Complex number: Difference between revisions
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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}}.]] | [[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>; | 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>; because no real number satisfies the above equation, {{mvar|i}} was called an [[imaginary number]] by [[René Descartes]]. Every complex number can be expressed in the form <math>a + bi</math>, where {{mvar|a}} and {{mvar|b}} are real numbers, {{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> | </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 | 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> | <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>. | has no real solution, because the square of a real number cannot be negative, but has the two nonreal complex solutions <math>x=-1+3i</math> and <math>x=-1-3i</math>. | ||
Addition, subtraction and multiplication of complex numbers | Addition, subtraction and multiplication of complex numbers are defined, taking advantage of 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]], allowing division by complex numbers other than zero. 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 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. | 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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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''}}. | 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''}}. | ||
The [[Set (mathematics)|set]] of all complex numbers is denoted by <math>\Complex</math> ([[blackboard bold]]) or {{math|'''C'''}} (upright bold). | The [[Set (mathematics)|set]] of all complex numbers is denoted by <math>\Complex</math> ([[blackboard bold]]) or {{math|'''C'''}} ([[Boldface|upright bold]]). | ||
In some disciplines such as electromagnetism and electrical engineering, {{mvar|j}} is used instead of {{mvar|i}}, as {{mvar|i}} frequently represents electric current,<ref name="Campbell_1911" /><ref name="Brown-Churchill_1996" /> and complex numbers are written as {{math|''a'' + ''bj''}} or {{math|''a'' + ''jb''}}. | In some disciplines such as electromagnetism and electrical engineering, {{mvar|j}} is used instead of {{mvar|i}}, as {{mvar|i}} frequently represents electric current,<ref name="Campbell_1911" /><ref name="Brown-Churchill_1996" /> and complex numbers are written as {{math|''a'' + ''bj''}} or {{math|''a'' + ''jb''}}. | ||
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===Multiplication{{anchor|Multiplication|Square}}=== | ===Multiplication{{anchor|Multiplication|Square}}=== | ||
[[File:complex_number_multiplication_visualisation.svg|thumb|Multiplication of complex numbers | [[File:complex_number_multiplication_visualisation.svg|thumb|Multiplication of complex numbers {{math|2−''i''}} and {{math|3+4''i''}} visualized with vectors]] | ||
The product of two complex numbers is computed as follows: | The product of two complex numbers is computed as follows: | ||
:<math>(a+bi) \cdot (c+di) = ac - bd + (ad+bc)i.</math> | :<math>(a+bi) \cdot (c+di) = ac - bd + (ad+bc)i.</math> | ||
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[[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.]] | [[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 defined as | The ''[[complex conjugate]]'' of the complex number {{math|1=''z'' = ''x'' + ''yi''}} is defined as | ||
<math>\overline z = x-yi.</math><ref>{{harvnb|Apostol|1981|pp=15–16}}</ref> It is also denoted by some authors by <math>z^*</math>. Geometrically, {{mvar|{{overline|z}}}} is the [[reflection symmetry|"reflection"]] of {{mvar|z}} about the real axis. Conjugating twice gives the original complex number: <math>\overline{\overline{z}}=z.</math> A complex number is real if and only if it equals its own conjugate. The [[unary operation]] of taking the complex conjugate of a complex number cannot be expressed by applying only the basic operations of addition, subtraction, multiplication and division. | <math>\overline z = x-yi.</math><ref>{{harvnb|Apostol|1981|pp=15–16}}</ref> It is also denoted by some authors by <math>z^*</math>. Geometrically, {{mvar|{{overline|z}}}} is the [[reflection symmetry|"reflection"]] of {{mvar|z}} about the real axis. Conjugating twice gives the original complex number: <math>\overline{\overline{z}}=z.</math> A complex number is real [[if and only if]] it equals its own conjugate. The [[unary operation]] of taking the complex conjugate of a complex number cannot be expressed by applying only the basic operations of addition, subtraction, multiplication and division. | ||
[[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.]] | ||
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is a ''non-negative real'' number. This allows to define the ''[[absolute value]]'' (or ''modulus'' or ''magnitude'') of ''z'' to be the square root{{sfn|Apostol|1981|p=18}} | is a ''non-negative real'' number. This allows to define the ''[[absolute value]]'' (or ''modulus'' or ''magnitude'') of ''z'' to be the square root{{sfn|Apostol|1981|p=18}} | ||
<math display="block">|z|=\sqrt{x^2+y^2}.</math> | <math display="block">|z|=\sqrt{x^2+y^2}.</math> | ||
By [[Pythagoras' theorem]], <math>|z|</math> is the distance from the origin to the point representing the complex number ''z'' in the complex plane. In particular, the [[unit circle|circle of radius one]] around the origin consists precisely of the numbers ''z'' such that <math>|z| = 1 </math>. If <math> z = x = x + 0i </math> is a real number, then <math> |z|= |x| </math>: its absolute value as a complex number and as a real number are equal. | By [[Pythagoras' theorem]], <math>|z|</math> is the distance from the origin to the point representing the complex number ''z'' in the complex plane. In particular, the [[unit circle|circle of radius one]] around the origin consists precisely of the numbers ''z'' such that <math>|z| = 1 </math>, known as the '''unit complex numbers'''.{{anchor|Unit}} If <math> z = x = x + 0i </math> is a real number, then <math> |z|= |x| </math>: its absolute value as a complex number and as a real number are equal. | ||
Using the conjugate, the [[multiplicative inverse|reciprocal]] of a nonzero complex number <math>z = x + yi</math> can be computed to be | Using the conjugate, the [[multiplicative inverse|reciprocal]] of a nonzero complex number <math>z = x + yi</math> can be computed to be | ||
