Point to Point Lee Model

2012-12-01T18:55:19Z

79.173.196.173: /* The frequency correction factor */

The '''history of mathematical notation'''<ref>[[Florian Cajori]]. A History of Mathematical Notations: Two Volumes in One. Cosimo, Inc., Dec 1, 2011</ref> includes the [[Provenance|commencement]], [[Progress (history)|progress]], and [[cultural diffusion]] of [[mathematical symbol]]s and the [[Conflict (process)|conflict]] the methods of notation confronted in a notation's move to [[popularity]] or inconspicuousness. [[Mathematical notation]]<ref>A Dictionary of Science, Literature, & Art, Volume 2. Edited by [[William Thomas Brande]], [[George William Cox]]. Pg [http://books.google.com/books?id=J5tPAAAAMAAJ&pg=PA683 683]</ref> comprises the [[symbol]]s used to write mathematical [[equation]]s and [[formula]]s. Notation generally implies a set of [[well-defined]] representations of quantities and symbols operators.<ref>Notation -- from Wolfram MathWorld http://mathworld.wolfram.com/Notation.html</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 host of symbols invented by mathematicians over the past several centuries.

The development of mathematical notation can be divided in stages.<ref>Diophantos of Alexandria: A Study in the History of Greek Algebra. By Sir Thomas Little Heath. Pg [http://books.google.com/books?id=ABkPAAAAIAAJ&pg=PA77 77].</ref><ref>Mathematics: Its Power and Utility. By Karl J. Smith. Pg [http://books.google.com/books?id=-0x2JszrkooC&pg=PA86 86].</ref> The "''[[rhetorical]]''" stage is where calculations are performed by words 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> The "''[[syncopated]]''" stage is where frequently used operations and quantities are represented by symbolic [[syntax|syntactical]] abbreviations. From ancient times through the post-classical age,<ref group="note">Or the Middle Ages.</ref> 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. The "''symbolic''" stage is where comprehensive systems of notation supersede rhetoric. Beginning in Italy in the 16th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day. This symbolic system was in use by medieval Indian mathematicians and in Europe since the middle of the 17th century,<ref>[[Solomon Gandz]]. "The Sources of al-Khowarizmi's Algebra"</ref> and has continued to develop in the [[contemporary era]].

The area of study known as the [[history of mathematics]] is primarily an investigation into the origin of discoveries in mathematics and, the focus here, the investigation into the mathematical methods and notation of the past.

{{details3|[[List of mathematical symbols]]|the list of symbols}}

==History==
{{see also|Timeline of mathematics|Foundations of mathematics}}
:''For more details on particular notations, see'': [[Arithmetic]] ([[History of arithmetic]]), [[Algebra]] ([[History of algebra]]), [[Geometry]] ([[History of geometry]]), [[Trigonometry]] ([[History of trigonometry]]), [[Calculus]] ([[History of calculus]])

{{History of science sidebar|startcollapsed=true}}

===Rhetorical stage===
{{expand section|date=July 2013}}
Although the history commences with that of the [[#Acrophonic and Milesian numeration|Ionian schools]], there is no doubt that those [[Ancient Greek]]s who paid attention to it were largely indebted to the previous investigations of the [[Ancient Egyptian]]s and [[Phoenicia|Ancient Phoenicians]]. Numerical notation distinctive feature, i.e. symbols having local as well as intrinsic values ([[arithmetic]]), implies a state of [[civilization]] at the period of its invention. Our knowledge of the mathematical attainments of these early peoples, to which this section is devoted, is imperfect and the following brief notes be regarded as a summary of the conclusions which seem most probable, and the history of mathematics begins with the symbolic sections.

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]].

There can be no doubt that most early peoples which have 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 Egyptians paid attention to geometry and numbers, and the Phoenicians to practical arithmetic, [[book-keeping]], [[navigation]], and land-surveying. The [[Knowledge transfer|results attained by these people seem to have been accessible]], under certain conditions, to travelers. 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.

: ''See also:'' ''[[Mensuration]] ([[Number]]s and [[Real number]]s)''

====Beginning of notation====
{{main|History of writing numbers}}

Written mathematics began with numbers expressed as [[tally marks]], with each tally representing a single unit. The numerical symbols consisted probably of strokes or notches cut in wood or stone, and intelligible alike to all nations.<ref group="note">Such characters, in fact, are preserved with little alteration in the [[Roman notation]], an account of which may be found in [[John Leslie (physicist)|John Leslie]]'s Philosophy of Arithmetic.</ref> For example, one notch in a bone represented one animal, or person, or anything else. 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: and Greek tradition uniformly assigned the special development of geometry to the Egyptians, and that of the [[number theory|science of numbers]]<ref group="note">[[Number theory]] is branch of pure mathematics devoted primarily to the [[Integer|study of the integers]]. Number theorists study [[prime number]]s as well as the properties of objects made out of integers (e.g., [[rational number]]s) or defined as generalizations of the [[integer]]s (e.g., [[algebraic integer]]s).</ref> either to the Egyptians or to the Phoenicians.

The [[Ancient Egyptians]] had a symbolic notation which was the [[numeration by Hieroglyphics]].<ref>Encyclopædia Americana. By Thomas Gamaliel Bradford. Pg [http://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 [http://books.google.com/books?id=GTgTnSGMukgC&pg=PA186 186]</ref> The [[Egyptian mathematics]] had a symbol 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 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 [[Mesopotamians]] had symbols for each power of ten.<ref>[http://www.metu.edu.tr/~beyaz/303/presentations/egyptMesopotamia.pdf Mathematics in Egypt and Mesopotamia]</ref> Later, they wrote their numbers in almost exactly the same way done in modern times. Instead of having symbols for each power of ten, they would just put the [[coefficient]] of that number. Each digit was at separated by only a space, but by the time of [[Alexander the Great]], they had created a symbol that represented zero and was a placeholder. The Mesopotamians also used a [[sexagesimal]] system, that is base sixty. It is this system that is used in modern times when measuring time and angles. Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s.<ref>Boyer, C. B. A History of Mathematics, 2nd ed. rev. by 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 script]], tablets were inscribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun. Some of these appear to be graded homework. 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 the Babylonian numerals also date back to this period.<ref>Duncan J. Melville (2003). [http://it.stlawu.edu/~dmelvill/mesomath/3Mill/chronology.html Third Millennium Chronology], ''Third Millennium Mathematics''. [[St. Lawrence University]].</ref>

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|regular]] [[Multiplicative inverse|reciprocal]] [[Twin prime|pairs]].<ref>{{cite book | authorlink = 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 √2 accurate to five 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 x 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.

{{see also|Ancient history|History of writing ancient numbers|History of science in early cultures}}

===Syncopated stage===
[[File:Domenico-Fetti Archimedes 1620.jpg|thumb|''[[Archimedes of Syracuse|Archimedes]] Thoughtful'' by [[Domenico Fetti|Fetti]] (1620)
----
The last words attributed to Archimedes are "[[Noli turbare circulos meos|Do not disturb my circles]]",<ref group="note">{{lang-el|μή μου τοὺς κύκλους τάραττε}}</ref> a reference to the circles in the mathematical drawing that he was studying when disturbed by the Roman soldier.]]

The history of mathematics cannot with certainty be traced back to any school or period before that of the Ionian Greeks, but 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 from its commencement to be deductive and scientific. 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 group="note">That is, <math>a^2 + b^2 = c^2</math>.</ref> The ancient mathematical texts are available with the prior mentioned Ancient Egyptians notation and with [[Plimpton 322]] (Babylonian mathematics c. 1900 BC). The study of mathematics as a subject in its own right begins in the 6th century BC with the [[Pythagoreans]], who coined the term "mathematics" from the ancient Greek ''μάθημα'' (''mathema''), meaning "subject of instruction".<ref>{{cite book|author=Heath|title=A Manual of Greek Mathematics|page=5}}</ref>

[[Plato]]'s influence was been 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 Mathematics''<ref name="Encyclopaedia Perthensi" /> is what treats of magnitude<ref group="note">[[Magnitude (mathematics)]], the relative size of an object ; [[Magnitude (vector)]], a term for the size or length of a vector; [[Scalar (mathematics)]], a quantity defined only by its magnitude; [[Euclidean vector]], a quantity defined by both its magnitude and its direction; [[Order of magnitude]], the class of scale having a fixed value ratio to the preceding class.</ref> or [[quantity]], absolutely and generally conferred, without regard to any species of particular magnitude, such as [[Arithmetic]] and [[Geometry]], In this sense, abstract mathematics is opposed to [[applied mathematics|mixed mathematics]]; wherein simple and abstract properties, and the relations of quantities primitively considered in mathematics, are applied to sensible objects, and by that means become intermixed with physical considerations; Such are [[Hydrostatics]], [[Optics]], [[Navigation]], &c.<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. 287 212 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 |accessdate=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 |author=O'Connor, J.J. and 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|accessdate= 2007-08-07| archiveurl= http://web.archive.org/web/20070715191704/http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html| archivedate= 15 July 2007 | deadurl= no}}</ref> He also defined the [[Archimedes spiral|spiral]] bearing his name, formulae for the [[volume]]s 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|Euclid's Elements
----
The prop. 31, 32 and 33 of the book of Euclid XI, which is located in vol. 2 of the manuscript, the sheets 207 to - 208 recto.]]
In the historical development of geometry, the steps in the abstraction of geometry were made by the ancient Greeks. [[Euclid's Elements]] being the earliest extant documentation of the axioms of plane geometry— though Proclus tells of an earlier [[Axiomatic system|axiomatisation]] by [[Hippocrates of Chios]].<ref>[http://www-gap.dcs.st-and.ac.uk/~history/Extras/Proclus_history_geometry.html Proclus' Summary]</ref> Euclid's ''Elements'' (c. 300 BC) is one of the oldest extant Greek mathematical treatises<ref group="note">[[Autolycus]]' [[On the Moving Sphere]] is another ancient mathematical manuscript of the time.</ref> and consisted of 13 books written in Alexandria; collecting theorems proven by other mathematicians, supplemented by some original work.<ref group="note">[[Proclus]], a Greek mathematician who lived several centuries after Euclid, wrote in his commentary of the Elements: "Euclid, who put together the Elements, collecting many of [[Eudoxus of Cnidus|Eudoxus]]' theorems, perfecting many of [[Theaetetus (mathematician)|Theaetetus]]', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors".</ref> The document is a successful collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. [[Euclid's lemma|Euclid's first theorem]] is a [[Lemma (mathematics)|lemma]] that possesses properties of [[prime number]]s. The influential thirteen books cover Euclidean geometry, geometric algebra, and the ancient Greek version of algebraic systems and elementary number theory. It was ubiquitous in the [[Quadrivium]] and is instrumental in the development of logic, mathematics, and science.

[[Diophantus of Alexandria]] was author of a series of books called ''[[Arithmetica]]'', many of which are now lost. These texts deal with solving [[algebraic equation]]s. [[Boethius]] provided a place for mathematics in the curriculum in the 6th century when he coined the term ''quadrivium'' to describe the study of arithmetic, geometry, astronomy, and music. He wrote ''De institutione arithmetica'', a free translation from the Greek of [[Nicomachus]]'s ''Introduction to Arithmetic''; ''De institutione musica'', also derived from Greek sources; and a series of excerpts from Euclid's ''Elements''. His works were theoretical, rather than practical, and were the basis of mathematical study until the recovery of Greek and Arabic mathematical works.<ref>Caldwell, John (1981) "The ''De Institutione Arithmetica'' and the ''De Institutione Musica''", pp. 135–54 in Margaret Gibson, ed., ''Boethius: His Life, Thought, and Influence,'' (Oxford: Basil Blackwell).</ref><ref>Folkerts, Menso, ''"Boethius" Geometrie II'', (Wiesbaden: Franz Steiner Verlag, 1970).</ref>

{{see also|Fundamental theorem of arithmetic}}
:''See also:'' ''[[Naive set theory]] and [[Set_theory#Axiomatic_set_theory|Axiomatic set theory]]''

====Acrophonic and Milesian numeration====

The [[ancient Greece|Greeks]] employed [[Attic numeration]],<ref>Mathematics and Measurement By Oswald Ashton Wentworth Dilk. Pg [http://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 vertical lines, as in the hieroglyphics. The symbol for five was the Greek letter Π (pi), which is the letter of the Greek word for five, ''pente''. (This is not to be confused with the constant π which is the ratio of the circumference of a circle to its diameter. Greek mathematicians did not have a formal name for that constant.) Numbers six through nine were ''pente'' with vertical lines next to it. Ten was represented by the letter (Δ) of the word for ten, ''deka'', one hundred by the letter from the word for hundred, etc.

