Editing Complex number (section)

==History==
{{See also|Negative number#History}}
The solution in [[nth root|radicals]] (without [[trigonometric functions]]) of a general [[cubic equation]], when all three of its roots are real numbers, contains the square roots of [[negative numbers]], a situation that cannot be rectified by factoring aided by the [[rational root test]], if the cubic is [[irreducible polynomial|irreducible]]; this is the so-called ''[[casus irreducibilis]]'' ("irreducible case"). This conundrum led Italian mathematician [[Gerolamo Cardano]] to conceive of complex numbers in around 1545 in his ''[[Ars Magna (Cardano book)|Ars Magna]]'',<ref>{{cite book|first=Morris |last= Kline|title=A history of mathematical thought, volume 1|page=253}}</ref> though his understanding was rudimentary; moreover, he later described complex numbers as being "as subtle as they are useless".<ref>{{Cite book|last=Jurij.|first=Kovič|url=http://worldcat.org/oclc/1080410598|title=Tristan Needham, Visual Complex Analysis, Oxford University Press Inc., New York, 1998, 592 strani|oclc=1080410598}}</ref> Cardano did use imaginary numbers, but described using them as "mental torture."<ref>O'Connor and Robertson (2016), "Girolamo Cardano."</ref> This was prior to the use of the graphical complex plane. Cardano and other Italian mathematicians, notably [[Scipione del Ferro]], in the 1500s created an algorithm for solving cubic equations which generally had one real solution and two solutions containing an imaginary number. Because they ignored the answers with the imaginary numbers, Cardano found them useless.<ref>Nahin, Paul J. An Imaginary Tale: The Story of √−1. Princeton: Princeton University Press, 1998.</ref>

Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every [[polynomial equation]] of degree one or higher. Complex numbers thus form an [[algebraically closed field]], where any polynomial equation has a [[Root of a function|root]].

Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician [[Rafael Bombelli]].<ref>{{cite book |last1=Katz |first1=Victor J. |title=A History of Mathematics, Brief Version |section= 9.1.4 |publisher=[[Addison-Wesley]] |isbn=978-0-321-16193-2 |year=2004}}</ref> A more abstract formalism for the complex numbers was further developed by the Irish mathematician [[William Rowan Hamilton]], who extended this abstraction to the theory of [[quaternions]].<ref>{{cite journal |last1=Hamilton |first1=Wm. |title=On a new species of imaginary quantities connected with a theory of quaternions |journal=Proceedings of the Royal Irish Academy |date=1844 |volume=2 |pages=424–434 |url=https://babel.hathitrust.org/cgi/pt?id=njp.32101040410779&view=1up&seq=454}}</ref>

The earliest fleeting reference to [[square root]]s of [[negative number]]s can perhaps be said to occur in the work of the Greek mathematician [[Hero of Alexandria]] in the 1st century [[AD]], where in his ''[[Hero of Alexandria#Bibliography|Stereometrica]]'' he considered, apparently in error, the volume of an impossible [[frustum]] of a [[pyramid]] to arrive at the term <math>\sqrt{81 - 144}</math> in his calculations, which today would simplify to <math>\sqrt{-63} = 3i\sqrt{7}</math>.{{efn|In the literature the imaginary unit often precedes the radical sign, even when preceded itself by an integer.<ref>{{cite book |title=Trigonometry |author1=Cynthia Y. Young |edition=4th |publisher=John Wiley & Sons |year=2017 |isbn=978-1-119-44520-3 |page=406 |url=https://books.google.com/books?id=476ZDwAAQBAJ}} [https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA406 Extract of page 406]</ref>}} Negative quantities were not conceived of in [[Hellenistic mathematics]] and Hero merely replaced the negative value by its positive <math>\sqrt{144 - 81} = 3\sqrt{7}.</math><ref>{{cite book |title=An Imaginary Tale: The Story of √−1 |last=Nahin |first=Paul J. |year=2007 |publisher=[[Princeton University Press]] |isbn=978-0-691-12798-9 |url=http://mathforum.org/kb/thread.jspa?forumID=149&threadID=383188&messageID=1181284 |access-date=20 April 2011 |archive-url=https://web.archive.org/web/20121012090553/http://mathforum.org/kb/thread.jspa?forumID=149&threadID=383188&messageID=1181284 |archive-date=12 October 2012 |url-status=live }}</ref>