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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 any <math>z \ne 0</math>, there are, in particular ''n'' distinct complex ''n''-th roots. For example, there are 4 fourth roots of 1, namely | 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 any <math>z \ne 0</math>, there are, in particular ''n'' distinct complex ''n''-th roots. For example, there are 4 fourth roots of 1, namely | ||
:<math>z_1 = 1, z_2 = i, z_3 = -1, z_4 = -i.</math> | :<math>z_1 = 1, z_2 = i, z_3 = -1, z_4 = -i.</math> | ||
In general there is ''no'' natural way of distinguishing one particular complex {{mvar|n}}th root of a complex number. (This is in contrast to the roots of a positive real number ''x'', which has a unique positive real ''n''-th root, which is therefore commonly referred to as ''the'' ''n''-th root of ''x''.) One refers to this situation by saying that the {{mvar|n}}th root is | In general there is ''no'' natural way of distinguishing one particular complex {{mvar|n}}th root of a complex number. (This is in contrast to the roots of a positive real number ''x'', which has a unique positive real ''n''-th root, which is therefore commonly referred to as ''the'' ''n''-th root of ''x''.) One refers to this situation by saying that the {{mvar|n}}th root is an [[multivalued function|{{mvar|n}}-valued function]] of {{mvar|z}}. | ||
===Fundamental theorem of algebra=== | ===Fundamental theorem of algebra=== | ||
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Because of this fact, <math>\Complex</math> is called an [[algebraically closed field]]. It is a cornerstone of various applications of complex numbers, as is detailed further below. | Because of this fact, <math>\Complex</math> is called an [[algebraically closed field]]. It is a cornerstone of various applications of complex numbers, as is detailed further below. | ||
There are various proofs of this theorem, by either analytic methods such as [[Liouville's theorem (complex analysis)|Liouville's theorem]], or [[topology|topological]] ones such as the [[winding number]], or a proof combining [[Galois theory]] and the fact that any real polynomial of ''odd'' degree has at least one real root. | There are various proofs of this theorem, by either analytic methods such as [[Liouville's theorem (complex analysis)|Liouville's theorem]], or [[topology|topological]] ones such as the [[winding number]], or a proof combining [[Galois theory]] and the fact that any real polynomial of ''odd'' degree has at least one real root. | ||
The field of complex numbers is defined as the (unique) algebraic [[extension field]] of the real numbers later in [[#Abstract algebraic definitions]]. | |||
==History== | ==History== | ||
{{See also|Negative number#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 | 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 {{lang|la|[[casus irreducibilis]]}} ({{gloss|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|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]]. | 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]]. | ||
Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician [[Rafael Bombelli]].<ref>{{cite book |last1=Katz |first1=Victor J. |title=A History of Mathematics, Brief Version |section= 9.1.4 |publisher=[[Addison-Wesley]] |isbn=978-0-321-16193-2 |year=2004}}</ref> A more abstract formalism for the complex numbers was further developed by the Irish mathematician [[William Rowan Hamilton]], who extended this abstraction to the theory of [[quaternions]].<ref>{{cite journal |last1=Hamilton |first1=Wm. |title=On a new species of imaginary quantities connected with a theory of quaternions |journal=Proceedings of the Royal Irish Academy |date=1844 |volume=2 |pages=424–434 |url=https://babel.hathitrust.org/cgi/pt?id=njp.32101040410779&view=1up&seq=454}}</ref> | Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician [[Rafael Bombelli]].<ref>{{cite book |last1=Katz |first1=Victor J. |author-link=Victor J. Katz |title=A History of Mathematics, Brief Version |section= 9.1.4 |publisher=[[Addison-Wesley]] |isbn=978-0-321-16193-2 |year=2004}}</ref> A more abstract formalism for the complex numbers was further developed by the Irish mathematician [[William Rowan Hamilton]], who extended this abstraction to the theory of [[quaternions]].<ref>{{cite journal |last1=Hamilton |first1=Wm. |title=On a new species of imaginary quantities connected with a theory of quaternions |journal=Proceedings of the Royal Irish Academy |date=1844 |volume=2 |pages=424–434 |url=https://babel.hathitrust.org/cgi/pt?id=njp.32101040410779&view=1up&seq=454}}</ref> | ||
The earliest fleeting reference to [[square root]]s of [[negative number]]s can perhaps be said to occur in the work of the Greek mathematician [[Hero of Alexandria]] in the 1st century [[AD]], where in his ''[[Hero of Alexandria#Bibliography|Stereometrica]]'' he considered, apparently in error, the volume of an impossible [[frustum]] of a [[pyramid]] to arrive at the term <math>\sqrt{81 - 144}</math> in his calculations, which today would simplify to <math>\sqrt{-63} = 3i\sqrt{7}</math>.{{efn|In the literature the imaginary unit often precedes the radical sign, even when preceded itself by an integer.<ref>{{cite book |title=Trigonometry |author1=Cynthia Y. Young |edition=4th |publisher=John Wiley & Sons |year=2017 |isbn=978-1-119-44520-3 |page=406 |url=https://books.google.com/books?id=476ZDwAAQBAJ}} [https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA406 Extract of page 406]</ref>}} Negative quantities were not conceived of in [[Hellenistic mathematics]] and Hero merely replaced the negative value by its positive <math>\sqrt{144 - 81} = 3\sqrt{7}.</math><ref>{{cite book |title=An Imaginary Tale: The Story of √−1 |last=Nahin |first=Paul J. |year=2007 |publisher=[[Princeton University Press]] |isbn=978-0-691-12798-9 |url=http://mathforum.org/kb/thread.jspa?forumID=149&threadID=383188&messageID=1181284 |access-date=20 April 2011 |archive-url=https://web.archive.org/web/20121012090553/http://mathforum.org/kb/thread.jspa?forumID=149&threadID=383188&messageID=1181284 |archive-date=12 October 2012 |url-status=live }}</ref> | The earliest fleeting reference to [[square root]]s of [[negative number]]s can perhaps be said to occur in the work of the Greek mathematician [[Hero of Alexandria]] in the 1st century [[AD]], where in his ''[[Hero of Alexandria#Bibliography|Stereometrica]]'' he considered, apparently in error, the volume of an impossible [[frustum]] of a [[pyramid]] to arrive at the term <math>\sqrt{81 - 144}</math> in his calculations, which today would simplify to <math>\sqrt{-63} = 3i\sqrt{7}</math>.