The [[Ionian numeration]] used their entire alphabet including three archaic letters. The numeral notation of the Greeks, though far less convenient than that now in use, was formed on a perfectly regular and scientific plan,<ref name="books.google.com">A dictionary of science, literature and art, ed. by W.T. Brande. Pg [http://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 system was totally inapplicable. The Greeks divided the twenty-four letters of their alphabet into three classes, and, by adding another symbol to each class, they had characters to represent the units, tens, and hundreds. ([[Jean Baptiste Joseph Delambre]]'s Astronomie Ancienne, t. ii.)

{| class=wikitable
|-
|[[Α]] (α)
|[[Β]] (β)
|[[Г]] (γ)
|[[Δ]] (δ)
|[[Ε]] (ε)
|[[Ϝ]] (ϝ)
|[[Z]] (ζ)
|[[H]] (η)
|[[θ]] (θ)
|[[I]] (ι)
|[[K]] (κ)
|[[Λ]] (λ)
|[[Μ]] (μ)
|[[Ν]] (ν)
|[[Ξ]] (ξ)
|[[Ο]] (ο)
|[[Π]] (π)
|[[Ϟ]] (ϟ)
|[[Ρ]] (ρ)
|[[Sigma|Σ]] (σ)
|[[Τ]] (τ)
|[[Υ]] (υ)
|[[Ф]] (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 upper case letters, the lower case 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. M (for μὐριοι, 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>

Greek mathematical reasoning was almost entirely [[geometry|geometric]] (albeit often used to reason about non-geometric subjects such as [[number theory]]), and hence the Greeks had no interest in [[algebra]]ic symbols. The great exception was [[Diophantus]] of [[Alexandria]], the great algebraist.<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. February 16, 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. An unknown number was called s.<ref>A History of Greek Mathematics: From Aristarchus to Diophantus. By Sir Thomas Little Heath. Pg [http://books.google.com/books?id=7DDQAAAAMAAJ&pg=PA456 456]</ref> 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>A History of Greek Mathematics: From Aristarchus to Diophantus. By Sir Thomas Little Heath. Pg [http://books.google.com/books?id=7DDQAAAAMAAJ&pg=PA458 458]</ref><ref group="note">The expression: <math>2x^4+3x^3-4x^2+5x-6</math> would be written as: SS2 C3 x5 M S4 u6 {{citation needed|date=July 2013}}.</ref>

====Chinese mathematical notation====
{{Main|Suzhou numerals}}
[[File:Huama numerals.svg|thumb|The numbers 0–9 in Chinese huāmǎ (花碼) numerals]]
The Chinese used numerals that look much like the tally system.<ref>The American Mathematical Monthly, Volume 16. Pg [http://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, the [[Numerals_in_Unicode#CJK_Suzhou_.28hu.C4.81m.C7.8E.29_numerals|huāmǎ system]] is only used for displaying prices in Chinese markets or on traditional handwritten invoices.

In the history of the Chinese, there were those who were familiar with the sciences of arithmetic, geometry, mechanics, optics, navigation, and astronomy. [[Mathematics in China]] emerged independently by the 11th century BC.<ref>[http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Chinese_overview.html Overview of Chinese mathematics. www-groups.dcs.st-and.ac.uk/~history/]</ref> It is indeed almost certain that the Chinese were acquainted with several geometrical or rather architectural implements;<ref group="note">such as the [[Ruler|rule]], [[Try square|square]], [[compasses]], [[spirit level|water level]] ([[reed level]]), and [[plumb-bob]].</ref> with mechanical machines;<ref group="note">such as the [[wheel]] and [[axle]]</ref> that they knew of the characteristic property of the magnetic needle; and were aware that astronomical events occurred in cycles. Chinese of that time had made attempts to classify or extend the rules of arithmetic or geometry which they knew, and to explain the causes of the phenomena with which they were acquainted beforehand. 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]].

[[File:Chounumerals.jpg|thumb|right|Counting rod numerals]]
[[Counting rods|Chinese mathematics]] made early contributions, including a [[place value system]].<ref>George Gheverghese Joseph, ''The Crest of the Peacock: Non-European Roots of Mathematics'',Penguin Books, London, 1991, pp.140—148</ref><ref>Georges Ifrah, ''Universalgeschichte der Zahlen'', Campus, Frankfurt/New York, 1986, pp.428—437</ref> The geometrical theorem known to the ancient Chinese were acquainted was applicable in certain cases (namely the ratio of sides).<ref group="note">The area of the square described on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the squares described on the sides</ref> It is that geometrical theorems which can be demonstrated in the quasi-experimental way of superposition were also known to them. In arithmetic their knowledge seems to have been confined to the art of calculation by means of the [[swan-pan]], and the power of expressing the results in writing. Our knowledge of the early attainments of the Chinese, slight though it is, is more complete than in the case of most of their contemporaries. It is thus instructive, and serves to illustrate the fact, that it can be known a nation may possess considerable skill in the applied arts with but our knowledge of the later mathematics on which those arts are founded can be scarce. Knowledge of Chinese mathematics before 254 BC is somewhat fragmentary, and even after this date the manuscript traditions are obscure. Dates centuries before the classical period are generally considered conjectural by Chinese scholars unless accompanied by verified archaeological evidence.

As in other early societies the focus was on [[astronomy]] in order to perfect the agricultural [[calendar]], and other practical tasks, and not on establishing [[formal systems]].The [[Chinese Board of Mathematics]] duties were confined to the annual preparation of an almanac, the dates and predictions in which it regulated. Ancient Chinese mathematicians did not develop an axiomatic approach, but made advances in algorithm development and algebra. The achievement of Chinese algebra reached its zenith in the 13th century, when [[Zhu Shijie]] invented method of four unknowns.

As a result of obvious linguistic and geographic barriers, as well as content, Chinese mathematics and that of the mathematics of the ancient Mediterranean world are presumed to have developed more or less independently up to the time when ''[[The Nine Chapters on the Mathematical Art]]'' reached its final form, while the ''[[Writings on Reckoning]]'' 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]]. The [[Pythagorean_theorem#History|Pythagorean theorem]] for example, [[Zhou Bi Suan Jing|has been attested]] to the time of the [[Duke of Zhou]]. Knowledge of [[Pascal's triangle]] has also been shown to have existed in China centuries before [[Blaise Pascal|Pascal]],<ref>[http://www.psupress.psu.edu/books/titles/0-271-01238-2.html Frank J. Swetz and T. I. Kao: Was Pythagoras Chinese?]</ref> such as by [[Shen Kuo]].

[[File:Shen Kua.JPG|thumb|Modern artist's impression of [[Shen Kuo]].]]
The state of [[trigonometry]] in China slowly began to change and advance during the Song Dynasty (960–1279), where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendarical science and astronomical calculations.<ref name="needham volume 3 109"/> The [[polymath]] Chinese scientist, mathematician and official [[Shen Kuo]] (1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs.<ref name="needham volume 3 109">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> Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the basis for [[spherical trigonometry]] developed in the 13th century by the mathematician and astronomer [[Guo Shoujing]] (1231–1316).<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">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><ref name="gauchet 151">[[Marcel Gauchet]], 151.</ref> The mathematical science of the Chinese would incorporate the work and teaching of Arab missionaries with knowledge of spherical trigonometry who had come to China in the course of the thirteenth century.

{{see also|Chinese numerals}}

====Indian mathematical notation====
Although the origin of our present system of numerical notation is ancient, there is no doubt that it was in use among the Hindus over two thousand years ago. The algebraic notation of the [[Indian mathematics|Indian mathematician]], [[Brahmagupta]], was syncopated. Addition was indicated by placing the numbers side by side, subtraction by placing a dot over the subtrahend, 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">Boyer, C. B. A History of Mathematics, 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989 ISBN 0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7). "China and India" p. 221. (cf., "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> 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>"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 http://www-history.mcs.st-and.ac.uk/HistTopics/Indian_numerals.html</ref>

{{see also|History of writing numbers}}

==== The Hindu-Arabic numerals and notations ====
{{Main|History of the Hindu-Arabic numeral system}}
[[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 actually started 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|Mohommed 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'', written about 825, along with the work of [[Al-Kindi]],<ref group="note">Al-Kindi also introduced [[cryptanalysis]] and [[frequency analysis]].</ref> were instrumental in spreading [[Indian mathematics]] and [[Hindu-Arabic numeral system|Indian 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'').

[[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> Al-Khwārizmī gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,<ref>Boyer, C. B. A History of Mathematics, 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989 ISBN 0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7). "The Arabic Hegemony" p. 230. (cf., "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was al-Khwārizmī's exposition that his readers must have had little difficulty in mastering the solutions.")</ref> and Al-Khwārizmī was to teach algebra in an [[Elementary algebra|elementary form]] and for its own sake.<ref>Gandz and Saloman (1936), ''The sources of Khwarizmi's algebra'', Osiris i, pp. 263–77: "In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers".</ref> Al-Khwārizmī also discussed the fundamental method of "[[Reduction (mathematics)|reduction]]" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which al-Khwārizmī originally described as ''al-jabr''.<ref name=Boyer-229>Boyer, C. B. A History of Mathematics, 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989 ISBN 0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7). "The Arabic Hegemony" p. 229. (cf., "It is not certain just what the terms ''al-jabr'' and ''muqabalah'' mean, but the usual interpretation is similar to that implied in the translation above. The word ''al-jabr'' presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word ''muqabalah'' is said to refer to "reduction" or "balancing" - that is, the cancellation of like terms on opposite sides of the equation.")</ref> His algebra was also no longer concerned "with a series of [[problem]]s to be resolved, but an [[Expository writing|exposition]] which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." Al-Khwārizmī also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."<ref name=Rashed-Armstrong>{{Cite book | last1=Rashed | first1=R. | last2=Armstrong | first2=Angela | year=1994 | title=The Development of Arabic Mathematics | publisher=[[Springer Science+Business Media|Springer]] | isbn=0-7923-2565-6 | oclc=29181926 | pages=11–12}}</ref>

[[Al-Karaji]], in his treatise ''al-Fakhri'', extends the methodology to incorporate integer powers and integer roots of unknown quantities.<ref group="note">Something close to a [[Mathematical proof|proof]] by [[mathematical induction]] appears in a book written by Al-Karaji around 1000 AD, who used it to prove the [[binomial theorem]], [[Pascal's triangle]], and the sum of [[integral]] [[Cube (algebra)|cubes]].</ref><ref>Victor J. Katz (1998). ''History of Mathematics: An Introduction'', pp. 255–59. [[Addison-Wesley]]. ISBN 0-321-01618-1.</ref> The [[historian]] of mathematics, F. Woepcke,<ref>F. Woepcke (1853). ''Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi''. [[Paris]].</ref> praised Al-Karaji for being "the first who introduced the [[theory]] of [[algebra]]ic [[calculus]]." Also in the 10th century, [[Abul Wafa]] translated the works of [[Diophantus]] into Arabic. [[Ibn al-Haytham]] would develop [[analytic geometry]]. Al-Haytham derived the formula for the sum of the fourth powers, using a method that is readily generalizable for determining the general formula for the sum of any integral powers. Al-Haytham performed an integration in order to find the volume of a [[paraboloid]], and was able to generalize his result for the integrals of [[polynomial]]s up to the [[Quartic polynomial|fourth degree]].<ref group="note">He thus came close to finding a general formula for the [[integral]]s of polynomials, but he was not concerned with any polynomials higher than the fourth degree.</ref><ref name=Katz>Victor J. Katz (1995), "Ideas of Calculus in Islam and India", ''Mathematics Magazine'' '''68''' (3): 163–74.</ref> In the late 11th century, [[Omar Khayyam]] would develop [[algebraic geometry]], wrote ''Discussions of the Difficulties in Euclid'',<ref group="note">a book about what he perceived as flaws in [[Euclid's Elements|Euclid's ''Elements'']], especially the [[parallel postulate]]</ref> and wrote on the general geometric solution to [[cubic equation]]s. [[Nasir al-Din Tusi]] (Nasireddin) made advances in [[spherical trigonometry]]. Muslim mathematicians during this period include the addition of the [[decimal point]] notation to the [[Arabic numerals]].