The impetus to study complex numbers as a topic in itself first arose in the 16th century when [[algebraic solution]]s for the roots of [[Cubic equation|cubic]] and [[Quartic equation|quartic]] [[polynomial]]s were discovered by Italian mathematicians ([[Niccolò Fontana Tartaglia]] and [[Gerolamo Cardano]]). It was soon realized (but proved much later)<ref name=Casus/> that these formulas, even if one were interested only in real solutions, sometimes required the manipulation of square roots of negative numbers. In fact, it was proved later that the use of complex numbers [[casus irreducibilis|is unavoidable]] when all three roots are real and distinct.{{efn|It has been proved that imaginary numbers necessarily appear in the cubic formula when the equation has three real, different roots by Pierre Laurent Wantzel in 1843, Vincenzo Mollame in 1890, Otto Hölder in 1891, and Adolf Kneser in 1892. Paolo Ruffini also provided an incomplete proof in 1799.——S. Confalonieri (2015)<ref name=Casus>{{cite book |title=The Unattainable Attempt to Avoid the Casus Irreducibilis for Cubic Equations: Gerolamo Cardano's De Regula Aliza |first=Sara |last=Confalonieri |publisher=Springer |year=2015 |pages=15–16 (note 26) |isbn=978-3658092757 }}</ref>}} However, the general formula can still be used in this case, with some care to deal with the ambiguity resulting from the existence of three cubic roots for nonzero complex numbers. Rafael Bombelli was the first to address explicitly these seemingly paradoxical solutions of cubic equations and developed the rules for complex arithmetic, trying to resolve these issues.

The term "imaginary" for these quantities was coined by [[René Descartes]] in 1637, who was at pains to stress their unreal nature:<ref>{{cite book |title=La Géométrie {{pipe}} The Geometry of René Descartes with a facsimile of the first edition |last=Descartes |first=René |author-link=René Descartes |year=1954 |orig-year=1637 |publisher=[[Dover Publications]] |isbn=978-0-486-60068-0 |url=https://archive.org/details/geometryofrenede00rend |access-date=20 April 2011 }}</ref>
{{blockquote|...&nbsp;sometimes only imaginary, that is one can imagine as many as I said in each equation, but sometimes there exists no quantity that matches that which we imagine.<br/>
[''...&nbsp;quelquefois seulement imaginaires c'est-à-dire que l'on peut toujours en imaginer autant que j'ai dit en chaque équation, mais qu'il n'y a quelquefois aucune quantité qui corresponde à celle qu'on imagine.'']}}
A further source of confusion was that the equation <math>\sqrt{-1}^2 = \sqrt{-1}\sqrt{-1} = -1</math> seemed to be capriciously inconsistent with the algebraic identity <math>\sqrt{a}\sqrt{b} = \sqrt{ab}</math>, which is valid for non-negative real numbers {{mvar|a}} and {{mvar|b}}, and which was also used in complex number calculations with one of {{mvar|a}}, {{mvar|b}} positive and the other negative. The incorrect use of this identity in the case when both {{mvar|a}} and {{mvar|b}} are negative, and the related identity <math display="inline">\frac{1}{\sqrt{a}} = \sqrt{\frac{1}{a}}</math>, even bedeviled [[Leonhard Euler]]. This difficulty eventually led to the convention of using the special symbol {{math|''i''}} in place of <math>\sqrt{-1}</math> to guard against this mistake.<ref>{{cite book |title=Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers |author1=Joseph Mazur |edition=reprinted |publisher=Princeton University Press |year=2016 |isbn=978-0-691-17337-5 |page=138 |url=https://books.google.com/books?id=O3CYDwAAQBAJ}} [https://books.google.com/books?id=O3CYDwAAQBAJ&pg=PA138 Extract of page 138]</ref><ref>{{cite book |title=Mathematical Fallacies and Paradoxes |author1=Bryan Bunch |edition=reprinted, revised |publisher=Courier Corporation |year=2012 |isbn=978-0-486-13793-3 |page=32 |url=https://books.google.com/books?id=jUTCAgAAQBAJ}} [https://books.google.com/books?id=jUTCAgAAQBAJ&pg=PA32 Extract of page 32]</ref> Even so, Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, ''[[Elements of Algebra]]'', he introduces these numbers almost at once and then uses them in a natural way throughout.