{{efn|In the literature the imaginary unit often precedes the radical sign, even when preceded itself by an integer.<ref>{{cite book |title=Trigonometry |author1=Cynthia Y. Young |edition=4th |publisher=John Wiley & Sons |year=2017 |isbn=978-1-119-44520-3 |page=406 |url=https://books.google.com/books?id=476ZDwAAQBAJ}} [https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA406 Extract of page 406]</ref>}} Negative quantities were not conceived of in [[Hellenistic mathematics]] and Hero merely replaced the negative value by its positive <math>\sqrt{144 - 81} = 3\sqrt{7}.</math><ref>{{cite book |title=An Imaginary Tale: The Story of √−1 |last=Nahin |first=Paul J. |year=2007 |publisher=[[Princeton University Press]] |isbn=978-0-691-12798-9 |url=http://mathforum.org/kb/thread.jspa?forumID=149&threadID=383188&messageID=1181284 |access-date=20 April 2011 |archive-url=https://web.archive.org/web/20121012090553/http://mathforum.org/kb/thread.jspa?forumID=149&threadID=383188&messageID=1181284 |archive-date=12 October 2012 |url-status=live }}</ref> | ||
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[[File:Circle_cos_sin.gif |thumb |upright=1.5 |Euler's formula relates the complex exponential function of an imaginary argument, which can be thought of as describing [[uniform circular motion]] in the complex plane, to the cosine and sine functions, geometrically its projections onto the real and imaginary axes, respectively.]] | [[File:Circle_cos_sin.gif |thumb |upright=1.5 |Euler's formula relates the complex exponential function of an imaginary argument, which can be thought of as describing [[uniform circular motion]] in the complex plane, to the cosine and sine functions, geometrically its projections onto the real and imaginary axes, respectively.]] | ||
In 1748, Euler went further and obtained [[Euler's formula]] of [[complex analysis]]:<ref>{{cite book |last1=Euler |first1= | In 1748, Euler went further and obtained [[Euler's formula]] of [[complex analysis]]:<ref>{{cite book |last1=Euler |first1=Leonhard |title=Introductio in Analysin Infinitorum |trans-title=Introduction to the Analysis of the Infinite |date=1748 |publisher=Marc Michel Bosquet & Co. |location=Lucerne, Switzerland |volume=1 |page=104 |url=https://books.google.com/books?id=jQ1bAAAAQAAJ&pg=PA104 |language=la}}</ref> | ||
<math display="block">e ^{i\theta } = \cos \theta + i\sin \theta </math> | <math display="block">e ^{i\theta } = \cos \theta + i\sin \theta </math> | ||
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by formally manipulating complex [[power series]] and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities. | by formally manipulating complex [[power series]] and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities. | ||
The idea of a complex number as a point in the complex plane was first described by [[Denmark|Danish]]–[[Norway|Norwegian]] | The idea of a complex number as a point in the complex plane was first described by [[Denmark|Danish]]–[[Norway|Norwegian]] mathematician [[Caspar Wessel]] in 1799,<ref>{{cite journal |last1=Wessel |first1=Caspar |title=Om Directionens analytiske Betegning, et Forsog, anvendt fornemmelig til plane og sphæriske Polygoners Oplosning |journal=Nye Samling af det Kongelige Danske Videnskabernes Selskabs Skrifter [New Collection of the Writings of the Royal Danish Science Society] |date=1799 |volume=5 |pages=469–518 |url=https://babel.hathitrust.org/cgi/pt?id=ien.35556000979690&view=1up&seq=561 |trans-title=On the analytic representation of direction, an effort applied in particular to the determination of plane and spherical polygons |language=da}}</ref> although it had been anticipated as early as 1685 in [[John Wallis|Wallis's]] ''A Treatise of Algebra''.<ref>{{cite book |last=Wallis |first=John |date=1685 |title=A Treatise of Algebra, Both Historical and Practical ... |url=https://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/H3GRV5AU/pageimg&start=291&mode=imagepath&pn=291|location=London, England |publisher=printed by John Playford, for Richard Davis |pages=264–273 }}</ref> | ||
Wessel's memoir appeared in the Proceedings of the [[Copenhagen Academy]] but went largely unnoticed. In 1806 [[Jean-Robert Argand]] independently issued a pamphlet on complex numbers and provided a rigorous proof of the [[Fundamental theorem of algebra#History|fundamental theorem of algebra]].<ref>{{cite book |last1=Argand |title=Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques |trans-title=Essay on a way to represent complex quantities by geometric constructions |date=1806 |publisher=Madame Veuve Blanc |location=Paris, France |url=http://www.bibnum.education.fr/mathematiques/geometrie/essai-sur-une-maniere-de-representer-des-quantites-imaginaires-dans-les-cons |language=fr}}</ref> [[Carl Friedrich Gauss]] had earlier published an essentially [[topology|topological]] proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1".<ref>Gauss, Carl Friedrich (1799) [https://books.google.com/books?id=g3VaAAAAcAAJ&pg=PP1 ''"Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse."''] [New proof of the theorem that any rational integral algebraic function of a single variable can be resolved into real factors of the first or second degree.] Ph.D. thesis, University of Helmstedt, (Germany). (in Latin)</ref> It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane,<ref name=Ewald>{{cite book |last=Ewald |first=William B. |date=1996 |title=From Kant to Hilbert: A Source Book in the Foundations of Mathematics |volume=1 |page=313 |publisher=Oxford University Press |isbn=9780198505358|url=https://books.google.com/books?id=rykSDAAAQBAJ&pg=PA313 |access-date=18 March 2020}}</ref> largely establishing modern notation and terminology:{{sfn|Gauss|1831}} | Wessel's memoir appeared in the Proceedings of the [[Copenhagen Academy]] but went largely unnoticed. In 1806 [[Jean-Robert Argand]] independently issued a pamphlet on complex numbers and provided a rigorous proof of the [[Fundamental theorem of algebra#History|fundamental theorem of algebra]].<ref>{{cite book |last1=Argand |title=Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques |trans-title=Essay on a way to represent complex quantities by geometric constructions |date=1806 |publisher=Madame Veuve Blanc |location=Paris, France |url=http://www.bibnum.education.fr/mathematiques/geometrie/essai-sur-une-maniere-de-representer-des-quantites-imaginaires-dans-les-cons |language=fr}}</ref> [[Carl Friedrich Gauss]] had earlier published an essentially [[topology|topological]] proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1".<ref>Gauss, Carl Friedrich (1799) [https://books.google.com/books?id=g3VaAAAAcAAJ&pg=PP1 ''"Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse."''] [New proof of the theorem that any rational integral algebraic function of a single variable can be resolved into real factors of the first or second degree.] Ph.D. thesis, University of Helmstedt, (Germany). (in Latin)</ref> It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane,<ref name=Ewald>{{cite book |last=Ewald |first=William B. |date=1996 |title=From Kant to Hilbert: A Source Book in the Foundations of Mathematics |volume=1 |page=313 |publisher=Oxford University Press |isbn=9780198505358|url=https://books.google.com/books?id=rykSDAAAQBAJ&pg=PA313 |access-date=18 March 2020}}</ref> largely establishing modern notation and terminology:{{sfn|Gauss|1831}} | ||
