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|<math>\ \vdots</math>
|<math>\ \vdots</math>
|-
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|<math>i</math> is a 4th<br/> [[root of unity]]
|<math>i</math> is a {{tmath|4}}th<br/> [[root of unity]]
 
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An '''imaginary number''' is the product of a [[real number]] and the [[imaginary unit]] {{mvar|i}},<ref group=note>{{mvar|j}} is usually used in engineering contexts where {{mvar|i}} has other meanings (such as electrical current)</ref> which is defined by its property {{math|1=''i''<sup>2</sup> = −1}}.<ref>
An '''imaginary number''' is the product of a [[real number]] and the [[imaginary unit]] {{mvar|i}},<ref group=note>{{mvar|j}} is usually used in engineering contexts where {{mvar|i}} has other meanings (such as electrical current).</ref> which is defined by its property {{math|1=''i''<sup>2</sup> = −1}}.<ref>
{{cite book
{{cite book
|chapter-url=https://books.google.com/books?id=SGVfGIewvxkC&pg=PA38
|chapter-url=https://books.google.com/books?id=SGVfGIewvxkC&pg=PA38
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|chapter=Chapter 2
|chapter=Chapter 2
}}
}}
</ref><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Imaginary Number|url=https://mathworld.wolfram.com/ImaginaryNumber.html|access-date=2020-08-10|website=mathworld.wolfram.com|language=en}}</ref> The [[square (algebra)|square]] of an imaginary number {{mvar|bi}} is {{math|−''b''<sup>2</sup>}}. For example, {{math|5''i''}} is an imaginary number, and its square is {{math|−25}}. The number [[0|zero]] is considered to be both real and imaginary.<ref>{{cite book|url=https://books.google.com/books?id=mqdzqbPYiAUC&pg=SA11-PA2|title=A Text Book of Mathematics Class XI|last=Sinha|first=K.C.|publisher=Rastogi Publications|year=2008|isbn=978-81-7133-912-9|edition=Second|page=11.2}}</ref>
</ref><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Imaginary Number|url=https://mathworld.wolfram.com/ImaginaryNumber.html|access-date=2020-08-10|website=mathworld.wolfram.com|language=en}}</ref> The [[Square (algebra)|square]] of an imaginary number {{mvar|bi}} is {{math|−''b''<sup>2</sup>}}. For example, {{math|5''i''}} is an imaginary number, and its square is {{math|−25}}. The number [[0|zero]] is considered to be both real and imaginary.<ref>{{cite book|url=https://books.google.com/books?id=mqdzqbPYiAUC&pg=SA11-PA2|title=A Text Book of Mathematics Class XI|last=Sinha|first=K.C.|publisher=Rastogi Publications|year=2008|isbn=978-81-7133-912-9|edition=Second|page=11.2}}</ref>


Originally coined in the 17th century by [[René Descartes]]<ref>{{cite book |title=Mathematical Analysis: Approximation and Discrete Processes |edition=illustrated |first1=Mariano |last1=Giaquinta |first2=Giuseppe |last2=Modica |publisher=Springer Science & Business Media |year=2004 |isbn=978-0-8176-4337-9 |page=121 |url=https://books.google.com/books?id=Z6q4EDRMC2UC}} [https://books.google.com/books?id=Z6q4EDRMC2UC&pg=PA121 Extract of page 121]</ref> as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of [[Leonhard Euler]] (in the 18th century) and [[Augustin-Louis Cauchy]] and [[Carl Friedrich Gauss]] (in the early 19th century).
Originally coined in the 17th century by [[René Descartes]]<ref>{{cite book |title=Mathematical Analysis: Approximation and Discrete Processes |edition=illustrated |first1=Mariano |last1=Giaquinta |first2=Giuseppe |last2=Modica |publisher=Springer Science & Business Media |year=2004 |isbn=978-0-8176-4337-9 |page=121 |url=https://books.google.com/books?id=Z6q4EDRMC2UC}} [https://books.google.com/books?id=Z6q4EDRMC2UC&pg=PA121 Extract of page 121]</ref> as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of [[Leonhard Euler]] in the 18th century, and [[Augustin-Louis Cauchy]] and [[Carl Friedrich Gauss]] in the early 19th century.