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<ref group="note">translated into Latin by [[Robert of Chester]]</ref> and the complete text of [[Euclid's Elements|Euclid's ''Elements'']].<ref group="note">translated in various versions by [[Adelard of Bath]], [[Herman of Carinthia]], and [[Gerard of Cremona]]</ref><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 the mathematical problem Fibonacci wrote in it about a population of rabbits. The growth of the population ended up being a [[Fibonacci sequence]], where a term is the sum of the two preceding terms.

[[Abū al-Hasan ibn Alī al-Qalasādī]] (1412–1482) was the last major medieval [[Mathematics in medieval Islam|Arab algebraist]], who improved on the [[Mathematical notation|algebraic notation]] earlier used by [[Ibn al-Yasamin|Ibn al-Yāsamīn]] in the 12th century{{Citation needed|date=May 2010}} and, in the [[Maghreb]], by [[Ibn al-Banna]] in the 13th century.<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>Boyer, C. B. A History of Mathematics, 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989 ISBN 0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7). "Revival and Decline of Greek Mathematics" p. 178 (cf., "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 to have symbols for these functions and was thus "the first steps toward the introduction of algebraic symbolism." He represented [[Table of mathematical symbols|mathematical symbols]] using characters from the [[Arabic alphabet]].<ref name=Qalasadi/>

===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) #
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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
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Period = from:1360 till:1962
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align:center textcolor:black fontsize:8 mark:(line,black) width:15

bar:  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:  color:age width:5
from:1360 till:1618 shift:(0,-10) text:Arithmetic
from:1618 till:1718 shift:(0,-10) text:Multiplicaton
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}}

[[File:Johannes Widmann-Mercantile Arithmetic 1489.jpg|thumbnail|The 1489 use of the [[plus and minus signs]] in print.]]
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 widely used arithmetic symbols are addition and subtraction, + and −. The [[plus sign]] was used by 1360 by [[Nicole Oresme]]<ref>Mathematical Magazine, Volume 1. Artemas Martin, 1887. [http://books.google.com/books?id=mG8yAQAAMAAJ&pg=PA124 Pg 124]</ref><ref group="note">His own personal use started around 1351.</ref> in his work ''[[Algorismus proportionum]]''.<ref>[http://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 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]]''.<ref>''Later [[early modern]] version'': [http://books.google.com/books?id=jdw2AAAAMAAJ 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. By [[Michael Walsh (1801)|Michael Walsh]]. [[Edmund M. Blunt]] (proprietor.), 1801.</ref> Widmann used the minus symbol with the plus symbol, to indicate deficit and surplus, respectively.<ref>Miller, Jeff. "Earliest Uses of Symbols of Operation." 4 June 2006. Gulf High School. 24 September 2006 <http://jeff560.tripod.com/operation.html>.</ref> In ''[[Summa de Arithmetica]]'',<ref group="note">Summa de Arithmetica: Geometria Proportioni et Proportionalita. ''Tr''. Sum of Arithmetic: Geometry in proportions and proportionality.</ref><ref>Arithmetical Books from the Invention of Printing to the Present Time. By [[Augustus De Morgan]]. p[http://books.google.com/books?id=YSUQAAAAYAAJ&pg=PA2 2].</ref> [[Luca Pacioli]] used symbols for [[plus and minus]] symbols and contained [[algebra]].<ref group="note">Much of the work originated from [[Piero Della Francesca]] whom he [[dict:appropriated|appropriated]] and [[dict:purloined|purloined]].</ref>

In the 15th century, [[Ghiyath al-Kashi]] computed the value of [[π]] to the 16th decimal place. Kashi also had an algorithm for calculating ''n''th roots.<ref group="note">This was a special case of the methods given many centuries later by [[Paolo Ruffini|Ruffini]] and [[William George Horner|Horner]].</ref> In 1533, [[Regiomontanus]]'s table of sines and cosines were published.<ref>{{cite book | last = Grattan-Guinness | first = Ivor | year = 1997 | title = The Rainbow of Mathematics: A History of the Mathematical Sciences | publisher = W.W. Norton | isbn = 0-393-32030-8}}</ref> [[Scipione del Ferro]] and [[Niccolò Fontana Tartaglia]] discovered solutions for [[cubic equation]]s. [[Gerolamo Cardano]] published them in his 1545 book ''[[Ars Magna (Gerolamo Cardano)|Ars Magna]]'', together with a solution for the [[quartic equation]]s, discovered by his student [[Lodovico Ferrari]]. The [[Nth root|radical]] symbol<ref group="note">That is, <math>\sqrt{~}</math>.</ref> for square root was introduced by [[Christoph Rudolff]].<ref group="note">Because, it is thought, it resembled a lowercase "r" (for "[[radix]]").</ref> [[Michael Stifel]]'s important work ''[[Arithmetica integra]]''<ref>[http://books.google.com/books?id=fndPsRv08R0C Arithmetica integra]. By [[Michael Stifel]], [[Philipp Melanchthon|Philipp Melanchton]]. [[Norimbergæ]]:[[Apud Iohan Petreium]], 1544.</ref> contained important innovations in mathematical notation. In 1556, [[Nicolo Tartaglia]] used parentheses for precedence grouping. In 1557 [[Robert Recorde]] published [[The Whetstone of Witte]] which used the equal sign (=) as well as plus and minus signs for the English reader. In 1564, [[Gerolamo Cardano]] analyzed [[game of chance|games of chance]] beginning the early stages of [[probability theory]]. In 1572 [[Rafael Bombelli]] published his ''L'Algebra'' 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. For navigation and accurate maps of large areas, [[trigonometry]] grew to be a major branch of mathematics. [[Bartholomaeus Pitiscus]] coin the word "trigonometry", publishing his ''Trigonometria'' in 1595.

[[John Napier]] is best known as the inventor of [[logarithms]]<ref group="note">Published in [[Description of the Marvelous Canon of Logarithms]]</ref><ref>The History of Mathematics By Anne Roone. [http://books.google.com/books?id=5O67iqeIHZ8C&pg=PA40 Pg 40]</ref> and made common the use of the [[decimal point]] in arithmetic and mathematics.<ref>[http://books.google.com/books?id=husGAAAAYAAJ Memoirs of John Napier of Merchiston]. By Mark Napier</ref><ref>[http://books.google.com/books?id=JUdkAAAAMAAJ An Account of the Life, Writings, and Inventions of John Napier, of Merchiston]. By David Stewart Erskine Earl of Buchan, Walter Minto</ref> After Napier, [[Edmund Gunter]] created the [[logarithmic scale]]s (lines, or rules) upon which [[slide rules]] are based, it was [[William Oughtred]] who used two such scales sliding by one another to perform direct [[multiplication]] and [[division (mathematics)|division]]; and he is credited as the inventor of the slide rule in 1622. In 1631 Oughtred introduced the multiplication sign (×) his proportionality sign,<ref group="note">That is, <div style="font-size:200%;">∷</div></ref> and abbreviations ''sin'' and ''cos'' for the [[sine]] and [[cosine]] functions.<ref>{{cite book | title = A History of Mathematics | author = [[Florian Cajori]] | year = 1919 | publisher = Macmillan | url = http://books.google.com/?id=bBoPAAAAIAAJ&pg=PA157}}</ref> [[Albert Girard]] also used the abbreviations 'sin', 'cos' and 'tan' for the [[trigonometric functions]] in his treatise.

[[Johannes Kepler]] was one of the pioneers of the mathematical applications of [[infinitesimals]].<ref group="note">see [[Law of Continuity]].</ref> [[René Descartes]] is credited as the father of [[analytical geometry]], the bridge between algebra and geometry,<ref group="note">Using [[Cartesian coordinates]] on the plane, the distance between two points (''x''1, ''y''1) and (''x''2, ''y''2) is defined by the formula: 
<math>d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2},\!</math> 
which can be viewed as a version of the [[Pythagorean theorem]].</ref> 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.<ref group="note">Further steps in abstraction were taken by [[Lobachevsky]], [[Bolyai]], [[Riemann]], and [[Carl Friedrich Gauss|Gauss]] who generalised the concepts of geometry to develop [[non-Euclidean geometry|non-Euclidean geometries]].</ref> [[Blaise Pascal]] influenced mathematics throughout his life. His ''Traité du triangle arithmétique'' ("Treatise on the Arithmetical Triangle") of 1653 described a convenient tabular presentation for [[binomial coefficient]]s.<ref group="note">Now called [[Pascal's triangle]].</ref> [[Pierre de Fermat]] and Blaise Pascal would investigate [[probability]].<ref group="note">For example, the "[[problem of points]]".</ref> [[John Wallis]] introduced the [[infinity symbol]].<ref group="note">That is, <math>{\infty}</math>.</ref> He similarly used this notation for infinitesimals.<ref group="note">For example, <math>\frac{1}{\infty}.</math></ref> In 1657, [[Christiaan Huygens]] published the treatise on probability, ''On Reasoning in Games of Chance''.<ref group="note">Original title, "''De ratiociniis in ludo aleae''"</ref><ref>[[Jan Gullberg]], Mathematics from the birth of numbers, W. W. Norton & Company; ISBN 978-0-393-04002-9 . pg 963-965,</ref>

[[Johann Rahn]] introduced the [[division symbol]] ([[obelus]]) and the [[therefore sign]] in 1659. [[William Jones (mathematician)|William Jones]] used π in ''[[Synopsis palmariorum mathesios]]''<ref>[http://books.google.com/books?id=b59EAAAAcAAJ Synopsis Palmariorum Matheseos]. By [[William Jones (mathematician)|William Jones]]. 1706. (Alt: [http://archive.org/details/SynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics Synopsis Palmariorum Matheseos: or, a New Introduction to the Mathematics]. archive.org.)</ref> in 1706 because it is the 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 [http://books.google.com/books?id=U1ovBsSRNscC&pg=PR18 18].</ref>

{{see also|Probability|Statistics|Notation in probability and statistics}}
{{see also|History of probability|History of statistics|Scientific revolution}}

===== Derivatives notation: Leibniz and Newton =====
{{Dablink|See also: [[Derivative#Newton's notation|Derivative: Newton's notation]], [[Derivative#Leibniz's notation|Derivative: Leibniz's notation]]}}

{|align=right
|-
! style="color:#black; background:#f8eaba; font-size:100%; text-align:center;" colspan="2"|[[Notation for differentiation|''Derivative'' notations]]
|-
|<gallery>
file:GodfreyKneller-IsaacNewton-1689.jpg|Sir Isaac Newton
file:Gottfried Wilhelm von Leibniz.jpg|Gottfried Wilhelm Leibniz
</gallery>
|}

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 the creators: that developed by [[Isaac Newton]] and the notation developed by [[Gottfried Leibniz]]. Leibniz's is the notation used most often today. Newton's was simply a dot or dash placed above the function.<ref group="note">For example, the derivative of the function ''x'' would be written as <math>\dot{x}</math>. The second derivative of ''x'' would be written as <math>\ddot{x}</math>, etc.</ref> 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 ''d'' as a prefix to indicate differentiation, and introduced the notation representing derivatives as if they were a special type of fraction.<ref group="note">For example, the derivative of the function ''x'' with respect to the variable ''t'' in Leibniz's notation would be written as <math>{ dx \over dt }</math>.</ref> This notation makes explicit the variable with respect to which the derivative of the function is taken. Leibniz also created the integral symbol.<ref group="note">That is, <math>\int_{-N}^{N} f(x)\, dx</math>.</ref> The symbol is an [[long s|elongated S]], representing the [[Latin]] word ''Summa'', meaning "sum". 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 elongated s, for sum.