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In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée,<ref>{{cite web| url = https://mathshistory.st-andrews.ac.uk/Biographies/Buee/| title = Adrien Quentin Buée (1745–1845): MacTutor}}</ref><ref>{{cite journal |last1=Buée |title=Mémoire sur les quantités imaginaires |journal=Philosophical Transactions of the Royal Society of London |date=1806 |volume=96 |pages=23–88 |doi=10.1098/rstl.1806.0003 |s2cid=110394048 |url=https://royalsocietypublishing.org/doi/pdf/10.1098/rstl.1806.0003 |trans-title=Memoir on imaginary quantities |language=fr}}</ref> [[C. V. Mourey|Mourey]],<ref>{{cite book |last1=Mourey |first1=C.V. |title=La vraies théore des quantités négatives et des quantités prétendues imaginaires |trans-title=The true theory of negative quantities and of alleged imaginary quantities |date=1861 |publisher=Mallet-Bachelier |location=Paris, France |url=https://archive.org/details/bub_gb_8YxKAAAAYAAJ |language=fr}} 1861 reprint of 1828 original.</ref> [[John Warren (mathematician)|Warren]],<ref>{{cite book |last1=Warren |first1=John |title=A Treatise on the Geometrical Representation of the Square Roots of Negative Quantities |date=1828 |publisher=Cambridge University Press |location=Cambridge, England |url=https://archive.org/details/treatiseongeomet00warrrich}}</ref><ref>{{cite journal |last1=Warren |first1=John |title=Consideration of the objections raised against the geometrical representation of the square roots of negative quantities |journal=Philosophical Transactions of the Royal Society of London |date=1829 |volume=119 |pages=241–254 |s2cid=186211638 |doi=10.1098/rstl.1829.0022 |doi-access=free }}</ref><ref>{{cite journal |last1=Warren |first1=John |title=On the geometrical representation of the powers of quantities, whose indices involve the square roots of negative numbers |journal=Philosophical Transactions of the Royal Society of London |date=1829 |volume=119 |pages=339–359 |s2cid=125699726 |doi=10.1098/rstl.1829.0031 |doi-access=free }}</ref> [[Jacques Frédéric Français|Français]] and his brother, [[Giusto Bellavitis|Bellavitis]].<ref>{{cite journal |last1=Français |first1=J.F. |title=Nouveaux principes de géométrie de position, et interprétation géométrique des symboles imaginaires |journal=Annales des mathématiques pures et appliquées |date=1813 |volume=4 |pages=61–71 |url=https://babel.hathitrust.org/cgi/pt?id=uc1.$c126478&view=1up&seq=69 |trans-title=New principles of the geometry of position, and geometric interpretation of complex [number] symbols |language=fr}}</ref><ref>{{cite book |title=Two Cultures |editor= Kim Williams |last1=Caparrini |first1=Sandro |chapter=On the Common Origin of Some of the Works on the Geometrical Interpretation of Complex Numbers |year=2000 |publisher=Birkhäuser |isbn=978-3-7643-7186-9 |page=139 |url=https://books.google.com/books?id=voFsJ1EhCnYC |chapter-url=https://books.google.com/books?id=voFsJ1EhCnYC&pg=PA139}}</ref> | In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée,<ref>{{cite web| url = https://mathshistory.st-andrews.ac.uk/Biographies/Buee/| title = Adrien Quentin Buée (1745–1845): MacTutor}}</ref><ref>{{cite journal |last1=Buée |title=Mémoire sur les quantités imaginaires |journal=Philosophical Transactions of the Royal Society of London |date=1806 |volume=96 |pages=23–88 |doi=10.1098/rstl.1806.0003 |s2cid=110394048 |url=https://royalsocietypublishing.org/doi/pdf/10.1098/rstl.1806.0003 |trans-title=Memoir on imaginary quantities |language=fr}}</ref> [[C. V. Mourey|Mourey]],<ref>{{cite book |last1=Mourey |first1=C.V. |title=La vraies théore des quantités négatives et des quantités prétendues imaginaires |trans-title=The true theory of negative quantities and of alleged imaginary quantities |date=1861 |publisher=Mallet-Bachelier |location=Paris, France |url=https://archive.org/details/bub_gb_8YxKAAAAYAAJ |language=fr}} 1861 reprint of 1828 original.</ref> [[John Warren (mathematician)|Warren]],<ref>{{cite book |last1=Warren |first1=John |title=A Treatise on the Geometrical Representation of the Square Roots of Negative Quantities |date=1828 |publisher=Cambridge University Press |location=Cambridge, England |url=https://archive.org/details/treatiseongeomet00warrrich}}</ref><ref>{{cite journal |last1=Warren |first1=John |title=Consideration of the objections raised against the geometrical representation of the square roots of negative quantities |journal=Philosophical Transactions of the Royal Society of London |date=1829 |volume=119 |pages=241–254 |s2cid=186211638 |doi=10.1098/rstl.1829.0022 |doi-access=free }}</ref><ref>{{cite journal |last1=Warren |first1=John |title=On the geometrical representation of the powers of quantities, whose indices involve the square roots of negative numbers |journal=Philosophical Transactions of the Royal Society of London |date=1829 |volume=119 |pages=339–359 |s2cid=125699726 |doi=10.1098/rstl.1829.0031 |doi-access=free }}</ref> [[Jacques Frédéric Français|Français]] and his brother, [[Giusto Bellavitis|Bellavitis]].<ref>{{cite journal |last1=Français |first1=J.F. |title=Nouveaux principes de géométrie de position, et interprétation géométrique des symboles imaginaires |journal=Annales des mathématiques pures et appliquées |date=1813 |volume=4 |pages=61–71 |url=https://babel.hathitrust.org/cgi/pt?id=uc1.$c126478&view=1up&seq=69 |trans-title=New principles of the geometry of position, and geometric interpretation of complex [number] symbols |language=fr}}</ref><ref>{{cite book |title=Two Cultures |editor= Kim Williams |last1=Caparrini |first1=Sandro |chapter=On the Common Origin of Some of the Works on the Geometrical Interpretation of Complex Numbers |year=2000 |publisher=Birkhäuser |isbn=978-3-7643-7186-9 |page=139 |url=https://books.google.com/books?id=voFsJ1EhCnYC |chapter-url=https://books.google.com/books?id=voFsJ1EhCnYC&pg=PA139}}</ref> | ||