In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 [[Abraham de Moivre]] noted that the identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be re-expressed by the following [[de Moivre's formula]]:

<math display=block>(\cos \theta + i\sin \theta)^{n} = \cos n \theta + i\sin n \theta. </math>

[[File:Circle_cos_sin.gif |thumb |upright=1.5 |Euler's formula relates the complex exponential function of an imaginary argument, which can be thought of as describing [[uniform circular motion]] in the complex plane, to the cosine and sine functions, geometrically its projections onto the real and imaginary axes, respectively.]]
In 1748, Euler went further and obtained [[Euler's formula]] of [[complex analysis]]:<ref>{{cite book |last1=Euler |first1=Leonard |title=Introductio in Analysin Infinitorum |trans-title=Introduction to the Analysis of the Infinite |date=1748 |publisher=Marc Michel Bosquet & Co. |location=Lucerne, Switzerland |volume=1 |page=104 |url=https://books.google.com/books?id=jQ1bAAAAQAAJ&pg=PA104 |language=la}}</ref>

<math display="block">e ^{i\theta } = \cos \theta + i\sin \theta </math>

by formally manipulating complex [[power series]] and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities.

The idea of a complex number as a point in the complex plane was first described by [[Denmark|Danish]]–[[Norway|Norwegian]] [[mathematician]] [[Caspar Wessel]] in 1799,<ref>{{cite journal |last1=Wessel |first1=Caspar |title=Om Directionens analytiske Betegning, et Forsog, anvendt fornemmelig til plane og sphæriske Polygoners Oplosning |journal=Nye Samling af det Kongelige Danske Videnskabernes Selskabs Skrifter [New Collection of the Writings of the Royal Danish Science Society] |date=1799 |volume=5 |pages=469–518 |url=https://babel.hathitrust.org/cgi/pt?id=ien.35556000979690&view=1up&seq=561 |trans-title=On the analytic representation of direction, an effort applied in particular to the determination of plane and spherical polygons |language=da}}</ref> although it had been anticipated as early as 1685 in [[John Wallis|Wallis's]] ''A Treatise of Algebra''.<ref>{{cite book |last=Wallis |first=John |date=1685 |title=A Treatise of Algebra, Both Historical and Practical ... |url=https://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/H3GRV5AU/pageimg&start=291&mode=imagepath&pn=291|location=London, England |publisher=printed by John Playford, for Richard Davis |pages=264–273 }}</ref>

Wessel's memoir appeared in the Proceedings of the [[Copenhagen Academy]] but went largely unnoticed. In 1806 [[Jean-Robert Argand]] independently issued a pamphlet on complex numbers and provided a rigorous proof of the [[Fundamental theorem of algebra#History|fundamental theorem of algebra]].<ref>{{cite book |last1=Argand |title=Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques |trans-title=Essay on a way to represent complex quantities by geometric constructions |date=1806 |publisher=Madame Veuve Blanc |location=Paris, France |url=http://www.bibnum.education.fr/mathematiques/geometrie/essai-sur-une-maniere-de-representer-des-quantites-imaginaires-dans-les-cons |language=fr}}</ref> [[Carl Friedrich Gauss]] had earlier published an essentially [[topology|topological]] proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of &minus;1".<ref>Gauss, Carl Friedrich (1799) [https://books.google.com/books?id=g3VaAAAAcAAJ&pg=PP1 ''"Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse."''] [New proof of the theorem that any rational integral algebraic function of a single variable can be resolved into real factors of the first or second degree.] Ph.D. thesis, University of Helmstedt, (Germany). (in Latin)</ref> It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane,<ref name=Ewald>{{cite book |last=Ewald |first=William B. |date=1996 |title=From Kant to Hilbert: A Source Book in the Foundations of Mathematics |volume=1 |page=313 |publisher=Oxford University Press |isbn=9780198505358|url=https://books.google.com/books?id=rykSDAAAQBAJ&pg=PA313 |access-date=18 March 2020}}</ref> largely establishing modern notation and terminology:{{sfn|Gauss|1831}}

