Difference between revisions of "Complex Numbers"

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has no real solution, since the square of a real number cannot be negative, but has the two nonreal complex solutions {{math|−1 + 3''i''}} and {{math|−1 − 3''i''}}.
 
has no real solution, since the square of a real number cannot be negative, but has the two nonreal complex solutions {{math|−1 + 3''i''}} and {{math|−1 − 3''i''}}.
  
Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule {{math|1=''i''<sup>2</sup> = −1}} combined with the associative, commutative and distributive laws. Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field that has the real numbers as a subfield. The complex numbers form also a real vector space of dimension two, with {{math|{{mset|1, ''i''}}}} as a standard basis.
+
Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule {{math|1=''i''<sup>2</sup> = −1}} combined with the associative, commutative and distributive laws. Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field that has the real numbers as a subfield. The complex numbers form also a real vector space of dimension two, with {{math|{1, ''i''}} 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 expressing in terms of complex numbers some geometric properties and constructions. For example, the real numbers form the real line which is identified to the horizontal axis of the complex plane. The complex numbers of absolute value one form the unit circle. The addition of a complex number is a translation in the complex plane, and the multiplication by a complex number is a similarity centered at the origin. The complex conjugation is the reflection symmetry with respect to the real axis. The complex absolute value is a Euclidean norm.  
 
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 expressing in terms of complex numbers some geometric properties and constructions. For example, the real numbers form the real line which is identified to the horizontal axis of the complex plane. The complex numbers of absolute value one form the unit circle. The addition of a complex number is a translation in the complex plane, and the multiplication by a complex number is a similarity centered at the origin. The complex conjugation is the reflection symmetry with respect to the real axis. The complex absolute value is a Euclidean norm.  
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==Definition==
 
==Definition==
  
[[File:Illustration of a complex number.svg|right|thumb|upright=1.05|An illustration of the complex number {{math|1=''z'' = ''x'' + ''iy''}} on the [[complex plane]]. The real part is {{mvar|x}}, and its imaginary part is {{mvar|y}}.]]
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[[File:Illustration of a complex number.svg|right|thumb|upright=1.05|An illustration of the complex number {{math|1=''z'' = ''x'' + ''iy''}} on the complex plane. The real part is {{mvar|x}}, and its imaginary part is {{mvar|y}}.]]
  
A complex number is a number of the form {{math|1=''a'' + ''bi''}}, where {{mvar|a}} and {{mvar|b}} are [[real numbers]], and {{math|''i''}} is an indeterminate satisfying {{math|1=''i''<sup>2</sup> = −1}}. For example, {{math|2 + 3''i''}} is a complex number.<ref>{{cite book|title=College algebra |url=https://archive.org/details/collegealgebrawi00axle |url-access=limited |last=Axler |first=Sheldon |page=[https://archive.org/details/collegealgebrawi00axle/page/n285 262]|publisher=Wiley|year=2010|isbn=9780470470770 }}</ref><ref name=":1" />
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A complex number is a number of the form {{math|1=''a'' + ''bi''}}, where {{mvar|a}} and {{mvar|b}} are real numbers, and {{math|''i''}} is an indeterminate satisfying {{math|1=''i''<sup>2</sup> = −1}}. For example, {{math|2 + 3''i''}} is a complex number.
  
This way, a complex number is defined as a [[polynomial]] with real coefficients in the single indeterminate {{math|''i''}}, for which the relation {{math|''i''<sup>2</sup> + 1 {{=}} 0}} is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials. The relation {{math|''i''<sup>2</sup> + 1 {{=}} 0}} induces the equalities {{math|''i''<sup>4''k''</sup> {{=}} 1, ''i''<sup>4''k''+1</sup> {{=}} ''i'', ''i''<sup>4''k''+2</sup> {{=}} −1,}} and {{math|''i''<sup>4''k''+3</sup> {{=}} −''i'',}} which hold for all integers {{mvar|k}}; these allow the reduction of any polynomial that results from the addition and multiplication of complex numbers to a linear polynomial in {{mvar|i}}, again of the form {{math|1=''a'' + ''bi''}} with real coefficients {{mvar|a, b.}}
+
This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate {{math|''i''}}, for which the relation {{math|''i''<sup>2</sup> + 1 = 0}} is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials. The relation {{math|''i''<sup>2</sup> + 1 = 0}} induces the equalities {{math|''i''<sup>4''k''</sup> = 1, ''i''<sup>4''k''+1</sup> = ''i'', ''i''<sup>4''k''+2</sup> = −1,}} and {{math|''i''<sup>4''k''+3</sup> = −''i'',}} which hold for all integers {{mvar|k}}; these allow the reduction of any polynomial that results from the addition and multiplication of complex numbers to a linear polynomial in {{mvar|i}}, again of the form {{math|1=''a'' + ''bi''}} with real coefficients {{mvar|a, b.}}
  
The real number {{mvar|a}} is called the ''real part'' of the complex number {{math|''a'' + ''bi''}}; the real number {{mvar|b}} is called its ''imaginary part''. To emphasize, the imaginary part does not include a factor {{mvar|i}}; that is, the imaginary part is {{mvar|b}}, not {{math|''bi''}}.<ref>{{cite book |last1=Spiegel |first1= M.R. |last2=Lipschutz |first2= S. |last3= Schiller |first3= J.J. |last4=Spellman |first4=D. |title= Complex Variables |edition=2nd |series=Schaum's Outline Series |publisher= McGraw Hill |isbn= 978-0-07-161569-3|date= 14 April 2009 }}</ref><ref>{{cite book |title=College Algebra and Trigonometry |edition=6 |first1=Richard N. |last1=Aufmann |first2=Vernon C. |last2=Barker |first3=Richard D. |last3=Nation |publisher=Cengage Learning |year=2007 |isbn=978-0-618-82515-8 |page=66 |chapter-url=https://books.google.com/books?id=g5j-cT-vg_wC&pg=PA66 |chapter=Chapter P}}</ref><ref name=":1" />
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The real number {{mvar|a}} is called the ''real part'' of the complex number {{math|''a'' + ''bi''}}; the real number {{mvar|b}} is called its ''imaginary part''. To emphasize, the imaginary part does not include a factor {{mvar|i}}; that is, the imaginary part is {{mvar|b}}, not {{math|''bi''}}.
  
Formally, the complex numbers are defined as the [[quotient ring]] of the [[polynomial ring]] in the indeterminate {{math|''i''}}, by the [[ideal (ring theory)|ideal]] generated by the polynomial {{math|''i''<sup>2</sup> + 1}} (see [[#Construction as a quotient field|below]]).<ref>{{harv|Bourbaki|1998|loc=§VIII.1}}</ref>
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Formally, the complex numbers are defined as the quotient ring of the polynomial ring in the indeterminate {{math|''i''}}, by the ideal generated by the polynomial {{math|''i''<sup>2</sup> + 1}}.
  
 
==Notation==
 
==Notation==
  
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. As with polynomials, it is common to write {{mvar|a}} for {{math|''a'' + 0''i''}} and {{math|''bi''}} for {{math|0 + ''bi''}}. Moreover, when the imaginary part is negative, that is, {{math|1=''b'' = −''{{!}}b{{!}}'' < 0}}, it is common to write {{math|''a'' − ''{{!}}b{{!}}i''}} instead of {{math|''a'' + (−''{{!}}b{{!}}'')''i''}}; for example, for {{math|1=''b'' = −4}}, {{math|3 − 4''i''}} can be written instead of {{math|3 + (−4)''i''}}.
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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. As with polynomials, it is common to write {{mvar|a}} for {{math|''a'' + 0''i''}} and {{math|''bi''}} for {{math|0 + ''bi''}}. Moreover, when the imaginary part is negative, that is, {{math|1=''b'' = −''{{!}}b{{!}}'' < 0}}, it is common to write {{math|''a'' − ''{{!}}b{{!}}i''}} instead of {{math|''a'' + (−''{{!}}b{{!}}'')''i''}}; for example, for {{math|1=''b'' = −4}}, {{math|3 − 4''i''}} can be written instead of {{math|3 + (−4)''i''}}.
  
Since the multiplication of the indeterminate {{math|''i''}} and a real is commutative in polynomials with real coefficients, the polynomial {{math|''a'' + ''bi''}} may be written as {{math|''a'' + ''ib''.}} This is often expedient for imaginary parts denoted by expressions, for example, when {{mvar|b}} is a radical.<ref>See {{harv|Ahlfors|1979}}.</ref>
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Since the multiplication of the indeterminate {{math|''i''}} and a real is commutative in polynomials with real coefficients, the polynomial {{math|''a'' + ''bi''}} may be written as {{math|''a'' + ''ib''.}} This is often expedient for imaginary parts denoted by expressions, for example, when {{mvar|b}} is a radical.
  