The English mathematician [[G.H. Hardy]] remarked that Gauss was the first mathematician to use complex numbers in "a really confident and scientific way" although mathematicians such as Norwegian [[Niels Henrik Abel]] and [[Carl Gustav Jacob Jacobi]] were necessarily using them routinely before Gauss published his 1831 treatise.<ref>{{cite book |title=An Introduction to the Theory of Numbers |last1=Hardy |first1=G.H. |last2=Wright |first2=E.M. |year=2000 |orig-year=1938 |publisher=[[Oxford University Press|OUP Oxford]] |isbn= 978-0-19-921986-5 |page=189 (fourth edition)}}</ref> | The English mathematician [[G.H. Hardy]] remarked that Gauss was the first mathematician to use complex numbers in "a really confident and scientific way" although mathematicians such as Norwegian [[Niels Henrik Abel]] and [[Carl Gustav Jacob Jacobi]] were necessarily using them routinely before Gauss published his 1831 treatise.<ref>{{cite book |title=[[An Introduction to the Theory of Numbers]] |last1=Hardy |first1=G.H. |last2=Wright |first2=E.M. |author-link2=E. M. Wright |year=2000 |orig-year=1938 |publisher=[[Oxford University Press|OUP Oxford]] |isbn= 978-0-19-921986-5 |page=189 (fourth edition)}}</ref> | ||
[[Augustin-Louis Cauchy]] and [[Bernhard Riemann]] together brought the fundamental ideas of [[#Complex analysis|complex analysis]] to a high state of completion, commencing around 1825 in Cauchy's case. | [[Augustin-Louis Cauchy]] and [[Bernhard Riemann]] together brought the fundamental ideas of [[#Complex analysis|complex analysis]] to a high state of completion, commencing around 1825 in Cauchy's case. | ||
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Later classical writers on the general theory include [[Richard Dedekind]], [[Otto Hölder]], [[Felix Klein]], [[Henri Poincaré]], [[Hermann Schwarz]], [[Karl Weierstrass]] and many others. Important work (including a systematization) in complex multivariate calculus has been started at beginning of the 20th century. Important results have been achieved by [[Wilhelm Wirtinger]] in 1927. | Later classical writers on the general theory include [[Richard Dedekind]], [[Otto Hölder]], [[Felix Klein]], [[Henri Poincaré]], [[Hermann Schwarz]], [[Karl Weierstrass]] and many others. Important work (including a systematization) in complex multivariate calculus has been started at beginning of the 20th century. Important results have been achieved by [[Wilhelm Wirtinger]] in 1927. | ||
==Abstract algebraic | ==Abstract algebraic definitions== | ||
While the above low-level definitions, including the addition and multiplication, accurately describe the complex numbers, there are other, equivalent approaches that reveal the abstract algebraic structure of the complex numbers more immediately. | While the above low-level definitions, including the addition and multiplication, accurately describe the complex numbers, there are other, equivalent approaches that reveal the abstract algebraic structure of the complex numbers more immediately. | ||
One formal definition of the complex numbers is that they form the quadratic [[extension field]] of the real numbers such that the polynomial <math>x^2+1=0</math> splits. Any two such fields are isomorphic; more generally, any non-trivial finite extension field of the reals is isomorphic to the complex field. | |||
===Construction as a quotient field=== | ===Construction as a quotient field=== | ||
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This function is [[surjective]] since every complex number can be obtained in such a way: the evaluation of a [[linear polynomial]] <math>a+bX</math> at <math>X = i</math> is <math>a+bi</math>. However, the evaluation of polynomial <math>X^2 + 1</math> at ''i'' is 0, since <math>i^2 + 1 = 0.</math> This polynomial is [[irreducible polynomial|irreducible]], i.e., cannot be written as a product of two linear polynomials. Basic facts of [[abstract algebra]] then imply that the [[Kernel (algebra)|kernel]] of the above map is an [[ideal (ring theory)|ideal]] generated by this polynomial, and that the quotient by this ideal is a field, and that there is an [[isomorphism]] | This function is [[surjective]] since every complex number can be obtained in such a way: the evaluation of a [[linear polynomial]] <math>a+bX</math> at <math>X = i</math> is <math>a+bi</math>. However, the evaluation of polynomial <math>X^2 + 1</math> at ''i'' is 0, since <math>i^2 + 1 = 0.</math> This polynomial is [[irreducible polynomial|irreducible]], i.e., cannot be written as a product of two linear polynomials. Basic facts of [[abstract algebra]] then imply that the [[Kernel (algebra)|kernel]] of the above map is an [[ideal (ring theory)|ideal]] generated by this polynomial, and that the quotient by this ideal is a field, and that there is an [[isomorphism]] | ||
:<math>\R[X] / (X^2 + 1) \stackrel \cong \to \C</math> | :<math>\R[X] / (X^2 + 1) \stackrel \cong \to \C</math> | ||
between the quotient ring and <math>\C</math>. Some authors take this as the definition of <math>\C</math>.<ref>{{harvnb|Bourbaki|1998|loc=§VIII.1}}</ref> | between the quotient ring and <math>\C</math>. Some authors take this as the definition of <math>\C</math>.<ref>{{harvnb|Bourbaki|1998|loc=§VIII.1}}</ref> This definition expresses <math>\C</math> as a [[quadratic algebra]]. | ||
Accepting that <math>\Complex</math> is algebraically closed, because it is an [[algebraic extension]] of <math>\mathbb{R}</math> in this approach, <math>\Complex</math> is therefore the [[algebraic closure]] of <math>\R.</math> | Accepting that <math>\Complex</math> is algebraically closed, because it is an [[algebraic extension]] of <math>\mathbb{R}</math> in this approach, <math>\Complex</math> is therefore the [[algebraic closure]] of <math>\R.</math> | ||
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The absolute value has three important properties: | The absolute value has three important properties: | ||
<math display=block> | z | \geq 0, \,</math> where <math> | z | = 0 \,</math> | <math display=block> | z | \geq 0, \,</math> where <math> | z | = 0 \,</math> if and only if <math> z = 0</math> | ||
<math display=block> | z + w | \leq | z | + | w | \,</math> ([[triangle inequality]]) | <math display=block> | z + w | \leq | z | + | w | \,</math> ([[triangle inequality]]) | ||
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===Complex logarithm=== | ===Complex logarithm=== | ||
{{main|Complex logarithm}} | {{main|Complex logarithm}} | ||
