Complex number: Difference between revisions

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At first glance this looks like nonsense. However, formal calculations with complex numbers show that the equation {{math|1=''z''³ = ''i''}} has solutions {{math|−''i''}}, {{math|{{sfrac|{{sqrt|3}}+''i''|2}}}} and {{math|{{sfrac|-{{sqrt|3}}+''i''|2}}}}. Substituting these in turn for {{math|{{sqrt|-1}}{{sup|1/3}}}} in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of {{math|1=''x''³ − ''x'' = 0}}. Of course this particular equation can be solved at sight but it does illustrate that when general formulas are used to solve cubic equations with real roots then, as later mathematicians showed rigorously,{{efn|It has been proved that imaginary numbers have necessarily to 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>}} the use of complex numbers [[casus irreducibilis|is unavoidable]]. [[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, although hewho was at pains to stress their imaginaryunreal nature<ref>{{cite book |title=La Géométrie &#124; 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>

{{quote|...&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.'']}}