abstraction

  mark:(line,black)  textcolor:black  fontsize:M
  bar:Events  color:filler  align:left
  at:1360 shift:(2,0) text:"[[Plus and minus signs|plus]]"
  at:1489 shift:(0,6) text:"[[Plus and minus signs|minus]]"
  at:1525 shift:(0,17) text:"[[nth root|radical]]"	
  at:1544 shift:(0,28) text:"[[Parentheses|parenth.]]"	
  at:1557 shift:(0,39) text:"[[equals sign|equals]]"	
  at:1618 shift:(0,3) text:"[[multiplication sign|multiply]]"	
  at:1628 shift:(0,14) text:"[[plus-minus sign|plus-minus]]"	
  at:1628 shift:(0,25) text:"[[Proportionality (mathematics)|proportion]]"	
  at:1629 shift:(0,36) text:"[[radical symbol|radical]]"	
  at:1631 shift:(0,47) text:"[[Inequality (mathematics)|inequality]]"	
  at:1636 shift:(0,58) text:"[[Subscript and superscript|superscript]]"	
  at:1637 shift:(0,69) text:"[[radical symbol|radical]]"
  at:1650 shift:(0,80) text:"[[percent sign|percent]]"		
  at:1655 shift:(0,91) text:"[[infinity sign|infinity]]"	
  at:1659 shift:(0,102) text:"[[division sign|division]]"	
  at:1670 shift:(0,113) text:[[Inequality signs|Inequality]]"	
  at:1675 shift:(0,124) text:[[differential sign|differential]]"	
  at:1675 shift:(0,135) text:[[integral sign|integral]]"	
  at:1684 shift:(0,146) text:[[Colon_(punctuation)|colon]]"	
  at:1698 shift:(0,157) text:[[middle dot|dot]]"	
  at:1718 shift:(0,5) text:[[Division (mathematics)|slash]]"	
  at:1734 shift:(0,15) text:[[inequality signs|inequality]]"	
  at:1755 shift:(0,25) text:[[summation]]"	
  at:1768 shift:(0,35) text:[[Proportionality (mathematics)|proportionality]]"
  at:1770 shift:(0,45) text:[[partial differential|differential]]"	
  at:1770 shift:(0,55) text:[[prime symbol|prime]]"
  at:1801 shift:(0,65) text:[[identity sign|identity]]"	
  at:1808 shift:(0,75) text:[[integral part|integral]]"	
  at:1808 shift:(0,85) text:[[factorial]]"	
  at:1812 shift:(0,95) text:[[Product sign|product]]"	
  at:1817 shift:(0,105) text:[[set inclusion|inclusion]]"	
  at:1841 shift:(0,115) text:[[Absolute value|abs.]]"	
  at:1841 shift:(0,125) text:[[Determinant|determ.]]"	
  at:1843 shift:(0,135) text:[[Matrix notation|line matrix]]"
  at:1846 shift:(0,145) text:"[[nabla symbol|nabla]]"	
  at:1888 shift:(0,17) text:"[[Union (set theory)|union]]~[[Intersection (set theory)|Intersection]]"
  at:1890 shift:(0,28) text:"[[set inclusion|inclusion]]"	
  at:1893 shift:(0,39) text:"[[Aleph number|aleph]]"	
  at:1894 shift:(0,50) text:"[[Element (mathematics)|membership]]"	
  at:1895 shift:(0,61) text:"[[curly brackets|braces]]"	
  at:1895 shift:(0,72) text:"[[Natural number|N]]"
  at:1897 shift:(0,83) text:"[[Existential quantification|existential]]"	
  at:1902 shift:(0,94) text:"[[cross product]]"	
  at:1902 shift:(0,105) text:"[[dot product]]"	
  at:1906 shift:(0,116) text:"[[logical disjunction|disjunction]]"	
  at:1909 shift:(0,127) text:"[[Matrix notation|parenth. matrix]]"	
  at:1913 shift:(0,138) text:"[[Matrix notation|box matrix]]"	
  at:1917 shift:(0,149) text:"[[Line integral|contour]]"	
  at:1930 shift:(0,160) text:"[[Integer|Z]], "	
  at:1930 shift:(12,160) text:"[[Rational number|Q]]"	
  at:1935 shift:(0,171) text:"[[universal quantifier|universals]]"
  at:1936 shift:(0,182) text:"[[Arrow (symbol)|arrow]]"	
  at:1939 shift:(0,193) text:"[[Empty set|empty]]"	
  at:1939 shift:(0,204) text:"[[Complex number|C]]"
  at:1940 shift:(0,215) text:"[[Function (mathematics)|arrow]]"
  at:1960 shift:(0,17) text:"[[end of proof|EoP]]"
  at:1960 shift:(3,5) text:"[[quabla|∎]]"
  at:1962 shift:(0,28) text:"[[integral]]"
</timeline>
{{Further|Table of mathematical symbols by introduction date}}

===Early arithmetic and multiplication===
{{See also|Early modern age|Probability|Statistics|Notation in probability and statistics|History of probability|History of statistics|Scientific revolution}}

The transition to symbolic algebra, where only symbols are used, can first be seen in the work of [[Ibn al-Banna' al-Marrakushi]] (1256–1321) and [[Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī]] (1412–1482).<ref>{{MacTutor|id=Al-Banna|title=al-Marrakushi ibn Al-Banna}}</ref><ref name=Gullberg>{{cite book |last1=Gullberg |first1=Jan |author-link=Jan Gullberg |title=Mathematics: From the Birth of Numbers |url=https://archive.org/details/mathematicsfromb1997gull |url-access=registration |date=1997 |publisher=W. W. Norton |isbn=0-393-04002-X |page=[https://archive.org/details/mathematicsfromb1997gull/page/298 298]}}</ref> Al-Qalasādī was the last major medieval [[Mathematics in medieval Islam|Arab algebraist]], who improved on the [[Mathematical notation|algebraic notation]] earlier used in the [[Maghreb]] by Ibn al-Banna.<ref name=Qalasadi>{{MacTutor Biography|id=Al-Qalasadi|title= Abu'l Hasan ibn Ali al Qalasadi}}</ref> In contrast to the syncopated notations of their predecessors, [[Diophantus]] and [[Brahmagupta]], which lacked symbols for [[Operation (mathematics)|mathematical operations]],<ref>{{cite book |last=Boyer |first=C. B. |title=A History of Mathematics |edition=2nd |editor=Uta C. Merzbach |place=New York |publisher=Wiley |year=1989 |isbn=0-471-09763-2 |chapter=Revival and Decline of Greek Mathematics |page=178 |quote="The chief difference between Diophantine syncopation and the modern algebraic notation is the lack of special symbols for operations and relations, as well as of the exponential notation."}}</ref> al-Qalasadi's algebraic notation was the first to have symbols for these functions and was thus "the first steps toward the introduction of algebraic symbolism". He represented mathematical symbols using characters from the [[Arabic alphabet]].<ref name=Qalasadi/>

