Complex number and Prank call: Difference between pages

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{{external links|April 2007}}

In [[mathematics]], a '''complex number''' is a [[number]] of the form

{{Primarysources|date=April 2007}}

:<math> a + bi \,</math>

A '''prank call,''' also known as a '''crank call''', '''hoax call''', '''phony call''', or '''phony phone call''' is a form of [[practical joke]] committed over the [[telephone]]. As with all practical jokes, prank calls are generally done for humorous effect, though there is a thin line between [[humor]] and [[harassment]], and the person receiving the call may not find it funny. Prank phone calls began to gain a nationwide following over a period of many years, as they gradually became a staple of the obscure and amusing cassette tapes traded amongst musicians, sound engineers, and media traders beginning in the late 1970s. Among the most famous and earliest recorded prank calls are the [[Tube Bar prank calls|Tube Bar]] tapes which centered around [[Louis "Red" Deutsch]], and the Lucius Tate phone calls. Comedian Jerry Lewis was an incorrigible phone prankster, and recordings of his hijinks, dating from the 1960s and possibly earlier, still circulate throughout the country to this day.

where ''a'' and ''b'' are [[real number]]s, and ''i'' is the [[imaginary unit]], with the property ''i'' ² = &minus;1. The real number ''a'' is called the ''[[real part]]'' of the complex number, and the real number ''b'' is the ''[[imaginary part]]''. When the imaginary part ''b'' is 0, the complex number is just the real number ''a''.

Even very prominent people have fallen victim to prank callers, as for example Queen [[Elizabeth II of the United Kingdom|Elizabeth II]],<ref>[http://news.bbc.co.uk/1/hi/uk/618065.stm bbc.co.uk article on Queen Elizabeth II prank call]</ref> who was fooled by Canadian DJ [[Pierre Brassard]] posing as Canadian Prime Minister [[Jean Chrétien]], asking her to record a speech in support of Canadian unity ahead of the [[1995 Quebec referendum]]. Two other particularly famous examples of prank calls were made by the [[Miami]]-based radio station [[Radio El Zol]]. In one, they telephoned Venezuelan president [[Hugo Chávez]] and spoke to him, pretending to be Cuban president [[Fidel Castro]].<ref>[http://news.bbc.co.uk/1/hi/world/americas/2637395.stm bbc.co.uk article on Hugo Chávez prank call]</ref> They later repeated the prank, except that they called Castro and pretended to be Chávez. El Zol was also fined by the [[Federal Communications Commission]] (FCC).

For example, 3 + 2''i'' is a ''complex number'', with real part ''3'' and imaginary part ''2''. If ''z = a + bi'', the real part (''a'') is denoted Re(''z''), and the imaginary part (''b'') is denoted Im(''z'').

Complex numbers can be added, subtracted, multiplied, and divided like real numbers, but they have additional elegant properties. For example, real numbers alone do not provide a solution for every [[polynomial]] algebraic equation with real coefficients, while complex numbers do (the [[fundamental theorem of algebra]]).

In some fields (in particular, [[electrical engineering]] and [[electronics]], where ''i'' is a symbol for [[Electric current|current]]), complex numbers are written as ''a'' + ''bj''.

Complex numebers are most commonly used in car washes, blow dryers, onboard computers, and factory machines.

==Prank calls in popular culture==

== Geometric interpretation of the operations on complex numbers==

Prank calls are generally done for the amusement of the pranksters themselves. Many pranksters record the calls to share the joke with an audience. Some performers such as the [[The Jerky Boys]] have made a name for themselves producing albums of their recorded prank calls. Other prank call performers, such as [[Bob Pranky - Prank Calls Unlimited]], [[Touch-Tone Terrorists]], [[Melba's Phone Militia|Brother Russell]], [[The Happy Telephone]] and

Consider a [[plane (mathematics)|plane]]. One point is the ''origin'', 0. Another point is the ''unity'', 1.

