Editing Expression (mathematics) (section)

== History ==
{{Broader|History of mathematics|History of mathematical notation}}
{{See also|History of the function concept}}

=== Early written mathematics ===
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| footer            = The [[Ishango bone]] at the [[Royal Belgian Institute of Natural Sciences|RBINS]]. A [[Babylonian tablet]] approximating the [[square root of 2]]. Problem 14 from the [[Moscow Mathematical Papyrus]].
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The earliest written mathematics likely began with [[tally marks]], where each mark represented one unit, carved into wood or stone. An example of early [[counting]] is the [[Ishango bone]], found near the [[Nile]] and dating back over [[Upper Paleolithic|20,000 years ago]], which is thought to show a six-month [[lunar calendar]].<ref name="Marshack2">Marshack, Alexander (1991). ''The Roots of Civilization'', Colonial Hill, Mount Kisco, NY.</ref> [[Ancient Egyptian mathematics|Ancient Egypt]] developed a symbolic system using [[hieroglyphics]], assigning symbols for powers of ten and using addition and subtraction symbols resembling legs in motion.<ref>Encyclopædia Americana. By Thomas Gamaliel Bradford. Pg [https://books.google.com/books?id=hrRPAAAAMAAJ&pg=PA314 314]</ref><ref>Mathematical Excursion, Enhanced Edition: Enhanced Webassign Edition By Richard N. Aufmann, Joanne Lockwood, Richard D. Nation, Daniel K. Cleg. Pg [https://books.google.com/books?id=GTgTnSGMukgC&pg=PA186 186]</ref> This system, recorded in texts like the [[Rhind Mathematical Papyrus]] (c. 2000–1800 BC), influenced other [[History of the Mediterranean region|Mediterranean cultures]]. In [[Mesopotamia]], a similar system evolved, with numbers written in a base-60 ([[sexagesimal]]) format on [[clay tablets]] written in [[Cuneiform]], a technique originating with the [[Sumerians]] around 3000 BC. This base-60 system persists today in measuring time and [[angle]]s.

=== Syncopated stage ===
The "syncopated" stage of mathematics introduced symbolic abbreviations for commonly used operations and quantities, marking a shift from purely [[Geometry|geometric]] reasoning. [[Ancient Greek mathematics]], largely geometric in nature, drew on [[Egyptian numerals|Egyptian numerical systems]] (especially [[Attic numerals]]),<ref>Mathematics and Measurement By Oswald Ashton Wentworth Dilk. Pg [https://books.google.com/books?id=AKJZvXOS7n4C&pg=PA14 14]</ref> with little interest in algebraic symbols, until the arrival of [[Diophantus]] of [[History of Alexandria|Alexandria]],<ref>[http://www.ms.uky.edu/~carl/ma330/projects/diophanfin1.html Diophantine Equations]. Submitted by: Aaron Zerhusen, Chris Rakes, & Shasta Meece. MA 330-002. Dr. Carl Eberhart. 16 February 1999.</ref> who pioneered a form of [[syncopated algebra]] in his ''[[Arithmetica]],'' which introduced symbolic manipulation of expressions.<ref>Boyer (1991). "Revival and Decline of Greek Mathematics". pp. 180-182. "In this respect it can be compared with the great classics of the earlier Alexandrian Age; yet it has practically nothing in common with these or, in fact, with any traditional Greek mathematics. It represents essentially a new branch and makes use of a different approach. Being divorced from geometric methods, it resembles Babylonian algebra to a large extent. But whereas Babylonian mathematicians had been concerned primarily with approximate solutions of determinate equations as far as the third degree, the Arithmetica of Diophantus (such as we have it) is almost entirely devoted to the exact solution of equations, both determinate and indeterminate. [...] Throughout the six surviving books of Arithmetica there is a systematic use of abbreviations for powers of numbers and for relationships and operations. An unknown number is represented by a symbol resembling the Greek letter ζ {\displaystyle \zeta } (perhaps for the last letter of arithmos). [...] It is instead a collection of some 150 problems, all worked out in terms of specific numerical examples, although perhaps generality of method was intended. There is no postulation development, nor is an effort made to find all possible solutions. In the case of quadratic equations with two positive roots, only the larger is give, and negative roots are not recognized. No clear-cut distinction is made between determinate and indeterminate problems, and even for the latter for which the number of solutions generally is unlimited, only a single answer is given. Diophantus solved problems involving several unknown numbers by skillfully expressing all unknown quantities, where possible, in terms of only one of them."</ref> His notation represented unknowns and powers symbolically, but without modern symbols for [[Relation (mathematics)|relations]] (such as [[Equality (mathematics)|equality]] or [[Inequality (mathematics)|inequality]]) or [[Exponentiation|exponents]].<ref name="Boyer">Boyer (1991). "Revival and Decline of Greek Mathematics". p. 178. "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> An unknown number was called <math>\zeta</math>.<ref>A History of Greek Mathematics: From Aristarchus to Diophantus.  By Sir Thomas Little Heath. Pg [[iarchive:bub gb 7DDQAAAAMAAJ/page/n472|456]]</ref> The square of <math>\zeta</math> was <math>\Delta^v</math>; the cube was <math>K^v</math>; the fourth power was <math>\Delta^v\Delta</math>; the fifth power was <math>\Delta K^v</math>; and <math>\pitchfork</math> meant to subtract everything on the right from the left.<ref>A History of Greek Mathematics: From Aristarchus to Diophantus.  By Sir Thomas Little Heath. Pg [[iarchive:bub gb 7DDQAAAAMAAJ/page/n474|458]]</ref> So for example, what would be written in modern notation as:
<math display="block">x^3 - 2x^2 + 10x -1,</math>
Would be written in Diophantus's syncopated notation as:
: <math>\Kappa^{\upsilon} \overline{\alpha} \; \zeta \overline{\iota} \;\, \pitchfork \;\, \Delta^{\upsilon} \overline{\beta} \; \Mu \overline{\alpha} \,\;</math>
In the 7th century, [[Brahmagupta]] used different colours to represent the unknowns in algebraic equations in the ''[[Brāhmasphuṭasiddhānta]]''. Greek and other ancient mathematical advances, were often trapped in cycles of bursts of creativity, followed by long periods of stagnation, but this began to change as knowledge spread in the [[early modern period]].