[[File:Johannes Widmann-Mercantile Arithmetic 1489.jpg|thumbnail|Early use of the [[plus and minus signs]] in print, by Widmann (1489)]]
The 14th century saw the development of new mathematical concepts to investigate a wide range of problems.<ref>Grant, Edward and John E. Murdoch (1987), eds., ''Mathematics and Its Applications to Science and Natural Philosophy in the Middle Ages'', (Cambridge: Cambridge University Press) {{ISBN|0-521-32260-X}}.</ref> The two most widely used arithmetic symbols are addition and subtraction, + and −. The [[plus sign]] was used starting around 1351 by [[Nicole Oresme]]<ref>Mathematical Magazine, Volume 1. Artemas Martin, 1887. [https://books.google.com/books?id=mG8yAQAAMAAJ&pg=PA124 Pg 124]</ref> and publicized in his work ''Algorismus proportionum'' (1360).<ref>[https://books.google.com/books?id=k0U1AQAAMAAJ Der Algorismus proportionum des Nicolaus Oresme]: Zum ersten Male nach der Lesart der Handschrift R.40.2. der Königlichen Gymnasial-bibliothek zu Thorn. [[Nicole Oresme]]. S. Calvary & Company, 1868.</ref> It is thought to be an abbreviation for "et", meaning "and" in Latin, in much the same way the [[ampersand]] sign also began as "et". Oresme at the [[University of Paris]] and the Italian [[Giovanni di Casali]] independently provided graphical demonstrations of the distance covered by a body undergoing uniformly accelerated motion, asserting that the area under the line depicting the constant acceleration and represented the total distance traveled.<ref>Clagett, Marshall (1961) ''The Science of Mechanics in the Middle Ages'', (Madison: University of Wisconsin Press), pp. 332–45, 382–91.</ref> The [[minus sign]] was used in 1489 by [[Johannes Widmann]] in ''Mercantile Arithmetic'' or ''Behende und hüpsche Rechenung auff allen Kauffmanschafft''.<ref>''Later, early modern version'': {{cite book |url=https://archive.org/details/anewsystemmerca04walsgoog/page/n5/mode/2up |title=A New System of Mercantile Arithmetic: Adapted to the Commerce of the United States, in Its Domestic and Foreign Relations with Forms of Accounts and Other Writings Usually Occurring in Trade |author=[[Michael Walsh (1801)|Michael Walsh]]. |publisher=[[Edmund M. Blunt]] |year=1801}}</ref> Widmann used the minus symbol with the plus symbol to indicate deficit and surplus, respectively.<ref>{{cite web |last=Miller |first=Jeff |title=Earliest Uses of Symbols of Operation |date= 4 June 2006 |work=Gulf High School |access-date=24 September 2006 |url=http://jeff560.tripod.com/operation.html}}</ref> In ''[[Summa de arithmetica|Summa de arithmetica, geometria, proportioni e proportionalità]]'',<ref>Arithmetical Books from the Invention of Printing to the Present Time. By [[Augustus De Morgan]]. p [https://archive.org/details/arithmeticalboo00morggoog/page/n24 2].</ref> [[Luca Pacioli]] used plus and minus symbols and algebra, though much of the work originated from [[Piero della Francesca]] whom he appropriated and purloined.{{fact|date=August 2024}}

The [[Nth root|radical]] symbol (√), for square root, was introduced by [[Christoph Rudolff]] in the early 1500s. [[Michael Stifel]]'s important work ''Arithmetica integra''<ref>[[iarchive:bub_gb_fndPsRv08R0C|Arithmetica integra]]. By [[Michael Stifel]], [[Philipp Melanchthon|Philipp Melanchton]]. [[Norimbergæ]]: [[Apud Iohan Petreium]], 1544.</ref> contained important innovations in mathematical notation. In 1556 [[Niccolò Fontana Tartaglia|Niccolò Tartaglia]] used parentheses for precedence grouping. In 1557 [[Robert Recorde]] published ''[[The Whetstone of Witte]]'', which introduced the equal sign (=), as well as plus and minus signs, to the English reader. In 1564 [[Gerolamo Cardano]] analyzed [[game of chance|games of chance]] beginning the early stages of [[probability theory]]. [[Rafael Bombelli]] published his ''L'Algebra'' (1572) in which he showed how to deal with the [[imaginary number|imaginary quantities]] that could appear in Cardano's formula for solving cubic equations. [[Simon Stevin]]'s book ''De Thiende'' ("The Art of Tenths"), published in Dutch in 1585, contained a systematic treatment of [[decimal notation]], which influenced all later work on the [[real number system]]. The [[new algebra]] (1591) of [[François Viète]] introduced the modern notational manipulation of algebraic expressions.

[[John Napier]] is best known as the inventor of [[logarithms]] (published in ''[[Description of the Marvelous Canon of Logarithms]]'')<ref>{{Cite book |last=Rooney |first=Anne |url=https://books.google.com/books?id=5O67iqeIHZ8C&pg=PA40 |title=The History of Mathematics |date=2012-07-15 |publisher=Rosen Publishing Group, Inc |isbn=978-1-4488-7369-2 |pages=40 |language=en}}</ref> and made common the use of the [[decimal point]] in arithmetic and mathematics.<ref>{{Cite book |last=Napier |first=Mark |url=https://archive.org/details/memoirsjohnnapi00napigoog |title=Memoirs of John Napier of Merchiston, his lineage, life, and times, with a history of the invention of logarithms |date=1834 |publisher=William Blackwood, Edinburgh, and Thomas Cadell, London }}</ref><ref>{{Cite book |last1=David Stewart Erskine Earl of Buchan |url=https://books.google.com/books?id=JUdkAAAAMAAJ |title=An Account of the Life, Writings, and Inventions of John Napier, of Merchiston |last2=Minto |first2=Walter |date=1787 |publisher=R. Morison, junr. |language=en}}</ref> After Napier, [[Edmund Gunter]] created the [[logarithmic scale]]s (lines, or rules); [[William Oughtred]] used two such scales sliding by one another to perform direct multiplication and division and is credited as the inventor of the [[slide rule]] in 1622. In 1631 Oughtred introduced the multiplication sign (×), his proportionality sign (∷), and abbreviations 'sin' and 'cos' for the [[sine]] and [[cosine]] functions.<ref>{{cite book | title = A History of Mathematics | first = Florian | last = Cajori | year = 1919 | publisher = Macmillan | url = https://archive.org/details/ahistorymathema02cajogoog| page = [https://archive.org/details/ahistorymathema02cajogoog/page/n169 157] | author-link = Florian Cajori }}</ref> [[Albert Girard]] also used the abbreviations 'sin', 'cos', and 'tan' for the trigonometric functions in his treatise.

[[René Descartes]] is credited as the father of [[analytical geometry]], the bridge between algebra and geometry, crucial to the discovery of [[infinitesimal calculus]] and [[mathematical analysis|analysis]]. In the 17th century, Descartes introduced [[Cartesian co-ordinates]] which allowed the development of analytic geometry, bringing the notation of equations to geometry. [[Blaise Pascal]] influenced mathematics throughout his life; for instance, his ''Traité du triangle arithmétique'' ("Treatise on the Arithmetical Triangle") (1653) described a convenient tabular presentation for [[binomial coefficient]]s, now called [[Pascal's triangle]]. [[John Wallis]] introduced the [[infinity symbol]] (∞) and also used this notation for [[infinitesimals]], for example, {{sfrac|1|∞}}.

[[Johann Rahn]] introduced the [[division sign]] (÷, an [[obelus]] variant repurposed) and the [[therefore sign]] (∴) in 1659. [[William Jones (mathematician)|William Jones]] used π in ''Synopsis palmariorum mathesios''<ref>[https://books.google.com/books?id=b59EAAAAcAAJ Synopsis Palmariorum Matheseos]. By [[William Jones (mathematician)|William Jones]]. 1706. (Alt: [https://archive.org/details/SynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics Synopsis Palmariorum Matheseos: or, a New Introduction to the Mathematics]. archive.org.)</ref> in 1706 because it is the initial letter of the Greek word ''perimetron'' (περιμετρον), which means [[perimeter]] in Greek. This usage was popularized in 1737 by Euler. In 1734, [[Pierre Bouguer]] used double horizontal bar below the [[inequality sign]].<ref>When Less is More: Visualizing Basic Inequalities. By Claudi Alsina, Roger B. Nelse. Pg [https://books.google.com/books?id=U1ovBsSRNscC&pg=PR18 18].</ref>

===Derivatives notation: Leibniz and Newton===
{{See also|Leibniz's notation|Leibniz–Newton calculus controversy}}

{{multiple image
 | header = [[Notation for differentiation|''Derivative'' notations]]
 | image1 = GodfreyKneller-IsaacNewton-1689.jpg
 | caption1 = Sir [[Isaac Newton]]
 | image2 = Gottfried Wilhelm Leibniz, Bernhard Christoph Francke.jpg
 | caption2 = [[Gottfried Wilhelm Leibniz]]
 | total_width = 300
}}
The study of [[linear algebra]] emerged from the study of [[determinant]]s, which were used to solve systems of [[linear equation]]s. Calculus had two main systems of notation, each created by one of its creators: that developed by [[Isaac Newton]] and that developed by [[Gottfried Leibniz]]. Leibniz's notation is used most often today.

Newton's notation was simply a dot or dash placed above the function. For example, the derivative of the function ''{{mvar|x}}'' would be written as <math>\dot{x}</math>. The second derivative of {{mvar|x}} would be written as <math>\ddot{x}</math>. In modern usage, this notation generally denotes derivatives of physical quantities with respect to time, and is used frequently in the science of [[mechanics]]. Leibniz, on the other hand, used the letter {{mvar|d}} as a prefix to indicate differentiation, and introduced the notation representing derivatives as if they were a special type of fraction. For example, the derivative of the function {{mvar|x}} with respect to the variable {{mvar|t}} in Leibniz's notation would be written as <math display="inline">{ dx \over dt }</math>. This notation makes explicit the variable with respect to which the derivative of the function is taken. Leibniz also created the integral symbol ({{math|∫|big=1}}). For example: <math display="inline">\int_{-N}^{N} f(x)\, dx</math>. When finding areas under curves, integration is often illustrated by dividing the area into infinitely many tall, thin rectangles, whose areas are added. Thus, the integral symbol is an [[long s|elongated S]], representing the [[Latin]] word ''summa'', meaning "sum".