{{further|Leibniz–Newton calculus controversy}}
{{see also|Leibniz's notation|History of calculus}}

==== High division operators and functions ====
{{see also|modern age}}
[[Letters of the alphabet]] in this time were to be used as symbols of [[quantity]]; and although much diversity existed with respect to the choice of letters, there were to be several [[Open standard|universally recognized rules]] in the following history.<ref name="books.google.com"/> Here thus in the history of equations the first letters of the alphabet were indicatively known as [[coefficients]], the last letters the [[dict:unknown|unknown term]]s (an ''[[Unknown_known#Unknown_knowns|incerti ordinis]]''). In [[algebraic geometry]], again, a similar rule was to be observed, the last letters of the alphabet there denoting the variable or current [[coordinates]]. Certain letters, such as <math>\pi</math>, <math>e</math>, [[etc.]], were by [[consensus gentium|universal consent]] appropriated as symbols of the frequently occurring numbers [[Pi|3.14159 ...]], and [[e (mathematical constant)|2.7182818 ....]],<ref group="note">See also: [[List of representations of e]]</ref> etc., and their use in any other acceptation was to be avoided as much as possible.<ref name="books.google.com"/> Letters, too, were to be employed as symbols of operation, and with them other previously menition arbitrary operation characters. The letters [[d|<math>d</math>]], [[long s|elongated <math>S</math>]] were to be appropriated as operative symbols in the [[differential calculus]] and [[integral calculus]], [[Delta (letter)|<math>\Delta</math>]] and ∑ in the [[calculus of differences]].<ref name="books.google.com"/> 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="books.google.com"/><ref group="note">Thus <math>f(x)</math> denotes the [[Result|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 were 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.</ref>

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]].<ref group="note">That is, <math>\Delta f(p) </math></ref> In 1750, [[Gabriel Cramer]] developed "''[[Cramer's Rule]]''" for solving [[linear system]]s. The "international [[mile]]" of 1760 international [[yard]]s is exactly 1609.344 [[metres]].<ref>1,760 yards × 0.9144 m/yard, according to the [http://www.legislation.gov.uk/ukpga/1985/72/schedules Weights & Measures Act 1985]. Schedule I, Part VI</ref> The [[kilometre]],<ref name="OED">{{cite book | title=The Compact Edition of the Oxford English Dictionary | publisher=Oxford University Press | year=1971 | pages=695}}</ref> a unit of [[length]], first appeared in English in 1810<ref>{{cite web | url=http://www.oed.com/view/Entry/103403 | title=The Oxford English Dictionary | accessdate=July 13, 2012}}</ref> By 1866, the "[[kilometer]]s per hour" [[Systems of measurement|compound unit]] of [[speed]] was in use in the US.<ref group="note">See also: [[Non-SI units mentioned in the SI]]</ref><ref>{{cite book | title=Journal of the Franklin Institute of the State of Pennsylvania for the Promotion of the Mechanic Arts | publisher=Franklin Institute | author=Frazer, John F. | year=1866 | location=Philadelphia | pages=314 | url=http://books.google.com/books?id=kKIqAQAAIAAJ&pg=PA314&lpg=PA314&dq=%22kilometers+per+hour%22&source=bl&ots=Osx40Oy1Na&sig=RQHVQQeN1oWkgnlX5rgjd6kowpM&hl=en&sa=X&ei=pET-T47dJ6782gWCxPnKDw&ved=0CDkQ6AEwAA#v=onepage&q=%22kilometers%20per%20hour%22&f=false | volume=LII | number=5 | month=November}}</ref>

=====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 ''[[e (mathematical constant)|e]]'' to represent the base of [[natural logarithm]]s. It is not known exactly why <math>e</math> was chosen, but it was probably because the four letters of the alphabet were already commonly used to represent variables and other constants. Euler used <math>\pi</math> to represent [[pi]] consistently. 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,<ref group="note">That is, <math>\sqrt{-1}</math></ref> although he earlier used it as an ''infinite number.'' <ref group="note">Today, the symbol created by [[John Wallis]], <math>\infty</math>, is used for infinity.</ref><ref group="note">As in, <math>\sum_{n=1}^\infty\frac{1}{n^2}</math></ref> For [[summation]], Euler used [[sigma]], [[Capital-sigma notation|Σ]].<ref group="note">[[Capital-sigma notation]] uses a symbol that compactly represents summation of many similar terms: the ''summation symbol'', ''[[Sigma|∑]]'', an enlarged form of the upright capital Greek letter [[Sigma (letter)|Sigma]]. 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''; ''ai'' 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 ''"i = m"'' under the summation symbol means that the index ''i'' starts out equal to ''m''. The index, ''i'', is incremented by 1 for each successive term, stopping when ''i'' = ''n''.</ref> For [[function (mathematics)|functions]], Euler used the notation <math>f(x)</math> to represent a function of <math>x</math>. In 1730, Euler wrote the [[gamma function]].<ref group="note">That is, <math>n!=\int_{0}^{1}(-\ln s)^{n}\,{\rm d}s\,,</math>. valid for n > 0.</ref> In 1736, Euler produces his paper on the [[Seven Bridges of Königsberg]]<ref>Euler, Leonhard, [http://www.math.dartmouth.edu/~euler/docs/originals/E053.pdf Solutio problematis ad geometriam situs pertinentis]</ref> regarding [[topology]].

The [[William Emerson (mathematician)|mathematician, William Emerson]]<ref>[http://books.google.com/books?id=x_82AAAAMAAJ The elements of geometry]. By William Emerson</ref> would develop the [[proportionality sign]].<ref group="note">That is, <div style="font-size:200%;">∝</div></ref><ref group="note">Proportionality is the [[ratio]] of one quantity to another, especially the ratio of a part compared to a whole. In a mathematical context, a proportion is the statement of equality between two ratios; See [[Proportionality (mathematics)]], the relationship of two variables whose ratio is constant. See also [[aspect ratio]], geometric proportions.</ref><ref>[http://books.google.com/books?id=YLxHMwEACAAJ The Doctrine of Proportion, Arithmetical and Geometrical. Together with a General Method of Arening by Proportional Quantities]. By William Emerson.</ref><ref>The Mathematical Correspondent. By George Baron. [http://books.google.com/books?id=IXkAAAAAMAAJ&pg=PA83 83]</ref> Much later in the abstract expressions of the value of various proportional phenomena, the [[parts-per notation]] would became 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.<ref group="note">The ''[[Curl (mathematics)|curly d]]'' or ''[[Jacobi's delta]]''.</ref> In 1771, [[Alexandre-Théophile Vandermonde]] deduced the importance of topological features when discussing the [[Knot theory|properties of knots]] related to the geometry of position. Between 1772 and 1788, [[Joseph-Louis Lagrange]] re-formulated the formulas and calculations of Classical "Newtonian" mechanics, called [[Lagrangian mechanics]]. The [[prime symbol]] for derivatives was also made by Lagrange.

{{rquote|right|But in our [[opinion]] [[truths]] of this kind should be drawn from notions rather than from notations.|Carl Friedrich Gauss<ref group="note">About the proof of [[Wilson's theorem]]. ''[[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 contributed [[function (mathematics)|functions]] of [[complex variable]]s, in [[geometry]], and on the convergence of [[series (mathematics)|series]]. He gave the satisfactory proofs of the [[fundamental theorem of algebra]] and of the [[quadratic reciprocity law]]. Gauss developed the theory of solving linear systems by using [[Gaussian elimination]], which was initially listed as an advancement in [[geodesy]].<ref>{{cite web|last=Vitulli|first=Marie|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|accessdate=2012-01-24}}</ref> He would also develop the [[product sign]]. Also in this time, [[Niels Henrik Abel]] and [[Évariste Galois]]<ref group="note">[[Galois theory]] and [[Galois geometry]] is named after him.</ref> conducted their work on the [[solvability of equations]], linking [[group theory]] and [[Field theory (mathematics)|field theory]].

After the 1800s, [[Christian Kramp]] would promote [[factorial]] notation during his research in generalized factorial function which applied to non-integers.<ref>Kramp biography http://www-history.mcs.st-and.ac.uk/Biographies/Kramp.html</ref> [[Joseph Diaz Gergonne]] introduced the [[set inclusion]] signs.<ref group="note">That is, "subset of" and "superset of"; This would later be redeveloped by [[Ernst Schröder]].</ref> [[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]].<ref group="note">A [[number theory|science of numbers]] that uses methods from [[mathematical analysis]] to solve problems about the integers.</ref> In 1828, Gauss proved his [[Theorema Egregium]] (''remarkable theorem'' in Latin), establishing property of surfaces. In the 1830s, [[George Green]] developed [[Green's function]]. In 1829. [[Carl Gustav Jacob Jacobi]] publishes [[Fundamenta nova theoriae functionum ellipticarum]] with his [[Elliptic function|elliptic]] [[theta function]]s. By 1841, [[Karl Weierstrass]], the "father of modern [[mathematical analysis|analysis]]", elaborated on the concept of [[absolute value]] and the [[determinant of a matrix]].

[[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: [http://books.google.com/books?id=Q8MKAAAAYAAJ Volume 1], [http://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>[http://books.google.com/books?id=tZdQAAAAYAAJ The collected mathematical papers of Arthur Cayley]. Volume 11. Page [http://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 [http://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">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>

{{rquote|right|[... The mathematical quaternion] has, or at least involves a reference to, four dimensions.|William Rowan Hamilton<ref group="note">quoted in [[Robert Percival Graves]]' "''Life of Sir William Rowan Hamilton''" (3 volumes, 1882, 1885, 1889)</ref>}}
[[William Rowan Hamilton]] would introduce the [[nabla symbol]]<ref group="note">That is, <math> \nabla</math> (or, later called ''del'', ∇)</ref> for [[vector differential]]s.<ref>The Words of Mathematics. By Steven Schwartzman. [http://books.google.com/books?id=SRw4PevE4zUC&pg=PA6 6].</ref><ref>Electro-Magnetism: Theory and Applications. By A. Pramanik. [http://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> Hamilton reformulated [[Newtonian mechanics]], now called [[Hamiltonian mechanics]]. This work has proven central to the modern study of classical field theories such as [[electromagnetism]]. This was also important to the development of [[quantum mechanics]].<ref group="note">See [[Hamiltonian (quantum mechanics)]].</ref> In mathematics, he is perhaps best known as the inventor of [[Classical Hamiltonian quaternions|quaternion notation]]<ref group="note">That is, [[History of quaternions|<math>i^2=j^2=k^2=ijk=-1</math>]]''</ref> and [[biquaternion]]s. Hamilton also introduced the word "[[tensor]]" in 1846.<ref>{{cite journal
|first=William Rowan |last=Hamilton
|title=On some Extensions of Quaternions
|url=http://www.emis.de/classics/Hamilton/ExtQuat.pdf
|journal=Philosophical Magazine
|year=1854–1855
|pages=492–499, 125–137, 261–269, 46–51, 280–290
|editor-first=David R.|editor-last=Wilkins
|issue=7–9
|issn=0302-7597
}}</ref><ref group="note">Though his use describes something different from what is now meant by a tensor. Namely, the [[norm (mathematics)|norm operation]] in a certain type of algebraic system (now known as a [[Clifford algebra]]).</ref> [[James Cockle (lawyer)|James Cockle]] would develop the [[tessarine]]s<ref group="note">That is, <math>t = w + x i + y j + z k, \quad w, x, y, z \in \mathbb{R}</math> where <math> i j = j i = k, \quad i^2 = -1, \quad j^2 = +1 .</math></ref> and, in 1849, [[coquaternion]]s. In 1848, [[James Joseph Sylvester]] introduced into [[matrix algebra]] the term [[Matrix (mathematics)|matrix]].<ref group="note">This is Latin for "womb".</ref>

=====Maxwell, Clifford, and Ricci notations=====
[[File:James Clerk Maxwell.png|thumbnail|right|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 |accessdate=25 March 2013 |publisher=IEEE Global History Network}}</ref> 
See also: [[History of Maxwell's equations]].]]
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 |authorlink=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 |format=PDF |journal=Philosophical Transactions of the Royal Society of London |volume=155 |pages=459–512 |year=1865}} (This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.)</ref> The method of calculation which it 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, he presented the ''[[Remarks on the mathematical classification of physical quantities]]''.<ref>Proceedings of the London Mathematical Society, Volume 3. [[London Mathematical Society]], 1871. [http://books.google.com/books?id=lekKAAAAYAAJ&pg=PA224 Pg. 224]</ref> Also in 1871, [[Richard Dedekind]] called a set of real or complex numbers which is closed under the four arithmetic operations a "field".