An imaginary number {{math|''bi''}} can be added to a real number {{mvar|a}} to form a [[complex number]] of the form {{math|''a'' + ''bi''}}, where the real numbers {{mvar|a}} and {{mvar|b}} are called, respectively, the ''real part'' and the ''imaginary part'' of the complex number.<ref>{{cite book |title= College Algebra: Enhanced Edition |edition= 6th |first1= Richard |last1= Aufmann |first2= Vernon C. |last2= Barker |first3= Richard |last3= Nation |publisher= Cengage Learning |year= 2009 |isbn= 978-1-4390-4379-0 |page= 66 |url= https://books.google.com/books?id=fjRa8Koq-RgC&pg=PA66}}</ref>
An imaginary number {{math|''bi''}} can be added to a real number {{mvar|a}} to form a [[complex number]] of the form {{math|''a'' + ''bi''}}, where the real numbers {{mvar|a}} and {{mvar|b}} are called, respectively, the ''real part'' and the ''imaginary part'' of the complex number.<ref>{{cite book |title= College Algebra: Enhanced Edition |edition= 6th |first1= Richard |last1= Aufmann |first2= Vernon C. |last2= Barker |first3= Richard |last3= Nation |publisher= Cengage Learning |year= 2009 |isbn= 978-1-4390-4379-0 |page= 66 |url= https://books.google.com/books?id=fjRa8Koq-RgC&pg=PA66}}</ref> Imaginary numbers are often called ''purely imaginary'' to distinguish them from complex numbers more generally; the set of all imaginary numbers is sometimes denoted {{tmath|i\R}}, where {{tmath|\R}} denotes the set of real numbers.