[http://www.leroymercer.com LEE ROY MERCER] have garnered a following as well. The [[television series|television show]] ''[[Crank Yankers]]'' is a series of real-life prank calls made by celebrities and re-enacted on-screen by [[puppet]]s for a humorous effect. As with the Bob Pranky calls, the Crank Yankers sometimes borrow ideas, or entire routines, from obscure releases by relatively unknown pranksters such as The Ballbusters, who are neither credited nor remunerated for their original fun{{Fact|date=April 2007}}.

==Prank calling and the Internet==

[[Image:Complex numbers addition.png|right|thumb|''X'' = ''A'' + ''B'']]

[[Michael Biggins]], an actor whose real name is Michael Bigansky and goes by the performance name of [[Blackout (entertainer)|Blackout]] was (as far as can be researched) the very first person to put original prank calls on the internet in a [[digital]] [[streaming]] (instantly playable) format. There may have been downloadable calls before this time, but not streaming. He was also the first person to host an internet talk radio show based primarily on prank calls on the now defunct [[Lycos Talk Radio network]]<ref>[http://www.findarticles.com/p/articles/mi_m0NEW/is_2001_March_26/ai_72284900 Lycos Talk Radio]</ref> Which was an internet based talk radio site using [http://www.wonderhorse.com Wonderhorse] internet broadcasting and [[teleconference]] software.<ref>[http://www.wonderhorse.com Wonderhorse internet broadcasting and teleconference software]</ref> Blackout started his site Blackout's Box in 1995 ''(cite: verifiable by a whois check on ___domain name blackout.com and also verifiable with web archives such as wayback machine and in [[newsgroups]] archive postings and several major print sources - see references section)'' and put his first pranks online using the test version of [[RealAudio]] Beta software (which has over time evolved into [[RealPlayer]]) on a 14,400 [[U.S. Robotics]] modem. This was long before [[mp3]] or [[SHOUTcast]] streaming or [[internet radio]] stations existed. Before this time, the only way for both prank callers and prank call fans to hear pranks was by exchanging underground tapes, or maybe hearing them on the radio, but many radio prank calls were fake or 'set up' because of the steep [[FCC]] fines that could be imposed on the station if caught. Blackout used [[reel to reel]] recorders in his high school radio station [[WKPX]]<ref>[http://www.wkpx.freeservers.com/history.html 88.5 WKPX]</ref> in Florida to record his first pranks, and then transferred them to ([[voice mail box]]es) - hence where he got the name 'Blackout's Box'. People would spread his [[voice mail box]] number around on [[BBS]]s ([[bulletin board systems]]) and other VMBs so that calls into Blackout's voice mail box to listen to his pranks would grow virally (before internet multimedia type viral spreading existed), and he would get requests from all over the U.S. for him to prank people. Blackout was finally able to preserve his pranks in a digital format when [[Digital audio tape|digital audio tape]] recorders became available. This allowed him to port them to computer hard drive without losing quality. They were then converted to the [[RealAudio]] format and uploaded via [[FTP]] to his site the first day [[RealAudio]] beta became available in 1995. It was quite a pain and time consuming process to do at the time, but the ability of having anyone in the world who had a modem and the RealAudio plugin be able to hear one a prank instantly and in [[real time]] with no downloading was a big step forward. The very first call he put on the internet was called ''The Rrrrrrrooksnitchzien Society''<ref>[http://www.blackout.com/blackout/rook28.ram The first Prank Call on the Internet: ''The Rrrrrrrooksnitchzien Society'']</ref> in which he kept [[411]] [[telephone operators]] going mad for a good half an hour. Blackout's pranks were known to be longer and more complex than your average quick prank and he gained quite a following and much international press because of this. Hardware, software, technology and bandwidth have grown exponentially since Blackout put that first call up on the internet and so has the flood of sites, shoutcast, icecast, and podcast stations devoted to prank calls. [[Blackout (entertainer)|Blackout]], while moving more into the acting and independent film scene, still hosts an interactive TV/RADIO show on Thursday nights and occasionally still does prank calls.