In 1878, [[William Kingdon Clifford]] publishes his [[Elements of Dynamic]].<ref>{{Internet Archive|elementsofdynami01clifiala|Books I, II, III (1878)}}; {{Internet Archive|elementsofdynami02clifiala|Book IV (1887)}}</ref> Clifford would develop [[split-biquaternion]]s,<ref group="note">That is, [[History of quaternions|<math>q = w + xi + yj + zk \!</math>]]''</ref> which he called ''algebraic motors''. Clifford [[Elimination theory|eliminated]] quaternion study by separating the [[dot product]] and [[cross product]] of two vectors from the complete quaternion notation.<ref group="note">Clifford intersected algebra with Hamilton's quaternions by replacing [[Hermann Grassmann]]'s rule ''epep'' = 0 by the rule ''epep'' = 1. For more details, see [[exterior algebra]].</ref> This approach made [[vector calculus]] available to engineers and others working in [[three dimensions]] and [[skeptical]] of the [[Phase (waves)|lead–lag effect]]<ref group="note">See: [[Phasor]], [[Group (mathematics)]], [[Signal velocity]], [[Polyphase system]], [[Harmonic oscillator]], and [[RLC series circuit]]</ref> in the [[Four-dimensional space|fourth dimension]].<ref group="note">Or the concept of a fourth spatial dimension. See also: [[Spacetime]], the unification of time and space as a four-dimensional [[Continuum (measurement)|continuum]]; and, [[Minkowski space]], the mathematical setting for special relativity.</ref> Between 1880 and 1887, [[Oliver Heaviside]] developed the [[operator theory|operational calculus]]<ref>[http://www.quadritek.com/bstj/vol01-1922/articles/bstj1-2-43.pdf The Heaviside Operational Calculus] www.quadritek.com/bstj/vol01-1922/articles/bstj1-2-43.pdf</ref> (involving the ''D'' notation for the [[differential operator]], which he is credited with creating), a method of solving differential equations by transforming them into ordinary [[algebraic equation]]s which caused a great deal of controversy when introduced, owing to the lack of [[rigour]] in his derivation of it.<ref group="note">He famously said, "Mathematics is an [[experimental science]], and [[Elementary definition|definitions]] do not come first, but later on." He was replying to criticism over his use of [[Operator (mathematics)|operators]] that were [[Elementary sentence|not clearly defined]]. On another occasion he stated somewhat more defensively, "I do not refuse my dinner simply because I do not understand the process of digestion."</ref> The common [[vector notation]] are used when working with vectors, which are [[vector (geometric)|spatial]] or more abstract members of [[vector space]]s. The [[angle notation]] (or [[phasor (sine waves)|phasor]] notation) is a notation used in electronics.

In 1881, [[Leopold Kronecker]] defined what he called a "domain of rationality", which is a [[field extension]] of the [[Field of rationals|field of rational numbers]] in modern terms.<ref>{{cite book | title=Galois Theory | volume=106 | series=Pure and Applied Mathematics | first=David A. | last=Cox | edition=2nd | publisher=John Wiley & Sons | year=2012 | isbn=1118218426 | page=348 }}</ref> In 1882, [[Hüseyin Tevfik Pasha]] wrote the book titled "Linear Algebra".<ref>[http://www.journals.istanbul.edu.tr/tr/index.php/oba/article/download/9103/8452 Hüseyin Tevfik Pasha: The Inventor of 'Linear Algebra']</ref><ref>http://archive.org/details/linearalgebra00tevfgoog</ref> [[Lord Kelvin]]'s [[Aether theories|aetheric]] [[atomic theory|atom theory]] (1860s) led [[Peter Guthrie Tait]], in 1885, to published a [[topology|topological]] table of knots with up to ten crossings know as the [[Tait conjectures]]. In 1893, [[Heinrich M. Weber]] gave the clear definition of an [[Finite field|abstract field]].<ref group="note"> See also: [[Field (mathematics)|Mathematic fields]] and [[Field extension]]</ref> [[Tensor calculus]] was developed by [[Gregorio Ricci-Curbastro]] between 1887–96, 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 |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|year=1900|edition=3rd|journal=Bull. Amer. Math. Soc.|year=1901|volume=7|issue=9|pages=399–408|url=http://www.ams.org/journals/bull/1901-07-09/S0002-9904-1901-00825-7/S0002-9904-1901-00825-7.pdf}}</ref>

=====From formula mathematics to tensors=====
{{rquote|right|The above proposition is occasionally useful.|Bertrand Russell
<ref group="note">Comment after the proof that 1+1=2, completed in Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell. Volume II, 1st edition (1912)</ref>}}
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]], [http://ia600302.us.archive.org/33/items/formulairedesmat00pean/formulairedesmat00pean.pdf 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]], the [[union sign]], the [[membership sign]] (is an element of), and [[existential quantifier]]<ref group="note">This raises questions of the [[existence theorem|pure existence theorems]].</ref> (there exists). Peano would pass to [[Bertrand Russell]] his work in 1900 at a Paris conference; it so impressed Russell that Russell 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.<ref group="note">Peano's ''Formulario Mathematico'', though less popular than Russell's work, continued through five editions. The fifth appeared in 1908 and included 4200 formulas and theorems.</ref> Ricci-Curbastro and [[Tullio Levi-Civita]] popularized the [[tensor index notation]] around 1900.<ref>{{citation|title=Méthodes de calcul différentiel absolu et leurs applications|last=Ricci|first=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|url=http://www.springerlink.com/content/u21237446l22rgg7/fulltext.pdf}}</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>
|}
At the beginning of this period, [[Felix Klein]]'s "[[Erlangen program]]" identified the underlying theme of various geometries, defining each of them as the study of [[Invariant (mathematics)|properties invariant]] under a given group of [[Symmetry|symmetries]]. This level of abstraction revealed connections between geometry and [[abstract algebra]]. [[Georg Cantor]]<ref group="note">Inventor of [[set theory]]</ref> would introduce the [[Aleph number|aleph symbol]] for [[cardinal number]]s of transfinite sets.<ref group="note">''Transfinite arithmetic'' is the generalization of [[elementary arithmetic]] to [[infinity|infinite]] quantities like [[infinite sets]]; See [[Transfinite number]]s, [[Transfinite induction]], and [[Transfinite interpolation]]. See also [[Ordinal arithmetic]].</ref> His notation for the cardinal numbers was the Hebrew letter <math>\aleph</math> ([[aleph number|aleph]]) with a natural number subscript; 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 which names an [[ordinal number]] according to some scheme which gives meaning to the language. [[Cantor's theorem|His theory]] created a [[Controversy over Cantor's theory|great deal of controversy]]. Cantor would, in his study of [[Fourier series]], consider point sets in [[Euclidean space]].

After the turn of the 20th century, [[Josiah Willard Gibbs]] would in [[physical chemistry]] introduce [[middle dot]] for [[dot product]] and the [[multiplication sign]] for [[cross product]]s. He would also supply notation for the scalar and vector products, which was introduced in ''[[Vector Analysis (Gibbs/Wilson)|Vector Analysis]]''. In 1904, [[Ernst Zermelo]] promotes [[axiom of choice]] and his proof of the [[well-ordering theorem]].<ref name="Zermelo, 1904">{{cite journal| first=Ernst| last=Zermelo| year=1904| url=http://gdz.sub.uni-goettingen.de/no_cache/en/dms/load/img/?IDDOC=28526 |format=reprint|title=Beweis, dass jede Menge wohlgeordnet werden kann| journal=Mathematische Annalen| volume=59| issue=4| pages=514–16|doi=10.1007/BF01445300}}</ref> Bertrand Russell would shortly afterward introduce [[logical disjunction]] ([[Logical disjunction|OR]]) in 1906. Also in 1906, Poincaré would publish ''[[On the Dynamics of the Electron]]''<ref>{{cite wikisource|title=On the Dynamics of the Electron (July)|last=|first=|year=|publisher=|page=|wspage=|scan=}}</ref> and [[Maurice Fréchet]] introduced [[metric space]].<ref>Fréchet, Maurice, "Sur quelques points du calcul fonctionnel", PhD dissertation, 1906</ref> Later, [[Gerhard Kowalewski]] and [[Cuthbert Edmund Cullis]]<ref>Matrices and determinoids, [http://books.google.com/books?id=_ArvAAAAMAAJ Volume 2]. By [[Cuthbert Edmund Cullis]].</ref><ref>[http://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>[http://books.google.com/books?id=NZt3wFzq5aoC An Introduction To The Modern Theory Of Equations]. By Florian Cajori.</ref> would successively introduce matrices notation, parenthetical matrix and box matrix notation respectively. After 1907, mathematicians<ref group="note">Such as [[Max Dehn]], [[James Waddell Alexander II|J. W. Alexander]], and others.</ref> studied knots from the point of view of the [[knot group]] and invariants from [[Homology (mathematics)|homology theory]].<ref group="note">Such as the [[Alexander polynomial]].</ref> In 1908, [[Joseph Wedderburn]]'s structure theorems were formulated for finite-dimensional [[algebra over a field|algebras over a field]]. Also in 1908, [[Ernst Zermelo]] proposed "definite" property and the first [[axiomatic set theory]], [[Zermelo set theory]]. In 1910 [[Ernst Steinitz]] published the influential paper ''Algebraic Theory of Fields''.<ref group="note">([[german language|German]]: Algebraische Theorie der Körper)</ref><ref group="note">In this paper Steinitz axiomatically studied the properties of fields and defined many important field theoretic concepts like [[prime field]], [[perfect field]] and the [[transcendence degree]] of a [[field extension]].</ref> In 1911, Steinmetz would publish [http://archive.org/details/theoryandcalcul12steigoog ''Theory and Calculation of Transient Electric Phenomena and Oscillations''].

[[File:Einstein 1921 by F Schmutzer.jpg|thumbnail|Albert Einstein in 1921]]
[[Albert Einstein]], in 1916, introduced the [[Einstein notation]]<ref group="note">The indices range over [[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 the convention to: 
<math> y = c_i x^i \,.</math> 
Upper indices are not [[Exponentiation|exponents]] but are indices of coordinates, [[coefficient]]s or [[basis vector]]s. 
See also: [[Ricci calculus]]</ref> which summed over a set of [[index notation|indexed terms]] in a formula, thus exerting notational brevity. [[Arnold Sommerfeld]] would create the [[contour integral]] sign in 1917. Also in 1917, [[Dimitry Mirimanoff]] proposes [[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>[http://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 [http://archive.org/details/fourlecturesonre00stei ''Four Lectures on Relativity and Space'']. Around 1924, [[Jan Arnoldus Schouten]] would develop 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.<ref group="note">[[Ricci calculus]] constitutes the rules of index notation and manipulation for [[tensors]] and [[tensor fields]]. See also: {{cite book |author=Synge J.L., Schild A.|publisher=first Dover Publications 1978 edition |title=Tensor Calculus |pages=6–108|year= 1949}}</ref><ref>{{cite book |pages=85–86, §3.5| author=J.A. Wheeler, C. Misner, K.S. Thorne| title=[[Gravitation (book)|Gravitation]]| publisher=W.H. Freeman & Co| year=1973 | isbn=0-7167-0344-0}}</ref><ref>{{cite book |author=R. Penrose| title=[[The Road to Reality]]| publisher= Vintage books| year=2007 | isbn=0-679-77631-1}}</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-dimmensional differential geometry)|language=german|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]] would describe 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]] would 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}}
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, Jan 1, 1998</ref><ref>The New American Encyclopedic Dictionary. Edited by Edward Thomas Roe, Le Roy Hooker, Thomas W. Handford. Pg [http://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> 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 substituted 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 [[falsity]], an [[insanity]].'')</ref>

[[Symbolic logic]] studies the purely formal properties of strings of symbols. The interest in this area springs from two sources. First, the notation used in symbolic logic can be seen as representing the words used in [[philosophical logic]]. Second, the rules for manipulating symbols found in symbolic logic can be implemented on a [[computer|computing machine]]. Symbolic logic is usually divided into two subfields, [[propositional logic]] and [[predicate logic]]. Other logics of interest include [[temporal logic]], [[modal logic]] and [[fuzzy logic]]. The area of symbolic logic called [[propositional logic]], also called ''propositional calculus'', studies the properties of sentences formed from [[logical constant|constants]]<ref group="note">Here a [[logical constant]] is a symbol in symbolic logic that has the same meaning in all models, such as the symbol "=" for "equals". A ''constant'', in a mathematical context, is a [[Mathematical constant|number that arises naturally in mathematics]], such as π or e; Such [[constant (mathematics)|mathematics constant]] value do not change. It can mean polynomial [[constant term]] (the term of degree 0) or the [[constant of integration]], a free parameter arising in integration. Related, the [[physical constant]] are a physical quantity generally believed to be universal and unchanging. [[Constant (programming)|Programming constants]] are a values that, unlike a variable, cannot be reassociated with a different value.</ref> and [[logical operator]]s. The corresponding logical operations are known, respectively, as [[logical conjunction|conjunction]], [[logical disjunction|disjunction]], [[material conditional]], [[biconditional]], and [[negation]]. These operators are denoted as [[reserved word|keywords]]<ref group="note">Though not an [[index term]], keywords are terms that represent information. A keyword is a word with special meaning (this is a semantic definition), while syntactically these are [[terminal symbol]]s in the phrase grammar. See [[reserved word]] for the related concept.</ref> and by symbolic notation.