 
The real part of a complex number {{mvar|z}} is denoted by {{math|Re(''z'')}}, <math>\mathcal{Re}(z)</math>, or <math>\mathfrak{R}(z)</math>; the imaginary part of a complex number {{mvar|z}} is denoted by {{math|Im(''z'')}}, <math>\mathcal{Im}(z)</math>, or <math>\mathfrak{I}(z).</math> For example,
 
The real part of a complex number {{mvar|z}} is denoted by {{math|Re(''z'')}}, <math>\mathcal{Re}(z)</math>, or <math>\mathfrak{R}(z)</math>; the imaginary part of a complex number {{mvar|z}} is denoted by {{math|Im(''z'')}}, <math>\mathcal{Im}(z)</math>, or <math>\mathfrak{I}(z).</math> For example,
 
<math display=block>  \operatorname{Re}(2 + 3i) = 2 \quad \text{ and } \quad  \operatorname{Im}(2 + 3i) = 3~.</math>
 
<math display=block>  \operatorname{Re}(2 + 3i) = 2 \quad \text{ and } \quad  \operatorname{Im}(2 + 3i) = 3~.</math>
  
The [[Set (mathematics)|set]] of all complex numbers is denoted by <math>\Complex</math> ([[blackboard bold]]) or {{math|'''C'''}} (upright bold).
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The set of all complex numbers is denoted by <math>\Complex</math> (blackboard bold) or {{math|'''C'''}} (upright bold).
  
In some disciplines, particularly in [[electromagnetism]] and [[electrical engineering]], {{mvar|j}} is used instead of {{mvar|i}} as {{mvar|i}} is frequently used to represent [[electric current]].<ref>{{cite book |last1=Brown |first1=James Ward |last2=Churchill |first2=Ruel V. |title=Complex variables and applications |year=1996 |publisher=McGraw-Hill |location=New York |isbn=978-0-07-912147-9 |edition=6th |page=2 |quote=In electrical engineering, the letter ''j'' is used instead of ''i''.}}</ref> In these cases, complex numbers are written as {{math|''a'' + ''bj''}}, or {{math|''a'' + ''jb''}}.
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In some disciplines, particularly in electromagnetism and electrical engineering, {{mvar|j}} is used instead of {{mvar|i}} as {{mvar|i}} is frequently used to represent electric current.
  
 
==Visualization==
 
==Visualization==
{{Main|Complex plane}}
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[[File:Complex number illustration.svg|thumb|right|A complex number {{mvar|z}}, as a point (black) and its position vector (blue)]]
 
[[File:Complex number illustration.svg|thumb|right|A complex number {{mvar|z}}, as a point (black) and its position vector (blue)]]
  
A complex number {{mvar|z}} can thus be identified with an [[ordered pair]] <math>(\Re (z),\Im (z))</math> of real numbers, which in turn may be interpreted as coordinates of a point in a two-dimensional space. The most immediate space is the Euclidean plane with suitable coordinates, which is then called ''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>{{efn| "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". — {{harv|Solomentsev|2001}} }}<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> named after [[Jean-Robert Argand]]. Another prominent space on which the coordinates may be projected is the two-dimensional surface of a sphere, which is then called [[Riemann sphere]].
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A complex number {{mvar|z}} can thus be identified with an ordered pair <math>(\Re (z),\Im (z))</math> of real numbers, which in turn may be interpreted as coordinates of a point in a two-dimensional space. The most immediate space is the Euclidean plane with suitable coordinates, which is then called ''complex plane'' or ''Argand diagram,'' named after Jean-Robert Argand. Another prominent space on which the coordinates may be projected is the two-dimensional surface of a sphere, which is then called Riemann sphere.
  
 
===Cartesian complex plane===
 
===Cartesian complex plane===
 
The definition of the complex numbers involving two arbitrary real values immediately suggests the use of Cartesian coordinates in the complex plane. The horizontal (''real'') axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical (''imaginary'') axis, with increasing values upwards.
 
The definition of the complex numbers involving two arbitrary real values immediately suggests the use of Cartesian coordinates in the complex plane. The horizontal (''real'') axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical (''imaginary'') axis, with increasing values upwards.
  
A charted number may be viewed either as the [[Wikt:coordinatize|coordinatized]] point or as a [[Vector (geometric)|position vector]] from the origin to this point. The coordinate values of a complex number {{mvar|z}} can hence be expressed in its ''Cartesian'', ''rectangular'', or ''algebraic'' form.
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A charted number may be viewed either as the coordinatized point or as a position vector from the origin to this point. The coordinate values of a complex number {{mvar|z}} can hence be expressed in its ''Cartesian'', ''rectangular'', or ''algebraic'' form.
  
Notably, the operations of addition and multiplication take on a very natural geometric character, when complex numbers are viewed as position vectors: addition corresponds to [[Euclidean vector#Addition and subtraction|vector addition]], while multiplication (see [[#Multiplication and division in polar form|below]]) corresponds to multiplying their magnitudes and adding the angles they make with the real axis. Viewed in this way, the multiplication of a complex number by {{math|''i''}} corresponds to rotating the position vector [[orientation (geometry)|counterclockwise]] by a quarter [[turn (geometry)|turn]] ([[right angle|90°]]) about the origin—a fact which can be expressed algebraically as follows:
+
Notably, the operations of addition and multiplication take on a very natural geometric character, when complex numbers are viewed as position vectors: addition corresponds to vector addition, while multiplication (see below) corresponds to multiplying their magnitudes and adding the angles they make with the real axis. Viewed in this way, the multiplication of a complex number by {{math|''i''}} corresponds to rotating the position vector counterclockwise by a quarter turn (90°) about the origin—a fact which can be expressed algebraically as follows:
 
<math display=block>(a + bi)\cdot i = ai + b(i)^2 = -b + ai .</math>
 
<math display=block>(a + bi)\cdot i = ai + b(i)^2 = -b + ai .</math>
  
===Polar complex plane=== <!-- [[Nth root]] links to this section -->
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===Polar complex plane===
{{Main|Polar coordinate system}}
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{{Redirect|Polar form|the higher-dimensional analogue|Polar decomposition}}
 
 
[[File:Complex number illustration modarg.svg|right|thumb|Argument {{mvar|φ}} and modulus {{mvar|r}} locate a point in the complex plane.]]
 
[[File:Complex number illustration modarg.svg|right|thumb|Argument {{mvar|φ}} and modulus {{mvar|r}} locate a point in the complex plane.]]
  
 
====Modulus and argument====
 
====Modulus and argument====
An alternative option for coordinates in the complex plane is the [[polar coordinate system]] that uses the distance of the point {{mvar|z}} from the [[origin (mathematics)|origin]] ({{mvar|O}}), and the angle subtended between the [[positive real axis]] and the line segment {{mvar|Oz}} in a counterclockwise sense. This leads to the polar form of complex numbers.
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An alternative option for coordinates in the complex plane is the polar coordinate system that uses the distance of the point {{mvar|z}} from the origin ({{mvar|O}}), and the angle subtended between the positive real axis and the line segment {{mvar|Oz}} in a counterclockwise sense. This leads to the polar form of complex numbers.
  
The ''[[absolute value]]'' (or ''modulus'' or ''magnitude'') of a complex number {{math|1=''z'' = ''x'' + ''yi''}} is<ref>See {{harv|Apostol|1981}}, page 18.</ref>
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The ''absolute value'' (or ''modulus'' or ''magnitude'') of a complex number {{math|1=''z'' = ''x'' + ''yi''}} is
 
<math display=block>r=|z|=\sqrt{x^2+y^2}.</math>
 
<math display=block>r=|z|=\sqrt{x^2+y^2}.</math>
 
If {{mvar|z}} is a real number (that is, if {{math|1=''y'' = 0}}), then {{math|1=''r'' = {{!}}''x''{{!}}}}. That is, the absolute value of a real number equals its absolute value as a complex number.
 
If {{mvar|z}} is a real number (that is, if {{math|1=''y'' = 0}}), then {{math|1=''r'' = {{!}}''x''{{!}}}}. That is, the absolute value of a real number equals its absolute value as a complex number.
  
By [[Pythagoras' theorem]], the absolute value of a complex number is the distance to the origin of the point representing the complex number in the [[complex plane]].
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By Pythagoras' theorem, the absolute value of a complex number is the distance to the origin of the point representing the complex number in the complex plane.
  