===Addition===

The ''sum'' of two points ''A'' and ''B'' is the point ''X'' = ''A''+''B'' such that the [[triangle]]s with vertices 0, ''A'', ''B'' and ''X'', ''B'', ''A'' are [[similarity (mathematics)#Similar triangles|similar]].

The Woodcreek Faction (http://www.youtube.com/thewoodcreekfaction) broke new ground by becoming pioneers of filming prank phone calls & posting them on the web. The films allow the prank caller to portray a character who is seen by the audience instead of heard. This format has inspired many other comedy teams. The Woodcreek Faction pranks, in particular, are very successful on YouTube as well as many similar sites.

[[Image:Complex numbers multiplication.png|right|thumb|''X'' = ''AB'']]

===Multiplication===

The ''product'' of two points ''A'' and ''B'' is the point ''X'' = ''AB'' such that the triangles with vertices 0, 1, ''A'', and ''0'', ''B'', ''X'' are similar.

A classic example of their work "Kung Fu Pizza Prank":

[[Image:Complex numbers conjugation.png|right|thumb|''X'' = ''A''*]]

http://www.youtube.com/watch?v=SKO_xHgj7Pw

===Conjugation===

The ''complex conjugate'' of a point ''A'' is a point ''X'' = ''A''* such that the triangles with vertices 0, 1, ''A'' and 0, 1, ''X'' are [[mirror image]] of each other.

==Online Prank Communities/Stations==

== Some properties ==

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===Real vector space===

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'''C''' is a two-dimensional real [[vector space]].

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Unlike the reals, complex numbers cannot be ordered in any way that is compatible with its arithmetic operations: '''C''' cannot be turned into an [[ordered field]].

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Ever since the opportunity has been available, there has been internet radio stations dedicated to prank calls. Most of them feature a so-called "rotation" of prank calls which is a constant broadcast of various prank calls submitted by the community, usually streamed from a [[SHOUTcast]] server host.

A lot of prank-call stations also feature live shows dedicated to prank calling - many of the infamous radio personalities from these communities like to spam their names on common sites like [[Wikipedia]] as an attempt to gain more listeners (see page history).

[[Linear transformation#Definition and first consequences|'''R'''-linear]] maps '''C''' → '''C''' have the general form

:<math>f(z)=az+b\overline{z}</math>

with complex coefficients ''a'' and ''b''. Only the first term is '''C'''-linear; also only the first term is [[Holomorphic function|holomorphic]]; the second term is real-differentiable, but does not satisfy the [[Cauchy-Riemann equations]].

Although prank call communities are still relatively small-scale compared to [[FM Radio|FM stations]] that feature live pranks, it is a growing community on the internet today and many new communities are developing.

The function

:<math>f(z)=az\,</math>

corresponds to rotations combined with scaling, while the function

:<math>f(z)=b\overline{z}</math>

corresponds to reflections combined with scaling.

==Examples==

===Solutions of polynomial equations===

Some examples of well-known prank calls are:

A ''root'' of the [[polynomial]] ''p'' is a complex number ''z'' such

that ''p''(''z'') = 0.

A most striking result is that all polynomials of

degree ''n'' with real or complex coefficients have exactly ''n''

complex roots (counting [[multiple roots of a polynomial|multiple roots]] according to their

multiplicity). This is known as the [[fundamental theorem of algebra]], and shows that the complex numbers are an [[algebraically closed field]].

:Caller: ''Do you have [[Prince Albert in a Can]]?''

Indeed, the complex number field is the [[algebraically closed field|algebraic closure]] of the real number field, and [[Cauchy]] constructed complex numbers in this way. It can be identified as the [[quotient ring]] of the [[polynomial]] [[ring (mathematics)|ring]] '''R'''[''X''] by the [[Ideal (ring theory)|ideal]] generated by the polynomial ''X''² + 1:

:Receiver: ''Yes, I do.''