==History==
==History==
{{Main|History of complex numbers}}
{{Main|History of complex numbers}}
[[File:Complex conjugate picture.svg|right|thumb|An illustration of the complex plane. The imaginary numbers are on the vertical coordinate axis.]]
[[File:Complex conjugate picture.svg|right|thumb|An illustration of the complex plane. The imaginary numbers are on the vertical coordinate axis.]]
Although the Greek [[mathematician]] and [[engineer]] [[Heron of Alexandria]] is noted as the first to present a calculation involving the square root of a negative number,<ref>{{cite book |title= Fivefold Symmetry |edition= 2 |first= István |last= Hargittai |publisher= World Scientific |year= 1992 |isbn= 981-02-0600-3 |page= 153 |url= https://books.google.com/books?id=-Tt37ajV5ZgC&pg=PA153}}</ref><ref>{{cite book |title= Complex Numbers: lattice simulation and zeta function applications |first= Stephen Campbell |last= Roy |publisher= Horwood |year= 2007 |isbn= 978-1-904275-25-1 |page= 1 |url= https://books.google.com/books?id=J-2BRbFa5IkC}}</ref> it was [[Rafael Bombelli]] who first set down the rules for multiplication of [[complex number]]s in 1572. The concept had appeared in print earlier, such as in work by [[Gerolamo Cardano]]. At the time, imaginary numbers and negative numbers were poorly understood and were regarded by some as fictitious or useless, much as zero once was. Many other mathematicians were slow to adopt the use of imaginary numbers, including [[René Descartes]], who wrote about them in his ''[[La Géométrie]]'' in which he coined the term ''imaginary'' and meant it to be derogatory.<ref>[[René Descartes|Descartes, René]], ''Discours de la méthode'' (Leiden, (Netherlands): Jan Maire, 1637), appended book: ''La Géométrie'', book three, p. 380. [http://gallica.bnf.fr/ark:/12148/btv1b86069594/f464.item.zoom From page 380:] ''"Au reste tant les vrayes racines que les fausses ne sont pas tousjours reelles; mais quelquefois seulement imaginaires; c'est a dire qu'on peut bien tousjours en imaginer autant que jay dit en chasque Equation; mais qu'il n'y a quelquefois aucune quantité, qui corresponde a celles qu'on imagine, comme encore qu'on en puisse imaginer trois en celle cy, x<sup>3</sup> – 6xx + 13x – 10 = 0, il n'y en a toutefois qu'une reelle, qui est 2, & pour les deux autres, quoy qu'on les augmente, ou diminue, ou multiplie en la façon que je viens d'expliquer, on ne sçauroit les rendre autres qu'imaginaires."'' (Moreover, the true roots as well as the false [roots] are not always real; but sometimes only imaginary [quantities]; that is to say, one can always imagine as many of them in each equation as I said; but there is sometimes no quantity that corresponds to what one imagines, just as although one can imagine three of them in this [equation], x<sup>3</sup> – 6xx + 13x – 10 = 0, only one of them however is real, which is 2, and regarding the other two, although one increase, or decrease, or multiply them in the manner that I just explained, one would not be able to make them other than imaginary [quantities].)</ref><ref name="Martinez">{{Citation |first= Albert A. |last= Martinez |title= Negative Math: How Mathematical Rules Can Be Positively Bent |location= Princeton |publisher= Princeton University Press |year= 2006 |isbn= 0-691-12309-8}}, discusses ambiguities of meaning in imaginary expressions in historical context.</ref> The use of imaginary numbers was not widely accepted until the work of [[Leonhard Euler]] (1707–1783) and [[Carl Friedrich Gauss]] (1777–1855). The geometric significance of complex numbers as points in a plane was first described by [[Caspar Wessel]] (1745–1818).<ref>{{cite book
Although the Greek [[mathematician]] and [[engineer]] [[Heron of Alexandria]] is noted as the first to present a calculation involving the square root of a negative number,<ref>{{cite book |title= Fivefold Symmetry |edition= 2 |first= István |last= Hargittai |publisher= World Scientific |year= 1992 |isbn= 981-02-0600-3 |page= 153 |url= https://books.google.com/books?id=-Tt37ajV5ZgC&pg=PA153}}</ref><ref>{{cite book |title= Complex Numbers: lattice simulation and zeta function applications |first= Stephen Campbell |last= Roy |publisher= Horwood |year= 2007 |isbn= 978-1-904275-25-1 |page= 1 |url= https://books.google.com/books?id=J-2BRbFa5IkC}}</ref> it was [[Rafael Bombelli]] who first set down the rules for multiplication of [[complex number]]s in 1572. The concept had appeared in print earlier, such as in ''Ars Magna'' (1545) by [[Gerolamo Cardano]].<ref>{{Cite web |last=Corry |first=Leo |date=2026-04-24 |title=Cardano and the solving of cubic and quartic equations |url=https://www.britannica.com/science/algebra/Cardano-and-the-solving-of-cubic-and-quartic-equations?.com |url-status=live |access-date=2026-05-22 |website=www.britannica.com}}</ref> At the time, imaginary numbers and negative numbers were poorly understood and were regarded by some as fictitious or useless, much as zero once was. Many other mathematicians were slow to adopt the use of imaginary numbers, including [[René Descartes]], who wrote about them in his ''[[La Géométrie]]'' in which he coined the term ''imaginary'' and meant it to be derogatory.<ref>[[René Descartes|Descartes, René]], ''Discours de la méthode'' (Leiden, (Netherlands): Jan Maire, 1637), appended book: ''La Géométrie'', book three, p. 380. [https://gallica.bnf.fr/ark:/12148/btv1b86069594/f464.item.zoom From page 380:] ''"Au reste tant les vrayes racines que les fausses ne sont pas tousjours reelles; mais quelquefois seulement imaginaires; c'est a dire qu'on peut bien tousjours en imaginer autant que jay dit en chasque Equation; mais qu'il n'y a quelquefois aucune quantité, qui corresponde a celles qu'on imagine, comme encore qu'on en puisse imaginer trois en celle cy, x<sup>3</sup> – 6xx + 13x – 10 = 0, il n'y en a toutefois qu'une reelle, qui est 2, & pour les deux autres, quoy qu'on les augmente, ou diminue, ou multiplie en la façon que je viens d'expliquer, on ne sçauroit les rendre autres qu'imaginaires."'' (Moreover, the true roots as well as the false [roots] are not always real; but sometimes only imaginary [quantities]; that is to say, one can always imagine as many of them in each equation as I said; but there is sometimes no quantity that corresponds to what one imagines, just as although one can imagine three of them in this [equation], x<sup>3</sup> – 6xx + 13x – 10 = 0, only one of them however is real, which is 2, and regarding the other two, even if one were to increase, or decrease, or multiply them in the manner that I just explained, one would not be able to make them other than imaginary [quantities].)</ref><ref name="Martinez">{{Citation |first= Albert A. |last= Martinez |title= Negative Math: How Mathematical Rules Can Be Positively Bent |location= Princeton |publisher= Princeton University Press |year= 2006 |isbn= 0-691-12309-8}}, discusses ambiguities of meaning in imaginary expressions in historical context.</ref> The use of imaginary numbers was not widely accepted until the work of [[Leonhard Euler]] (1707–1783) and [[Carl Friedrich Gauss]] (1777–1855). The geometric significance of complex numbers as points in a plane was first described by [[Caspar Wessel]] (1745–1818).<ref>{{cite book
  |title= A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space
  |title= A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space
  |first= Boris Abramovich
  |first= Boris Abramovich
Line 62: Line 61:
</ref>
</ref>