===High division operators and functions===
At this time, letters of the alphabet were to be used as symbols of [[quantity]]; and although much diversity existed with respect to the choice of letters, there came to be several [[Open standard|universally recognized rules]].<ref name="Brande" /> Here thus in the history of equations the first letters of the alphabet became indicatively known as [[coefficients]], while the last letters as unknown terms (an ''[[Unknown known#Unknown knowns|incerti ordinis]]''). In [[algebraic geometry]], again, a similar rule was to be observed: the last letters of the alphabet came to denote the variable or current [[coordinates]]. Certain letters were by [[consensus gentium|universal consent]] appropriated as symbols for the frequently occurring numbers (such as <math>\pi</math> for [[Pi|3.14159...]] and {{Mvar|e}} for [[e (mathematical constant)|2.7182818...]]), and other uses were to be avoided as much as possible.<ref name="Brande" /> Letters, too, were to be employed as symbols of operation, and with them other previously mentioned arbitrary operation characters. The letters {{Mvar|d}} and [[long s|elongated S]] were to be appropriated as operative symbols in [[differential calculus]] and [[integral calculus]], and [[Delta (letter)|<math>\Delta</math>]] and <math>\Sigma</math> in the [[calculus of differences]].<ref name="Brande"/> In [[Functional (mathematics)|functional notation]], a letter, as a symbol of operation, is combined with another which is regarded as a symbol of [[Quantity#Quantity in mathematics|quantity]].<ref name="Brande"/>

Thus, <math>f(x)</math> denotes the mathematical result of the performance of the operation <math>f</math> upon the [[Mathematical object|subject]] <math>x</math>. If upon this result the same operation is repeated, the new result would be expressed by <math>f[f(x)]</math>, or more concisely by <math>f^2(x)</math>, and so on. The quantity <math>x</math> itself regarded as the result of the same operation <math>f</math> upon some other function; the proper symbol for which is, by analogy, <math>f^{-1} (x)</math>. Thus <math>f</math> and <math>f^{-1}</math> are symbols of [[Multiplicative inverse|inverse operation]]s, the former cancelling the effect of the latter on the subject <math>x</math>. <math>f(x)</math> and <math>f^{-1} (x)</math> in a similar manner are termed [[inverse function]]s.

Beginning in 1718, Thomas Twinin used the [[Slash (punctuation)|division slash]] ([[Slash (punctuation)|solidus]]), deriving it from the earlier Arabic [[horizontal fraction bar]]. [[Pierre-Simon Laplace|Pierre-Simon, Marquis de Laplace]] developed the widely used [[Laplace operator|Laplacian differential operator]] (e.g. <math>\Delta f(p) </math>). In 1750, [[Gabriel Cramer]] developed [[Cramer's Rule]] for solving linear systems.

====Euler and prime notations====
[[File:Euler's signature.png|thumbnail|Leonhard Euler's signature]]
[[Leonhard Euler]] was one of the most prolific mathematicians in history, and also a prolific inventor of canonical notation. [[Contributions of Leonhard Euler to mathematics|His contributions]] include his use of {{Mvar|[[E (mathematical constant)|e]]}} to represent the base of [[natural logarithm]]s. It is not known exactly why {{Mvar|e}} was chosen, but it was probably because the first four letters of the alphabet were already commonly used to represent variables and other constants. Euler consistently used <math>\pi</math> to represent [[pi]]. The use of <math>\pi</math> was suggested by [[William Jones (mathematician)|William Jones]], who used it as shorthand for [[perimeter]]. Euler used <math>i</math> to represent the square root of negative one (<math display="inline">\sqrt{-1}</math>) although he earlier used it as an ''infinite number.'' Today, the symbol created by [[John Wallis]], <math>\infty</math>, is used for infinity, as in e.g. <math display="inline">\sum_{n=1}^\infty\frac{1}{n^2}</math>. For [[summation]], Euler used an enlarged form of the upright capital Greek letter [[Sigma (letter)|sigma]] (Σ), known as [[capital-sigma notation]]. This is defined as:

{{Equation box 1|indent =:|equation =<math>\sum_{i=m}^n a_i = a_m + a_{m+1} + a_{m+2} +\cdots+ a_{n-1} + a_n. </math>|cellpadding= 6|border = 0|border colour = black|background colour=white}}

where ''i'' represents the index of summation; ''a<sub>i</sub>'' is an indexed variable representing each successive term in the series; ''m'' is the lower bound of summation, and ''n'' is the upper bound of summation. The term "''i = m''" under the summation symbol means that the index ''i'' starts equal to ''m''. The index, ''i'', is incremented by 1 for each successive term, stopping when ''i'' = ''n''.

For [[function (mathematics)|functions]], Euler used the notation <math>f(x)</math> to represent a function of <math>x</math>.

The [[William Emerson (mathematician)|mathematician William Emerson]]<ref>{{Cite book |last=Emerson |first=William |url=https://archive.org/details/elementsgeometr00emergoog |title=The elements of geometry |date=1794 |publisher=F. Wingrave |location=London}}</ref> developed the [[proportionality sign]] (∝). [[Proportionality (mathematics)|Proportionality]] is the [[ratio]] of one quantity to another, and the sign is used to indicate the ratio between two variables is constant.<ref>{{Cite book |last=Emerson |first=William |url=https://books.google.com/books?id=YLxHMwEACAAJ |title=The Doctrine of Proportion, Arithmetical and Geometrical. Together with a General Method of Arening by Proportional Quantities |date=1763 |language=}}</ref><ref>{{Cite book |last=Baron |first=George |url=https://books.google.com/books?id=IXkAAAAAMAAJ&pg=PA83 |title=The Mathematical Correspondent: Containing New Eludications, Discoveries, and Improvements, in Various Branches of the Mathematics |date=1804 |publisher=Sage and Clough |pages=83 |language=en}}</ref> Much later in the abstract expressions of the value of various proportional phenomena, the [[parts-per notation]] would become useful as a set of pseudo-units to describe small values of miscellaneous [[Dimensionless quantity|dimensionless quantities]]. [[Marquis de Condorcet]], in 1768, advanced the [[partial differential]] sign (∂), known as the ''curly d'' or ''[[Jacobi's delta]]''. The [[prime symbol]] (′) for derivatives was made by [[Joseph-Louis Lagrange]].

{{rquote|right|But in our [[opinion]] [[truths]] of this kind should be drawn from notions rather than from notations.|Carl Friedrich Gauss, writing about the proof of [[Wilson's theorem]]<ref>''[[Disquisitiones Arithmeticae]]'' (1801) Article 76</ref>}}

====Gauss, Hamilton, and matrix notations====
At the turn of the 19th century, [[Carl Friedrich Gauss]] developed the [[Identity (mathematics)|identity sign]] for [[congruence relation]] and, in [[quadratic reciprocity]], the [[Floor and ceiling functions|integral part]]. Gauss developed functions of [[complex variable]]s, functions of geometry, and functions for the convergence of [[series (mathematics)|series]]. He devised satisfactory proofs of the [[fundamental theorem of algebra]] and the [[quadratic reciprocity law]]. Gauss developed the [[Gaussian elimination]] method of solving linear systems, which was initially listed as an advancement in [[geodesy]].<ref>{{cite web|last=Vitulli |first=Marie |author-link= Marie A. Vitulli |title=A Brief History of Linear Algebra and Matrix Theory |url=http://darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html |work=Department of Mathematics |publisher=University of Oregon |access-date=2012-01-24  |archive-url=https://web.archive.org/web/20120910034016/http://darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html |archive-date=2012-09-10 }}</ref> He would also develop the [[product sign]] (<math display="inline">\textstyle \prod</math>).

In the 1800s, [[Christian Kramp]] promoted [[factorial]] notation during his research in generalized factorial function which applied to non-integers.<ref>{{cite web|url=http://www-history.mcs.st-and.ac.uk/Biographies/Kramp.html |title=Kramp biography |publisher=History.mcs.st-and.ac.uk |access-date=2014-06-24}}</ref> [[Joseph Diaz Gergonne]] introduced the [[set inclusion]] signs (⊆, ⊇), later redeveloped by [[Ernst Schröder (mathematician)|Ernst Schröder]]. [[Peter Gustav Lejeune Dirichlet]] developed [[Dirichlet L-function|Dirichlet ''L''-functions]] to give the proof of [[Dirichlet's theorem on arithmetic progressions]] and began [[analytic number theory]]. In 1829, [[Carl Gustav Jacob Jacobi]] published ''[[Fundamenta nova theoriae functionum ellipticarum]]'' with his [[Elliptic function|elliptic]] [[theta function]]s.

