Complex number: Difference between revisions

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</ref><ref name=":0">{{cite web |title=Comprehensive List of Algebra Symbols |date=2020-03-25 |website=Math Vault |language=en-US |url=https://mathvault.ca/hub/higher-math/math-symbols/algebra-symbols/ |access-date=2020-08-12}}</ref><ref name=":1">{{cite web |title=Complex Numbers |website=www.mathsisfun.com |url=https://www.mathsisfun.com/numbers/complex-numbers.html |access-date=2020-08-12}}</ref><ref>{{cite web |title=Complex Numbers |website=Brilliant Math & Science Wiki |url=https://brilliant.org/wiki/complex-numbers/ |access-date=2020-08-12}}</ref>{{efn| "Complex numbers, as much as reals, and perhaps even more, find a unity with nature that is truly remarkable. It is as though Nature herself is as impressed by the scope and consistency of the complex-number system as we are ourselves, and has entrusted to these numbers the precise operations of her world at its minutest scales." — R. Penrose (2016, <!-- [https://books.google.com/books?id=VWTNCwAAQBAJ&pg=PA73 ] --> p.&nbsp;73)<ref>{{cite book |first=Roger |last=Penrose |year=2016 |title=The Road to Reality: A complete guide to the laws of the universe |edition=reprint |publisher=Random House |isbn=978-1-4464-1820-8 |pages=72–73 |url=https://books.google.com/books?id=VWTNCwAAQBAJ&pg=PA73}}</ref> }}

Complex numbers allow solutions to certainall equations[[polynomial equation]]s, even those that have no solutions in real numbers. More precisely, the [[fundamental theorem of algebra]] asserts that every polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation

:<math>(x+1)^2 = -9</math>

has no real solution, since the square of a real number cannot be negative. Complex numbers, however,but provide a solution to this problem. The idea is to [[field extension|extend]]has the realtwo numbersnonreal with an [[Indeterminate (variable)|indeterminate]] {{math|''i''}} (sometimes called the imaginary unit) taken to satisfy the relation {{math|1=''i''² = −1}}, so thatcomplex solutions to equations like the preceding one can be found. In this case, the solutions are {{math|−1 + 3''i''}} and {{math|−1 − 3''i''}}, as can be verified using the fact that {{math|1=''i''² = −1}}:.

:<math>((-1+3i)+1)^2 = (3i)^2 = \left(3^2\right)\left(i^2\right) = 9(-1) = -9,</math>

:<math>((-1-3i)+1)^2 = (-3i)^2 = (-3)^2\left(i^2\right) = 9(-1) = -9.</math>

According to the [[fundamental theorem of algebra]], all [[polynomial equation]]s with real or complex coefficients in a single variable have a solution in complex numbers. In contrast, some polynomial equations with real coefficients have no solution in real numbers. The 16th-century Italian mathematician [[Gerolamo Cardano]] is credited with introducing complex numbers in his attempts to find solutions to [[cubic equation]]s.<ref>{{cite book |last1=Burton |first1=David M. |title=The History of Mathematics |publisher=[[McGraw-Hill]] |___location=New York |edition= 3rd |page=294 |isbn=978-0-07-009465-9 |year=1995}}</ref>

Formally, the complex number system can be defined as the [[field extension|algebraic extension]] of the ordinary real numbers by an imaginary number {{mvar|i}}.<ref name=Bourbaki-topology>{{cite book |last=Bourbaki |first=Nicolas |author-link=Nicolas Bourbaki |title=General Topology |publisher=Springer-Verlag}}</ref>{{rp|at=§VIII.1}} This means that complex numbers can be added, subtracted and multiplied as polynomials in the variable {{mvar|i}}, under the rule that {{math|''i''² {{=}} −1}}. Furthermore, complex numbers can also be divided by nonzero complex numbers.<ref name=":1"/> Overall, the complex number system is a [[field (mathematics)|field]].