Editing Complex number (section)

{{Short description|Number with a real and an imaginary part}}
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[[File:A plus bi.svg|thumb|upright=1.15|right|A complex number can be visually represented as a pair of numbers {{math|(''a'', ''b'')}} forming a vector 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}}
</ref><ref>"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.", {{harvnb|Penrose|2005|loc=pp.72–73 |url=https://books.google.com/books?id=VWTNCwAAQBAJ&pg=PA73}}.</ref>

Complex numbers allow solutions to all [[polynomial equation]]s, even those that have no solutions in real numbers. More precisely, the [[fundamental theorem of algebra]] asserts that every non-constant 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, because the square of a real number cannot be negative, but has the two nonreal complex solutions <math>-1+3i</math> and <math>-1-3i</math>.

Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule <math>i^{2}=-1</math> along with the [[associative law|associative]], [[commutative law|commutative]], and [[distributive law]]s. Every nonzero complex number has a [[multiplicative inverse]]. This makes the complex numbers a [[field (mathematics)|field]] with the real numbers as a subfield. Because of these properties, {{tmath|1=a + bi = a + ib}}, and which form is written depends upon convention and style considerations.

The complex numbers also form a [[real vector space]] of [[Two-dimensional space|dimension two]], with <math>\{1,i\}</math> as a [[standard basis]]. This standard basis makes the complex numbers a [[Cartesian plane]], called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely some geometric objects and operations can be expressed in terms of complex numbers. For example, the real numbers form the [[real line]], which is pictured as the horizontal axis of the complex plane, while real multiples of <math>i</math> are the vertical axis. A complex number can also be defined by its geometric [[Polar coordinate system|polar coordinates]]: the radius is called the [[absolute value]] of the complex number, while the angle from the positive real axis is called the argument of the complex number. The complex numbers of absolute value one form the [[unit circle]]. Adding a fixed complex number to all complex numbers defines a [[translation (geometry)|translation]] in the complex plane, and multiplying by a fixed complex number is a [[similarity (geometry)|similarity]] centered at the origin (dilating by the absolute value, and rotating by the argument). The operation of [[complex conjugation]] is the [[reflection symmetry]] with respect to the real axis. 

The complex numbers form a rich structure that is simultaneously an [[algebraically closed field]], a [[commutative algebra (structure)|commutative algebra]] over the reals, and a [[Euclidean vector space]] of dimension two. 
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