[[Matrix notation]] would be more fully developed by [[Arthur Cayley]] in his three papers, on subjects which had been suggested by reading the ''[[Mécanique analytique]]''<ref>Mécanique analytique: [[iarchive:mcaniqueanalyti06lagrgoog|Volume 1]], [https://books.google.com/books?id=hclDyo8wE44C Volume 2]. By [[Joseph Louis Lagrange]]. Ms. [[Ve Courcier]], 1811.</ref> of Lagrange and some of the works of Laplace. Cayley defined [[matrix multiplication]] and [[matrix inverse]]s. Cayley used a single letter to denote a matrix,<ref>[https://books.google.com/books?id=tZdQAAAAYAAJ The collected mathematical papers of Arthur Cayley]. Volume 11. Page [https://books.google.com/books?id=tZdQAAAAYAAJ&pg=PA243 243].</ref> thus treating a matrix as an aggregate object. He also realized the connection between matrices and determinants,<ref>Historical Encyclopedia of Natural and Mathematical Sciences, Volume 1. By Ari Ben-Menahem. Pg [https://books.google.com/books?id=9tUrarQYhKMC&pg=PA2070 2070].</ref> and wrote "There would be many things to say about this theory of matrices which should, it seems to me, precede the theory of determinants."<ref name="Vitulli, Marie">[[Marie A. Vitulli|Vitulli, Marie]]. "A Brief History of Linear Algebra and Matrix Theory". Department of Mathematics. University of Oregon. Originally at: darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html</ref>

[[William Rowan Hamilton]] introduced the [[nabla symbol]] (<math> \nabla</math> or, later called ''del'', ∇) for [[vector differential]]s.<ref>The Words of Mathematics. By Steven Schwartzman. [https://archive.org/details/wordsofmathemati0000schw/page/6 6].</ref><ref>Electro-Magnetism: Theory and Applications. By A. Pramanik. [https://books.google.com/books?id=gnEEwy12S5cC&pg=PA38 38]</ref> This was previously used by Hamilton as a general-purpose [[Operator (mathematics)|operator sign]].<ref>[http://homepages.math.uic.edu/~hanson/math210/nabla98symbols.html History of Nabla and Other Math Symbols]. homepages.math.uic.edu/~hanson.</ref> <math>\hat{H}</math>, <math>H</math>, and <math>\check{H}</math> are used for the [[Hamiltonian (quantum mechanics)|Hamiltonian operator in quantum mechanics]] and <math>\mathcal{H}</math> (or ℋ ) for the Hamiltonian function in classical [[Hamiltonian mechanics]]. In mathematics, Hamilton is perhaps best known as the inventor of [[Classical Hamiltonian quaternions|quaternion notation]] and [[biquaternion]]s.

====Maxwell, Clifford, and Ricci notations====
[[File:James Clerk Maxwell.png|thumbnail|[[James Clerk Maxwell]]
----
Maxwell's most prominent achievement was to formulate a [[Maxwell's equations|set of equations]] that united previously unrelated observations, experiments, and equations of [[electricity]], [[magnetism]], and [[optics]] into a consistent theory.<ref>{{cite web |title=James Clerk Maxwell |url=http://www.ieeeghn.org/wiki/index.php/James_Clerk_Maxwell |access-date=25 March 2013 |publisher=IEEE Global History Network}}</ref>]]

In 1864 [[James Clerk Maxwell]] reduced all of the then-current knowledge of electromagnetism into a linked set of [[differential equation]]s with 20 equations in 20 variables, contained in ''[[A Dynamical Theory of the Electromagnetic Field]]''.<ref name=ADTEF>{{cite journal |doi=10.1098/rstl.1865.0008 |last=Maxwell |first=James Clerk |author-link=James Clerk Maxwell |title=A dynamical theory of the electromagnetic field | url=http://upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf |journal=Philosophical Transactions of the Royal Society of London |volume=155 |pages=459–512 |year=1865 |bibcode=1865RSPT..155..459M|s2cid=186207827 }} (This article accompanied an 8 December 1864 presentation by Maxwell to the Royal Society.)</ref> (See [[Maxwell's equations]].) The method of calculation that is necessary to employ was given by Lagrange, and afterwards developed, with some modifications, by [[Hamilton's equations]]. It is usually referred to as [[Hamilton's principle]]; when the equations in the original form are used, they are known as [[Lagrange's equations]]. In 1871 [[Richard Dedekind]] defined a [[field (mathematics)|field]] to be a set of real or complex numbers which is closed under the four arithmetic operations. In 1873 Maxwell presented ''[[A Treatise on Electricity and Magnetism]]''.

In 1878 [[William Kingdon Clifford]] published his ''[[Elements of Dynamic]]''.<ref>{{Internet Archive|elementsofdynami01clifiala|Books I, II, III (1878)}}; {{Internet Archive|elementsofdynami02clifiala|Book IV (1887)}}</ref> Clifford developed [[split-biquaternion]]s (e.g. [[History of quaternions|<math>q = w + xi + yj + zk </math>]]) which he called ''algebraic motors''. Clifford obviated quaternion study by separating the [[dot product]] and [[cross product]] of two vectors from the complete quaternion notation.

The common [[vector notation]]s are used when working with [[vector (geometric)|spatial]] vectors or more abstract members of [[vector space]]s, while [[angle notation]] (or [[phasor (sine waves)|phasor]] notation) is a notation used in [[electronics]].

[[Lord Kelvin]]'s [[Aether theories|aetheric]] [[atomic theory|atom theory]] (1860s) led [[Peter Guthrie Tait]], in 1885, to publish a [[topology|topological]] table of knots with up to ten crossings known as the [[Tait conjectures]]. [[Tensor calculus]] was developed by [[Gregorio Ricci-Curbastro]] between 1887 and 1896, presented in 1892 under the title ''Absolute differential calculus'',<ref>{{cite journal
 |first=G. |last=Ricci Curbastro
 |title=Résumé de quelques travaux sur les systèmes variables de fonctions associés à une forme différentielle quadratique
 |journal=Bulletin des Sciences Mathématiques
 |volume=2
 |pages=167–189
 |year=1892
 |issue=16
}}</ref> and the contemporary usage of "tensor" was stated by [[Woldemar Voigt]] in 1898.<ref>{{cite book |first=Woldemar |last=Voigt |title=Die fundamentalen physikalischen Eigenschaften der Krystalle in elementarer Darstellung |url=https://archive.org/details/bub_gb__Ps4AAAAMAAJ |publisher=Von Veit |place=Leipzig |year=1898}}</ref> In 1895, [[Henri Poincaré]] published ''[[Analysis Situs (paper)|Analysis Situs]]''.<ref>Poincaré, Henri, "Analysis situs", Journal de l'École Polytechnique ser 2, 1 (1895) pp. 1–123</ref> In 1897, [[Charles Proteus Steinmetz]] would publish [[:openlibrary:books/OL7218906M/Theory and calculation of alternating current phenomena|''Theory and Calculation of Alternating Current Phenomena'']], with the assistance of Ernst J. Berg.<ref>{{cite journal|last=Whitehead|first=John B. Jr.|title=Review: ''Alternating Current Phenomena'', by C. P. Steinmetz|journal=Bull. Amer. Math. Soc.|year=1901|volume=7|issue=9|pages=399–408|url=https://www.ams.org/journals/bull/1901-07-09/S0002-9904-1901-00825-7/S0002-9904-1901-00825-7.pdf|doi=10.1090/s0002-9904-1901-00825-7|doi-access=free}}</ref>

====From formula mathematics to tensors====
In 1895 [[Giuseppe Peano]] issued his ''[[Formulario mathematico]]'',<ref>
There are many editions. Here are two:
* (French) Published 1901 by Gauthier-Villars, Paris.  230p. [[:openlibrary:works/OL15255022W/Formulaire des mathematiques|OpenLibrary OL15255022W]], [https://archive.org/details/formulairedesmat00pean PDF].
* (Italian) Published 1960 by Edizione cremonese, Roma. 463p. [[:openlibrary:books/OL16587658M/Formulario mathematico|OpenLibrary OL16587658M]].</ref> an effort to digest mathematics into terse text based on special symbols. He would provide a definition of a [[vector space]] and [[linear map]]. He would also introduce the [[Intersection (set theory)|intersection sign]] (<math>\cap</math>), the [[union sign]] (<math>\cup</math>), the [[Element (mathematics)|membership sign]] (∈), and [[existential quantifier]] (∃). Peano would pass to [[Bertrand Russell]] his work in 1900 at a Paris conference; it so impressed Russell that he too was taken with the drive to render mathematics more concisely. The result was ''[[Principia Mathematica]]'' written with [[Alfred North Whitehead]]. This treatise marks a watershed in modern literature where symbol became dominant. Peano's ''Formulario Mathematico'', though less popular than Russell's work, continued through five editions. The fifth appeared in 1908 and included 4,200 formulas and theorems.

