Editing Number (section)

===Irrational numbers {{anchor|History of irrational numbers}}===
{{further|History of irrational numbers}}
The earliest known use of irrational numbers was in the [[Indian mathematics|Indian]] [[Sulba Sutras]] composed between 800 and 500&nbsp;BC.<ref>{{Cite book |editor-last=Selin |editor-first=Helaine |editor-link=Helaine Selin |title=Mathematics across cultures: the history of non-Western mathematics |publisher=Kluwer Academic Publishers |year=2000 |page=451 |isbn=0-7923-6481-3}}</ref>{{Better source needed|reason=Source may be unreliable it garbles both the history and the mathematics. Source only says the mathematics in the Shulba Sutras "leads to the concept of irrational numbers". Since good approximations of irrational numbers appeared in earlier times, it's not clear what special role is being claimed for the Shulba Sutras in the history of irrational numbers. Also, should page reference be to p. 412 rather than p. 451?|date=September 2020}} The first existence proofs of irrational numbers is usually attributed to [[Pythagoras]], more specifically to the [[Pythagoreanism|Pythagorean]] [[Hippasus|Hippasus of Metapontum]], who produced a (most likely geometrical) proof of the irrationality of the [[square root of 2]]. The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction. However, Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. He could not disprove their existence through logic, but he could not accept irrational numbers, and so, allegedly and frequently reported, he sentenced Hippasus to death by drowning, to impede spreading of this disconcerting news.<ref>{{cite book |title=Harvard Studies in Classical Philology |chapter=Horace and the Monuments: A New Interpretation of the Archytas ''Ode'' |author=Bernard Frischer |editor=D.R. Shackleton Bailey |editor-link=D. R. Shackleton Bailey |page=83 |publisher=Harvard University Press |year=1984 |isbn=0-674-37935-7}}</ref>{{Better source needed|reason=Hippasus is mentioned only briefly in passing in this work. Entire books have been written on Pythagoras and Pythagoreanism; surely a reference could be provide to one of those? But any serious work will say that everything in this paragraph is unreliable myth, and some is outright modern fabrication, e.g. Pythagoras sentencing Hippasus to death.|date=September 2020}}

The 16th century brought final European acceptance of negative integral and fractional numbers. By the 17th century, mathematicians generally used decimal fractions with modern notation. It was not, however, until the 19th century that mathematicians separated irrationals into algebraic and transcendental parts, and once more undertook the scientific study of irrationals. It had remained almost dormant since [[Euclid]]. In 1872, the publication of the theories of [[Karl Weierstrass]] (by his pupil E. Kossak), [[Eduard Heine]],<ref>Eduard Heine, [[doi:10.1515/crll.1872.74.172|"Die Elemente der Functionenlehre"]], ''[Crelle's] Journal für die reine und angewandte Mathematik'', No. 74 (1872): 172–188.</ref> [[Georg Cantor]],<ref>Georg Cantor, [[doi:10.1007/BF01446819|"Ueber unendliche, lineare Punktmannichfaltigkeiten", pt. 5]], ''Mathematische Annalen'', 21, 4 (1883‑12): 545–591.</ref> and [[Richard Dedekind]]<ref>Richard Dedekind, ''[https://books.google.com/books?id=n-43AAAAMAAJ Stetigkeit & irrationale Zahlen] {{Webarchive|url=https://web.archive.org/web/20210709184745/https://books.google.ca/books?id=n-43AAAAMAAJ |date=2021-07-09 }}'' (Braunschweig: Friedrich Vieweg & Sohn, 1872). Subsequently published in: ''———, Gesammelte mathematische Werke'', ed. Robert Fricke, Emmy Noether & Öystein Ore (Braunschweig: Friedrich Vieweg & Sohn, 1932), vol. 3, pp. 315–334.</ref> was brought about. In 1869, [[Charles Méray]] had taken the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method was completely set forth by [[Salvatore Pincherle]] (1880), and Dedekind's has received additional prominence through the author's later work (1888) and endorsement by [[Paul Tannery]] (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a [[Dedekind cut|cut (Schnitt)]] in the system of [[real number]]s, separating all [[rational number]]s into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, [[Leopold Kronecker|Kronecker]],<ref>L. Kronecker, [[doi:10.1515/crll.1887.101.337|"Ueber den Zahlbegriff"]], ''[Crelle's] Journal für die reine und angewandte Mathematik'', No. 101 (1887): 337–355.</ref> and Méray.

The search for roots of [[Quintic equation|quintic]] and higher degree equations was an important development, the [[Abel–Ruffini theorem]] ([[Paolo Ruffini (mathematician)|Ruffini]] 1799, [[Niels Henrik Abel|Abel]] 1824) showed that they could not be solved by [[nth root|radicals]] (formulas involving only arithmetical operations and roots). Hence it was necessary to consider the wider set of [[algebraic numbers]] (all solutions to polynomial equations). [[Évariste Galois|Galois]] (1832) linked polynomial equations to [[group theory]] giving rise to the field of [[Galois theory]].

[[Simple continued fraction]]s, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of [[Euler]],<ref>Leonhard Euler, "Conjectura circa naturam aeris, pro explicandis phaenomenis in atmosphaera observatis", ''Acta Academiae Scientiarum Imperialis Petropolitanae'', 1779, 1 (1779): 162–187.</ref> and at the opening of the 19th&nbsp;century were brought into prominence through the writings of [[Joseph Louis Lagrange]]. Other noteworthy contributions have been made by Druckenmüller (1837), Kunze (1857), Lemke (1870), and Günther (1872). Ramus<ref>Ramus, "Determinanternes Anvendelse til at bes temme Loven for de convergerende Bröker", in: ''Det Kongelige Danske Videnskabernes Selskabs naturvidenskabelige og mathematiske Afhandlinger'' (Kjoebenhavn: 1855), p. 106.</ref> first connected the subject with [[determinant]]s, resulting, with the subsequent contributions of Heine,<ref>Eduard Heine, [[doi:10.1515/crll.1859.56.87|"Einige Eigenschaften der ''Lamé''schen Funktionen"]], ''[Crelle's] Journal für die reine und angewandte Mathematik'', No. 56 (Jan. 1859): 87–99 at 97.</ref> [[August Ferdinand Möbius|Möbius]], and Günther,<ref>Siegmund Günther, ''Darstellung der Näherungswerthe von Kettenbrüchen in independenter Form'' (Erlangen: Eduard Besold, 1873); ———, "Kettenbruchdeterminanten", in: ''Lehrbuch der Determinanten-Theorie: Für Studirende'' (Erlangen: Eduard Besold, 1875), c. 6, pp. 156–186.</ref> in the theory of {{Lang|de|Kettenbruchdeterminanten}}.