Editing Gerolamo Cardano (section)

== Mathematics ==
[[File:Gerolamo Cardano (colour).jpg|thumb|right|200px|Portrait of Cardano on display at the School of Mathematics and Statistics, [[University of St Andrews]]]]

Gerolamo Cardano was the first European mathematician to make systematic use of negative numbers.<ref>[[Isaac Asimov]], ''Asimov on Numbers'', published by [[Pocket Books]], a division of [[Simon & Schuster]], 1966, 1977, page 119.</ref> He published with attribution the solution of [[Scipione del Ferro]] to the [[cubic equation]] and the solution of Cardano's student [[Lodovico Ferrari]] to the [[quartic equation]] in his 1545 book ''[[Ars Magna (Cardano book)|Ars Magna]]'', an influential work on algebra. The solution to one particular case of the cubic equation <math>ax^3+bx+c=0</math><ref>{{cite book|last=Burton|first=David|title=The History of Mathematics: An Introduction|edition=7th (2010)|publisher= [[McGraw-Hill]]|location=New York}}</ref> (in modern notation) had been communicated to him in 1539 by [[Niccolò Fontana Tartaglia]] (who later claimed that Cardano had sworn not to reveal it, and engaged Cardano in a decade-long dispute) in the form of a poem,<ref>V.J. Katz, ''A History of Mathematics: An Introduction'', 3rd edn. (Boston: Pearson Education, 2009).</ref> but del Ferro's solution predated Tartaglia's.<ref name=":0" /> In his exposition, he acknowledged the existence of what are now called [[imaginary number]]s, although he did not understand their properties, described for the first time by his Italian contemporary [[Rafael Bombelli]]. In ''Opus novum de proportionibus'' he introduced the [[binomial coefficient]]s and the [[binomial theorem]].

Cardano was chronically short of money and kept himself solvent by being an accomplished gambler and [[chess]] player. His book about games of chance, ''Liber de ludo aleae'' ("Book on Games of Chance"), written around 1564,<ref name="LudoYear">In Chapter 20 of ''Liber de Ludo Aleae'' he describes a personal experience from 1526 and then adds that "thirty-eight years have passed" [elapsis iam annis triginta octo]. This sentence is written by Cardano around 1564, age 63.</ref> but not published until 1663, contains the first systematic treatment of [[probability]],<ref>Katz, ibid., p. 488</ref> as well as a section on effective cheating methods. He used the game of throwing dice to understand the basic concepts of probability. He demonstrated the efficacy of defining [[odds]] as the ratio of favourable to unfavourable outcomes (which implies that the probability of an [[Event (probability theory)|event]] is given by the ratio of favourable outcomes to the total number of possible outcomes).<ref>[http://www.columbia.edu/~pg2113/index_files/Gorroochurn-Some%20Laws.pdf Some laws and problems in classical probability and how Cardano anticipated them Gorrochum, P. ''Chance''magazine 2012]</ref> He was also aware of the multiplication rule for independent events but was not certain about what values should be multiplied.<ref>Katz, ibid., p. 488</ref>