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== Nike Free Tr Fit 2 "> ==
{{redirect2|Imaginary Number|Imaginary numbers|the 2013 album by [[The Maine (band)|The Maine]]|Imaginary Numbers (The Maine EP)}}
{| class="wikitable" style="float: right; margin-left: 1em; text-align: center;"
|-
|{{math|...}} (repeats the pattern<br />from blue area)
|-
|{{math|1=''i''<sup>−3</sup> = ''i''}}
|-
|{{math|1=''i''<sup>−2</sup> = −1}}
|-
|{{math|1=''i''<sup>−1</sup> = −''i''}}
|-
|style="background:#cedff2;" | {{math|1=''i''<sup>0</sup> = 1}}
|-
|style="background:#cedff2;" | {{math|1=''i''<sup>1</sup> = i}}
|-
|style="background:#cedff2;" | {{math|1=''i''<sup>2</sup> = −1}}
|-
|style="background:#cedff2;" | {{math|1=''i''<sup>3</sup> = −i}}
|-
|{{math|1=''i''<sup>4</sup> = 1}}
|-
|{{math|1=''i''<sup>5</sup> = ''i''}}
|-
|{{math|1=''i''<sup>6</sup> = −1}}
|-
|{{math|1=''i''<sup>''n''</sup> = ''i''<sup>n(mod 4)</sup>}}<br />(see [[modular arithmetic|modulus]])
|}


An '''imaginary number''' is a number that can be written as a [[real number]] multiplied by the [[imaginary unit]] {{mvar|i}}, which is defined by its property {{math|1=''i''<sup>2</sup> = −1}}.<ref>{{citation
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</ref> The [[square (algebra)|square]] of an imaginary number {{mvar|bi}} is -''b''<sup>2</sup>. For example, {{math|5''i''}} is an imaginary number, and its square is −25. Except for 0 (which is both real and imaginary<ref>{{cite book |last=Sinha |first=K.C. |title=A Text Book of Mathematics XI |url=http://books.google.com/books?id=mqdzqbPYiAUC |publisher=Rastogi Publications |isbn=8171339123 |page=11.2}}[http://books.google.com/books?id=mqdzqbPYiAUC&pg=SA11-PA2 Extract of page 11.2]</ref>), imaginary numbers produce negative real numbers when squared.
 
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An imaginary number {{mvar|bi}} can be added to a real number {{mvar|a}} to form a [[complex number]] of the form {{math|''a'' + ''bi''}}, where {{mvar|a}} and {{mvar|b}} are called, respectively, the ''real part'' and the ''imaginary part'' of the complex number. Imaginary numbers can therefore be thought of as complex numbers whose real part is zero. The name "imaginary number" was coined in the 17th century as a derogatory term, as such numbers were regarded by some as fictitious or useless. The term "imaginary number" now means simply a complex number with a [[real part]] equal to 0, that is, a number of the form ''bi''.
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{{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.]]
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Although Greek mathematician and engineer [[Heron of Alexandria]] is noted as the first to have conceived these numbers,<ref>{{cite book |title=Fivefold symmetry |edition=2nd |first1=István |last1=Hargittai |publisher=World Scientific |year=1992 |isbn=981-02-0600-3 |page=153 |url=http://books.google.com/books?id=-Tt37ajV5ZgC}}, [http://books.google.com/books?id=-Tt37ajV5ZgC&pg=PA153 Extract of page 153]</ref><ref>{{cite book |title=Complex numbers: lattice simulation and zeta function applications |first1=Stephen Campbell |last1=Roy |publisher=Horwood |year=2007 |isbn=1-904275-25-7 |page=1 |url=http://books.google.com/books?id=J-2BRbFa5IkC}}</ref> [[Rafael Bombelli]] first set down the rules for multiplication of complex numbers in 1572. The concept had appeared in print earlier, for instance in work by [[Gerolamo Cardano]].  At the time, such numbers were poorly understood and regarded by some as fictitious or useless, much as zero and the negative numbers once were. Many other mathematicians were slow to adopt the use of imaginary numbers, including [[René Descartes]], who wrote about them in his ''[[La Géométrie]]'', where the term ''imaginary'' was used and meant to be derogatory.<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
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In 1843 a mathematical physicist, [[William Rowan Hamilton]], extended the idea of an axis of imaginary numbers in the plane to a three-dimensional space of [[quaternion#Definition|quaternion imaginaries]].
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With the development of [[quotient ring]]s of [[polynomial ring]]s, the concept behind an imaginary number became more substantial, but then one also finds other imaginary numbers such as the j of [[tessarine]]s which has a square of +1. This idea first surfaced with the articles by [[James Cockle (lawyer)|James Cockle]] beginning in 1848.<ref>James Cockle (1848) "On Certain Functions Resembling Quaternions and on a New Imaginary in Algebra", London-Dublin-Edinburgh [[Philosophical Magazine]], series 3, 33:435&ndash;9 and Cockle (1849) "On a New Imaginary in Algebra", Philosophical Magazine 34:37&ndash;47</ref>
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==Geometric interpretation==
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[[File:90-Degree Rotations in the Complex Plane.png|thumb|90-degree rotations in the complex plane]]