Ricci-Curbastro and [[Tullio Levi-Civita]] popularized the [[tensor index notation]] around 1900.<ref>{{citation|title=Méthodes de calcul différentiel absolu et leurs applications|last1=Ricci|first1=Gregorio|author-link=Gregorio Ricci-Curbastro|last2=Levi-Civita|first2=Tullio|journal=Mathematische Annalen|publisher=Springer|volume=54|issue=1–2|date=March 1900|pages=125–201|doi=10.1007/BF01454201|s2cid=120009332|url=https://zenodo.org/record/1428270}}</ref>

===Mathematical logic and abstraction===
{|align=right
|-
! style="color:#black; background:#f8eaba; font-size:100%; text-align:center;" colspan="2"|Abstraction
|-
|<gallery>
file:Felix_Klein.jpeg| Felix Klein
file:Georg Cantor2.jpg| Georg Cantor
</gallery>
|}
[[Georg Cantor]] introduced [[Aleph number]]s, so named because they use the aleph symbol (א) with natural number subscripts to denote [[cardinality]] in infinite sets. For the ordinals he employed the Greek letter ω ([[omega]]). This notation is still in use today in [[ordinal notation]] of a finite sequence of symbols from a finite alphabet that names an [[ordinal number]] according to some scheme which gives meaning to the language.

After the turn of the 20th century, [[Josiah Willard Gibbs]] introduced into [[physical chemistry]] the [[middle dot]] for [[dot product]] and the [[multiplication sign]] for [[cross product]]s. He also supplied notation for the scalar and vector products, which were introduced in ''[[Vector Analysis (Gibbs/Wilson)|Vector Analysis]]''. Bertrand Russell shortly afterward introduced [[logical disjunction]] (or) in 1906. [[Gerhard Kowalewski]] and [[Cuthbert Edmund Cullis]]<ref>{{cite book
| last1=Cullis | first1=Cuthbert Edmund | author-link1=Cuthbert Edmund Cullis
| title=Matrices and determinoids
| volume=2
| date=March 2013
| publisher=Cambridge University Press
| isbn=978-1-107-62083-4}}</ref><ref>[https://books.google.com/books?id=Sp0KAAAAYAAJ Can be assigned a given matrix]: About a class of matrices. (Gr. Ueber eine Klasse von Matrizen: die sich einer gegebenen Matrix zuordnen lassen.) by [[Issai Schur|Isay Schur]]</ref><ref>[https://archive.org/details/anintroductiont01cajogoog An Introduction To The Modern Theory Of Equations]. By Florian Cajori.</ref> introduced and helped standardized matrices notation, and parenthetical matrix and box matrix notation, respectively.

[[File:Einstein 1921 by F Schmutzer - restoration.jpg|thumbnail|Albert Einstein (1921)]]
[[Albert Einstein]], in 1916, introduced [[Einstein notation]], which summed over a set of [[index notation|indexed terms]] in a formula, thus exerting notational brevity. For example, for indices ranging over the [[Set (mathematics)|set]] {{math|{1, 2, 3}}},
:<math> y = \sum_{i=1}^3 c_i x^i = c_1 x^1 + c_2 x^2 + c_3 x^3 </math>
is reduced by convention to:
:<math> y = c_i x^i \,.</math><br />
Upper indices are not [[Exponentiation|exponents]] but are indices of coordinates, [[coefficient]]s, or [[basis vector]]s.

In 1917 [[Arnold Sommerfeld]] created the [[contour integral]] sign, and [[Dimitry Mirimanoff]] proposed the [[axiom of regularity]]. In 1919, [[Theodor Kaluza]] would solve [[general relativity]] equations using [[Five-dimensional space|five dimensions]], the results would have electromagnetic equations emerge.<ref>Proceedings of the Prussian Academy of Sciences (1918). Pg 966.</ref> This would be published in 1921 in "Zum Unitätsproblem der Physik".<ref>[https://archive.org/details/sitzungsberichte1921preussi Sitzungsberichte der Preussischen Akademie der Wissenschaften (1918)]
(Tr. Proceedings of the Prussian Academy of Sciences (1918)). archive.org; See also: [[Kaluza–Klein theory]].</ref> In 1922, [[Abraham Fraenkel]] and [[Thoralf Skolem]] independently proposed replacing the [[axiom schema of specification]] with the [[axiom schema of replacement]]. Also in 1922, [[Zermelo–Fraenkel set theory]] was developed. In 1923, Steinmetz would publish [https://archive.org/details/fourlecturesonre00stei ''Four Lectures on Relativity and Space'']. Around 1924, [[Jan Arnoldus Schouten]] developed the modern notation and formalism for the [[Ricci calculus]] framework during the absolute differential calculus applications to [[general relativity]] and [[differential geometry]] in the early twentieth century. Ricci calculus constitutes the rules of index notation and manipulation for [[tensors]] and [[tensor fields]].<ref>{{cite book |author1=Synge J.L. |author2=Schild A. |publisher=first Dover Publications 1978 edition |title=Tensor Calculus |pages=6–108|year= 1949}}</ref><ref>{{cite book |pages=[https://archive.org/details/gravitation00misn_003/page/n109 85]–86, §3.5|author1=J.A. Wheeler |author2=C. Misner |author3=K.S. Thorne | title=Gravitation| publisher=W.H. Freeman & Co| year=1973 | isbn=0-7167-0344-0|title-link=Gravitation (book) }}</ref><ref>{{cite book |author=R. Penrose| title=The Road to Reality| publisher= Vintage books| year=2007 | isbn=978-0-679-77631-4| title-link=The Road to Reality}}</ref><ref>{{cite book|last=Schouten|first=Jan A.|title=Der Ricci-Kalkül – Eine Einführung in die neueren Methoden und Probleme der mehrdimensionalen Differentialgeometrie (Ricci Calculus – An introduction in the latest methods and problems in multi-dimensional differential geometry)|language=de|year=1924|series=Grundlehren der mathematischen Wissenschaften|volume=10|editor= R. Courant|publisher=Springer Verlag|location=Berlin|url=http://resolver.sub.uni-goettingen.de/purl?PPN373339186}}</ref> In 1925, [[Enrico Fermi]] described a [[Fermi–Dirac statistics|system comprising many identical particles that obey the Pauli exclusion principle]], afterwards developing a [[diffusion equation]] ([[Fermi age equation]]). In 1926, [[Oskar Klein]] develop the [[Kaluza–Klein theory]]. In 1928, [[Emil Artin]] abstracted [[ring theory]] with [[Artinian ring]]s. In 1933, [[Andrey Kolmogorov]] introduces the [[Kolmogorov axioms]]. In 1937, [[Bruno de Finetti]] deduced the "[[Coherence (philosophical gambling strategy)|operational subjective]]" [[concept]].