=== Symbolic stage and early arithmetic ===
[[File:Johannes_Widmann-Mercantile_Arithmetic_1489.jpg|thumb|275x275px|The 1489 use of the [[plus and minus signs]] in print.]]
The transition to fully symbolic algebra began with [[Ibn al-Banna' al-Marrakushi]] (1256–1321) and [[Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī]], (1412–1482) who introduced symbols for operations using [[Arabic script|Arabic characters]].<ref>{{MacTutor|id=Al-Banna|title=al-Marrakushi ibn Al-Banna}}</ref><ref name="Gullberg2">{{cite book |last1=Gullberg |first1=Jan |author-link=Jan Gullberg |url=https://archive.org/details/mathematicsfromb1997gull |title=Mathematics: From the Birth of Numbers |date=1997 |publisher=W. W. Norton |isbn=0-393-04002-X |page=[https://archive.org/details/mathematicsfromb1997gull/page/298 298] |url-access=registration}}</ref><ref name="Qalasadi2">{{MacTutor Biography|id=Al-Qalasadi|title=Abu'l Hasan ibn Ali al Qalasadi}}</ref>  The [[plus sign]] (+) appeared around 1351 with [[Nicole Oresme]],<ref>[https://books.google.com/books?id=k0U1AQAAMAAJ Der Algorismus proportionum des Nicolaus Oresme]: Zum ersten Male nach der Lesart der Handschrift R.40.2. der Königlichen Gymnasial-bibliothek zu Thorn. [[Nicole Oresme]]. S. Calvary & Company, 1868.</ref> likely derived from the Latin ''et'' (meaning "and"), while the minus sign (−) was first used in 1489 by [[Johannes Widmann]].<ref>''Later [[early modern]] version'': [[iarchive:anewsystemmerca04walsgoog|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> [[Luca Pacioli]] included these symbols in his works, though much was based on earlier contributions by [[Piero della Francesca]]. The [[radical symbol]] (√) for [[square root]] was introduced by [[Christoph Rudolff]] in the 1500s, and [[parentheses]] for [[Precedence (mathematics)|precedence]] by [[Niccolò Tartaglia]] in 1556. [[François Viète]]’s ''New Algebra'' (1591) formalized modern symbolic manipulation. The [[multiplication sign]] (×) was first used by [[William Oughtred]] and the [[division sign]] (÷) by [[Johann Rahn]].

[[René Descartes]] further advanced algebraic symbolism in ''[[La Géométrie]]'' (1637), where he introduced the use of letters at the end of the alphabet (x, y, z) for [[Variable (mathematics)|variables]], along with the [[Cartesian coordinate system]], which bridged algebra and geometry.<ref>{{harvnb|Descartes|2006|loc=p.1xiii}} "This short work marks the moment at which algebra and geometry ceased being separate."</ref> [[Isaac Newton]] and [[Gottfried Wilhelm Leibniz]] independently developed [[Calculus of variations|calculus]] in the late 17th century, with [[Leibniz's notation]] becoming the standard.