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.<ref group="note">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]]. See also: [[Propositional calculus#propcalc table|Basic and Derived Argument Forms]].</ref> Predicate logic, originally called ''predicate calculus'', expands on propositional logic by the introduction of [[Variable (mathematics)|variables]]<ref group="note">Usually denoted by ''x'', ''y'', ''z'', or other lowercase letters 
Here a symbols that represents a quantity in a mathematical expression, a [[Variable (mathematics)|mathematical variable]] as used in many sciences. 
Variables can be symbolic name associated with a value and whose associated value may be changed, known in computer science as a [[Variable (computer science)|variable reference]]. A ''variable'' can also be the [[operationalization|operationalized]] way in which the attribute is represented for further [[data processing]] (eg., a logical set of attributes). See also: [[Dependent and independent variables]] in statistics.</ref> and by sentences containing variables, called [[Predicate (mathematical logic)|predicates]].<ref group="note">Usually denoted by an uppercase letter followed by a list of variables, such as P(''x'') or Q(''y'',''z'') 
Here a [[Predicate (mathematical logic)|mathematical logic predicate]], a fundamental concept in first-order logic. [[Predicate (grammar)|Grammatical predicates]] are grammatical components of a sentence. 
Related is the [[syntactic predicate]] in parser technology which are guidelines for the parser process. In computer programming, a [[branch predication]] allows a choice to execute or not to execute a given instruction based on the content of a machine register.</ref> In addition, predicate logic allows [[quantifiers]].<ref group="note">Representing ALL and EXISTS</ref> With these [[logic symbol]]s and additional [[quantifier]]s from predicate logic,<ref group="note">e.g. ∃ for "there exists" and ∀ for "for all"</ref> [[Validity|valid]] [[Mathematical proof|proofs]] [[Argumentation_theory#Mathematical_argumentation|can be made]] that are [[Absurdity|irrationally artificial]],<ref group="note">See also: [[Dialetheism]], [[Contradiction]], and [[Paradox]]</ref> but syntactical.<ref group="note">Related, [[dict:facetious|facetious]] [[abstract nonsense]] describes certain kinds of arguments and methods related to category theory which resembles comical [[Non sequitur (literary device)|literary non sequitur devices]] (not [[Non sequitur (logic)|illogical non sequiturs]]).</ref>

{{details3|[[Mathematical logic|Symbolic logic]]|applications of formal logic to mathematics}}
{{See also|History of logic|Order theory}}
{{see also|Table of logic symbols|Logic alphabet}}

=====Gödel incompleteness notation=====
{{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 his [[incompleteness theorems]],<ref group="note">[[Gödel's incompleteness theorems]] shows that [[Hilbert's program]] to find a complete and consistent set of [[axiom]]s for all [[mathematics]] is impossible, giving a contested negative answer to [[Hilbert's second problem]]</ref> [[Kurt Gödel]] created an alternative to the symbols normally used in logic. He used [[Gödel number]]s, which were numbers that represented operations with set numbers, and variables with the prime numbers greater than 10. With Gödel numbers, logic statements can be broken down into a number sequence. Gödel then took this one step farther, taking the ''n'' prime numbers and putting them to the power of the numbers in the sequence. These numbers were then multiplied together to get the final product, giving every logic statement its own number.<ref>Casti, John L. ''5 Golden Rules''. New York: MJF Books, 1996.</ref><ref group="note">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 ten prime numbers are found and these are: <math>\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29\}</math>. Then, the Gödel numbers are made the powers of the respective primes and multiplied, giving: <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>.</ref>
{{see also|Proof sketch for Gödel's first incompleteness theorem}}

==== Contemporary notation and topics ====
{{see also|Contemporary era}}

=====Early 20th Century notation=====
Abstraction of notation is an ongoing process and the historical development of many mathematical topics exhibits a progression from the concrete to the abstract. Various [[set notation]]s would be developed for fundamental object [[Set (mathematics)|sets]]. Around 1924, [[David Hilbert]] and [[Richard Courant]] would publish "[[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.<ref group="note">The Klein–Gordon equation is: {{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}}</ref> 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
| authorlink= 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 | pages=243–265
| doi=10.1098/rspa.1927.0039
| bibcode=1927RSPSA.114..243D
}}</ref> In 1928, the [[relativistic wave equation|relativistic]] [[Dirac equation]] is formulated by Dirac to explain the behavior of the relativistically moving [[electron]].<ref group="note">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]].</ref> Dirac would 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]], [[Werner Heisenberg]] and an elegant formulation of quantum electrodynamics due to [[Enrico Fermi]],<ref name=fermi>
{{cite journal
| author=E. Fermi
| authorlink= Enrico Fermi
| year=1932
| title=Quantum Theory of Radiation
| journal=[[Reviews of Modern Physics]]
| volume=4 | 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]])<ref group="note">That is, 
{{Equation box 1|indent =:|equation =<math>\partial_\mu(\partial^\mu A^\nu - \partial^\nu A^\mu)+\left(\frac{mc}{\hbar}\right)^2 A^\nu=0</math> |cellpadding= 6|border = 0|border colour = black|background colour=white}}</ref> for the vector [[meson]] theory of [[nuclear force]]s and the [[relativistic quantum field equations]]. [[John Archibald Wheeler]] in 1937 develops [[S-matrix]]. Studies by [[Felix Bloch]] with [[Arnold Nordsieck]],<ref name=bloch>{{cite journal
| author1=F. Bloch
| authorlink1= Felix Bloch
| author2=A. Nordsieck
| authorlink2= Arnold Nordsieck
| year=1937
| title=Note on the Radiation Field of the Electron
| journal=[[Physical Review]]
| volume=52 | 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
| authorlink= Victor Weisskopf
| year=1939
| title=On the Self-Energy and the Electromagnetic Field of the Electron
| journal=[[Physical Review]]
| volume=56 | 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
| authorlink= Robert Oppenheimer
| year=1930
| title=Note on the Theory of the Interaction of Field and Matter
| journal=[[Physical Review]]
| volume=35 | pages=461–477
| doi=10.1103/PhysRev.35.461
| bibcode=1930PhRv...35..461O
}}</ref> At higher orders in the series infinities emerged, 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 for integer numbers sets was created by [[Edmund Landau]]. [[Nicolas Bourbaki]] would create the double-struck capital Q for rational numbers sets. In 1935, Gerhard Gentzen made [[universal quantifier]]s. In 1936, [[Tarski's undefinability theorem]] is stated by [[Alfred Tarski]] and proved.<ref group="note">The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system.</ref> In 1938, [[Kurt Gödel|Gödel]] proposes the [[constructible universe]] in the paper "''[http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1077160/ The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis]''". [[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 for [[complex number]] sets.

Around the 1930s, [[Voigt notation]]<ref group="note">Named to honor Voigt's 1898 work.</ref> would be developed for [[multilinear algebra]] as a way to represent a [[symmetric tensor]] by reducing its order. [[Schoenflies notation|Schönflies notation]]<ref group="note">Named after [[Arthur Moritz Schoenflies]]</ref> 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|year=1933|pages=462–474}}</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, e.g., →, was developed for [[function notation]] in 1936 to denote images of specific elements<ref group="note">See [[Galois connection]]s.</ref> by [[Øystein Ore]].<ref group="note">Oystein Ore would also write, "[http://books.google.com/books?id=Sl_6BPp7S0AC Number Theory and Its History]".</ref> Later, in 1940, it would take it's the present form, e.g., ''f: X → Y'', through the work of [[Witold Hurewicz]]. [[Werner Heisenberg]], in 1941, proposes the [[S-matrix theory]] of particle interactions.

[[File:Diracb.jpg|thumbnail|right|Paul Dirac, pictured here, 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}}<ref group="note"><math>\langle\phi|\psi\rangle</math></ref> consisting of a left part, ⟨''φ''|, and a right part, |''ψ''⟩. The notation was introduced in 1939 by [[Paul Dirac]]<ref name=Dirac>{{cite news
|author=PAM 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}}
</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 [[dict:ado|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
|author=Steven Weinberg
|year=1964
|title=The quantum theory of fields, Volume 2
|publisher=''Cambridge University Press, 1995''
|isbn=0-521-55001-7
|url=http://books.google.com/books?id=3ws6RJzqisQC&lpg=PA358&dq=%22Dirac%20Slash%22&pg=PA358#v=onepage&q&f=false
|pages=358
}}</ref>) was developed by [[Richard Feynman]] for the study of [[Fermionic field#Dirac fields|Dirac fields]] in [[quantum field theory]].

In 1948, [[Valentine Bargmann]] and [[Eugene Wigner]] proposed the [[relativistic wave equations|relativistic]] [[Bargmann–Wigner equations]] to describe [[free particle]]s and the equations are in the form of multi-component [[spinor field]] [[wavefunction]]s. In 1950, [[W. V. D. Hodge|William Vallance Douglas Hodge]] presents "[[Hodge conjecture|The topological invariants of algebraic varieties]]" at the Proceedings of the International Congress of Mathematicians. Between 1954 and 1957, [[Eugenio Calabi]] worked on the [[Calabi conjecture]] for [[Kähler metric]]s and the development of [[Calabi–Yau manifold]]s. In 1957, [[Tullio Regge]] formulated the mathematical property of potential scattering in the [[Schrödinger equation]].<ref group="note">That the scattering amplitude can be thought of as an analytic function of the angular momentum, and that the position of the poles determine power-law growth rates of the amplitude in the purely mathematical region of large values of the cosine of the scattering angle.</ref> [[Stanley Mandelstam]], along with Regge, did the initial development of the [[Regge theory]] of strong interaction phenomenology. In 1958, [[Murray Gell-Mann]] and [[Richard Feynman]], along with [[George Sudarshan]] and [[Robert Marshak]], deduce the [[chiral|chiral structures]] of the [[weak interaction]] in physics. [[Geoffrey Chew]], along with others, would promote matrix notation for the [[strong interaction]], 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=====
In the modern mathematics of [[special relativity]], [[electromagnetism]] and [[Wave|wave theory]], the [[d'Alembert operator]]<ref group="note">That is, <math>\scriptstyle\Box</math></ref><ref group="note">Also known as the [[d'Alembertian]] or [[wave operator]].</ref> is the [[Laplace operator]] of [[Minkowski space]]. The [[Levi-Civita symbol]]<ref group="note">Also known as, "[[permutation symbol]]" (see: [[permutation]]), "[[antisymmetric symbol]]" (see: [[antisymmetric]]), or "[[alternating symbol]]"</ref> is used in [[tensor calculus]].