:<math> \mathbb{C} = \mathbb{R}[ X ] / ( X^2 + 1). \,</math>

:Caller: '' Well let him out!''

This is indeed a field because ''X''² + 1 is [[irreducible polynomial|irreducible]], hence generating a [[maximal ideal]], in '''R'''[''X'']. The image of ''X'' in this quotient ring becomes the imaginary unit ''i''.

:Caller: ''Hello! Is your refrigerator'' (''Nose'', ''Toilet'', or ''Water'' in some variants) ''running?''

===Algebraic characterization===

:Receiver: ''Yes, it is.''

The field '''C''' is ([[up to]] field [[isomorphism]]) [[characterization (mathematics)|characterized]] by the following three facts:

:Caller: ''Then you'd better go catch it!''

* its [[characteristic (algebra)|characteristic]] is 0

* its [[transcendence degree]] over the [[prime field]] is the [[cardinality of the continuum]]

* it is [[algebraically closed]]

:Caller: ''Is Mrs. Wall there?''

Consequently, '''C''' contains many proper subfields which are isomorphic to '''C'''. Another consequence of this characterization is that the [[Galois group]] of '''C''' over the rational numbers is enormous, with cardinality equal to [[Beth two|that of the power set of the continuum]].

:Receiver: ''No.''

:Caller: ''Is Mr. Wall there?''

:Receiver: ''No.''

:Caller: ''Are there any Walls there?''

:Receiver: ''No.''

:Caller: ''Then how does your roof stay up?''

===''The Simpsons''===

During the early years of ''[[The Simpsons]]'', a popular [[Running gag|recurring gag]] involved Bart making prank calls to [[Moe's Tavern]]. This bit was inspired by the infamous [[Tube Bar prank calls]] of the 1970s. The calls usually followed a set pattern: Bart would ask for a non-existent person, Moe would shout loudly for the person Bart asked for, Moe catching on only after the bar (usually) erupts in uproarious laughter, and Moe threatening violent revenge upon catching the perpetrator. Moe never seemed to realize that it was Bart who made the call. Once Bart even told Moe that he made prank calls and Moe still did not catch on: Bart: "Well I make prank phone calls." Moe (in a happy voice one uses when talking to children): "Good for you."

"People" whom Bart has asked for include:

===Characterization as a topological field===

* I.P. Freely - (''I pee freely'')

As noted above, the algebraic characterization of '''C''' fails to capture some of its most important properties. These properties, which underpin the foundations of [[complex analysis]], arise from the [[topology]] of '''C'''. The following properties characterize '''C''' as a [[topological ring|topological field]]:

* Maya Buttreeks - (''My butt reeks'')

*'''C''' is a field.

* Jacques Strappe – (''jock strap'')

*'''C''' contains a subset ''P'' of nonzero elements satisfying:

* Ivanna Tinkle – (''I wanna tinkle'')

**''P'' is closed under addition, multiplication and taking inverses.

* Heywood U. Cuddleme – (''Hey, would you cuddle me?'')

**If x and y are distinct elements of ''P'', then either ''x-y'' or ''y-x'' is in ''P''

* Amanda Huggenkiss – (''A man to hug and kiss'')

**If ''S'' is any nonempty subset of ''P'', then ''S+P=x+P'' for some ''x'' in '''C'''.

* Mike Rotch – (''My crotch'')

*'''C''' has a nontrivial involutive automorphism ''x->x*'', fixing ''P'' and such that ''xx*'' is in ''P'' for any nonzero ''x'' in '''C'''.

* Al Coholic – (''Alcoholic'')

* Bea O'Problem – (''B.O. problem'')

* Seymour Butz – (''See more butts'')

* Anita Bath – (''I need a bath'')

* Homer Sexual – (''A homosexual'')

* Lee V. Mediately - (''Leave immediately'')

* Eura Snotball - (''You're a snotball'')

* G.I. Manidiot - (''Gee, I'm an idiot'')

* Oliver Clothesoff (''All of her clothes off!'')