In 1843, [[William Rowan Hamilton]] extended the idea of an axis of imaginary numbers in the plane to a four-dimensional space of [[quaternion#Definition|quaternion imaginaries]] in which three of the dimensions are analogous to the imaginary numbers in the complex field.
In 1843, [[William Rowan Hamilton]] extended the idea of an axis of imaginary numbers in the plane to a four-dimensional space of [[Quaternion#Definition|quaternion imaginaries]] in which three of the dimensions are analogous to the imaginary numbers in the complex field.


==Geometric interpretation==
==Geometric interpretation==
[[File:Rotations on the complex plane.svg|thumb|90-degree rotations in the [[complex plane]]]]
[[File:Rotations on the complex plane.svg|thumb|{{math|90}}-degree rotations in the [[complex plane]]]]


Geometrically, imaginary numbers are found on the vertical axis of the [[Complex plane|complex number plane]], which allows them to be presented [[perpendicular]] to the real axis. One way of viewing imaginary numbers is to consider a standard [[number line]] positively increasing in magnitude to the right and negatively increasing in magnitude to the left. At 0 on the {{mvar|x}}-axis, a {{mvar|y}}-axis can be drawn with "positive" direction going up; "positive" imaginary numbers then increase in magnitude upwards, and "negative" imaginary numbers increase in magnitude downwards. This vertical axis is often called the "imaginary axis"<ref name=Meier>{{cite book|url=https://books.google.com/books?id=bWAi22IB3lkC|title=Electric Power Systems – A Conceptual Introduction|last=von Meier|first=Alexandra|publisher=[[John Wiley & Sons]]|date=2006|access-date=2022-01-13|pages=61–62|isbn=0-471-17859-4}}</ref> and is denoted <math>i \mathbb{R},</math> <math>\mathbb{I},</math> or {{math|ℑ}}.<ref>{{cite book|chapter=5. Meaningless marks on paper|title=Clash of Symbols – A Ride Through the Riches of Glyphs|last1=Webb|first1=Stephen|publisher=[[Springer Science+Business Media]]|date=2018|pages=204–205|doi=10.1007/978-3-319-71350-2_5|isbn=978-3-319-71350-2}}</ref>
Geometrically, imaginary numbers are found on the vertical axis of the [[Complex plane|complex number plane]], which allows them to be presented [[perpendicular]] to the real axis. One way of viewing imaginary numbers is: Firstly, to consider a standard [[number line]], i.e., a horizontal {{mvar|x}}-axis on which positive real numbers increase in magnitude to the right and negative real numbers increase in magnitude to the left. Secondly, at {{math|0}} on the {{mvar|x}}-axis, to consider a vertical {{mvar|y}}-axis with "positive" direction going up; then, "positive" imaginary numbers increase in magnitude upwards and "negative" imaginary numbers increase in magnitude downwards. This vertical axis is often called the "imaginary axis".<ref name=Meier>{{cite book|url=https://books.google.com/books?id=bWAi22IB3lkC|title=Electric Power Systems – A Conceptual Introduction|last=von Meier|first=Alexandra|publisher=[[John Wiley & Sons]]|date=2006|access-date=2022-01-13|pages=61–62|isbn=0-471-17859-4}}</ref>