The ''[[arg (mathematics)|argument]]'' of {{mvar|z}} (in many applications referred to as the "phase" {{mvar|φ}})<ref name=":2" /> is the angle of the [[radius]] {{mvar|Oz}} with the positive real axis, and is written as {{math|arg ''z''}}. As with the modulus, the argument can be found from the rectangular form {{mvar|x + yi}}<ref>{{cite book
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The ''argument'' of {{mvar|z}} (in many applications referred to as the "phase" {{mvar|φ}}) is the angle of the radius {{mvar|Oz}} with the positive real axis, and is written as {{math|arg ''z''}}. As with the modulus, the argument can be found from the rectangular form {{mvar|x + yi}} by applying the inverse tangent to the quotient of imaginary-by-real parts. By using a half-angle identity, a single branch of the arctan suffices to cover the range of the {{math|arg}}-function, (−''π'', ''π''], and avoids a more subtle case-by-case analysis
|title=Complex Variables: Theory And Applications
 
|edition=2nd
 
|chapter=Chapter 1
 
|first1=H.S.
 
|last1=Kasana
 
|publisher=PHI Learning Pvt. Ltd
 
|year=2005
 
|isbn=978-81-203-2641-5
 
|page=14
 
|chapter-url=https://books.google.com/books?id=rFhiJqkrALIC&pg=PA14}}</ref>—by applying the inverse tangent to the quotient of imaginary-by-real parts. By using a half-angle identity, a single branch of the arctan suffices to cover the range of the {{math|arg}}-function, {{open-closed|−''π'', ''π''}}, and avoids a more subtle case-by-case analysis
 
  
 
<math display=block>\varphi = \arg (x+yi) = \begin{cases}
 
<math display=block>\varphi = \arg (x+yi) = \begin{cases}
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  \end{cases}</math>
 
  \end{cases}</math>
  
Normally, as given above, the [[principal value]] in the interval {{open-closed|−{{mvar|π}}, {{mvar|π}}}} is chosen. If the arg value is negative, values in the range {{open-closed|−{{mvar|π}}, {{mvar|π}}}} or {{closed-open|0, 2{{mvar|π}}}} can be obtained by adding {{math|2''π''}}.<!--don't change this into π. Doing so produces *another* complex number.--> The value of {{mvar|φ}} is expressed in [[radian]]s in this article. It can increase by any integer multiple of {{math|2''π''}} and still give the same angle, viewed as subtended by the rays of the positive real axis and from the origin through {{mvar|z}}. Hence, the arg function is sometimes considered as [[Multivalued function|multivalued]]. The polar angle for the complex number 0 is indeterminate, but arbitrary choice of the polar angle&nbsp;0 is common.
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Normally, as given above, the principal value in the interval (−{{mvar|π}}, {{mvar|π}}] is chosen. If the arg value is negative, values in the range (−{{mvar|π}}, {{mvar|π}}] or [0, 2{{mvar|π}}) can be obtained by adding {{math|2''π''}}.<!--don't change this into π. Doing so produces *another* complex number.--> The value of {{mvar|φ}} is expressed in radians in this article. It can increase by any integer multiple of {{math|2''π''}} and still give the same angle, viewed as subtended by the rays of the positive real axis and from the origin through {{mvar|z}}. Hence, the arg function is sometimes considered as multivalued. The polar angle for the complex number 0 is indeterminate, but arbitrary choice of the polar angle&nbsp;0 is common.
  
The value of {{mvar|φ}} equals the result of [[atan2]]:
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The value of {{mvar|φ}} equals the result of atan2:
 
<math display=block>\varphi = \operatorname{atan2}\left(\operatorname{Im}(z),\operatorname{Re}(z) \right).</math>
 
<math display=block>\varphi = \operatorname{atan2}\left(\operatorname{Im}(z),\operatorname{Re}(z) \right).</math>
  
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<math display=block> z = r(\cos \varphi + i\sin \varphi ).</math>
 
<math display=block> z = r(\cos \varphi + i\sin \varphi ).</math>
  
Using [[Euler's formula]] this can be written as
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Using Euler's formula this can be written as
 
<math display=block>z = r e^{i \varphi} \text{ or } z = r \exp i \varphi.</math>
 
<math display=block>z = r e^{i \varphi} \text{ or } z = r \exp i \varphi.</math>
  
Using the {{math|[[Cis (mathematics)|cis]]}} function, this is sometimes abbreviated to
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Using the {{math|cis}} function, this is sometimes abbreviated to
 
<math display=block> z = r \operatorname\mathrm{cis} \varphi. </math>
 
<math display=block> z = r \operatorname\mathrm{cis} \varphi. </math>
  
In [[angle notation]], often used in [[electronics]] to represent a [[Phasor (sine waves)|phasor]] with amplitude {{mvar|r}} and phase {{mvar|φ}}, it is written as<ref>
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In angle notation, often used in electronics to represent a phasor with amplitude {{mvar|r}} and phase {{mvar|φ}}, it is written as  
{{cite book
 
|first1=James William |last1=Nilsson
 
|first2=Susan A.      |last2=Riedel
 
|year=2008
 
|title=Electric circuits  |edition=8th  |page=338
 
|chapter=Chapter 9
 
|publisher=Prentice Hall
 
|isbn=978-0-13-198925-2
 
|chapter-url=https://books.google.com/books?id=sxmM8RFL99wC&pg=PA338
 
}}
 
</ref>
 
 
<math display=block>z = r \angle \varphi . </math>
 
<math display=block>z = r \angle \varphi . </math>
  
 
===Complex graphs===
 
===Complex graphs===
{{main|Domain coloring|Riemann surface}}
 
[[File:Complex-plot.png|right|thumb|A color wheel graph of the expression
 
{{math|{{sfrac|(''z''<sup>2</sup> − 1)(''z'' − 2 − ''i'')<sup>2</sup>|''z''<sup>2</sup> + 2 + 2''i''}}}}]]
 
When visualizing [[complex analysis|complex functions]], both a complex input and output are needed. Because each complex number is represented in two dimensions, visually graphing a complex function would require the perception of a [[four dimensional space]], which is possible only in projections. Because of this, other ways of visualizing complex functions have been designed.
 
  
In [[domain coloring]] the output dimensions are represented by color and brightness, respectively. Each point in the complex plane as domain is ''ornated'', typically with ''color'' representing the argument of the complex number, and ''brightness'' representing the magnitude. Dark spots mark moduli near zero, brighter spots are farther away from the origin, the gradation may be discontinuous, but is assumed as monotonous. The colors often vary in steps of {{sfrac|{{pi}}|3}} for {{math|0}} to {{math|2{{pi}}}} from red, yellow, green, cyan, blue, to magenta. These plots are called [[Domain coloring|color wheel graphs]]. This provides a simple way to visualize the functions without losing information. The picture shows zeros for {{math|±1, (2 + ''i'')}} and poles at {{math{{radic|−2 −2 ''i''  }} .}}
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[[File:Complex-plot.png|right|thumb|A color wheel graph of the expression <math>\frac{(z^2 - 1)(z - 2 - i)^2}{z^2 + 2 + 2i}</math>]]
 +
When visualizing complex functions, both a complex input and output are needed. Because each complex number is represented in two dimensions, visually graphing a complex function would require the perception of a four dimensional space, which is possible only in projections. Because of this, other ways of visualizing complex functions have been designed.
 +
 
 +
In domain coloring the output dimensions are represented by color and brightness, respectively. Each point in the complex plane as domain is ''ornated'', typically with ''color'' representing the argument of the complex number, and ''brightness'' representing the magnitude. Dark spots mark moduli near zero, brighter spots are farther away from the origin, the gradation may be discontinuous, but is assumed as monotonous. The colors often vary in steps of <math>\frac{\pi}{3}</math> for {{math|0}} to <math>2\pi</math> from red, yellow, green, cyan, blue, to magenta. These plots are called color wheel graphs. This provides a simple way to visualize the functions without losing information. The picture shows zeros for {{math|±1, (2 + ''i'')}} and poles at <math>\sqrt{-2-2\text{i}}</math>
  
[[Riemann surface]]s are another way to visualize complex functions.{{explain|date=December 2018}} Riemann surfaces can be thought of as [[Deformation theory|deformations]] of the complex plane; while the horizontal axes represent the real and imaginary inputs, the single vertical axis only represents either the real or imaginary output. However, Riemann surfaces are built in such a way that rotating them 180 degrees shows the imaginary output, and vice versa. Unlike domain coloring, Riemann surfaces can represent [[multivalued function]]s like {{math|{{radic|''z''}}}}.
+
Riemann surfaces are another way to visualize complex functions. Riemann surfaces can be thought of as deformations of the complex plane; while the horizontal axes represent the real and imaginary inputs, the single vertical axis only represents either the real or imaginary output. However, Riemann surfaces are built in such a way that rotating them 180 degrees shows the imaginary output, and vice versa. Unlike domain coloring, Riemann surfaces can represent multivalued functions like <math>\sqrt{z}</math>.
  