Geometrically, imaginary numbers are found on the vertical axis of the [[complex plane|complex number plane]], allowing 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 this {{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" and is denoted {{math|''i''ℝ}}, <math>\scriptstyle\mathbb{I}</math>, or simply {{math|ℑ}}.
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In this representation, multiplication by&nbsp;–1 corresponds to a [[rotation]] of 180 degrees about the origin. Multiplication by {{mvar|i}} corresponds to a 90-degree rotation in the "positive" direction (i.e., counterclockwise), and the equation {{math|1=''i''<sup>2</sup> = -1}} is interpreted as saying that if we apply two 90-degree rotations about the origin, the net result is a single 180-degree rotation. Note that a 90-degree rotation in the "negative" direction (i.e. clockwise) also satisfies this interpretation. This reflects the fact that {{mvar|-i}} also solves the equation {{math|1=''x''<sup>2</sup> = -1}}. In general, multiplying by a complex number is the same as rotating around the origin by the complex number's [[Arg (mathematics)|argument]], followed by a scaling by its magnitude.
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Care must be used in multiplying square roots of negative numbers. For example,<ref>{{Citation | last1=Maxwell | first1=E. A. | title=Fallacies in mathematics | publisher=[[Cambridge University Press]] | mr=0099907 | year=1959 |url=http://books.google.com/books?id=zNvvoFEzP8IC&lpg=PP1&dq=fallacies%20in%20mathematics&pg=PA37}}.
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Chapter VI, §I.2</ref> the following reasoning is incorrect:


:<math>-1 = i^2 = \sqrt{-1}\sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{1} = 1 </math>
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The [[Mathematical fallacy|fallacy]] is that the rule {{math|1={{sqrt|''x''}}{{sqrt|''y''}} = {{sqrt|''xy''}}}}, where the principal value of the square root is taken in each instance, is generally valid only if at least one of the two numbers {{mvar|x}} or {{mvar|y}} is positive, which is not the case here.
  <li>[http://www.juegosetnicos.com.ar/spip.php?article49&lang=fr/ http://www.juegosetnicos.com.ar/spip.php?article49&lang=fr/]</li>
 
 
==See also==
  <li>[http://cerisier.info/spip.php?article20/ http://cerisier.info/spip.php?article20/]</li>
* [[Imaginary unit]]
 
* [[de Moivre's formula]]
  <li>[http://www.thomassankara.net/spip.php?article709&lang=fr&date=2009-05/ http://www.thomassankara.net/spip.php?article709&lang=fr&date=2009-05/]</li>
* [[NaN]] (Not a number)
 
* [[Octonion]]
  <li>[http://www.cpefound.org.php5-14.dfw1-1.websitetestlink.com/node/13#comment-93728 http://www.cpefound.org.php5-14.dfw1-1.websitetestlink.com/node/13#comment-93728]</li>
* [[Quaternion]]
 
 
  <li>[http://www.metransparent.com/spip.php?article20626&lang=ar&id_forum=34132/ http://www.metransparent.com/spip.php?article20626&lang=ar&id_forum=34132/]</li>
==References==
 
{{reflist}}
</ul>
 
==Bibliography==
* {{Citation |first=Paul |last=Nahin |title=An Imaginary Tale: the Story of the Square Root of −1 |location=Princeton |publisher=Princeton University Press |year=1998 |isbn=0-691-02795-1 }}, explains many applications of imaginary expressions.
 
==External links==
* [http://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.
* [http://www.bbc.co.uk/programmes/b00tt6b2 In our time: Imaginary numbers] Discussion of imaginary numbers on BBC Radio 4.
* [http://www.bbc.co.uk/radio4/science/5numbers4.shtml 5Numbers programme 4] BBC Radio 4 programme
 
{{Number Systems}}
 
{{DEFAULTSORT:Imaginary Number}}
[[Category:Complex numbers| ]]
 
{{Link FA|lmo}}

Latest revision as of 14:40, 31 December 2014

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