[[File:ComplexExpStrips.svg|right|thumb|The exponential function maps complex numbers ''z'' differing by a multiple of <math>2\pi i</math> to the same complex number ''w''.]] | [[File:ComplexExpStrips.svg|right|thumb|The exponential function maps complex numbers ''z'' differing by a multiple of <math>2\pi i</math> to the same complex number ''w''.]] | ||
For any positive real number ''t'', there is a unique real number ''x'' such that <math>\exp(x) = t</math>. This leads to the definition of the [[natural logarithm]] as the [[inverse function|inverse]] | For any positive real number ''t'', there is a unique real number ''x'' such that <math>\exp(x) = t</math>. This leads to the definition of the [[natural logarithm]] as the [[inverse function|inverse]] | ||
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:<math>f(z) = \overline z</math> | :<math>f(z) = \overline z</math> | ||
is differentiable as a function <math>\R^2 \to \R^2</math>, but is ''not'' complex differentiable. | is differentiable as a function <math>\R^2 \to \R^2</math>, but is ''not'' complex differentiable. | ||
A real differentiable function is complex differentiable | A real differentiable function is complex differentiable if and only if it satisfies the [[Cauchy–Riemann equations]], which are sometimes abbreviated as | ||
:<math>\frac{\partial f}{\partial \overline z} = 0.</math> | :<math>\frac{\partial f}{\partial \overline z} = 0.</math> | ||
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====Triangles==== | ====Triangles==== | ||
Every triangle has a unique [[Steiner inellipse]] – an [[ellipse]] inside the triangle and tangent to the midpoints of the three sides of the triangle. The [[Focus (geometry)|foci]] of a triangle's Steiner inellipse can be found as follows, according to [[Marden's theorem]]:<ref>{{cite journal |last1=Kalman|first1=Dan|title=An Elementary Proof of Marden's Theorem |url=http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3338&pf=1 |journal=[[American Mathematical Monthly]] |volume=115 |issue=4 |pages=330–38 |year=2008a |doi=10.1080/00029890.2008.11920532 |s2cid=13222698 |issn=0002-9890 |access-date=1 January 2012 |archive-url=https://web.archive.org/web/20120308104622/http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3338&pf=1 |archive-date=8 March 2012|url-status=live}}</ref><ref>{{cite journal |last1=Kalman |first1=Dan |title=The Most Marvelous Theorem in Mathematics |url=http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=1663 |journal=[[Journal of Online Mathematics and Its Applications]] |year=2008b |access-date=1 January 2012|archive-url=https://web.archive.org/web/20120208014954/http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=1663 |archive-date=8 February 2012 |url-status=live}}</ref> Denote the triangle's vertices in the complex plane as {{math|1=''a'' = ''x''<sub>''A''</sub> + ''y''<sub>''A''</sub>''i''}}, {{math|1=''b'' = ''x''<sub>''B''</sub> + ''y''<sub>''B''</sub>''i''}}, and {{math|1=''c'' = ''x''<sub>''C''</sub> + ''y''<sub>''C''</sub>''i''}}. Write the [[cubic equation]] <math>(x-a)(x-b)(x-c)=0</math>, take its derivative, and equate the (quadratic) derivative to zero. Marden's theorem says that the solutions of this equation are the complex numbers denoting the locations of the two foci of the Steiner inellipse. | Every triangle has a unique [[Steiner inellipse]] – an [[ellipse]] inside the triangle and tangent to the midpoints of the three sides of the triangle. The [[Focus (geometry)|foci]] of a triangle's Steiner inellipse can be found as follows, according to [[Marden's theorem]]:<ref>{{cite journal |last1=Kalman|first1=Dan|title=An Elementary Proof of Marden's Theorem |url=http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3338&pf=1 |journal=[[American Mathematical Monthly]] |volume=115 |issue=4 |pages=330–38 |year=2008a |doi=10.1080/00029890.2008.11920532 |s2cid=13222698 |issn=0002-9890 |access-date=1 January 2012 |archive-url=https://web.archive.org/web/20120308104622/http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3338&pf=1 |archive-date=8 March 2012|url-status=live|url-access=subscription }}</ref><ref>{{cite journal |last1=Kalman |first1=Dan |title=The Most Marvelous Theorem in Mathematics |url=http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=1663 |journal=[[Journal of Online Mathematics and Its Applications]] |year=2008b |access-date=1 January 2012|archive-url=https://web.archive.org/web/20120208014954/http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=1663 |archive-date=8 February 2012 |url-status=live}}</ref> Denote the triangle's vertices in the complex plane as {{math|1=''a'' = ''x''<sub>''A''</sub> + ''y''<sub>''A''</sub>''i''}}, {{math|1=''b'' = ''x''<sub>''B''</sub> + ''y''<sub>''B''</sub>''i''}}, and {{math|1=''c'' = ''x''<sub>''C''</sub> + ''y''<sub>''C''</sub>''i''}}. Write the [[cubic equation]] <math>(x-a)(x-b)(x-c)=0</math>, take its derivative, and equate the (quadratic) derivative to zero. Marden's theorem says that the solutions of this equation are the complex numbers denoting the locations of the two foci of the Steiner inellipse. | ||
===Algebraic number theory=== | ===Algebraic number theory=== | ||
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===Analytic number theory=== | ===Analytic number theory=== | ||
{{main|Analytic number theory}} | {{main|Analytic number theory}} | ||
Analytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can be regarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoretic information in complex-valued functions. For example, the [[Riemann zeta function]] {{math|ζ(''s'')}} is related to the distribution of [[prime number]]s. | Analytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can be regarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoretic information in complex-valued functions. For example, the [[Riemann zeta function]] {{math|ζ(''s'')}} is related to the distribution of [[prime number]]s. | ||
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====Electromagnetism and electrical engineering==== | ====Electromagnetism and electrical engineering==== | ||
{{Main|Alternating current}} | {{Main|Alternating current}} | ||
In [[electrical engineering]], the [[Fourier transform]] is used to analyze varying [[electric current]]s and [[voltage]]s. The treatment of [[resistor]]s, [[capacitor]]s, and [[inductor]]s can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the [[Electrical impedance|impedance]]. This approach is called [[phasor]] calculus. | In [[electrical engineering]], the [[Fourier transform]] is used to analyze varying [[electric current]]s and [[voltage]]s. The treatment of [[resistor]]s, [[capacitor]]s, and [[inductor]]s can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the [[Electrical impedance|impedance]]. This approach is called [[phasor]] calculus. | ||