After the full [[Lorentz covariance]] formulations that were finite at any order in a perturbation series of quantum electrodynamics, [[Sin-Itiro Tomonaga]], [[Julian Schwinger]] and [[Richard Feynman]] were jointly awarded with a [[Nobel prize in physics]] in 1965.<ref name=nobel65>{{cite web | title = The Nobel Prize in Physics 1965 | publisher = Nobel Foundation | url = http://nobelprize.org/nobel_prizes/physics/laureates/1965/index.html|accessdate=2008-10-09}}</ref> Their contributions, and those of [[Freeman Dyson]], were about covariant and [[gauge invariant]] formulations of quantum electrodynamics that allow computations of observables at any order of [[Perturbation theory (quantum mechanics)|perturbation theory]]. Feynman's mathematical technique, based on his [[Feynman diagram|diagrams]], initially seemed very different from the field-theoretic, [[Operator (physics)|operator]]-based approach of Schwinger and Tomonaga, but [[Freeman Dyson]] later showed that the two approaches were equivalent. [[Renormalization]], the need to attach a physical meaning at certain divergences appearing in the theory through [[integral]]s, has subsequently become one of the fundamental aspects of [[quantum field theory]] and has come to be seen as a criterion for a theory's general acceptability. Quantum electrodynamics has served as the model and template for subsequent quantum field theories. [[Peter Higgs]], [[Jeffrey Goldstone]], and others, [[Sheldon Glashow]], [[Steven Weinberg]] and [[Abdus Salam]] independently showed how the [[weak nuclear force]] and quantum electrodynamics could be merged into a single [[electroweak force]]. In the late 1960s, the [[particle zoo]] was composed of the then known [[elementary particle]]s before the discovery of [[quarks]].

[[File:Standard Model of Elementary Particles.svg|thumbnail|Standard model of elementary particles.
----
The [[fermion|fundamental fermions]] and the [[Boson|fundamental bosons]]. (c.2008)<ref group="note">Note that "[[dict:mass|masses]]" (eg., the coherent non-definite body shape) of [[particles]] are periodically ''reevaluated'' by the [[scientific community]]. The values may have been adjusted; [[dict:adjustment|adjustment]] by operations carried out on instruments in order that it provides given indications corresponding to given values of the [[measurand]]. In engineering, mathematics, and geodesy, the optimal [[parameter]] such estimation of a [[mathematical model]] so as to [[best fit]] a [[data set]].</ref> Based on the [[Proprietary format|proprietary publication]], [[Review of Particle Physics]].<ref group="note">For the [[Consensus decision-making|consensus]], see [[Particle Data Group]].</ref>]]

A step towards the [[Standard Model]] was [[Sheldon Lee Glashow|Sheldon Glashow]]'s discovery, in 1960, of a way to combine the [[electromagnetism|electromagnetic]] and [[weak interaction]]s.<ref>
{{cite journal
| author=S.L. Glashow
| year=1961
| title=Partial-symmetries of weak interactions
| journal=[[Nuclear Physics (journal)|Nuclear Physics]]
| volume=22 | pages=579–588
| doi=10.1016/0029-5582(61)90469-2
|bibcode = 1961NucPh..22..579G }}</ref> In 1967, [[Steven Weinberg]]<ref>
{{cite journal
| author=S. Weinberg
| year=1967
| title=A Model of Leptons
| journal=[[Physical Review Letters]]
| volume=19 | pages=1264–1266
| doi=10.1103/PhysRevLett.19.1264
| bibcode=1967PhRvL..19.1264W
}}</ref> and [[Abdus Salam]]<ref>
{{cite conference
| author=A. Salam
| editor=N. Svartholm
| year=1968
| booktitle=Elementary Particle Physics: Relativistic Groups and Analyticity
| pages=367
| conference=[[Nobel Symposium|Eighth Nobel Symposium]]
| publisher=[[Almquvist and Wiksell]]
| location=Stockholm
}}</ref> incorporated the [[Higgs mechanism]]<ref>
{{cite journal
| author=F. Englert, R. Brout
| year=1964
| title=Broken Symmetry and the Mass of Gauge Vector Mesons
| journal=[[Physical Review Letters]]
| volume=13 | pages=321–323
| doi=10.1103/PhysRevLett.13.321
| bibcode=1964PhRvL..13..321E
}}</ref><ref>
{{cite journal
| author=P.W. Higgs
| year=1964
| title=Broken Symmetries and the Masses of Gauge Bosons
| journal=[[Physical Review Letters]]
| volume=13 | pages=508–509
| doi=10.1103/PhysRevLett.13.508
| bibcode=1964PhRvL..13..508H
}}</ref><ref>
{{cite journal
| author=G.S. Guralnik, C.R. Hagen, T.W.B. Kibble
| year=1964
| title=Global Conservation Laws and Massless Particles
| journal=[[Physical Review Letters]]
| volume=13 | pages=585–587
| doi=10.1103/PhysRevLett.13.585
| bibcode=1964PhRvL..13..585G
}}</ref> into Glashow's [[electroweak theory]], giving it its modern form. The Higgs mechanism is believed to give rise to the [[mass]]es of all the [[elementary particle]]s in the Standard Model. This includes the masses of the [[W and Z bosons]], and the masses of the [[fermion]]s - i.e. the [[quark]]s and [[lepton]]s. Also in 1967, [[Bryce DeWitt]] published [[Wheeler–DeWitt equation|his equation]] under the name "''[[Wheeler–DeWitt equation|Einstein–Schrödinger equation]]''" (later renamed the "''[[John Archibald Wheeler|Wheeler]]–DeWitt equation''").<ref>http://www.physics.drexel.edu/~vkasli/phys676/Notes%20for%20a%20brief%20history%20of%20quantum%20gravity%20-%20Carlo%20Rovelli.pdf</ref> In 1969, [[Yoichiro Nambu]], [[Holger Bech Nielsen]], and [[Leonard Susskind]] descried space and time in [[String theory|terms of strings]]. In 1970, [[Pierre Ramond]] develop two-dimensional supersymmetries. [[Michio Kaku]] and [[Keiji Kikkawa]] would afterwards formulate string variations. In 1972, [[Michael Artin]], [[Alexandre Grothendieck]], [[Jean-Louis Verdier]] propose the [[Grothendieck universe]].<ref>{{cite conference | first = Nicolas | last = Bourbaki | authorlink = Nicolas Bourbaki | year = 1972 | title = Univers | booktitle = Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 1 (Lecture notes in mathematics '''269''') | editor = [[Michael Artin]], [[Alexandre Grothendieck]], [[Jean-Louis Verdier]], eds. | publisher = [[Springer Science+Business Media|Springer-Verlag]] | location = Berlin; New York | language = French | pages = 185–217 | url = http://library.msri.org/books/sga/sga/4-1/4-1t_185.html}}</ref>

After the [[Neutral current|neutral weak currents]] caused by {{SubatomicParticle|Z boson}} boson exchange [[Gargamelle|were discovered]] at [[CERN]] in 1973,<ref>
{{cite journal
|author=F.J. Hasert ''et al.''
|year=1973
|title=Search for elastic muon-neutrino electron scattering
|journal=[[Physics Letters B]]
|volume=46 |page=121
|doi=10.1016/0370-2693(73)90494-2
|bibcode = 1973PhLB...46..121H }}</ref><ref>
{{cite journal
|author=F.J. Hasert ''et al.''
|year=1973
|title=Observation of neutrino-like interactions without muon or electron in the gargamelle neutrino experiment
|journal=[[Physics Letters B]]
|volume=46 |page=138
|doi=10.1016/0370-2693(73)90499-1
|bibcode = 1973PhLB...46..138H }}</ref><ref>
{{cite journal
|author=F.J. Hasert ''et al.''
|year=1974
|title=Observation of neutrino-like interactions without muon or electron in the Gargamelle neutrino experiment
|journal=[[Nuclear Physics B]]
|volume=73 |page=1
|doi=10.1016/0550-3213(74)90038-8
|bibcode = 1974NuPhB..73....1H }}</ref><ref>
{{cite web
|author=D. Haidt
|date=4 October 2004
|title=The discovery of the weak neutral currents
|url=http://cerncourier.com/cws/article/cern/29168
|work=[[CERN Courier]]
|accessdate=2008-05-08
}}</ref> the electroweak theory became widely accepted and Glashow, Salam, and Weinberg shared the 1979 [[Nobel Prize in Physics]] for discovering it. The theory of the [[strong interaction]], to which many contributed, acquired its modern form around 1973–74. With the establishment of [[quantum chromodynamics]], a finalized a set of fundamental and exchange particles, which allowed for the establishment of a "[[Standard Model|standard model]]" based on the mathematics of [[Gauge theory|gauge invariance]], which successfully described all forces except for gravity, and which remains generally accepted within the domain to which it is designed to be applied. In the late 1970s, [[William Thurston]] introduced [[hyperbolic geometry]] into the [[Knot theory|study of knots]] with the [[geometrization conjecture|hyperbolization theorem]]. The [[orbifold notation]] system, invented by Thurston, has been developed for representing types of [[symmetry groups]] in two-dimensional spaces of constant curvature. In 1978, [[Shing-Tung Yau]] deduced that the [[Calabi conjecture]] have [[Ricci-flat manifold|Ricci flat]] metrics. In 1979, [[Daniel Friedan]] showed that the equations of motions of [[string theory]] are abstractions of [[Einstein equations]] of [[General Relativity]].

The [[first superstring revolution]] is composed of mathematical equations developed between 1984 and 1986. In 1984, [[Vaughan Jones]] deduced the [[Jones polynomial]] and subsequent contributions from [[Edward Witten]], [[Maxim Kontsevich]], and others, revealed deep connections between knot theory and mathematical methods in [[statistical mechanics]] and quantum field theory. According to [[string theory]], all particles in the "particle zoo" have a common ancestor, namely a [[String (physics)|vibrating string]]. In 1985, [[Philip Candelas]], [[Gary Horowitz]],<ref>“Gary Horowitz” http://web.physics.ucsb.edu/~gary/</ref> [[Andrew Strominger]], and Edward Witten would publish "Vacuum configurations for superstrings"<ref>Nuclear Physics B 258: 46–74, Bibcode:1985NuPhB.258...46C, doi:10.1016/0550-3213(85)90602-9</ref> Later, the [[tetrad formalism]] ([[tetrad (index notation)|tetrad index notation]]) would be 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.<ref group="note">A locally defined set of four linearly independent [[vector field]]s called a [[Tetrad (general relativity)|tetrad]]</ref><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]] would propose [[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 would also introduce [[abstract index notation]].<ref group="note">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.</ref> In 1995, Edward Witten suggested [[M-theory]] and subsequently used it to explain some observed [[Duality (mathematics)|dualities]], initiating the [[second superstring revolution]].<ref group="note">See also: [[String theory landscape]] and [[Swampland (physics)|Swampland]]</ref>

[[File:John H Conway 2005 (cropped).jpg|thumb|John H Conway, prolific mathematician of notation.]]
[[John Horton Conway|John Conway]] would further 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]] and [[Norman Johnson (mathematician)|Norman Johnson]] more comprehensively defined it.