====Mathematical symbolism====
{{See also|Category theory|Model theory|Table of logic symbols|Logic alphabet}}
Mathematical abstraction began as a process of extracting the underlying [[essence]] of a mathematical concept,<ref>Robert B. Ash. A Primer of Abstract Mathematics. Cambridge University Press, 1 Jan 1998</ref><ref>The New American Encyclopedic Dictionary. Edited by Edward Thomas Roe, Le Roy Hooker, Thomas W. Handford. Pg [https://books.google.com/books?id=KhQLAQAAMAAJ&pg=PA34 34]</ref> removing any dependence on real world objects with which it might originally have been connected,<ref>The Mathematical Principles of Natural Philosophy, Volume 1. By Sir Isaac Newton, John Machin. Pg 12.</ref> and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent [[phenomena]]. Two abstract areas of modern mathematics are [[category theory]] and [[model theory]]. Bertrand Russell<ref>In The Scientific Outlook (1931)</ref> once said, "Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say." Though, one can substitute mathematics for real world objects, and wander off through equation after equation, and can build a concept structure which has no relation to reality.<ref>Mathematics simplified and made attractive: or, The laws of motion explained. By Thomas Fisher. Pg 15. (cf. ''But an abstraction not founded upon, and not consonant with [[Nature (philosophy)|Nature]] and ''([[Logical truth|Logical]])'' [[Truth]], would be a [[Deception|falsity]], an [[insanity]].'')</ref>

Some of the introduced mathematical logic notation during this time included the set of symbols used in [[Boolean algebra (logic)|Boolean algebra]]. This was created by [[George Boole]] in 1854. Boole himself did not see logic as a branch of mathematics, but it has come to be encompassed anyway. Symbols found in Boolean algebra include <math>\land</math> (and), <math>\lor</math> (or), and <math>\lnot</math> (not). With these symbols, and letters to represent different [[truth value]]s, one can make logical statements such as <math>a\lor\lnot a=1</math>, that is "(''a'' is true or ''a'' is not true) is true", meaning it is true that ''a'' is either true or not true (i.e. false). Boolean algebra has many practical uses as it is, but it also was the start of what would be a large set of symbols to be used in logic. Most of these symbols can be found in [[propositional calculus]], a formal system described as <math>\mathcal{L} = \mathcal{L}\ (\Alpha,\ \Omega,\ \Zeta,\ \Iota)</math>. <math>\Alpha</math> is the set of elements, such as the ''a'' in the example with Boolean algebra above. <math>\Omega</math> is the set that contains the subsets that contain operations, such as <math>\lor</math> or <math>\land</math>. <math>\Zeta</math> contains the [[inference rules]], which are the rules dictating how inferences may be logically made, and <math>\Iota</math> contains the [[axioms]]. Predicate logic, originally called ''predicate calculus'', expands on propositional logic by the introduction of [[Variable (mathematics)|variables]], usually denoted by ''x'', ''y'', ''z'', or other lowercase letters, and by sentences containing variables, called [[Predicate (mathematical logic)|predicates]]. These are usually denoted by an uppercase letter followed by a list of variables, such as P(''x'') or Q(''y'',''z''). Predicate logic uses special symbols for [[Quantifiers (logic)|quantifiers]]: ∃ for "there exists" and ∀ for "for all".

====Gödel incompleteness notation====
{{See also|Proof sketch for Gödel's first incompleteness theorem}}

{{rquote|right|To every [[Omega-consistent|ω-consistent]] recursive class κ of ''formulae'' there correspond recursive ''class signs r'', such that neither ''v'' Gen ''r'' nor Neg (''v'' Gen ''r'') belongs to Flg (κ) (where ''v'' is the ''free variable'' of ''r'').| Kurt Gödel<ref>Proposition VI, ''[[On Formally Undecidable Propositions of Principia Mathematica and Related Systems|On Formally Undecidable Propositions in ''Principia Mathematica'' and Related Systems I]]'' (1931)</ref>}}
While proving [[Gödel's incompleteness theorems|his incompleteness theorems]], [[Kurt Gödel]] created an alternative to the symbols normally used in logic. He used [[Gödel number]]s—numbers assigned to represent mathematical operations—and variables with the prime numbers greater than 10. With Gödel numbers, a logic statement can be broken down into a number sequence. By taking the ''n'' prime numbers to the power of the Gödel numbers in the sequence, and then multiplying the terms together, a unique final product is generated. In this way, every logic statement can be encoded as its own number.<ref>Casti, John L. ''5 Golden Rules''. New York: MJF Books, 1996.</ref>

For example, take the statement "There exists a number ''x'' such that it is not ''y''". Using the symbols of propositional calculus, this would become
:<math>(\exists x)(x=\lnot y)</math>.

If the Gödel numbers replace the symbols, it becomes:

:<math>\{8, 4, 11, 9, 8, 11, 5, 1, 13, 9\}</math>.

There are ten numbers, so the first ten prime numbers are used:

:<math>\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29\}</math>.

Then, each prime is raised to the power of the corresponding Gödel number, and multiplied:

:<math>2^8\times3^4\times5^{11}\times7^9\times11^8\times13^{11}\times17^5\times19^1\times23^{13}\times29^9</math>.

The resulting number is approximately <math>3.096262735\times10^{78}</math>.

===Contemporary notation and topics===
====Early 20th-century notation====
The abstraction of notation is an ongoing process. The historical development of many mathematical topics exhibits a progression from the concrete to the abstract. Throughout 20th century, various [[set notation]]s were developed for fundamental object [[Set (mathematics)|sets]]. Around 1924, [[David Hilbert]] and [[Richard Courant]] published ''[[Methoden der mathematischen Physik|Methods of mathematical physics. Partial differential equations]]''.<ref>Gr. ''Methoden Der Mathematischen Physik''</ref> In 1926, [[Oskar Klein]] and [[Walter Gordon (physicist)|Walter Gordon]] proposed the [[Klein–Gordon equation]] to describe relativistic particles:

{{Equation box 1|indent =:|equation =<math> \frac {1}{c^2} \frac{\partial^2}{\partial t^2} \psi - \nabla^2 \psi + \frac {m^2 c^2}{\hbar^2} \psi = 0. </math> |cellpadding= 6|border = 0|border colour = black|background colour=white}}

The first formulation of a [[quantum mechanics|quantum theory]] describing radiation and matter interaction is due to [[Paul Adrien Maurice Dirac]], who, during 1920, was first able to compute the coefficient of spontaneous emission of an [[atom]].<ref name=dirac>
{{cite journal
 | author=P.A.M. Dirac
 | author-link= Paul Adrien Maurice Dirac
 | year=1927
 | title=The Quantum Theory of the Emission and Absorption of Radiation
 | journal=[[Proceedings of the Royal Society of London A]]
 | volume=114 | issue= 767
 | pages=243–265
 | doi=10.1098/rspa.1927.0039
| bibcode=1927RSPSA.114..243D
| doi-access=free
 }}</ref> In 1928, the [[relativistic wave equation|relativistic]] [[Dirac equation]] was formulated by Dirac to explain the behavior of the relativistically moving [[electron]]. The Dirac equation in the form originally proposed by Dirac is:

{{Equation box 1|indent =:|equation = <math>\left(\beta mc^2 + \sum_{k = 1}^3 \alpha_k p_k \, c\right) \psi (\mathbf{x},t) = i \hbar \frac{\partial\psi(\mathbf{x},t) }{\partial t} </math>|cellpadding= 6|border = 0|border colour = black|background colour=white}}

where, {{math|ψ {{=}} ψ('''x''', ''t'')}} is the [[wave function]] for the [[electron]], {{math|'''x'''}} and {{math|''t''}} are the space and time coordinates, {{math|''m''}} is the [[rest mass]] of the electron, {{math|''p''}} is the [[momentum]] (understood to be the [[momentum operator]] in the [[Schrödinger equation|Schrödinger theory]]), {{math|''c''}} is the [[speed of light]], and {{math|''ħ'' {{=}} ''h''/2''π''}} is the reduced [[Planck constant]]. Dirac described the quantification of the electromagnetic field as an ensemble of [[harmonic oscillator]]s with the introduction of the concept of [[creation and annihilation operators]] of particles. In the following years, with contributions from [[Wolfgang Pauli]], [[Eugene Wigner]], [[Pascual Jordan]], and [[Werner Heisenberg]], and an elegant formulation of quantum electrodynamics due to [[Enrico Fermi]],<ref name=fermi>
{{cite journal
 | author=E. Fermi
 | author-link= Enrico Fermi
 | year=1932
 | title=Quantum Theory of Radiation
 | journal=[[Reviews of Modern Physics]]
 | volume=4 | issue= 1
 | pages=87–132
 | doi=10.1103/RevModPhys.4.87
| bibcode=1932RvMP....4...87F
}}
</ref> physicists came to believe that, in principle, it would be possible to perform any computation for any physical process involving photons and charged particles.