 
==Relations and operations==
 
==Relations and operations==
  
 
===Equality===
 
===Equality===
Complex numbers have a similar definition of equality to real numbers; two complex numbers {{math|''a''<sub>1</sub> + ''b''<sub>1</sub>''i''}} and {{math|''a''<sub>2</sub> + ''b''<sub>2</sub>''i''}} are equal [[if and only if]] both their real and imaginary parts are equal, that is, if {{math|1=''a''<sub>1</sub> = ''a''<sub>2</sub>}} and {{math|1=''b''<sub>1</sub> = ''b''<sub>2</sub>}}. Nonzero complex numbers written in [[polar form]] are equal if and only if they have the same magnitude and their arguments differ by an integer multiple of {{math|2''π''}}.
+
Complex numbers have a similar definition of equality to real numbers; two complex numbers {{math|''a''<sub>1</sub> + ''b''<sub>1</sub>''i''}} and {{math|''a''<sub>2</sub> + ''b''<sub>2</sub>''i''}} are equal if and only if both their real and imaginary parts are equal, that is, if {{math|1=''a''<sub>1</sub> = ''a''<sub>2</sub>}} and {{math|1=''b''<sub>1</sub> = ''b''<sub>2</sub>}}. Nonzero complex numbers written in polar form are equal if and only if they have the same magnitude and their arguments differ by an integer multiple of {{math|2''π''}}.
  
 
===Ordering===
 
===Ordering===
Unlike the real numbers, there is no natural ordering of the complex numbers. In particular, there is no [[linear ordering]] on the complex numbers that is compatible with addition and multiplication – the complex numbers cannot have the structure of an ordered field. This is e.g. because every non-trivial sum of squares in an [[ordered field#nontrivialSquareSum|ordered field]] is {{math|≠ 0}}, and {{math|1=''i''<sup>2</sup> + 1<sup>2</sup> = 0}} is a non-trivial sum of squares.
+
Unlike the real numbers, there is no natural ordering of the complex numbers. In particular, there is no linear ordering on the complex numbers that is compatible with addition and multiplication – the complex numbers cannot have the structure of an ordered field. This is e.g. because every non-trivial sum of squares in an ordered field is {{math|≠ 0}}, and {{math|1=''i''<sup>2</sup> + 1<sup>2</sup> = 0}} is a non-trivial sum of squares.
 
Thus, complex numbers are naturally thought of as existing on a two-dimensional plane.
 
Thus, complex numbers are naturally thought of as existing on a two-dimensional plane.
  
 
===Conjugate===
 
===Conjugate===
{{See also|Complex conjugate}}
 
[[File:Complex conjugate picture.svg|right|thumb|upright=0.8|Geometric representation of {{mvar|z}} and its conjugate {{mvar|{{overline|z}}}} in the complex plane]]
 
The ''[[complex conjugate]]'' of the complex number {{math|1=''z'' = ''x'' + ''yi''}} is given by {{math|''x'' − ''yi''}}. It is denoted by either {{mvar|{{overline|z}}}} or {{math|''z''*}}.<ref>For the former notation, See {{harv|Apostol|1981}}, pages 15–16.</ref> This [[unary operation]] on complex numbers cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division.
 
  
Geometrically, {{mvar|{{overline|z}}}} is the [[reflection symmetry|"reflection"]] of {{mvar|z}} about the real axis. Conjugating twice gives the original complex number
+
[[File:Complex conjugate picture.svg|right|thumb|upright=0.8|Geometric representation of {{mvar|z}} and its conjugate <math>\overline{z}</math> in the complex plane]]
 +
The ''complex conjugate'' of the complex number {{math|1=''z'' = ''x'' + ''yi''}} is given by {{math|''x'' − ''yi''}}. It is denoted by either <math>\overline{z}</math> or {{math|''z''*}}. This unary operation on complex numbers cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division.
 +
 
 +
Geometrically, <math>\overline{z}</math> is the "reflection" of {{mvar|z}} about the real axis. Conjugating twice gives the original complex number
 
<math display=block>\overline{\overline{z}}=z,</math>
 
<math display=block>\overline{\overline{z}}=z,</math>
  
which makes this operation an [[involution (mathematics)|involution]]. The reflection leaves both the real part and the magnitude of {{mvar|z}} unchanged, that is
+
which makes this operation an involution. The reflection leaves both the real part and the magnitude of {{mvar|z}} unchanged, that is
 
<math display=block>\operatorname{Re}(\overline{z}) = \operatorname{Re}(z)\quad</math> and <math>\quad |\overline{z}| = |z|.</math>
 
<math display=block>\operatorname{Re}(\overline{z}) = \operatorname{Re}(z)\quad</math> and <math>\quad |\overline{z}| = |z|.</math>
  
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<math display=block>\operatorname{Im}(\overline{z}) = -\operatorname{Im}(z)\quad \text{ and } \quad \operatorname{arg} \overline{z} \equiv -\operatorname{arg} z \pmod {2\pi}.</math>
 
<math display=block>\operatorname{Im}(\overline{z}) = -\operatorname{Im}(z)\quad \text{ and } \quad \operatorname{arg} \overline{z} \equiv -\operatorname{arg} z \pmod {2\pi}.</math>
  
For details on argument and magnitude, see the section on [[#Polar form|Polar form]].
+
For details on argument and magnitude, see the section on Polar form.
  
The product of a complex number {{math|''z'' {{=}} ''x'' + ''yi''}} and its conjugate is known as the ''[[absolute square]]''. It is always a non-negative real number and equals the square of the magnitude of each:
+
The product of a complex number {{math|''z'' = ''x'' + ''yi''}} and its conjugate is known as the ''absolute square''. It is always a non-negative real number and equals the square of the magnitude of each:
 
<math display=block>z\cdot \overline{z} = x^2 + y^2 = |z|^2 = |\overline{z}|^2.</math>
 
<math display=block>z\cdot \overline{z} = x^2 + y^2 = |z|^2 = |\overline{z}|^2.</math>
  
This property can be used to convert a fraction with a complex denominator to an equivalent fraction with a real denominator by expanding both numerator and denominator of the fraction by the conjugate of the given denominator. This process is sometimes called "[[rationalisation (mathematics)|rationalization]]" of the denominator (although the denominator in the final expression might be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator.
+
This property can be used to convert a fraction with a complex denominator to an equivalent fraction with a real denominator by expanding both numerator and denominator of the fraction by the conjugate of the given denominator. This process is sometimes called "rationalization" of the denominator (although the denominator in the final expression might be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator.
  
 
The real and imaginary parts of a complex number {{mvar|z}} can be extracted using the conjugation:
 
The real and imaginary parts of a complex number {{mvar|z}} can be extracted using the conjugation:
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<math display=block>\overline{z\cdot w} = \overline{z} \cdot\overline{w},\quad \overline{z/w} = \overline{z}/\overline{w}.</math>
 
<math display=block>\overline{z\cdot w} = \overline{z} \cdot\overline{w},\quad \overline{z/w} = \overline{z}/\overline{w}.</math>
  
Conjugation is also employed in [[inversive geometry]], a branch of geometry studying reflections more general than ones about a line. In the [[Network analysis (electrical circuits)|network analysis of electrical circuits]], the complex conjugate is used in finding the equivalent impedance when the [[maximum power transfer theorem]] is looked for.
+
Conjugation is also employed in inversive geometry, a branch of geometry studying reflections more general than ones about a line. In the network analysis of electrical circuits, the complex conjugate is used in finding the equivalent impedance when the maximum power transfer theorem is looked for.
  
 
===Addition and subtraction===
 
===Addition and subtraction===
 
[[File:Vector Addition.svg|right|thumb|Addition of two complex numbers can be done geometrically by constructing a parallelogram.]]
 
[[File:Vector Addition.svg|right|thumb|Addition of two complex numbers can be done geometrically by constructing a parallelogram.]]
  
Two complex numbers {{mvar|a}} and {{mvar|b}} are most easily [[addition|added]] by separately adding their real and imaginary parts of the summands. That is to say:
+
Two complex numbers {{mvar|a}} and {{mvar|b}} are most easily added by separately adding their real and imaginary parts of the summands. That is to say:
 
<math display=block>a + b =(x+yi) + (u+vi) = (x+u) + (y+v)i.</math>
 
<math display=block>a + b =(x+yi) + (u+vi) = (x+u) + (y+v)i.</math>
Similarly, [[subtraction]] can be performed as
+
Similarly, subtraction can be performed as
 
<math display=block>a - b =(x+yi) - (u+vi) = (x-u) + (y-v)i.</math>
 
<math display=block>a - b =(x+yi) - (u+vi) = (x-u) + (y-v)i.</math>
  
Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers {{mvar|a}} and {{mvar|b}}, interpreted as points in the complex plane, is the point obtained by building a [[parallelogram]] from the three vertices {{mvar|O}}, and the points of the arrows labeled {{mvar|a}} and {{mvar|b}} (provided that they are not on a line). Equivalently, calling these points {{mvar|A}}, {{mvar|B}}, respectively and the fourth point of the parallelogram {{mvar|X}} the [[triangle]]s {{mvar|OAB}} and {{mvar|XBA}} are [[Congruence (geometry)|congruent]]. A visualization of the subtraction can be achieved by considering addition of the negative [[subtrahend]].
+
Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers {{mvar|a}} and {{mvar|b}}, interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices {{mvar|O}}, and the points of the arrows labeled {{mvar|a}} and {{mvar|b}} (provided that they are not on a line). Equivalently, calling these points {{mvar|A}}, {{mvar|B}}, respectively and the fourth point of the parallelogram {{mvar|X}} the triangles {{mvar|OAB}} and {{mvar|XBA}} are congruent. A visualization of the subtraction can be achieved by considering addition of the negative subtrahend.
  