In this representation, multiplication by&nbsp;{{mvar|i}} corresponds to a counterclockwise [[rotation]] of 90 degrees about the origin, which is a quarter of a circle. Multiplication by&nbsp;{{math|−''i''}} corresponds to a clockwise rotation of 90 degrees about the origin. Similarly, multiplying by a purely imaginary number {{mvar|bi}}, with {{mvar|b}} a real number, both causes a counterclockwise rotation about the origin by 90 degrees and scales the answer by a factor of {{mvar|b}}. When {{math|''b'' < 0}}, this can instead be described as a clockwise rotation by 90 degrees and a scaling by {{math|{{abs|''b''}}}}.<ref>{{cite book|url=https://books.google.com/books?id=_2sS4mC0p-EC&pg=PA10|title=Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality|last=Kuipers|first=J. B.|publisher=[[Princeton University Press]]|date=1999|access-date=2022-01-13|pages=10–11|isbn=0-691-10298-8}}</ref>
In this representation, multiplication by&nbsp;{{mvar|i}} corresponds to a counterclockwise [[rotation]] of {{math|90}} degrees about the origin, which is a quarter of a turn. Multiplication by&nbsp;{{math|−''i''}} corresponds to a clockwise rotation of {{math|90}} degrees about the origin. Similarly, multiplying by a purely imaginary number {{mvar|bi}}, with {{mvar|b}} a real positive number, both causes a counterclockwise rotation about the origin by {{math|90}} degrees and scales the answer by a factor of {{mvar|b}}. When {{math|''b'' < 0}}, this can instead be described as a clockwise rotation by {{math|90}} degrees and a scaling by {{math|{{abs|''b''}}}}.<ref>{{cite book|url=https://books.google.com/books?id=_2sS4mC0p-EC&pg=PA10|title=Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality|last=Kuipers|first=J. B.|publisher=[[Princeton University Press]]|date=1999|access-date=2022-01-13|pages=10–11|isbn=0-691-10298-8}}</ref>


==Square roots of negative numbers==
==Square roots of negative numbers==
Care must be used when working with imaginary numbers that are expressed as the [[principal value]]s of the [[square root]]s of [[negative number]]s.<ref>{{cite book |title=An Imaginary Tale: The Story of "i" [the square root of minus one] |first1=Paul J. |last1=Nahin |publisher=Princeton University Press |year=2010 |isbn=978-1-4008-3029-9 |page=12 |url=https://books.google.com/books?id=PflwJdPhBlEC}} [https://books.google.com/books?id=PflwJdPhBlEC&pg=PA12 Extract of page 12]</ref> For example, if {{mvar|x}} and {{mvar|y}} are both positive real numbers, the following chain of equalities appears reasonable at first glance:
{{Classification of numbers}}
: <math>\textstyle
Care must be used when working with imaginary numbers that are expressed as the [[principal value]]s of the [[square root]]s of [[negative number]]s.<ref>{{cite book |title=An Imaginary Tale: The Story of "i" [the square root of minus one] |first1=Paul J. |last1=Nahin |publisher=Princeton University Press |year=2010 |isbn=978-1-4008-3029-9 |page=12 |url=https://books.google.com/books?id=PflwJdPhBlEC}} [https://books.google.com/books?id=PflwJdPhBlEC&pg=PA12 Extract of page 12]</ref> For example, the second equality in
\sqrt{x \cdot y \vphantom{t}}
:<math>\textstyle
=\sqrt{(-x) \cdot (-y)}
\sqrt{6}
\mathrel{\stackrel{\text{ (fallacy) }}{=}} \sqrt{-x\vphantom{ty}} \cdot \sqrt{-y\vphantom{ty}}
=\sqrt{(-2) \cdot (-3)}
= i\sqrt{x\vphantom{ty}} \cdot i\sqrt{y\vphantom{ty}}
\mathrel{\stackrel{\text{ (invalid) }}{=}} \sqrt{-2} \cdot \sqrt{-3}
= -\sqrt{x \cdot y \vphantom{ty}}\,.
= i\sqrt{2} \cdot i\sqrt{3}
= -\sqrt{6}\,
</math>
</math>
 
is invalid: the identity <math>\sqrt{xy} = \sqrt{x} \sqrt{y}</math> for nonnegative real numbers does not always hold for the principal branch of the complex square root function.
But the result is clearly nonsense. The step where the square root was broken apart was illegitimate. (See [[Mathematical fallacy]].)