In 1931, [[Alexandru Proca]] developed the [[Proca equation]] ([[Euler–Lagrange equation]]) for the vector [[meson]] theory of [[nuclear force]]s and the [[relativistic quantum field equations]]. [[John Archibald Wheeler]] in 1937 developed the [[S-matrix]]. Studies by [[Felix Bloch]] with [[Arnold Nordsieck]],<ref name=bloch>{{cite journal
 | author1=F. Bloch
 | author-link1= Felix Bloch
 | author2=A. Nordsieck
 | author-link2= Arnold Nordsieck
 | year=1937
 | title=Note on the Radiation Field of the Electron
 | journal=[[Physical Review]]
 | volume=52 | issue= 2
 | pages=54–59
 | doi=10.1103/PhysRev.52.54
| bibcode=1937PhRv...52...54B
}}</ref> and [[Victor Weisskopf]],<ref name=weisskopf>{{cite journal
 | author=V. F. Weisskopf
 | author-link= Victor Weisskopf
 | year=1939
 | title=On the Self-Energy and the Electromagnetic Field of the Electron
 | journal=[[Physical Review]]
 | volume=56 | issue= 1
 | pages=72–85
 | doi=10.1103/PhysRev.56.72
| bibcode=1939PhRv...56...72W
}}</ref> in 1937 and 1939, revealed that such computations were reliable only at a first order of [[Perturbation theory (quantum mechanics)|perturbation theory]], a problem already pointed out by [[Robert Oppenheimer]].<ref name=oppenheimer>{{cite journal
 | author=R. Oppenheimer
 | author-link= Robert Oppenheimer
 | year=1930
 | title=Note on the Theory of the Interaction of Field and Matter
 | journal=[[Physical Review]]
 | volume=35 | issue= 5
 | pages=461–477
 | doi=10.1103/PhysRev.35.461
| bibcode=1930PhRv...35..461O
}}</ref> Infinities emerged at higher orders in the series, making such computations meaningless and casting serious doubts on the internal consistency of the theory itself. With no solution for this problem known at the time, it appeared that a fundamental incompatibility existed between [[special relativity]] and [[quantum mechanics]].

In the 1930s, the double-struck capital Z (<math>\mathbb{Z}</math>) for integer number sets was created by [[Edmund Landau]]. [[Nicolas Bourbaki]] created the double-struck capital Q (<math>\mathbb{Q}</math>) for rational number sets. In 1935 Gerhard Gentzen made [[universal quantifier]]s. [[André Weil]] and [[Nicolas Bourbaki]] would develop the [[empty set]] sign (∅) in 1939. That same year, [[Nathan Jacobson]] would coin the double-struck capital C (<math>\mathbb{C}</math>) for [[complex number]] sets.

Around the 1930s, [[Voigt notation]] (so named to honor Voigt's 1898 work) would be developed for [[multilinear algebra]] as a way to represent a [[symmetric tensor]] by reducing its order. [[Schoenflies notation|Schönflies notation]] became one of two conventions used to describe [[point group]]s (the other being [[Hermann–Mauguin notation]]). Also in this time, [[van der Waerden notation]]<ref>{{cite journal|title=Spinoranalyse|author=Van der Waerden B.L.|journal=Nachr. Ges. Wiss. Göttingen Math.-Phys.|volume=1929|year=1929|pages=100–109}}</ref><ref>{{cite journal|title=Geometry of two-component Spinors|author=Veblen O.|journal=Proc. Natl. Acad. Sci. USA|volume=19|issue=4|year=1933|pages=462–474|doi=10.1073/pnas.19.4.462 |pmc=1086023|bibcode=1933PNAS...19..462V|pmid=16577541|doi-access=free}}</ref> became popular for the usage of two-component [[spinor]]s ([[Weyl spinor]]s) in four spacetime dimensions. [[Arend Heyting]] would introduce [[Heyting algebra]] and [[Heyting arithmetic]].

The arrow (→) was developed for [[function notation]] in 1936 by [[Øystein Ore]] to denote images of specific elements and to denote [[Galois connection]]s. Later, in 1940, it took its present form (''f: X→Y'') through the work of [[Witold Hurewicz]]. [[Werner Heisenberg]], in 1941, proposed the [[S-matrix theory]] of particle interactions.

[[File:P.A.M. Dirac at the blackboard.jpg|thumbnail|[[Paul Dirac]] made fundamental contributions to the early development of both quantum mechanics and [[quantum electrodynamics]].]]
[[Bra–ket notation]] ([[Dirac notation]]) is a standard notation for describing [[quantum state]]s, composed of [[bracket|angle brackets]] and [[vertical bar]]s. It can also be used to denote abstract [[vector space|vectors]] and [[linear functional]]s. It is so called because the [[inner product]] (or [[dot product]] on a complex vector space) of two states is denoted by a {{langle}}bra|ket{{rangle}}: <math>\langle\phi|\psi\rangle</math>. The notation was introduced in 1939 by [[Paul Dirac]],<ref name=Dirac>{{cite journal
|first=P.A.M.
|last=Dirac
|authorlink=Paul Dirac
|title=A new notation for quantum mechanics
|journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]]
|year=1939
|volume=35|issue=3|pages=416–418|doi=10.1017/S0305004100021162
|url=http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2031476|bibcode=1939PCPS...35..416D|s2cid=121466183
|url-access=subscription
}}
</ref> though the notation has precursors in [[Hermann Grassmann|Grassmann]]'s use of the notation [''φ''|''ψ''] for his inner products nearly 100 years previously.<ref name=Grassmann>{{cite book
|author=H. Grassmann
|title=Extension Theory
|series=History of Mathematics Sources
|publisher=American Mathematical Society, London Mathematical Society, 2000 translation by Lloyd C. Kannenberg
|year=1862}}</ref>

Bra–ket notation is widespread in [[quantum mechanics]]: almost every phenomenon that is explained using quantum mechanics—including a large portion of [[modern physics]]—is usually explained with the help of bra–ket notation. The notation establishes an encoded abstract representation-independence, producing a versatile specific representation (e.g., ''x'', or ''p'', or [[eigenfunction]] base) without much ado, or excessive reliance on, the [[Nature (philosophy)|nature]] of the [[linear space]]s involved. The overlap expression ⟨''φ''|''ψ''⟩ is typically interpreted as the [[probability amplitude]] for the [[Quantum state|state]] ''[[ψ]]'' to [[wavefunction collapse|collapse]] into the state ''[[Φ#Use as a symbol|ϕ]]''. The [[Feynman slash notation]] (Dirac slash notation<ref>{{citation
 |first=Steven
 |last=Weinberg
 |year=1964
 |title=The quantum theory of fields, Volume 2
 |publisher=Cambridge University Press, 1995
 |isbn=0-521-55001-7
 |url=https://books.google.com/books?id=3ws6RJzqisQC&q=%22Dirac%20Slash%22&pg=PA358
 |page=358
}}</ref>) was developed by [[Richard Feynman]] for the study of [[Dirac field]]s in [[quantum field theory]].

[[Geoffrey Chew]], along with others, would promote matrix notation for the [[strong interaction]] in particle physics, and the associated [[Bootstrapping (physics)|bootstrap principle]], in 1960. In the 1960s, [[set-builder notation]] was developed for describing a [[Set (mathematics)|set]] by stating the properties that its members must satisfy. Also in the 1960s, tensors are abstracted within [[category theory]] by means of the concept of [[monoidal category]]. Later, [[multi-index notation]] eliminates conventional notions used in [[multivariable calculus]], [[partial differential equation]]s, and the theory of [[distribution (mathematics)|distributions]], by abstracting the concept of an integer [[index notation|index]] to an ordered [[tuple]] of indices.

====Modern mathematical notation====
{{See also|Approximation theory|Universal property|Tensor algebra|Free algebra|Abstract algebra}}
In the modern mathematics of [[special relativity]], [[electromagnetism]], and [[Wave|wave theory]], the [[d'Alembert operator]] (<math>\scriptstyle\Box</math>) is the [[Laplace operator]] of [[Minkowski space]]. The [[Levi-Civita symbol]] ({{Mvar|ε}}), also known as the permutation symbol, is used in [[tensor calculus]].

[[Feynman diagrams]] are used in particle physics, equivalent to the [[Operator (physics)|operator]]-based approach of [[Sin-Itiro Tomonaga]] and [[Julian Schwinger]]. The [[orbifold notation]] system, invented by [[William Thurston]], has been developed for representing types of [[symmetry groups]] in two-dimensional spaces of constant curvature.