 
===Multiplication and square===
 
===Multiplication and square===
The rules of the [[distributive property]], the [[commutative property|commutative properties]] (of addition and multiplication), and the defining property {{math|1=''i''<sup>2</sup> = −1}} apply to complex numbers. It follows that  
+
The rules of the distributive property, the commutative properties (of addition and multiplication), and the defining property {{math|1=''i''<sup>2</sup> = −1}} apply to complex numbers. It follows that  
 
<math display=block>(x+yi)\, (u+vi)= (xu - yv) + (xv + yu)i.</math>
 
<math display=block>(x+yi)\, (u+vi)= (xu - yv) + (xv + yu)i.</math>
  
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===Reciprocal and division===
 
===Reciprocal and division===
Using the conjugation, the [[Multiplicative inverse|reciprocal]] of a nonzero complex number {{math|1=''z'' = ''x'' + ''yi''}} can always be broken down to
+
Using the conjugation, the reciprocal of a nonzero complex number {{math|1=''z'' = ''x'' + ''yi''}} can always be broken down to
 
<math display=block>\frac{1}{z}=\frac{\overline{z}}{z \overline{z}} = \frac{\overline{z}}{|z|^2}=\frac{\overline{z}}{x^2+y^2}=\frac{x}{x^2+y^2} -\frac{y}{x^2+y^2}i,</math>
 
<math display=block>\frac{1}{z}=\frac{\overline{z}}{z \overline{z}} = \frac{\overline{z}}{|z|^2}=\frac{\overline{z}}{x^2+y^2}=\frac{x}{x^2+y^2} -\frac{y}{x^2+y^2}i,</math>
  
 
since ''non-zero'' implies that {{math|''x''<sup>2</sup> + ''y''<sup>2</sup>}} is greater than zero.
 
since ''non-zero'' implies that {{math|''x''<sup>2</sup> + ''y''<sup>2</sup>}} is greater than zero.
  
This can be used to express a division of an arbitrary complex number {{math|''w'' {{=}} ''u'' + ''vi''}} by a non-zero complex number {{mvar|z}} as
+
This can be used to express a division of an arbitrary complex number {{math|''w'' = ''u'' + ''vi''}} by a non-zero complex number {{mvar|z}} as
 
<math display=block>\frac {w}{z}= w\cdot \frac {1}{z}= (u+vi)\cdot \left(\frac{x}{x^2+y^2} -\frac{y}{x^2+y^2}i\right)= \frac{(ux+vy)+(vx-uy)i} {x^2+y^2}.</math>
 
<math display=block>\frac {w}{z}= w\cdot \frac {1}{z}= (u+vi)\cdot \left(\frac{x}{x^2+y^2} -\frac{y}{x^2+y^2}i\right)= \frac{(ux+vy)+(vx-uy)i} {x^2+y^2}.</math>
  
 
===Multiplication and division in polar form===
 
===Multiplication and division in polar form===
[[File:Complex multi.svg|right|thumb|Multiplication of {{math|2 + ''i''}} (blue triangle) and {{math|3 + ''i''}} (red triangle). The red triangle is rotated to match the vertex of the blue one and stretched by [[square root of 5|√5]], the length of the [[hypotenuse]] of the blue triangle.]]
+
[[File:Complex multi.svg|right|thumb|Multiplication of {{math|2 + ''i''}} (blue triangle) and {{math|3 + ''i''}} (red triangle). The red triangle is rotated to match the vertex of the blue one and stretched by √5, the length of the hypotenuse of the blue triangle.]]
Formulas for multiplication, division and exponentiation are simpler in polar form than the corresponding formulas in Cartesian coordinates. Given two complex numbers {{math|1=''z''<sub>1</sub> = ''r''<sub>1</sub>(cos ''φ''<sub>1</sub> + ''i'' sin ''φ''<sub>1</sub>)}} and {{math|1=''z''<sub>2</sub> = ''r''<sub>2</sub>(cos ''φ''<sub>2</sub> + ''i'' sin ''φ''<sub>2</sub>)}}, because of the [[trigonometric identities]]
+
Formulas for multiplication, division and exponentiation are simpler in polar form than the corresponding formulas in Cartesian coordinates. Given two complex numbers {{math|1=''z''<sub>1</sub> = ''r''<sub>1</sub>(cos ''φ''<sub>1</sub> + ''i'' sin ''φ''<sub>1</sub>)}} and {{math|1=''z''<sub>2</sub> = ''r''<sub>2</sub>(cos ''φ''<sub>2</sub> + ''i'' sin ''φ''<sub>2</sub>)}}, because of the trigonometric identities
 
<math display=block>\begin{alignat}{4}
 
<math display=block>\begin{alignat}{4}
 
\cos a \cos b & - \sin a \sin b \,& = \,& \cos(a + b) {} \\
 
\cos a \cos b & - \sin a \sin b \,& = \,& \cos(a + b) {} \\
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<math display=block>z_1 z_2 = r_1 r_2 (\cos(\varphi_1 + \varphi_2) + i \sin(\varphi_1 + \varphi_2)).</math>
 
<math display=block>z_1 z_2 = r_1 r_2 (\cos(\varphi_1 + \varphi_2) + i \sin(\varphi_1 + \varphi_2)).</math>
In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. For example, multiplying by {{math|''i''}} corresponds to a quarter-[[turn (geometry)|turn]] counter-clockwise, which gives back {{math|1=''i''<sup>2</sup> = −1}}. The picture at the right illustrates the multiplication of
+
In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. For example, multiplying by {{math|''i''}} corresponds to a quarter-turn counter-clockwise, which gives back {{math|1=''i''<sup>2</sup> = −1}}. The picture at the right illustrates the multiplication of
 
<math display=block>(2+i)(3+i)=5+5i. </math>
 
<math display=block>(2+i)(3+i)=5+5i. </math>
Since the real and imaginary part of {{math|5 + 5''i''}} are equal, the argument of that number is 45 degrees, or {{math|''π''/4}} (in [[radian]]). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are [[arctan]](1/3) and arctan(1/2), respectively. Thus, the formula
+
Since the real and imaginary part of {{math|5 + 5''i''}} are equal, the argument of that number is 45 degrees, or {{math|''π''/4}} (in radians). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are arctan(1/3) and arctan(1/2), respectively. Thus, the formula
 
<math display=block>\frac{\pi}{4} = \arctan\left(\frac{1}{2}\right) + \arctan\left(\frac{1}{3}\right) </math>
 
<math display=block>\frac{\pi}{4} = \arctan\left(\frac{1}{2}\right) + \arctan\left(\frac{1}{3}\right) </math>
holds. As the [[arctan]] function can be approximated highly efficiently, formulas like this – known as [[Machin-like formulas]] – are used for high-precision approximations of [[pi|{{pi}}]].
+
holds. As the arctan function can be approximated highly efficiently, formulas like this – known as Machin-like formulas – are used for high-precision approximations of <math>\pi</math>.
  
 
Similarly, division is given by
 
Similarly, division is given by
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===Square root===
 
===Square root===
{{see also|Square root#Square roots of negative and complex numbers|l1=Square roots of negative and complex numbers}}
+
 
 
The square roots of {{math|''a'' + ''bi''}} (with {{math|''b'' ≠ 0}}) are <math> \pm (\gamma + \delta i)</math>, where
 
The square roots of {{math|''a'' + ''bi''}} (with {{math|''b'' ≠ 0}}) are <math> \pm (\gamma + \delta i)</math>, where
  
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<math display=block>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}},</math>
 
<math display=block>\delta = (\sgn b)\sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}},</math>
  
where {{math|sgn}} is the [[sign function|signum]] function. This can be seen by squaring <math> \pm (\gamma + \delta i)</math> to obtain {{math|''a'' + ''bi''}}.<ref>{{cite book
+
where {{math|sgn}} is the signum function. This can be seen by squaring <math> \pm (\gamma + \delta i)</math> to obtain {{math|''a'' + ''bi''}}.
|title=Handbook of mathematical functions with formulas, graphs, and mathematical tables
 
|first1=Milton
 
|last1=Abramowitz
 
|first2=Irene A.
 