<blockquote>If one formerly contemplated this subject from a false point of view and therefore found a mysterious darkness, this is in large part attributable to clumsy terminology. Had one not called +1, −1, <math>\sqrt{-1}</math> positive, negative, or imaginary (or even impossible) units, but instead, say, direct, inverse, or lateral units, then there could scarcely have been talk of such darkness.</blockquote>

In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée,<ref>{{cite web| url = https://mathshistory.st-andrews.ac.uk/Biographies/Buee/| title = Adrien Quentin Buée (1745–1845): MacTutor}}</ref><ref>{{cite journal |last1=Buée |title=Mémoire sur les quantités imaginaires |journal=Philosophical Transactions of the Royal Society of London |date=1806 |volume=96 |pages=23–88 |doi=10.1098/rstl.1806.0003 |s2cid=110394048 |url=https://royalsocietypublishing.org/doi/pdf/10.1098/rstl.1806.0003 |trans-title=Memoir on imaginary quantities |language=fr}}</ref> [[C. V. Mourey|Mourey]],<ref>{{cite book |last1=Mourey |first1=C.V. |title=La vraies théore des quantités négatives et des quantités prétendues imaginaires |trans-title=The true theory of negative quantities and of alleged imaginary quantities |date=1861 |publisher=Mallet-Bachelier |location=Paris, France |url=https://archive.org/details/bub_gb_8YxKAAAAYAAJ |language=fr}}  1861 reprint of 1828 original.</ref> [[John Warren (mathematician)|Warren]],<ref>{{cite book |last1=Warren |first1=John |title=A Treatise on the Geometrical Representation of the Square Roots of Negative Quantities |date=1828 |publisher=Cambridge University Press |location=Cambridge, England |url=https://archive.org/details/treatiseongeomet00warrrich}}</ref><ref>{{cite journal |last1=Warren |first1=John |title=Consideration of the objections raised against the geometrical representation of the square roots of negative quantities |journal=Philosophical Transactions of the Royal Society of London |date=1829 |volume=119 |pages=241–254 |s2cid=186211638 |doi=10.1098/rstl.1829.0022 |doi-access=free }}</ref><ref>{{cite journal |last1=Warren |first1=John |title=On the geometrical representation of the powers of quantities, whose indices involve the square roots of negative numbers |journal=Philosophical Transactions of the Royal Society of London |date=1829 |volume=119 |pages=339–359 |s2cid=125699726 |doi=10.1098/rstl.1829.0031 |doi-access=free }}</ref> [[Jacques Frédéric Français|Français]] and his brother, [[Giusto Bellavitis|Bellavitis]].<ref>{{cite journal |last1=Français |first1=J.F. |title=Nouveaux principes de géométrie de position, et interprétation géométrique des symboles imaginaires |journal=Annales des mathématiques pures et appliquées |date=1813 |volume=4 |pages=61–71 |url=https://babel.hathitrust.org/cgi/pt?id=uc1.$c126478&view=1up&seq=69 |trans-title=New principles of the geometry of position, and geometric interpretation of complex [number] symbols |language=fr}}</ref><ref>{{cite book |title=Two Cultures |editor= Kim Williams |last1=Caparrini |first1=Sandro |chapter=On the Common Origin of Some of the Works on the Geometrical Interpretation of Complex Numbers |year=2000 |publisher=Birkhäuser |isbn=978-3-7643-7186-9 |page=139 |url=https://books.google.com/books?id=voFsJ1EhCnYC |chapter-url=https://books.google.com/books?id=voFsJ1EhCnYC&pg=PA139}}</ref>

