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[[File:A plus bi.svg|thumb|upright=1.15|right|A complex number {{math|''z''}} can be visually represented as a pair of numbers {{math|(''a'', ''b'')}} forming a [[vector (geometric)|position vector]] (blue) or a point (red) on a diagram called an Argand diagram, representing the complex plane. ''Re'' is the real axis, ''Im'' is the imaginary axis, and {{mvar|i}} is the "imaginary unit", that satisfies {{math|1=''i''<sup>2</sup> = −1}}.]]
In mathematics, a '''complex number''' is an element of a [[number system]] that extends the [[real number]]s with a specific element denoted {{mvar|i}}, called the [[imaginary unit]] and satisfying the equation <math>i^{2}= -1</math>; every complex number can be expressed in the form <math>a + bi</math>, where {{mvar|a}} and {{mvar|b}} are real numbers. Because no real number satisfies the above equation, {{mvar|i}} was called an [[imaginary number]] by [[René Descartes]]. For the complex number {{nowrap|<math>a+bi</math>,}} {{mvar|a}} is called the '''{{visible anchor|real part}}''', and {{mvar|b}} is called the '''{{visible anchor|imaginary part}}'''. The set of complex numbers is denoted by either of the symbols <math>\mathbb C</math> or {{math|'''C'''}}. Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world.<ref>For an extensive account of the history of "imaginary" numbers, from initial skepticism to ultimate acceptance, see {{cite book |last=Bourbaki |first=Nicolas |author-link=Nicolas Bourbaki |year=1998 |title=Elements of the History of Mathematics |chapter=Foundations of Mathematics § Logic: Set theory |pages=18–24 |publisher=Springer}}
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A complex number {{mvar|z}} can be identified with the [[ordered pair]] of real numbers <math>(\Re (z),\Im (z))</math>, which may be interpreted as coordinates of a point in a Euclidean plane with standard coordinates, which is then called the ''[[complex plane]]'' or ''[[Argand diagram]].''<ref>{{cite book |last=Pedoe |first=Dan |author-link=Daniel Pedoe |title=Geometry: A comprehensive course |publisher=Dover |year=1988 |isbn=978-0-486-65812-4}}</ref><ref name=":2">{{Cite web |last=Weisstein |first=Eric W. |title=Complex Number |url=https://mathworld.wolfram.com/ComplexNumber.html |access-date=2020-08-12 |website=mathworld.wolfram.com}}</ref>{{efn| {{harvnb|Solomentsev|2001}}: "The plane <math>\R^2</math> whose points are identified with the elements of <math>\Complex</math> is called the complex plane ... The complete geometric interpretation of complex numbers and operations on them appeared first in the work of C. Wessel (1799). The geometric representation of complex numbers, sometimes called the 'Argand diagram', came into use after the publication in 1806 and 1814 of papers by J.R. Argand, who rediscovered, largely independently, the findings of Wessel".}} The horizontal axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical axis, with increasing values upwards.
A real number {{mvar|a}} can be regarded as a complex number {{math|''a'' + 0''i''}}, whose imaginary part is 0. A purely imaginary number {{math|''bi''}} is a complex number {{math|0 + ''bi''}}, whose real part is zero. It is common to write {{math|1=''a'' + 0''i'' = ''a''}}, {{math|1=0 + ''bi'' = ''bi''}}, and {{math|1=''a'' + (−''b'')''i'' = ''a'' − ''bi''}}; for example, {{math|1=3 + (−4)''i'' = 3 − 4''i''}}.
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