The [[tetrad formalism]] ([[tetrad (index notation)|tetrad index notation]]) was introduced as an approach to [[general relativity]] that replaces the choice of a [[coordinate basis]] by the less restrictive choice of a local basis for the tangent bundle (a locally defined set of four linearly independent [[vector field]]s called a [[Tetrad (general relativity)|tetrad]]).<ref>{{citation | last1=De Felice|first1=F.|last2=Clarke|first2=C.J.S. |title=Relativity on Curved Manifolds| year=1990|page=133}}</ref>

In the 1990s, [[Roger Penrose]] proposed [[Penrose graphical notation]] ([[tensor diagram notation]]) as a, usually handwritten, visual depiction of [[multilinear function]]s or [[tensor]]s.<ref>"Quantum invariants of knots and 3-manifolds" by V. G. Turaev (1994), page 71</ref> Penrose also introduced [[abstract index notation]]. His usage of the Einstein summation was in order to offset the inconvenience in describing [[tensor contraction|contractions]] and [[covariant derivative|covariant differentiation]] in modern abstract tensor notation, while maintaining explicit [[covariance]] of the expressions involved.{{fact|date=August 2024}}

[[File:John H Conway 2005 (cropped).jpg|thumb|[[John Horton Conway|John Conway]], prolific mathematician of notation]]
[[John Horton Conway|John Conway]] furthered various notations, including the [[Conway chained arrow notation]], the [[Conway notation (knot theory)|Conway notation of knot theory]], and the [[Conway polyhedron notation]]. The [[Coxeter notation]] system classifies symmetry groups, describing the angles between with fundamental reflections of a [[Coxeter group]]. It uses a bracketed notation, with modifiers to indicate certain subgroups. The notation is named after [[H. S. M. Coxeter]]; [[Norman Johnson (mathematician)|Norman Johnson]] more comprehensively defined it.

[[Combinatorics|Combinatorial]] [[LCF notation]], devised by [[Joshua Lederberg]] and extended by [[Harold Scott MacDonald Coxeter]] and [[Robert Frucht]], was developed for the representation of [[cubic graph]]s that are [[Hamiltonian path|Hamiltonian]].<ref>{{citation
 | last1 = Pisanski | first1 = Tomaž | author1-link = Tomaž Pisanski
 | last2 = Servatius | first2 = Brigitte | author2-link = Brigitte Servatius
 | contribution = 2.3.2 Cubic graphs and LCF notation
 | isbn = 978-0-8176-8364-1
 | page = 32
 | publisher = Springer
 | title = Configurations from a Graphical Viewpoint
 | url = https://books.google.com/books?id=bnh2zkuTZr4C&pg=PA32
 | year = 2013}}</ref><ref>{{citation|last=Frucht|first=R.|title=A canonical representation of trivalent Hamiltonian graphs|journal=[[Journal of Graph Theory]]|volume=1|pages=45–60|year=1976|issue=1|doi=10.1002/jgt.3190010111}}</ref> The [[cycle notation]] is the convention for writing down a [[permutation]] in terms of its constituent [[cyclic permutation|cycles]].<ref>Fraleigh 2002:89; Hungerford 1997:230</ref> This is also called [[circular notation]] and the permutation called a ''cyclic'' or ''circular'' permutation.<ref>Dehn, Edgar. Algebraic Equations, Dover. 1930:19</ref>

====Computers and markup notation====
{{Main|History of computing|Timeline of computing}}
{{See also|Symbolic computation|Symbolic dynamics|Computational complexity theory|Mathematical markup language|MathML|Basic Linear Algebra Subprograms|Numerical linear algebra|List of numerical libraries|List of numerical-analysis software|DOT language|Lisp (programming language)|Object-oriented programming|Earley algorithm}}

In 1931, [[IBM]] produces the [[List of IBM products|IBM 601 Multiplying Punch]]; it is an electromechanical machine that could read two numbers, up to eight digits long, from a card and punch their product onto the same card.<ref>{{cite web|url=http://www.columbia.edu/cu/computinghistory/601.html |title=The IBM 601 Multiplying Punch |publisher=Columbia.edu |access-date=2014-06-24}}</ref> In 1934, [[Wallace Eckert]] used a rigged IBM 601 Multiplying Punch to automate the integration of differential equations.<ref>{{cite web|url=http://www.columbia.edu/cu/computinghistory/switch.html |title=Interconnected Punched Card Equipment |publisher=Columbia.edu |date=1935-10-24 |access-date=2014-06-24}}</ref>

In 1962, [[Kenneth E. Iverson]] developed an integral part notation, which became known as Iverson notation, that developed into [[APL (programming language)|APL]].<ref name=sourcebook>{{citation|editor-last=McDonnell|editor-first=Eugene|editor-link=Eugene McDonnell|title=A Source Book in APL, Introduction|publisher=APL Press|date=1981|url=https://code.jsoftware.com/wiki/Doc/A_Source_Book_in_APL|access-date=19 April 2016}}</ref> In the 1970s within [[computer architecture]], [[Quote notation]] was developed for a representing number system of [[rational number]]s. Also in this decade, the [[Z notation]] (just like the [[APL (programming language)|APL language]], long before it) uses many non-[[ASCII]] symbols, the specification includes suggestions for rendering the Z notation symbols in ASCII and in [[LaTeX]]. There are presently various [[C mathematical functions]] (Math.h) and numerical [[Library (computing)|libraries]] used to perform [[numerical analysis|numerical]] calculations in [[software development]]. These calculations can be handled by [[symbolic execution]]s—analyzing a program to determine what inputs cause each part of a program to execute. [[Mathematica]] and [[SymPy]] are examples of computational software programs based on [[symbolic mathematics]].

==References and citations==
;General
*[[Florian Cajori]] (1929) ''A History of Mathematical Notations'', 2 vols. Dover reprint in 1 vol., 1993. {{isbn|0-486-67766-4}}.
;Citations
{{reflist}}

==Further reading==
;General:
* [https://books.google.com/books?id=U9zuAAAAMAAJ A Short Account of the History of Mathematics]. By [[Walter William Rouse Ball]].
* [https://books.google.com/books?id=fYS4AAAAIAAJ A Primer of the History of Mathematics]. By Walter William Rouse Ball.
* [https://archive.org/details/cu31924002955130 A History of Elementary Mathematics]: With Hints on Methods of Teaching. By Florian Cajori.
* [https://archive.org/details/ahistoryelement01cajogoog A History of Elementary Mathematics]. By Florian Cajori.
* [https://archive.org/details/ahistorymathema01cajogoog A History of Mathematics]. By Florian Cajori.
* [https://archive.org/details/bub_gb_9d8DAAAAMAAJ A Short History of Greek Mathematics]. By [[James Gow (scholar)|James Gow]].
* [https://books.google.com/books?id=it3UAAAAMAAJ On the Development of Mathematical Thought During the Nineteenth Century]. By [[John Theodore Merz]].
* [https://archive.org/details/newmathematicalp00barluoft A New Mathematical and Philosophical Dictionary]. By Peter Barlow.
* [https://books.google.com/books?id=e1UuAAAAYAAJ Historical Introduction to Mathematical Literature]. By [[George Abram Miller]]
* [https://archive.org/details/abriefhistoryma01smitgoog A Brief History of Mathematics]. By [[Karl Fink]], [[Wooster Woodruff Beman]], [[David Eugene Smith]]
* [https://books.google.com/books?id=yUJBAAAAYAAJ History of Modern Mathematics]. By David Eugene Smith.
* [https://books.google.com/books?id=SlUNAAAAYAAJ History of modern mathematics]. By David Eugene Smith, [[Mansfield Merriman]].

;Other
* Principia Mathematica, [https://archive.org/details/principlesofmath01russ Volume 1] & [https://books.google.com/books?id=sbTVAAAAMAAJ Volume 2]. By Alfred North Whitehead, Bertrand Russell.
* [https://archive.org/details/bub_gb_exwAAAAAQAAJ The Mathematical Principles of Natural Philosophy], Volume 1, Issue 1. By Sir Isaac Newton, Andrew Motte, William Davis, John Machin, William Emerson.
* [https://archive.org/details/generalinvestig00hiltgoog General investigations of curved surfaces of 1827 and 1825]. By Carl Friedrich Gaus.

==External links==
* [http://www.stephenwolfram.com/publications/mathematical-notation-past-future Mathematical Notation: Past and Future]
* [https://web.archive.org/web/20060821061824/http://www.roma.unisa.edu.au/07305/symbols.htm History of Mathematical Notation]
* [http://jeff560.tripod.com/mathsym.html Earliest Uses of Mathematical Notation]
* [https://web.archive.org/web/20121104103101/http://www.files.chem.vt.edu/chem-dept/field/numbers.htm Finger counting]. files.chem.vt.edu.
* [https://www.math.ucdavis.edu/~anne/WQ2007/mat67-Common_Math_Symbols.pdf Some Common Mathematical Symbols and Abbreviations (with History)]. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling.

{{History of mathematics}}

[[Category:History of mathematics|Mathematical notation]]
[[Category:Mathematical notation]]
[[Category:Articles which contain graphical timelines]]