* Will U. P. Onme (''Will you pee on me?'')

* Hugh Jass (''Huge ass'')

* Ollie Tabooger (''I'll eat a booger'')

* Ahmed Adoudi (''I made a doodie'')

One backfire on this formula was a call to "Hugh Jass" (''huge ass''), as there turned out to be a person in the bar named Hugh Jass. Another backfire was when Homer was running the bar and didn't know how to carry out the prank when Bart asked for Ollie Tabooger (''I'll eat a booger''). A third was a time where [[Mr. Burns]] called Moe's by mistake while looking for [[Smithers]], and was threatened by Moe who thought it was a prank call. Finally, in a flashback scene to Homer and Marge's youth, Marge tries to call Homer (whom she believes goes by the name "Elvis Jagger Abdul-Jabbar" because of his shyness), only to get Moe to threaten her when she asks for his name. After hanging up, Moe mutters "And that's the origin of that!"

Given these properties, one can then define a topology on '''C''' by taking the sets

*<math>B(x,p) = \{y | p - (y-x)(y-x)^*\in P\}</math>

as a [[base (topology)|base]], where ''x'' ranges over '''C''', and ''p'' ranges over ''P''.

Interestingly, in the video game [[The Simpsons: Bart vs. the Space Mutants]] for NES and Genesis, you can make prank calls to Moe by putting a coin in the phone by Moe's Tavern. Examples of what Bart can say are I.M. Adope (''I am a dope'') and Stu Piddidiot. (''stupid idiot'')

To see that these properties characterize '''C''' as a [[topological ring|topological field]], one notes that ''P'' ∪ {0} ∪ ''-P'' is an ordered [[Dedekind completion|Dedekind-complete]] field and thus can be identified with the [[real number]]s '''R''' by a unique field isomorphism. The last property is easily seen to imply that the [[Galois group]] over the real numbers is of order two, completing the characterization.

===Futurama===

[[Lev Semenovich Pontryagin|Pontryagin]] has shown that the only [[connected space|connected]] [[locally compact]] [[topological ring|topological fields]] are '''R''' and '''C'''. This gives another characterization of '''C''' as a topological field, since '''C''' can be distinguished from '''R''' by noting the nonzero complex numbers are [[connected space|connected]] whereas the nonzero real numbers are not.

A prank call leads to [[Fry (futurama)|Fry]]'s delivery of a pizza to a cryogenic lab, which sets the whole series in motion. The name used is I.C. Wiener (Meaning "I see Wiener" or "Icey wiener"), referring to the men frozen inside of the cryogenic chambers.

Fry also adopts a dog after receiving a prank call asking for a pizza to be delivered to a Seymour Asses (see more asses). Fry then names the dog Seymour following the prank call.

==Complex analysis==

{{details|Complex analysis}}

==Reaction==

The study of functions of a complex variable is known as [[complex analysis]] and has enormous practical use in [[applied mathematics]] as well as in other branches of mathematics. Often, the most natural proofs for statements in [[real analysis]] or even [[number theory]] employ techniques from complex analysis (see [[prime number theorem]] for an example). Unlike real functions which are commonly represented as two dimensional graphs, complex functions have four dimensional graphs

In the United States, prank calls are easily traced through [[Caller ID]]. It is possible that caller ID can reduce the number of prank calls. However, most telephone companies currently permit callers to withhold caller ID if they do not wish the called party to know their identity. In North America, there is a way to disable the victim's Caller ID by dialing *67 before dialing.

and may usefully be illustrated by color coding a three dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.