==See also==
==See also==
Line 87: Line 86:
* [[Dual number]]
* [[Dual number]]
* [[Split-complex number]]
* [[Split-complex number]]
{{Classification of numbers}}


==Notes==
==Notes==

Latest revision as of 10:53, 29 May 2026

"Imaginary Numbers" redirects here. For the 2013 EP by The Maine, see Imaginary Numbers (EP).

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The powers of i
are cyclic:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \vdots}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ i^{-2} = -1\phantom{i}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ i^{-1} = -i\phantom1}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \ i^{0}\ = \phantom-1\phantom{i}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \ i^{1}\ = \phantom-i\phantom1}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \ i^{2}\ = -1\phantom{i}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \ i^{3}\ = -i\phantom1}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \ i^{4}\ = \phantom-1\phantom{i}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \ i^{5}\ = \phantom-i\phantom1}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \vdots}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} is a Template:Tmathth
root of unity

An imaginary number is the product of a real number and the imaginary unit i,[note 1] which is defined by its property i2 = −1.[1][2] The square of an imaginary number bi is −b2. For example, 5i is an imaginary number, and its square is −25. The number zero is considered to be both real and imaginary.[3]

Originally coined in the 17th century by René Descartes[4] as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of Leonhard Euler in the 18th century, and Augustin-Louis Cauchy and Carl Friedrich Gauss in the early 19th century.

An imaginary number bi can be added to a real number a to form a complex number of the form a + bi, where the real numbers a and b are called, respectively, the real part and the imaginary part of the complex number.[5] Imaginary numbers are often called purely imaginary to distinguish them from complex numbers more generally; the set of all imaginary numbers is sometimes denoted Template:Tmath, where Template:Tmath denotes the set of real numbers.

History

Main article: History of complex numbers
File:Complex conjugate picture.svg
An illustration of the complex plane. The imaginary numbers are on the vertical coordinate axis.

Although the Greek mathematician and engineer Heron of Alexandria is noted as the first to present a calculation involving the square root of a negative number,[6][7] it was Rafael Bombelli who first set down the rules for multiplication of complex numbers in 1572. The concept had appeared in print earlier, such as in Ars Magna (1545) by Gerolamo Cardano.[8] At the time, imaginary numbers and negative numbers were poorly understood and were regarded by some as fictitious or useless, much as zero once was. Many other mathematicians were slow to adopt the use of imaginary numbers, including René Descartes, who wrote about them in his La Géométrie in which he coined the term imaginary and meant it to be derogatory.[9][10] The use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855). The geometric significance of complex numbers as points in a plane was first described by Caspar Wessel (1745–1818).[11]

In 1843, William Rowan Hamilton extended the idea of an axis of imaginary numbers in the plane to a four-dimensional space of quaternion imaginaries in which three of the dimensions are analogous to the imaginary numbers in the complex field.

Geometric interpretation

File:Rotations on the complex plane.svg
90-degree rotations in the complex plane

Geometrically, imaginary numbers are found on the vertical axis of the complex number plane, which allows them to be presented perpendicular to the real axis. One way of viewing imaginary numbers is: Firstly, to consider a standard number line, i.e., a horizontal x-axis on which positive real numbers increase in magnitude to the right and negative real numbers increase in magnitude to the left. Secondly, at 0 on the x-axis, to consider a vertical y-axis with "positive" direction going up; then, "positive" imaginary numbers increase in magnitude upwards and "negative" imaginary numbers increase in magnitude downwards. This vertical axis is often called the "imaginary axis".[12]

In this representation, multiplication by i corresponds to a counterclockwise rotation of 90 degrees about the origin, which is a quarter of a turn. Multiplication by −i corresponds to a clockwise rotation of 90 degrees about the origin. Similarly, multiplying by a purely imaginary number bi, with b a real positive number, both causes a counterclockwise rotation about the origin by 90 degrees and scales the answer by a factor of b. When b < 0, this can instead be described as a clockwise rotation by 90 degrees and a scaling by |b|.[13]

Square roots of negative numbers

Template:Classification of numbers Care must be used when working with imaginary numbers that are expressed as the principal values of the square roots of negative numbers.[14] For example, the second equality in

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textstyle \sqrt{6} =\sqrt{(-2) \cdot (-3)} \mathrel{\stackrel{\text{ (invalid) }}{=}} \sqrt{-2} \cdot \sqrt{-3} = i\sqrt{2} \cdot i\sqrt{3} = -\sqrt{6}\, }

is invalid: the identity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{xy} = \sqrt{x} \sqrt{y}} for nonnegative real numbers does not always hold for the principal branch of the complex square root function.