The English mathematician [[G.H. Hardy]] remarked that Gauss was the first mathematician to use complex numbers in "a really confident and scientific way" although mathematicians such as Norwegian [[Niels Henrik Abel]] and [[Carl Gustav Jacob Jacobi]] were necessarily using them routinely before Gauss published his 1831 treatise.<ref>{{cite book |title=An Introduction to the Theory of Numbers |last1=Hardy |first1=G.H. |last2=Wright |first2=E.M. |year=2000 |orig-year=1938 |publisher=[[Oxford University Press|OUP Oxford]] |isbn= 978-0-19-921986-5 |page=189 (fourth edition)}}</ref>

[[Augustin-Louis Cauchy]] and [[Bernhard Riemann]] together brought the fundamental ideas of [[#Complex analysis|complex analysis]] to a high state of completion, commencing around 1825 in Cauchy's case.

The common terms used in the theory are chiefly due to the founders. Argand called {{math|cos ''φ'' + ''i'' sin ''φ''}} the ''direction factor'', and <math>r = \sqrt{a^2 + b^2}</math> the ''modulus'';{{efn| {{harvnb|Argand|1814|p=204}} defines the modulus of a complex number but he doesn't name it:<br/>''"Dans ce qui suit, les accens, indifféremment placés, seront employés pour indiquer la grandeur absolue des quantités qu'ils affectent; ainsi, si <math>a = m + n\sqrt{-1}</math>, <math>m</math> et <math>n</math> étant réels, on devra entendre que <math>a_'</math> ou <math>a' = \sqrt{m^2 + n^2}</math>."''<br/>[In what follows, accent marks, wherever they're placed, will be used to indicate the absolute size of the quantities to which they're assigned; thus if <math>a = m + n\sqrt{-1}</math>, <math>m</math> and <math>n</math> being real, one should understand that <math>a_'</math> or <math>a' = \sqrt{m^2 + n^2}</math>.]<br/>
{{harvnb|Argand|1814|p=208}} defines and names the ''module'' and the ''direction factor'' of a complex number:  ''"...&nbsp;<math>a = \sqrt{m^2 + n^2}</math> pourrait être appelé le ''module'' de <math>a + b  \sqrt{-1}</math>, et représenterait la ''grandeur absolue'' de la ligne <math>a + b  \sqrt{-1}</math>, tandis que l'autre facteur, dont le module est l'unité, en représenterait la direction."''<br/>[...&nbsp;<math>a = \sqrt{m^2 + n^2}</math> could be called the ''module'' of <math>a + b  \sqrt{-1}</math> and would represent the ''absolute size'' of the line <math>a + b  \sqrt{-1}\,,</math> (Argand represented complex numbers as vectors.) whereas the other factor [namely, <math>\tfrac{a}{\sqrt{a^2 + b^2}} + \tfrac{b}{\sqrt{a^2 + b^2}} \sqrt{-1} </math>], whose module is unity [1], would represent its direction.]}}<ref>{{cite web |author=Jeff Miller |date=Sep 21, 1999 |title=MODULUS |url=http://members.aol.com/jeff570/m.html|archive-url=https://web.archive.org/web/19991003034827/http://members.aol.com/jeff570/m.html |work=Earliest Known Uses of Some of the Words of Mathematics (M) |archive-date=1999-10-03 |url-status=usurped}}</ref> Cauchy (1821) called {{math|cos ''φ'' + ''i'' sin ''φ''}} the ''reduced form'' (l'expression réduite)<ref>{{cite book |last=Cauchy |first=Augustin-Louis |date=1821 |title=Cours d'analyse de l'École royale polytechnique |url=https://archive.org/details/coursdanalysede00caucgoog/page/n209/mode/2up |location=Paris, France |publisher=L'Imprimerie Royale |volume=1 |page=183 |language=fr }}</ref> and apparently introduced the term ''argument''; Gauss used {{math|''i''}} for <math>\sqrt{-1}</math>,{{efn| Gauss writes:<ref>{{harvnb|Gauss|1831|p=96}}</ref> ''"Quemadmodum scilicet arithmetica sublimior in quaestionibus hactenus pertractatis inter solos numeros integros reales versatur, ita theoremata circa residua biquadratica tunc tantum in summa simplicitate ac genuina venustate resplendent, quando campus arithmeticae ad quantitates ''imaginarias'' extenditur, ita ut absque restrictione ipsius obiectum constituant numeri formae ''a + bi'', denotantibus ''i'', pro more quantitatem imaginariam <math>\sqrt{-1}</math>, atque ''a, b'' indefinite omnes numeros reales integros inter -<math>\infty</math> et +<math>\infty</math>."'' [Of course just as the higher arithmetic has been investigated so far in problems only among real integer numbers, so theorems regarding biquadratic residues then shine in greatest simplicity and genuine beauty, when the field of arithmetic is extended to ''imaginary'' quantities, so that, without restrictions on it, numbers of the form ''a + bi'' — ''i'' denoting by convention the imaginary quantity <math>\sqrt{-1}</math>, and the variables ''a, b'' [denoting] all real integer numbers between <math>-\infty</math> and <math>+\infty</math> — constitute an object.]}} introduced the term ''complex number'' for {{math|''a'' + ''bi''}},{{efn|Gauss:<ref>{{harvnb|Gauss|1831|p=96}}</ref> ''"Tales numeros vocabimus numeros integros complexos, ita quidem, ut reales complexis non opponantur, sed tamquam species sub his contineri censeantur."'' [We will call such numbers [namely, numbers of the form ''a + bi'' ] "complex integer numbers", so that real [numbers] are regarded not as the opposite of complex [numbers] but [as] a type [of number that] is, so to speak, contained within them.]}} and called {{math|''a''<sup>2</sup> + ''b''<sup>2</sup>}} the ''norm''.{{efn|Gauss:<ref>{{harvnb|Gauss|1831|p=98}}</ref> ''"Productum numeri complexi per numerum ipsi conjunctum utriusque ''normam'' vocamus. Pro norma itaque numeri realis, ipsius quadratum habendum est."'' [We call a "norm" the product of a complex number [for example, ''a + ib'' ] with its conjugate [''a - ib'' ].  Therefore the square of a real number should be regarded as its norm.]}} The expression ''direction coefficient'', often used for {{math|cos ''φ'' + ''i'' sin ''φ''}}, is due to Hankel (1867),<ref>{{cite book |last=Hankel |first=Hermann |date=1867 |title=Vorlesungen über die complexen Zahlen und ihre Functionen |trans-title=Lectures About the Complex Numbers and Their Functions |url=https://books.google.com/books?id=754KAAAAYAAJ&pg=PA71 |location=Leipzig, [Germany] |publisher=Leopold Voss |volume=1 |page=71 |language=de }}  From p. 71:  ''"Wir werden den Factor (''cos'' φ + i ''sin'' φ) haüfig den ''Richtungscoefficienten'' nennen."'' (We will often call the factor (cos φ + i sin φ) the "coefficient of direction".)</ref> and ''absolute value,'' for ''modulus,'' is due to Weierstrass.

Later classical writers on the general theory include [[Richard Dedekind]], [[Otto Hölder]], [[Felix Klein]], [[Henri Poincaré]], [[Hermann Schwarz]], [[Karl Weierstrass]] and many others. Important work (including a systematization) in complex multivariate calculus has been started at beginning of the 20th century. Important results have been achieved by [[Wilhelm Wirtinger]] in 1927.