|last2=Stegun
 
|publisher=Courier Dover Publications
 
|year=1964
 
|isbn=978-0-486-61272-0
 
|page=17
 
|url=https://books.google.com/books?id=MtU8uP7XMvoC
 
|access-date=16 February 2016
 
|archive-url=https://web.archive.org/web/20160423180235/https://books.google.com/books?id=MtU8uP7XMvoC
 
|archive-date=23 April 2016
 
|url-status=live
 
}}, [http://www.math.sfu.ca/~cbm/aands/page_17.htm Section 3.7.26, p.&nbsp;17] {{Webarchive|url=https://web.archive.org/web/20090910094533/http://www.math.sfu.ca/~cbm/aands/page_17.htm |date=10 September 2009 }}</ref><ref>{{cite book
 
|title=Classical Algebra: its nature, origins, and uses
 
|first1=Roger
 
|last1=Cooke
 
|publisher=John Wiley and Sons
 
|year=2008
 
|isbn=978-0-470-25952-8
 
|page=59
 
|url=https://books.google.com/books?id=lUcTsYopfhkC
 
|access-date=16 February 2016
 
|archive-url=https://web.archive.org/web/20160424023007/https://books.google.com/books?id=lUcTsYopfhkC
 
|archive-date=24 April 2016
 
|url-status=live
 
}}, [https://books.google.com/books?id=lUcTsYopfhkC&pg=PA59 Extract: page 59] {{Webarchive|url=https://web.archive.org/web/20160423183239/https://books.google.com/books?id=lUcTsYopfhkC&pg=PA59 |date=23 April 2016 }}</ref> Here <math>\sqrt{a^2 + b^2}</math> is called the [[absolute value|modulus]] of {{math|''a'' + ''bi''}}, and the square root sign indicates the square root with non-negative real part, called the '''principal square root'''; also <math>\sqrt{a^2 + b^2}= \sqrt{z\overline{z}},</math> where {{math|''z'' {{=}} ''a'' + ''bi''}}.<ref>See {{harv|Ahlfors|1979}}, page 3.</ref>
 
  
 
===Exponential function===
 
===Exponential function===
The [[exponential function]] <math>\exp \colon \Complex \to \Complex ; z \mapsto \exp z </math> can be defined for every complex number {{mvar|z}} by the [[power series]]
+
The exponential function <math>\exp \colon \Complex \to \Complex ; z \mapsto \exp z </math> can be defined for every complex number {{mvar|z}} by the power series
 
<math display=block>\exp z= \sum_{n=0}^\infty \frac {z^n}{n!},</math>
 
<math display=block>\exp z= \sum_{n=0}^\infty \frac {z^n}{n!},</math>
which has an infinite [[radius of convergence]].
+
which has an infinite radius of convergence.
  
The value at {{math|1}} of the exponential function is [[Euler's number]]
+
The value at {{math|1}} of the exponential function is Euler's number
 
<math display=block>e = \exp 1 = \sum_{n=0}^\infty \frac1{n!}\approx 2.71828.</math>
 
<math display=block>e = \exp 1 = \sum_{n=0}^\infty \frac1{n!}\approx 2.71828.</math>
 
If {{mvar|z}} is real, one has  
 
If {{mvar|z}} is real, one has  
 
<math>\exp z=e^z.</math>
 
<math>\exp z=e^z.</math>
[[Analytic continuation]] allows extending this equality for every complex value of {{mvar|z}}, and thus to define the complex exponentiation with base {{mvar|e}} as
+
Analytic continuation allows extending this equality for every complex value of {{mvar|z}}, and thus to define the complex exponentiation with base {{mvar|e}} as
 
<math display=block>e^z=\exp z.</math>
 
<math display=block>e^z=\exp z.</math>
  
 
====Functional equation====
 
====Functional equation====
The exponential function satisfies the [[functional equation]] <math>e^{z+t}=e^ze^t.</math>
+
The exponential function satisfies the functional equation <math>e^{z+t}=e^ze^t.</math>
This can be proved either by comparing the power series expansion of both members or by applying [[analytic continuation]] from the restriction of the equation to real arguments.
+
This can be proved either by comparing the power series expansion of both members or by applying analytic continuation from the restriction of the equation to real arguments.
  
 
====Euler's formula====
 
====Euler's formula====
[[Euler's formula]] states that, for any real number {{mvar|y}},
+
Euler's formula states that, for any real number {{mvar|y}},
 
<math display=block>e^{iy} = \cos y + i\sin y .</math>
 
<math display=block>e^{iy} = \cos y + i\sin y .</math>
  
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===Complex logarithm===
 
===Complex logarithm===
In the real case, the [[natural logarithm]] can be defined as the [[inverse function|inverse]]
+
In the real case, the natural logarithm can be defined as the inverse  
<math>\ln \colon \R^+ \to \R ; x \mapsto \ln x </math> of the exponential function. For extending this to the complex domain, one can start from Euler's formula. It implies that, if a complex number <math>z\in \Complex^\times</math> is written in [[polar form]]
+
<math>\ln \colon \R^+ \to \R ; x \mapsto \ln x </math> of the exponential function. For extending this to the complex domain, one can start from Euler's formula. It implies that, if a complex number <math>z\in \Complex^\times</math> is written in polar form
 
<math display=block> z = r(\cos \varphi + i\sin \varphi )</math>
 
<math display=block> z = r(\cos \varphi + i\sin \varphi )</math>
 
with <math>r, \varphi \in \R ,</math> then with
 
with <math>r, \varphi \in \R ,</math> then with
 
<math display=block> \ln z = \ln r + i \varphi </math>
 
<math display=block> \ln z = \ln r + i \varphi </math>
as [[complex logarithm]] one has a proper inverse:
+
as complex logarithm one has a proper inverse:
 
<math display=block> \exp \ln z = \exp(\ln r + i \varphi ) = r \exp i \varphi = r(\cos \varphi + i\sin \varphi ) = z .</math>
 
<math display=block> \exp \ln z = \exp(\ln r + i \varphi ) = r \exp i \varphi = r(\cos \varphi + i\sin \varphi ) = z .</math>
  
 
However, because cosine and sine are periodic functions, the addition of an integer multiple of {{math|2''π''}} to {{mvar|φ}} does not change {{mvar|z}}. For example, {{math|1=''e''<sup>''iπ''</sup> = ''e''<sup>3''iπ''</sup> = −1}} , so both {{mvar|iπ}} and {{math|3''iπ''}} are possible values for the natural logarithm of {{math|−1}}.
 
However, because cosine and sine are periodic functions, the addition of an integer multiple of {{math|2''π''}} to {{mvar|φ}} does not change {{mvar|z}}. For example, {{math|1=''e''<sup>''iπ''</sup> = ''e''<sup>3''iπ''</sup> = −1}} , so both {{mvar|iπ}} and {{math|3''iπ''}} are possible values for the natural logarithm of {{math|−1}}.
  
Therefore, if the complex logarithm is not to be defined as a [[multivalued function]]
+
Therefore, if the complex logarithm is not to be defined as a multivalued function
 
<math display=block> \ln z = \left\{ \ln r + i (\varphi + 2\pi k) \mid k \in \Z \right\},</math>
 
<math display=block> \ln z = \left\{ \ln r + i (\varphi + 2\pi k) \mid k \in \Z \right\},</math>
one has to use a [[branch cut]] and to restrict the [[codomain]], resulting in the [[bijective]] function
+
one has to use a branch cut and to restrict the codomain, resulting in the bijective function
 
<math display=block>\ln \colon \; \Complex^\times \; \to \; \; \; \R^+ + \; i \, \left(-\pi, \pi\right] .</math>
 
<math display=block>\ln \colon \; \Complex^\times \; \to \; \; \; \R^+ + \; i \, \left(-\pi, \pi\right] .</math>
  
If <math>z \in \Complex \setminus \left( -\R_{\ge 0} \right)</math> is not a non-positive real number (a positive or a non-real number), the resulting [[principal value]] of the complex logarithm is obtained with {{math|−''π'' < ''φ'' < ''π''}}. It is an [[analytic function]] outside the negative real numbers, but it cannot be prolongated to a function that is continuous at any negative real number <math>z \in -\R^+ </math>, where the principal value is {{math|1=ln ''z'' = ln(−''z'') + ''iπ''}}.{{efn|However for another inverse function of the complex exponential function (and not the above defined principal value), the branch cut could be taken at any other [[Line (geometry)#Ray|ray]] thru the origin.}}
+
If <math>z \in \Complex \setminus \left( -\R_{\ge 0} \right)</math> is not a non-positive real number (a positive or a non-real number), the resulting principal value of the complex logarithm is obtained with {{math|−''π'' < ''φ'' < ''π''}}. It is an analytic function outside the negative real numbers, but it cannot be prolongated to a function that is continuous at any negative real number <math>z \in -\R^+ </math>, where the principal value is {{math|1=ln ''z'' = ln(−''z'') + ''iπ''}}.
  
 
===Exponentiation===
 
===Exponentiation===
Line 295: Line 244:
 
where {{math|ln}} denotes the natural logarithm.
 
where {{math|ln}} denotes the natural logarithm.
  
It seems natural to extend this formula to complex values of {{mvar|x}}, but there are some difficulties resulting from the fact that the complex logarithm is not really a function, but a [[multivalued function]].
+
It seems natural to extend this formula to complex values of {{mvar|x}}, but there are some difficulties resulting from the fact that the complex logarithm is not really a function, but a multivalued function.
  
