Editing Expression (mathematics) (section)

=== 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.