See also

Notes

  1. ↑ j is usually used in engineering contexts where i has other meanings (such as electrical current).

References

  1. ↑ Uno Ingard, K. (1988). "Chapter 2". Fundamentals of Waves and Oscillations. Cambridge University Press. p. 38. ISBN 0-521-33957-X.
  2. ↑ Weisstein, Eric W. "Imaginary Number". mathworld.wolfram.com. Retrieved 2020-08-10.
  3. ↑ Sinha, K.C. (2008). A Text Book of Mathematics Class XI (Second ed.). Rastogi Publications. p. 11.2. ISBN 978-81-7133-912-9.
  4. ↑ Giaquinta, Mariano; Modica, Giuseppe (2004). Mathematical Analysis: Approximation and Discrete Processes (illustrated ed.). Springer Science & Business Media. p. 121. ISBN 978-0-8176-4337-9. Extract of page 121
  5. ↑ Aufmann, Richard; Barker, Vernon C.; Nation, Richard (2009). College Algebra: Enhanced Edition (6th ed.). Cengage Learning. p. 66. ISBN 978-1-4390-4379-0.
  6. ↑ Hargittai, István (1992). Fivefold Symmetry (2 ed.). World Scientific. p. 153. ISBN 981-02-0600-3.
  7. ↑ Roy, Stephen Campbell (2007). Complex Numbers: lattice simulation and zeta function applications. Horwood. p. 1. ISBN 978-1-904275-25-1.
  8. ↑ Corry, Leo (2026-04-24). "Cardano and the solving of cubic and quartic equations". www.britannica.com. Retrieved 2026-05-22.
  9. ↑ Descartes, René, Discours de la méthode (Leiden, (Netherlands): Jan Maire, 1637), appended book: La Géométrie, book three, p. 380. From page 380: "Au reste tant les vrayes racines que les fausses ne sont pas tousjours reelles; mais quelquefois seulement imaginaires; c'est a dire qu'on peut bien tousjours en imaginer autant que jay dit en chasque Equation; mais qu'il n'y a quelquefois aucune quantité, qui corresponde a celles qu'on imagine, comme encore qu'on en puisse imaginer trois en celle cy, x3 – 6xx + 13x – 10 = 0, il n'y en a toutefois qu'une reelle, qui est 2, & pour les deux autres, quoy qu'on les augmente, ou diminue, ou multiplie en la façon que je viens d'expliquer, on ne sçauroit les rendre autres qu'imaginaires." (Moreover, the true roots as well as the false [roots] are not always real; but sometimes only imaginary [quantities]; that is to say, one can always imagine as many of them in each equation as I said; but there is sometimes no quantity that corresponds to what one imagines, just as although one can imagine three of them in this [equation], x3 – 6xx + 13x – 10 = 0, only one of them however is real, which is 2, and regarding the other two, even if one were to increase, or decrease, or multiply them in the manner that I just explained, one would not be able to make them other than imaginary [quantities].)
  10. ↑ Martinez, Albert A. (2006), Negative Math: How Mathematical Rules Can Be Positively Bent, Princeton: Princeton University Press, ISBN 0-691-12309-8, discusses ambiguities of meaning in imaginary expressions in historical context.
  11. ↑ Rozenfeld, Boris Abramovich (1988). "Chapter 10". A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space. Springer. p. 382. ISBN 0-387-96458-4.
  12. ↑ von Meier, Alexandra (2006). Electric Power Systems – A Conceptual Introduction. John Wiley & Sons. pp. 61–62. ISBN 0-471-17859-4. Retrieved 2022-01-13.
  13. ↑ Kuipers, J. B. (1999). Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality. Princeton University Press. pp. 10–11. ISBN 0-691-10298-8. Retrieved 2022-01-13.
  14. ↑ Nahin, Paul J. (2010). An Imaginary Tale: The Story of "i" [the square root of minus one]. Princeton University Press. p. 12. ISBN 978-1-4008-3029-9. Extract of page 12

Bibliography

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