 
It follows that if {{mvar|z}} is as above, and if {{mvar|t}} is another complex number, then the ''exponentiation'' is the multivalued function
 
It follows that if {{mvar|z}} is as above, and if {{mvar|t}} is another complex number, then the ''exponentiation'' is the multivalued function
Line 301: Line 250:
  
 
====Integer and fractional exponents====
 
====Integer and fractional exponents====
{{Visualisation complex number roots|1=upright=1.35}}
+
 
If, in the preceding formula, {{mvar|t}} is an integer, then the sine and the cosine are independent of {{mvar|k}}. Thus, if the exponent {{mvar|n}} is an integer, then {{math|''z''{{sup|''n''}}}} is well defined, and the exponentiation formula simplifies to [[de Moivre's formula]]:
+
If, in the preceding formula, {{mvar|t}} is an integer, then the sine and the cosine are independent of {{mvar|k}}. Thus, if the exponent {{mvar|n}} is an integer, then <math>z^n</math> is well defined, and the exponentiation formula simplifies to de Moivre's formula:
 
<math display=block> z^{n}=(r(\cos \varphi + i\sin \varphi ))^n = r^n \, (\cos n\varphi + i \sin n \varphi).</math>
 
<math display=block> z^{n}=(r(\cos \varphi + i\sin \varphi ))^n = r^n \, (\cos n\varphi + i \sin n \varphi).</math>
  
The {{mvar|n}} [[nth root|{{mvar|n}}th roots]] of a complex number {{mvar|z}} are given by
+
The {{mvar|n}} {{mvar|n}}th roots of a complex number {{mvar|z}} are given by
 
<math display=block>z^{1/n} = \sqrt[n]r \left( \cos \left(\frac{\varphi+2k\pi}{n}\right) + i \sin \left(\frac{\varphi+2k\pi}{n}\right)\right)</math>
 
<math display=block>z^{1/n} = \sqrt[n]r \left( \cos \left(\frac{\varphi+2k\pi}{n}\right) + i \sin \left(\frac{\varphi+2k\pi}{n}\right)\right)</math>
 
for {{math|0 ≤ ''k'' ≤ ''n'' − 1}}. (Here <math>\sqrt[n]r</math> is the usual (positive) {{mvar|n}}th root of the positive real number {{mvar|r}}.) Because sine and cosine are periodic, other integer values of {{mvar|k}} do not give other values.
 
for {{math|0 ≤ ''k'' ≤ ''n'' − 1}}. (Here <math>\sqrt[n]r</math> is the usual (positive) {{mvar|n}}th root of the positive real number {{mvar|r}}.) Because sine and cosine are periodic, other integer values of {{mvar|k}} do not give other values.
  
While the {{mvar|n}}th root of a positive real number {{mvar|r}} is chosen to be the ''positive'' real number {{mvar|c}} satisfying {{math|1=''c''<sup>''n''</sup> = ''r''}}, there is no natural way of distinguishing one particular complex {{mvar|n}}th root of a complex number. Therefore, the {{mvar|n}}th root is a [[multivalued function|{{mvar|n}}-valued function]] of {{mvar|z}}. This implies that, contrary to the case of positive real numbers, one has   
+
While the {{mvar|n}}th root of a positive real number {{mvar|r}} is chosen to be the ''positive'' real number {{mvar|c}} satisfying {{math|1=''c''<sup>''n''</sup> = ''r''}}, there is no natural way of distinguishing one particular complex {{mvar|n}}th root of a complex number. Therefore, the {{mvar|n}}th root is a {{mvar|n}}-valued function of {{mvar|z}}. This implies that, contrary to the case of positive real numbers, one has   
 
<math display=block>(z^n)^{1/n} \ne z,</math>
 
<math display=block>(z^n)^{1/n} \ne z,</math>
 
since the left-hand side consists of {{mvar|n}} values, and the right-hand side is a single value.
 
since the left-hand side consists of {{mvar|n}} values, and the right-hand side is a single value.

Latest revision as of 10:06, 3 November 2021

A complex number can be visually represented as a pair of numbers (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 i is the "imaginary unit" that satisfies i2 = −1.

In mathematics, a complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a symbol, called the imaginary unit, that satisfies the equation i2 = −1. Because no real number satisfies this equation, i was called an imaginary number by René Descartes. For the complex number a + bi, a is called the real part and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.

Complex numbers allow solutions to all polynomial equations, 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 has no real solution, since the square of a real number cannot be negative, but has the two nonreal complex solutions −1 + 3i and −1 − 3i.

Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule i2 = −1 combined with the associative, commutative and distributive laws. Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field that has the real numbers as a subfield. The complex numbers form also a real vector space of dimension two, with {1, i 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 expressing in terms of complex numbers some geometric properties and constructions. For example, the real numbers form the real line which is identified to the horizontal axis of the complex plane. The complex numbers of absolute value one form the unit circle. The addition of a complex number is a translation in the complex plane, and the multiplication by a complex number is a similarity centered at the origin. The complex conjugation is the reflection symmetry with respect to the real axis. The complex absolute value is a Euclidean norm.

In summary, the complex numbers form a rich structure that is simultaneously an algebraically closed field, a commutative algebra over the reals, and a Euclidean vector space of dimension two.

Definition

An illustration of the complex number z = x + iy on the complex plane. The real part is x, and its imaginary part is y.

A complex number is a number of the form a + bi, where a and b are real numbers, and i is an indeterminate satisfying i2 = −1. For example, 2 + 3i is a complex number.

This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation {{{1}}} is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials. The relation {{{1}}} induces the equalities {{{1}}} and {{{1}}} which hold for all integers k; these allow the reduction of any polynomial that results from the addition and multiplication of complex numbers to a linear polynomial in i, again of the form a + bi with real coefficients a, b.

The real number a is called the real part of the complex number a + bi; the real number b is called its imaginary part. To emphasize, the imaginary part does not include a factor i; that is, the imaginary part is b, not bi.

Formally, the complex numbers are defined as the quotient ring of the polynomial ring in the indeterminate i, by the ideal generated by the polynomial i2 + 1.

Notation

A real number a can be regarded as a complex number a + 0i, whose imaginary part is 0. A purely imaginary number bi is a complex number 0 + bi, whose real part is zero. As with polynomials, it is common to write a for a + 0i and bi for 0 + bi. Moreover, when the imaginary part is negative, that is, b = −|b| < 0, it is common to write a|b|i instead of a + (−|b|)i; for example, for b = −4, 3 − 4i can be written instead of 3 + (−4)i.

Since the multiplication of the indeterminate i and a real is commutative in polynomials with real coefficients, the polynomial a + bi may be written as a + ib. This is often expedient for imaginary parts denoted by expressions, for example, when b is a radical.

The real part of a complex number z is denoted by Re(z), , or ; the imaginary part of a complex number z is denoted by Im(z), , or For example,

The set of all complex numbers is denoted by (blackboard bold) or C (upright bold).

In some disciplines, particularly in electromagnetism and electrical engineering, j is used instead of i as i is frequently used to represent electric current.

Visualization

A complex number z, as a point (black) and its position vector (blue)

A complex number z can thus be identified with an ordered pair of real numbers, which in turn may be interpreted as coordinates of a point in a two-dimensional space. The most immediate space is the Euclidean plane with suitable coordinates, which is then called complex plane or Argand diagram, named after Jean-Robert Argand. Another prominent space on which the coordinates may be projected is the two-dimensional surface of a sphere, which is then called Riemann sphere.

Cartesian complex plane

The definition of the complex numbers involving two arbitrary real values immediately suggests the use of Cartesian coordinates in the complex plane. The horizontal (real) axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical (imaginary) axis, with increasing values upwards.

A charted number may be viewed either as the coordinatized point or as a position vector from the origin to this point. The coordinate values of a complex number z can hence be expressed in its Cartesian, rectangular, or algebraic form.

Notably, the operations of addition and multiplication take on a very natural geometric character, when complex numbers are viewed as position vectors: addition corresponds to vector addition, while multiplication (see below) corresponds to multiplying their magnitudes and adding the angles they make with the real axis. Viewed in this way, the multiplication of a complex number by i corresponds to rotating the position vector counterclockwise by a quarter turn (90°) about the origin—a fact which can be expressed algebraically as follows:

Polar complex plane

Argument φ and modulus r locate a point in the complex plane.

Modulus and argument

An alternative option for coordinates in the complex plane is the polar coordinate system that uses the distance of the point z from the origin (O), and the angle subtended between the positive real axis and the line segment Oz in a counterclockwise sense. This leads to the polar form of complex numbers.

The absolute value (or modulus or magnitude) of a complex number z = x + yi is

If z is a real number (that is, if y = 0), then r = |x|. That is, the absolute value of a real number equals its absolute value as a complex number.

By Pythagoras' theorem, the absolute value of a complex number is the distance to the origin of the point representing the complex number in the complex plane.

