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{{pp-pc1}}
{| class="wikitable" style="float: right; margin-left: 1em; text-align: center;"
! The powers of {{mvar|i}}<br/> are cyclic:
|-
|<math>\ \vdots</math>
|All powers of {{math|''i''}} assume values<br />from blue area
|-
|<math>\ i^{-2} = -1\phantom{i}</math>
|{{math|1=''i''<sup>−3</sup> = ''i''}}
|-
|{{<math|>\ i^{-1} ='' -i''<sup>−2\phantom1</supmath> = −1}}
|-
|style="background:#e1edfd;" | <math>\ \ i^{0}\ = \phantom-1\phantom{i}</math>
|{{math|1=''i''<sup>−1</sup> = −''i''}}
|-
|style="background:#cedff2e1edfd;" | {{<math|>\ \ i^{1}\ ='' \phantom-i''<sup>0\phantom1</supmath> = 1}}
|-
|style="background:#cedff2e1edfd;" | {{math|1=''i''<sup>1</supmath>\ \ i^{2}\ = ''-1\phantom{i''}}</math>
|-
|style="background:#cedff2e1edfd;" | {{<math|1>\ \ i^{3}\ ='' -i''<sup>2\phantom1</supmath> = −1}}
|-
|<math>\ \ i^{4}\ = \phantom-1\phantom{i}</math>
|style="background:#cedff2;" | {{math|1=''i''<sup>3</sup> = −''i''}}
|-
|<math>\ \ i^{5}\ = \phantom-i\phantom1</math>
|{{math|1=''i''<sup>4</sup> = 1}}
|-
|<math>\ \vdots</math>
|{{math|1=''i''<sup>5</sup> = ''i''}}
|-
|{{<math|1=''>i''<sup/math>6 is a 4th<br/sup> =[[root of −1}}unity]]
 
|-
|{{math|''i''}} is a 4th [[root of unity]]
|}
 
An '''imaginary number''' is the product of a [[real number]] multiplied byand 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
|chapter-url=https://books.google.com/books?id=SGVfGIewvxkC&pg=PA38
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|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>
 
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).
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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>
 
In this representation, multiplication by&nbsp;{{mathmvar|–1i}} corresponds to a counterclockwise [[rotation]] of 18090 degrees about the origin, which is a halfquarter of a circle. Multiplication by&nbsp;{{math|−''i''}} corresponds to a counterclockwiseclockwise [[rotation]] of 90 degrees about the origin. Similarly, whichmultiplying isby a quarterpurely ofimaginary anumber circle.{{mvar|bi}}, Bothwith these{{mvar|b}} numbersa arereal rootsnumber, ofboth <math>1</math>:causes <math>(-1)^2=1</math>,a <math>i^4=1</math>.counterclockwise Inrotation about the fieldorigin ofby complex90 numbers,degrees forand everyscales <math>the nanswer \inby \mathbb{N}a </math>,factor <math>1</math>of has{{mvar|b}}. nth rootsWhen <{{math>|''b'' \varphi_n</math> 0}}, meaningthis <math>can \varphi_n^ninstead =1be </math>,described calledas [[roota ofclockwise unity|rootsrotation ofby unity]].90 Multiplyingdegrees byand thea firstscaling <by {{math|{{abs|''b''}}}}.<ref>{{cite n<book|url=https:/math>th/books.google.com/books?id=_2sS4mC0p-EC&pg=PA10|title=Quaternions rootand ofRotation unitySequences: causesA aPrimer rotationwith ofApplications <math>to \frac{360}{n}</math>Orbits, degreesAerospace, aboutand theVirtual originReality|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>
 
Multiplying by a complex number is the same as rotating around the origin by the complex number's [[Argument (complex analysis)|argument]], followed by a scaling by its magnitude.<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==
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:
: <math>\textstyle
: <math>6=\sqrt{36}=\sqrt{(-4)(-9)} \ne \sqrt{-4}\sqrt{-9} = (2i)(3i) = 6 i^2 = -6.</math>
\sqrt{x \cdot y \vphantom{t}}
That is sometimes written as:
=\sqrt{(-x) \cdot (-y)}
:<math>-1 = i^2 = \sqrt{-1}\sqrt{-1} \stackrel{\text{ (fallacy) }}{=} \sqrt{(-1)(-1)} = \sqrt{1} = 1.</math>
\mathrel{\stackrel{\text{ (fallacy) }}{=}} \sqrt{-x\vphantom{ty}} \cdot \sqrt{-y\vphantom{ty}}
The [[mathematical fallacy|fallacy]] occurs as the equality <math>\sqrt{xy} = \sqrt{x}\sqrt{y}</math> fails when the variables are not suitably constrained. In that case, the equality fails to hold as the numbers are both negative, which can be demonstrated by:
:<math>\sqrt{-x}\sqrt{-y} = i \sqrt{x} \ i \sqrtvphantom{yty}} =\cdot i^2 \sqrt{x} \sqrt{y} = -\sqrtvphantom{xyty} \neq \sqrt{xy},</math>
= -\sqrt{x \cdot y \vphantom{ty}}\,.
where both {{mvar|x}} and {{mvar|y}} are positive real numbers.
</math>
 
But the result is clearly nonsense. The step where the square root was broken apart was illegitimate. (See [[Mathematical fallacy]].)
 
==See also==
* [[Octonion]]
* [[−1]]
{{Classification of numbers}}
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* [https://www.math.toronto.edu/mathnet/answers/imagexist.html How can one show that imaginary numbers really do exist?] – an article that discusses the existence of imaginary numbers.
* [https://www.bbc.co.uk/radio4/science/5numbers4.shtml 5Numbers programme 4] BBC Radio 4 programme
* [http://www2.dsu.nodak.edu/users/mberg/Imaginary/imaginary.htm Why Use Imaginary Numbers?] {{Webarchive|url=https://web.archive.org/web/20190825172656/http://www2.dsu.nodak.edu/users/mberg/Imaginary/imaginary.htm |date=2019-08-25 }} Basic Explanation and Uses of Imaginary Numbers
 
{{Complex numbers}}
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