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====Quantum mechanics==== | ====Quantum mechanics==== | ||
The complex number field is intrinsic to the [[mathematical formulations of quantum mechanics]], where complex [[Hilbert space]]s provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the [[Schrödinger equation]] and Heisenberg's [[matrix mechanics]] – make use of complex numbers. | The complex number field is intrinsic to the [[mathematical formulations of quantum mechanics]], where complex [[Hilbert space]]s provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the [[Schrödinger equation]] and Heisenberg's [[matrix mechanics]] – make use of complex numbers.<ref>{{cite book | ||
|last=Penrose | |||
|first=Roger | |||
|author-link=Roger Penrose | |||
|title=The Road to Reality: A Complete Guide to the Laws of the Universe | |||
|publisher=Jonathan Cape | |||
|location=London | |||
|year=2004 | |||
|isbn=0-224-04447-8 | |||
}}</ref> | |||
====Relativity==== | ====Relativity==== | ||
In [[special relativity]] and [[general relativity]], some formulas for the metric on [[spacetime]] become simpler if one takes the time component of the spacetime continuum to be imaginary. (This approach is no longer standard in classical relativity, but is [[Wick rotation|used in an essential way]] in [[quantum field theory]].) Complex numbers are essential to [[spinor]]s, which are a generalization of the [[tensor]]s used in relativity. | In [[special relativity]] and [[general relativity]], some formulas for the metric on [[spacetime]] become simpler if one takes the time component of the spacetime continuum to be imaginary. (This approach is no longer standard in classical relativity, but is [[Wick rotation|used in an essential way]] in [[quantum field theory]].) Complex numbers are essential to [[spinor]]s, which are a generalization of the [[tensor]]s used in relativity.<ref>{{cite book | ||
|last=Penrose | |||
|first=Roger | |||
|author-link=Roger Penrose | |||
|title=The Road to Reality: A Complete Guide to the Laws of the Universe | |||
|publisher=Jonathan Cape | |||
|location=London | |||
|year=2004 | |||
|isbn=0-224-04447-8 | |||
}}</ref> | |||
==Characterizations, generalizations and related notions== | ==Characterizations, generalizations and related notions== | ||
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* [[Eisenstein integer]] | * [[Eisenstein integer]] | ||
* [[Geometric algebra#Unit pseudoscalars|Geometric algebra]] (which includes the complex plane as the 2-dimensional [[Spinor#Two dimensions|spinor]] subspace <math>\mathcal{G}_2^+</math>) | * [[Geometric algebra#Unit pseudoscalars|Geometric algebra]] (which includes the complex plane as the 2-dimensional [[Spinor#Two dimensions|spinor]] subspace <math>\mathcal{G}_2^+</math>) | ||
{{Classification of numbers}} | |||
==Notes== | ==Notes== | ||
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==References== | ==References== | ||
<references> | |||
<ref name="Campbell_1911">{{cite journal |title=Cisoidal oscillations |author-link=George Ashley Campbell |author-first=George Ashley |author-last=Campbell |journal=[[Proceedings of the American Institute of Electrical Engineers]] |publisher=[[American Institute of Electrical Engineers]] |volume=XXX |issue=1–6 |date=April 1911 |doi=10.1109/PAIEE.1911.6659711 |s2cid=51647814 |pages=789–824 [Fig. 13 on p. 810] |url=https://ia800708.us.archive.org/view_archive.php?archive=/28/items/crossref-pre-1923-scholarly-works/10.1109%252Fpaiee.1910.6660428.zip&file=10.1109%252Fpaiee.1911.6659711.pdf |access-date=2023-06-24 |quote-page=789 |quote=The use of ''i'' (or Greek ''ı'') for the imaginary symbol is nearly universal in mathematical work, which is a very strong reason for retaining it in the applications of mathematics in electrical engineering. Aside, however, from the matter of established conventions and facility of reference to mathematical literature, the substitution of the symbol ''j'' is objectionable because of the vector terminology with which it has become associated in engineering literature, and also because of the confusion resulting from the divided practice of engineering writers, some using ''j'' for +''i'' and others using ''j'' for −''i''.}}</ref> | <ref name="Campbell_1911">{{cite journal |title=Cisoidal oscillations |author-link=George Ashley Campbell |author-first=George Ashley |author-last=Campbell |journal=[[Proceedings of the American Institute of Electrical Engineers]] |publisher=[[American Institute of Electrical Engineers]] |volume=XXX |issue=1–6 |date=April 1911 |doi=10.1109/PAIEE.1911.6659711 |s2cid=51647814 |pages=789–824 [Fig. 13 on p. 810] |bibcode=1911PAIEE..30d.789C |url=https://ia800708.us.archive.org/view_archive.php?archive=/28/items/crossref-pre-1923-scholarly-works/10.1109%252Fpaiee.1910.6660428.zip&file=10.1109%252Fpaiee.1911.6659711.pdf |access-date=2023-06-24 |quote-page=789 |quote=The use of ''i'' (or Greek ''ı'') for the imaginary symbol is nearly universal in mathematical work, which is a very strong reason for retaining it in the applications of mathematics in electrical engineering. Aside, however, from the matter of established conventions and facility of reference to mathematical literature, the substitution of the symbol ''j'' is objectionable because of the vector terminology with which it has become associated in engineering literature, and also because of the confusion resulting from the divided practice of engineering writers, some using ''j'' for +''i'' and others using ''j'' for −''i''.}}</ref> | ||
<ref name="Brown-Churchill_1996">{{cite book |author-last1=Brown |author-first1=James Ward |author-last2=Churchill |author-first2=Ruel V. |title=Complex variables and applications |date=1996 |publisher=[[McGraw-Hill]] |location=New York, USA |isbn=978-0-07-912147-9 |edition=6 |page=2 |quote-page=2 |quote=In electrical engineering, the letter ''j'' is used instead of ''i''.}}</ref> | <ref name="Brown-Churchill_1996">{{cite book |author-last1=Brown |author-first1=James Ward |author-last2=Churchill |author-first2=Ruel V. |title=Complex variables and applications |date=1996 |publisher=[[McGraw-Hill]] |location=New York, USA |isbn=978-0-07-912147-9 |edition=6 |page=2 |quote-page=2 |quote=In electrical engineering, the letter ''j'' is used instead of ''i''.}}</ref> | ||
</references> | |||
{{refbegin}} | {{refbegin}} | ||
