Complex number: Difference between revisions

Revision as of 21:17, 5 August 2025

Revision as of 19:50, 8 August 2025

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===Multiplication{{anchor|Multiplication|Square}}===

[[File:complex_number_multiplication_visualisation.svg|thumb|Multiplication of complex numbers 2−𝑖2−''i'' and 3+4𝑖''i'' visualised with vectors]]

The product of two complex numbers is computed as follows:

:<math>(a+bi) \cdot (c+di) = ac - bd + (ad+bc)i.</math>

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==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]].