[[Combinatorics|Combinatorial]] [[LCF notation]]<ref group="note">Devised by [[Joshua Lederberg]] and extended by [[Harold Scott MacDonald Coxeter|Coxeter]] and [[Robert Frucht|Frucht]]</ref> has been 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
| contribution = 2.3.2 Cubic graphs and LCF notation
| isbn = 9780817683641
| page = 32
| publisher = Springer
| title = Configurations from a Graphical Viewpoint
| url = http://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 [[cycle (mathematics)|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>

{{see also|Approximation theory|Universal property}}
{{see also|Tensor algebra|Free algebra|Abstract algebra}}

=====Computers and markup notation=====
{{main|History of computing|Timeline of computing}}
{{see also|Symbolic computation|Symbolic dynamics|Computational complexity theory}}

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 8 digits long, from a card and punch their product onto the same card.<ref>[http://www.columbia.edu/cu/computinghistory/601.html The IBM 601 Multiplying Punch]</ref> In 1934, [[Wallace Eckert]] used a rigged IBM 601 Multiplying Punch to automate the integration of differential equations.<ref>[http://www.columbia.edu/cu/computinghistory/switch.html Interconnected Punched Card Equipment]</ref> In 1936, [[Alan Turing]] publishes "[http://classes.soe.ucsc.edu/cmps210/Winter11/Papers/turing-1936.pdf On Computable Numbers, With an Application to the Entscheidungsproblem]".<ref>Proceedings of the London Mathematical Society 42 (2)</ref><ref group="note">And, in 1938, "On Computable Numbers, with an Application to the Entscheidungsproblem: A correction" (Proceedings of the London Mathematical Society, 2 (1937) 43 (6): 544–6, doi:10.1112/plms/s2-43.6.544).</ref> [[John von Neumann]], pioneer of the digital computer and of computer science,<ref group="note">Among von Neumann's other contributions include the application of [[operator theory]] to [[quantum mechanics]], in the development of [[functional analysis]], and on various forms of [[operator theory]].</ref> in 1945, writes the [[unfinished work|incomplete]] ''[[First Draft of a Report on the EDVAC]]''. In 1962, [[Kenneth E. Iverson]] developed an integral part notation that became known as [[Iverson Notation]] for manipulating arrays that he taught to his students, and described in his book ''[[A Programming Language]]''. In 1970, [[Edgar F. Codd|E.F. Codd]] proposed [[relational algebra]] as a [[relational model|relational model of data]] for [[database query language]]s. In 1971, [[Stephen Cook]] publishes "[[P versus NP problem|The complexity of theorem proving procedures]]"<ref>{{Cite book|last=Cook|first=Stephen|authorlink=Stephen Cook|year=1971|chapter=The complexity of theorem proving procedures|chapterurl=http://portal.acm.org/citation.cfm?coll=GUIDE&dl=GUIDE&id=805047|title=Proceedings of the Third Annual ACM Symposium on Theory of Computing|pages=151–158}}</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 libraries]]. They are [[Library (computing)|libraries]] used in [[software development]] for performing [[numerical analysis|numerical]] calculations. These calculations can be handled by [[symbolic execution]]s; analyzing a program to determine what inputs cause each part of a program to execute.

:''See also'': ''[[Internet shorthand notation|ASCII math notation]] and [[MathML]]''
{{see also|Basic Linear Algebra Subprograms|Numerical linear algebra}}
{{see also|List of numerical libraries|List of numerical analysis software}}
:''See also'': ''[[DOT language]], [[Lisp (programming language)]], [[Object-oriented programming]], and [[Earley algorithm]]''

=== Future of mathematical notation ===
{{Main|Future of mathematics}}
[[File:Calabi yau.jpg|thumb|A section of a quintic Calabi–Yau three-fold ([[3D projection]]); recalling ''[[Lord_Kelvin#Kelvin.27s_vortex_theory_of_the_atom|atomic vortex theory]].''
----
See [[history of knot theory]] (emphasis on [[atom]]s); Additionally, the [[mechanical explanations of gravitation]] ([[gravitation]] theory).]]
In the history of mathematical notation, ideographic symbol notation has come full circle with the rise of computer visualization systems. The notations can be applied to abstract visualizations, such as for rendering some projections of a [[Calabi-Yau]] [[manifold]]. Examples of [[Information visualization|abstract visualization]] which properly belong to the mathematical imagination can be found in [[computer graphics]]. The need for such models abounds, for example, when the measures for the subject of study are actually [[random variable]]s and not really ordinary [[mathematical function]]s.

==See also==

{{Portal|Mathematics}}
;Main relevance: [[Abuse of notation]], [[Well-formed formula]], [[Big O notation]] ([[L-notation]]), [[Dowker notation]], [[Hungarian notation]], [[Infix notation]], [[Positional notation]], [[Polish notation]] ([[Reverse Polish notation]]), [[Sign-value notation]], [[Subtractive notation]], [[infix notation]]

; Numbers and quantities: [[List of numbers]], [[Irrational number|Irrational and suspected irrational numbers]], [[Euler–Mascheroni constant|γ]], [[Apéry's constant|ζ(3)]], [[Square root of 2|√2]], [[Square root of 3|√3]], [[Square root of 5|√5]], [[Golden ratio|φ]], [[Plastic number|ρ]], [[Silver ratio|δ''S'']], [[Feigenbaum constants|α]], [[E (mathematical constant)|''e'']], [[Pi|π]], [[Feigenbaum constants|δ]], [[Physical constant]]s, [[speed of light|''c'']], [[vacuum permittivity|''ε''0]], [[Planck's constant|''h'']], [[gravitational constant|''G'']], [[Greek letters used in mathematics, science, and engineering]]

;General relevance: [[Order of operations]], [[Scientific notation]] ([[Engineering notation]]), [[Actuarial notation]]

;Dot notation: [[Chemical formula|Chemical notation]] ([[Lewis dot notation]] ([[Electron dot notation]])), [[Dot-decimal notation]]

;Arrow notation: [[Knuth's up-arrow notation]], [[infinitary combinatorics]] (Arrow notation (Ramsey theory))

;Geometries: [[Projective geometry]], [[Affine geometry]], [[Finite geometry]]


;Lists and outlines: [[Outline of mathematics]] ([[List of mathematics history topics|Mathematics history topics]] and [[Lists of mathematics topics|Mathematics topics]] ([[List of mathematics categories|Mathematics categories]])), [[List of mathematical theories|Mathematical theories]] ( [[List of first-order theories|First-order theories]], [[List of theorems|Theorems]] and [[List of disproved mathematical ideas|Disproved mathematical ideas]]), [[List of mathematical proofs|Mathematical proofs]] ([[List of incomplete proofs|Incomplete proofs]]), [[List of mathematical identities|Mathematical identities]], [[List of mathematical series|Mathematical series]], [[List of mathematics reference tables|Mathematics reference tables]], [[List of mathematical logic topics|Mathematical logic topics]], [[List of mathematics-based methods|Mathematics-based methods]], [[List of mathematical functions|Mathematical functions]], [[List of transforms|Transforms]] and [[List of operators|Operators]], [[List of points in mathematics|Points in mathematics]], [[List of mathematical shapes|Mathematical shapes]], [[List of knots|Knots]] ([[List of prime knots|Prime knots]] and [[List of mathematical knots and links|Mathematical knots and links]]), [[List of inequalities|Inequalities]], [[List of mathematical concepts named after places|Mathematical concepts named after places]], [[List of mathematical topics in classical mechanics|Mathematical topics in classical mechanics]], [[List of mathematical topics in quantum theory|Mathematical topics in quantum theory]], [[List of mathematical topics in relativity|Mathematical topics in relativity]], [[List of string theory topics|String theory topics]], [[List of unsolved problems in mathematics|Unsolved problems in mathematics]], [[List of mathematical jargon|Mathematical jargon]], [[List of mathematical examples|Mathematical examples]], [[List of mathematical abbreviations|Mathematical abbreviations]]

;Misc.: [[Hilbert's problems]], [[Mathematical coincidence]], [[Chess notation]], [[Line notation]], [[Musical notation]] ([[Dotted note]]), [[Whyte notation]], [[Dice notation]], [[recursive categorical syntax]]

;People: [[List of mathematicians|Mathematicians]] ([[List of amateur mathematicians|Amateur mathematicians]] and [[List of female mathematicians|Female mathematicians]]), [[Thomas Bradwardine]], [[Thomas Harriot]], [[Felix Hausdorff]], [[Gaston Julia]], [[Helge von Koch]], [[Paul Lévy (mathematician)|Paul Lévy]], [[Aleksandr Lyapunov]], [[Benoit Mandelbrot]], [[Lewis Fry Richardson]], [[Wacław Sierpiński]], [[Saunders Mac Lane]], [[Paul Cohen (mathematician)|Paul Cohen]], [[Gottlob Frege]], [[G. S. Carr]], [[Robert Recorde]], [[Bartel Leendert van der Waerden]], [[G. H. Hardy]], [[E. M. Wright]], [[James R. Newman]], [[Carl Gustav Jacob Jacobi]], [[Roger Joseph Boscovich]], [[Eric W. Weisstein]], [[List of mathematical probabilists|Mathematical probabilists]], [[List of statisticians|Statisticians]]

{{Abel Prize laureates}}
{{Fields medalists}}
{{Guy Medal}}
{{Navbox Kenneth O. May Prize laureates}}
{{Nevanlinna Prize winners}}
{{Schock Prize laureates}}
{{Wolf Prize in Mathematics}}

==Further reading==
;General:
* [http://books.google.com/books?id=U9zuAAAAMAAJ A Short Account of the History of Mathematics]. By [[Walter William Rouse Ball]].
* [http://books.google.com/books?id=fYS4AAAAIAAJ A Primer of the History of Mathematics]. By Walter William Rouse Ball.
* [http://books.google.com/books?id=rh0qAAAAYAAJ A History of Elementary Mathematics]: With Hints on Methods of Teaching. By Florian Cajori.
* [http://books.google.com/books?id=UicJAAAAIAAJ A History of Elementary Mathematics]. By Florian Cajori.
* [http://books.google.com/books?id=kqQPAAAAYAAJ A History of Mathematics]. By Florian Cajori.
* [http://books.google.com/books?id=9d8DAAAAMAAJ A Short History of Greek Mathematics]. By [[James Gow (Author)|James Gow]].
* [http://books.google.com/books?id=it3UAAAAMAAJ On the Development of Mathematical Thought During the Nineteenth Century]. By [[John Theodore Merz]].
* [http://books.google.com/books?id=Y-csAAAAYAAJ A New Mathematical and Philosophical Dictionary]. By Peter Barlow.
* [http://books.google.com/books?id=e1UuAAAAYAAJ Historical Introduction to Mathematical Literature]. By [[George Abram Miller]]
* [http://books.google.com/books?id=DMjlAAAAMAAJ A Brief History of Mathematics]. By [[Karl Fink]], [[Wooster Woodruff Beman]], [[David Eugene Smith]]
* [http://books.google.com/books?id=yUJBAAAAYAAJ History of Modern Mathematics]. By David Eugene Smith.
* [http://books.google.com/books?id=SlUNAAAAYAAJ History of modern mathematics]. By David Eugene Smith, [[Mansfield Merriman]].

;Other
* Principia Mathematica, [http://books.google.com/books?id=yN9LAAAAMAAJ Volume 1] & [http://books.google.com/books?id=sbTVAAAAMAAJ Volume 2]. By Alfred North Whitehead, Bertrand Russell.
* [http://books.google.com/books?id=exwAAAAAQAAJ The Mathematical Principles of Natural Philosophy], Volume 1, Issue 1. By Sir Isaac Newton, Andrew Motte, William Davis, John Machin, William Emerson.
* [http://books.google.com/books?id=SYJsAAAAMAAJ General investigations of curved surfaces of 1827 and 1825]. By Carl Friedrich Gaus.

==Notes==
{{Reflist|2|group=note}}

==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|2}}

==External links==
;Notation
* [http://www.stephenwolfram.com/publications/recent/mathml/mathml2.html Mathematical Notation: Past and Future]
* [http://www.roma.unisa.edu.au/07305/symbols.htm History of Mathematical Notation]
* [http://jeff560.tripod.com/mathsym.html Earliest Uses of Mathematical Notation]
* [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.
;General
* [[Eric W. Weisstein]], {{URL|http://mathworld.wolfram.com|mathworld.wolfram.com}} [[Wolfram Research, Inc.]]
* [[Encyclopedia of Mathematics]] [http://www.encyclopediaofmath.org/index.php/Main_Page encyclopediaofmath.org]
* History of Mathematics - Main Page http://www.math.tamu.edu/~dallen/masters/hist_frame.htm
* MacTutor History of Mathematics http://www-history.mcs.st-and.ac.uk/
* Mathematics History http://library.thinkquest.org/22584/

{{DEFAULTSORT:History Of Mathematical Notation}}
[[Category:History of mathematics|Mathematical notation]]
[[Category:Mathematical notation]]
[[Category:Articles which contain graphical timelines]]

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