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* {{cite book |last=Apostol |first=Tom |author-link=Tom Apostol |year=1981 |title=Mathematical analysis |publisher=Addison-Wesley}} | * {{cite book |last=Apostol |first=Tom |author-link=Tom Apostol |year=1981 |title=Mathematical analysis |publisher=Addison-Wesley}} | ||
* {{cite book |last1=Aufmann |first1=Richard N. |title=College Algebra and Trigonometry |last2=Barker |first2=Vernon C. |last3=Nation |first3=Richard D. |publisher=Cengage Learning |year=2007 |isbn=978-0-618-82515-8 |edition=6 |url=https://books.google.com/books?id=g5j-cT-vg_wC&pg=PA66}} | * {{cite book |last1=Aufmann |first1=Richard N. |title=College Algebra and Trigonometry |last2=Barker |first2=Vernon C. |last3=Nation |first3=Richard D. |publisher=Cengage Learning |year=2007 |isbn=978-0-618-82515-8 |edition=6 |url=https://books.google.com/books?id=g5j-cT-vg_wC&pg=PA66}} | ||
* {{cite book |ref=none |last=Conway |first=John B. |title=Functions of One Complex Variable I |year=1986 |publisher=Springer |isbn=978-0-387-90328-6}} | * {{cite book |ref=none |last=Conway |first=John B. |author-link=John B. Conway |title=Functions of One Complex Variable I |year=1986 |publisher=Springer |isbn=978-0-387-90328-6}} | ||
* {{cite book |last=Derbyshire |first=John |author-link=John Derbyshire |year=2006 |title=Unknown Quantity: A real and imaginary history of algebra |publisher=Joseph Henry Press |isbn=978-0-309-09657-7 |url=https://archive.org/details/isbn_9780309096577}} | * {{cite book |last=Derbyshire |first=John |author-link=John Derbyshire |year=2006 |title=Unknown Quantity: A real and imaginary history of algebra |publisher=Joseph Henry Press |isbn=978-0-309-09657-7 |url=https://archive.org/details/isbn_9780309096577}} | ||
* {{cite book |ref=none |last1=Joshi |first1=Kapil D. |title=Foundations of Discrete Mathematics |publisher=[[John Wiley & Sons]] |location=New York |isbn=978-0-470-21152-6 |year=1989}} | * {{cite book |ref=none |last1=Joshi |first1=Kapil D. |title=Foundations of Discrete Mathematics |publisher=[[John Wiley & Sons]] |location=New York |isbn=978-0-470-21152-6 |year=1989}} | ||
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{{refbegin}} | {{refbegin}} | ||
* {{cite journal |last=Argand |date=1814 |title=Reflexions sur la nouvelle théorie des imaginaires, suives d'une application à la demonstration d'un theorème d'analise |journal=Annales de mathématiques pures et appliquées |volume=5 |pages=197–209 |url=https://babel.hathitrust.org/cgi/pt?id=uc1.$c126479&view=1up&seq=209 |trans-title=Reflections on the new theory of complex numbers, followed by an application to the proof of a theorem of analysis |language=fr}} | * {{cite journal |last=Argand |date=1814 |title=Reflexions sur la nouvelle théorie des imaginaires, suives d'une application à la demonstration d'un theorème d'analise |journal=Annales de mathématiques pures et appliquées |volume=5 |pages=197–209 |url=https://babel.hathitrust.org/cgi/pt?id=uc1.$c126479&view=1up&seq=209 |trans-title=Reflections on the new theory of complex numbers, followed by an application to the proof of a theorem of analysis |language=fr}} | ||
* {{cite book |ref=none |last=Bourbaki |first=Nicolas |author-link=Nicolas Bourbaki |title= Elements of the history of mathematics |chapter= Foundations of mathematics § logic: set theory |publisher= Springer |year= 1998}} | * {{cite book |ref=none |last=Bourbaki |first=Nicolas |author-link=Nicolas Bourbaki |title=Elements of the history of mathematics |chapter=Foundations of mathematics § logic: set theory |publisher=Springer |year=1998}} | ||
* {{cite book |ref=none |last1=Burton |first1=David M. |title=The History of Mathematics |publisher=[[McGraw-Hill]] |location=New York |edition= 3rd |isbn=978-0-07-009465-9 |year=1995}} | * {{cite book |ref=none |last1=Burton |first1=David M. |title=The History of Mathematics |publisher=[[McGraw-Hill]] |location=New York |edition=3rd |isbn=978-0-07-009465-9 |year=1995}} | ||
* {{cite journal |last=Gauss |first=C. F. |date= 1831 |title=Theoria residuorum biquadraticorum. Commentatio secunda. |trans-title=Theory of biquadratic residues. Second memoir. |url=https://babel.hathitrust.org/cgi/pt?id=mdp.39015073697180&view=1up&seq=283 |journal=Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores |volume=7 |pages=89–148 |language=la |author-link= Carl Friedrich Gauss}} | * {{cite journal |last=Gauss |first=C. F. |date=1831 |title=Theoria residuorum biquadraticorum. Commentatio secunda. |trans-title=Theory of biquadratic residues. Second memoir. |url=https://babel.hathitrust.org/cgi/pt?id=mdp.39015073697180&view=1up&seq=283 |journal=Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores |volume=7 |pages=89–148 |language=la |author-link=Carl Friedrich Gauss}} | ||
* {{cite book |ref=none |last1=Katz |first1=Victor J. |title=A History of Mathematics, Brief Version |publisher=[[Addison-Wesley]] |isbn=978-0-321-16193-2 |year=2004}} | * {{cite book |ref=none |last1=Katz |first1=Victor J. |author-link=Victor J. Katz |title=A History of Mathematics, Brief Version |publisher=[[Addison-Wesley]] |isbn=978-0-321-16193-2 |year=2004}} | ||
* {{cite book |ref=none |title=An Imaginary Tale: The Story of <math>\scriptstyle\sqrt{-1}</math> |first=Paul J. |last=Nahin |publisher=Princeton University Press |isbn=978-0-691-02795-1 |year=1998}} — A gentle introduction to the history of complex numbers and the beginnings of complex analysis. | * {{cite book |ref=none |title=An Imaginary Tale: The Story of <math>\scriptstyle\sqrt{-1}</math> |first=Paul J. |last=Nahin |publisher=Princeton University Press |isbn=978-0-691-02795-1 |year=1998}} — A gentle introduction to the history of complex numbers and the beginnings of complex analysis. | ||
* {{cite book |first1=H. D. |last1= Ebbinghaus |first2=H. |last2= Hermes |first3=F. |last3=Hirzebruch |first4=M. |last4=Koecher |first5=K. |last5= Mainzer |first6=J. |last6= Neukirch |first7=A. |last7=Prestel |first8=R. |last8=Remmert |title=Numbers |publisher=Springer |isbn=978-0-387-97497-2 |edition=hardcover |year=1991}} — An advanced perspective on the historical development of the concept of number. | * {{cite book |first1=H. D. |last1=Ebbinghaus |first2=H. |last2=Hermes |first3=F. |last3=Hirzebruch |first4=M. |last4=Koecher |first5=K. |last5=Mainzer |first6=J. |last6=Neukirch |first7=A. |last7=Prestel |first8=R. |last8=Remmert |title=Numbers |publisher=Springer |isbn=978-0-387-97497-2 |edition=hardcover |year=1991}} — An advanced perspective on the historical development of the concept of number. | ||
{{refend}} | {{refend}} | ||