The argument of z (in many applications referred to as the "phase" φ) is the angle of the radius Oz with the positive real axis, and is written as arg z. As with the modulus, the argument can be found from the rectangular form x + yi by applying the inverse tangent to the quotient of imaginary-by-real parts. By using a half-angle identity, a single branch of the arctan suffices to cover the range of the arg-function, (−π, π], and avoids a more subtle case-by-case analysis

Normally, as given above, the principal value in the interval (−π, π] is chosen. If the arg value is negative, values in the range (−π, π] or [0, 2π) can be obtained by adding 2π. The value of φ is expressed in radians in this article. It can increase by any integer multiple of 2π and still give the same angle, viewed as subtended by the rays of the positive real axis and from the origin through z. Hence, the arg function is sometimes considered as multivalued. The polar angle for the complex number 0 is indeterminate, but arbitrary choice of the polar angle 0 is common.

The value of φ equals the result of atan2:

Together, r and φ give another way of representing complex numbers, the polar form, as the combination of modulus and argument fully specify the position of a point on the plane. Recovering the original rectangular co-ordinates from the polar form is done by the formula called trigonometric form

Using Euler's formula this can be written as

Using the cis function, this is sometimes abbreviated to

In angle notation, often used in electronics to represent a phasor with amplitude r and phase φ, it is written as

Complex graphs

A color wheel graph of the expression

When visualizing complex functions, both a complex input and output are needed. Because each complex number is represented in two dimensions, visually graphing a complex function would require the perception of a four dimensional space, which is possible only in projections. Because of this, other ways of visualizing complex functions have been designed.

In domain coloring the output dimensions are represented by color and brightness, respectively. Each point in the complex plane as domain is ornated, typically with color representing the argument of the complex number, and brightness representing the magnitude. Dark spots mark moduli near zero, brighter spots are farther away from the origin, the gradation may be discontinuous, but is assumed as monotonous. The colors often vary in steps of for 0 to from red, yellow, green, cyan, blue, to magenta. These plots are called color wheel graphs. This provides a simple way to visualize the functions without losing information. The picture shows zeros for ±1, (2 + i) and poles at

Riemann surfaces are another way to visualize complex functions. Riemann surfaces can be thought of as deformations of the complex plane; while the horizontal axes represent the real and imaginary inputs, the single vertical axis only represents either the real or imaginary output. However, Riemann surfaces are built in such a way that rotating them 180 degrees shows the imaginary output, and vice versa. Unlike domain coloring, Riemann surfaces can represent multivalued functions like .

Relations and operations

Equality

Complex numbers have a similar definition of equality to real numbers; two complex numbers a1 + b1i and a2 + b2i are equal if and only if both their real and imaginary parts are equal, that is, if a1 = a2 and b1 = b2. Nonzero complex numbers written in polar form are equal if and only if they have the same magnitude and their arguments differ by an integer multiple of 2π.

Ordering

Unlike the real numbers, there is no natural ordering of the complex numbers. In particular, there is no linear ordering on the complex numbers that is compatible with addition and multiplication – the complex numbers cannot have the structure of an ordered field. This is e.g. because every non-trivial sum of squares in an ordered field is ≠ 0, and i2 + 12 = 0 is a non-trivial sum of squares. Thus, complex numbers are naturally thought of as existing on a two-dimensional plane.

Conjugate

Geometric representation of z and its conjugate in the complex plane

The complex conjugate of the complex number z = x + yi is given by xyi. It is denoted by either or z*. This unary operation on complex numbers cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division.

Geometrically, is the "reflection" of z about the real axis. Conjugating twice gives the original complex number

which makes this operation an involution. The reflection leaves both the real part and the magnitude of z unchanged, that is

and

The imaginary part and the argument of a complex number z change their sign under conjugation

For details on argument and magnitude, see the section on Polar form.

The product of a complex number {{{1}}} and its conjugate is known as the absolute square. It is always a non-negative real number and equals the square of the magnitude of each:

This property can be used to convert a fraction with a complex denominator to an equivalent fraction with a real denominator by expanding both numerator and denominator of the fraction by the conjugate of the given denominator. This process is sometimes called "rationalization" of the denominator (although the denominator in the final expression might be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator.

The real and imaginary parts of a complex number z can be extracted using the conjugation:

Moreover, a complex number is real if and only if it equals its own conjugate.

Conjugation distributes over the basic complex arithmetic operations:

Conjugation is also employed in inversive geometry, a branch of geometry studying reflections more general than ones about a line. In the network analysis of electrical circuits, the complex conjugate is used in finding the equivalent impedance when the maximum power transfer theorem is looked for.

Addition and subtraction

Addition of two complex numbers can be done geometrically by constructing a parallelogram.

Two complex numbers a and b are most easily added by separately adding their real and imaginary parts of the summands. That is to say:

Similarly, subtraction can be performed as

Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers a and b, interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices O, and the points of the arrows labeled a and b (provided that they are not on a line). Equivalently, calling these points A, B, respectively and the fourth point of the parallelogram X the triangles OAB and XBA are congruent. A visualization of the subtraction can be achieved by considering addition of the negative subtrahend.

Multiplication and square

The rules of the distributive property, the commutative properties (of addition and multiplication), and the defining property i2 = −1 apply to complex numbers. It follows that

In particular,

Reciprocal and division

Using the conjugation, the reciprocal of a nonzero complex number z = x + yi can always be broken down to

since non-zero implies that x2 + y2 is greater than zero.

This can be used to express a division of an arbitrary complex number {{{1}}} by a non-zero complex number z as

Multiplication and division in polar form

Multiplication of 2 + i (blue triangle) and 3 + i (red triangle). The red triangle is rotated to match the vertex of the blue one and stretched by √5, the length of the hypotenuse of the blue triangle.

Formulas for multiplication, division and exponentiation are simpler in polar form than the corresponding formulas in Cartesian coordinates. Given two complex numbers z1 = r1(cos φ1 + i sin φ1) and z2 = r2(cos φ2 + i sin φ2), because of the trigonometric identities

we may derive

In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. For example, multiplying by i corresponds to a quarter-turn counter-clockwise, which gives back i2 = −1. The picture at the right illustrates the multiplication of
Since the real and imaginary part of 5 + 5i are equal, the argument of that number is 45 degrees, or π/4 (in radians). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are arctan(1/3) and arctan(1/2), respectively. Thus, the formula
holds. As the arctan function can be approximated highly efficiently, formulas like this – known as Machin-like formulas – are used for high-precision approximations of .

Similarly, division is given by

Square root

The square roots of a + bi (with b ≠ 0) are , where

and

where sgn is the signum function. This can be seen by squaring to obtain a + bi.

Exponential function

The exponential function can be defined for every complex number z by the power series

which has an infinite radius of convergence.

The value at 1 of the exponential function is Euler's number

If z is real, one has Analytic continuation allows extending this equality for every complex value of z, and thus to define the complex exponentiation with base e as

Functional equation

The exponential function satisfies the functional equation This can be proved either by comparing the power series expansion of both members or by applying analytic continuation from the restriction of the equation to real arguments.

Euler's formula

Euler's formula states that, for any real number y,

The functional equation implies thus that, if x and y are real, one has

which is the decomposition of the exponential function into its real and imaginary parts.

Complex logarithm

In the real case, the natural logarithm can be defined as the inverse of the exponential function. For extending this to the complex domain, one can start from Euler's formula. It implies that, if a complex number is written in polar form

with then with
as complex logarithm one has a proper inverse:

However, because cosine and sine are periodic functions, the addition of an integer multiple of 2π to φ does not change z. For example, e = e3 = −1 , so both and 3 are possible values for the natural logarithm of −1.

Therefore, if the complex logarithm is not to be defined as a multivalued function

one has to use a branch cut and to restrict the codomain, resulting in the bijective function

If is not a non-positive real number (a positive or a non-real number), the resulting principal value of the complex logarithm is obtained with π < φ < π. It is an analytic function outside the negative real numbers, but it cannot be prolongated to a function that is continuous at any negative real number , where the principal value is ln z = ln(−z) + .

Exponentiation

If x > 0 is real and z complex, the exponentiation is defined as

where ln denotes the natural logarithm.

It seems natural to extend this formula to complex values of x, but there are some difficulties resulting from the fact that the complex logarithm is not really a function, but a multivalued function.

It follows that if z is as above, and if t is another complex number, then the exponentiation is the multivalued function

Integer and fractional exponents

If, in the preceding formula, t is an integer, then the sine and the cosine are independent of k. Thus, if the exponent n is an integer, then is well defined, and the exponentiation formula simplifies to de Moivre's formula:

The n nth roots of a complex number z are given by

for 0 ≤ kn − 1. (Here is the usual (positive) nth root of the positive real number r.) Because sine and cosine are periodic, other integer values of k do not give other values.

While the nth root of a positive real number r is chosen to be the positive real number c satisfying cn = r, there is no natural way of distinguishing one particular complex nth root of a complex number. Therefore, the nth root is a n-valued function of z. This implies that, contrary to the case of positive real numbers, one has

since the left-hand side consists of n values, and the right-hand side is a single value.

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