Imaginary number: Difference between revisions

|<math>i</math> is a 4th<br/> [[root of unity]]

 

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''² = −1}}.<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''²}}. 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).

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>

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, 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, 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

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.

[[File:Rotations on the complex plane.svg|thumb|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>

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>

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

\sqrt{x \cdot y \vphantom{t}}

=\sqrt{(-x) \cdot (-y)}

\mathrel{\stackrel{\text{ (fallacy) }}{=}} \sqrt{-x\vphantom{ty}} \cdot \sqrt{-y\vphantom{ty}}

= i\sqrt{x\vphantom{ty}} \cdot i\sqrt{y\vphantom{ty}}

= -\sqrt{x \cdot y \vphantom{ty}}\,.

 

But the result is clearly nonsense. The step where the square root was broken apart was illegitimate. (See [[Mathematical fallacy]].)