Sometimes the joke can be taken too far, especially if the prankster succeeds in making his victim believe the scenario is real. Prank call comedian [[Jim Florentine]] (who mainly takes incoming calls from [[telemarketer]]s and turns the tables by performing pranks on them) has had the police called on him on more than one occasion for taking his jokes too far. During one call, Florentine tells an [[Agency (law)|insurance agent]] that, rather than pay to keep an elderly woman alive, he is going to go to the hospital and smother her with a pillow.<ref>''Terrorizing Telemarketers III'', Jim Florentine</ref> After the call, the agent called [[9-1-1]] and gave them Florentine's number and the address on file, and the police arrived at his home with guns drawn. However, when the police showed up and discovered it was actually a prank, the officer simply asked, "Don't you think you're a little old for this?"<ref>liner notes, ''Terrorizing Telemarketers III'', Jim Florentine</ref>

==Applications==

Complex numbers are most commonly used in nuclear weapons, biotechnologies, explosives, jet engines, the number of hydrogen atoms in xenon lamps, and mainly in new handheld electronics such as PDA's and certain laptops made from 2003 and newer. The words "real" and "imaginary" were meaningful when complex numbers were used mainly as an aid in manipulating "real" numbers, with only the "real" part directly describing the world. Later applications, and especially the discovery of quantum mechanics, showed that nature has no preference for "real" numbers and its most ''real'' descriptions often require complex numbers, the "imaginary" part being just as physical as the "real" part.

===ControlAs theory=A Crime==

Prank calls range from annoying hang-ups to false calls to [[emergency services]] or [[bomb threat]]s. Prank calls that waste the time of emergency services are a [[criminal offense]] in most countries.

In [[control theory]], systems are often transformed from the [[time ___domain]] to the [[frequency ___domain]] using the [[Laplace transform]]. The system's [[pole (complex analysis)|poles]] and [[zero (complex analysis)|zeros]] are then analyzed in the ''complex plane''. The [[root locus]], [[Nyquist plot]], and [[Nichols plot]] techniques all make use of the complex plane.

One such hoax call occurred in [[Perth, Western Australia|Perth]], [[Australia]], on [[New Year's Eve]] [[2002]], when a drunken teenager called the new anti-terrorist hotline to report a bomb threat against the New Year's Eve fireworks celebration.<ref>[http://www.ag.gov.au/agd/WWW/attorneygeneralHome.nsf/Page/Media_Releases_2003_January_2003_WA_man_charged_over_hoax_hotline_call_(1_January_2003) Perth, Australia bomb threat hoax]</ref> The threat was taken seriously, and the celebrations were about to be canceled, when police discovered that no such threat existed. The teen was arrested for deliberate false reporting.

In the root locus method, it is especially important whether the [[pole (complex analysis)|poles]] and [[zero (complex analysis)|zeros]] are in the left or right half planes, i.e. have real part greater than or less than zero. If a system has poles that are

*in the right half plane, it will be [[unstable]],

*all in the left half plane, it will be [[stability|stable]],

*on the imaginary axis, it will have [[marginal stability]].

If a system has zeros in the right half plane, it is a [[nonminimum phase]] system.

Tension was also caused in December 2005, when a Catholic Church-owned radio station in [[Spain]] ([[COPE (radio station)|COPE]]) played a prank on [[Bolivia]]n president-elect [[Evo Morales]]. The hoaxer pretended to be Spanish Prime Minister [[José Luis Rodríguez Zapatero]], congratulating Morales on his election<ref>[http://www.quepasa.com/english/news/latinamerica/Zapatero.Morales.prank/406542.html Prank call to Evo Morales]</ref> and saying things like, "I imagine the only one not to have called you was George Bush. I've been here two years and he still hasn't called me".<ref>[http://www.informativos.telecinco.es/cope/zapatero/broma_morales/dn_17521.htm Transcript of call (in Spanish)]</ref> The Bolivian government protested to Spain, and the real Zapatero called Morales and apologized. The Spanish government in turn summoned the [[papal nuncio]] in protest.

===Signal analysis===

Complex numbers are used in [[signal analysis]] and other fields as a convenient description for periodically varying signals. The absolute value |''z''| is interpreted as the [[amplitude]] and the argument arg(''z'') as the [[phase (waves)|phase]] of a [[sine wave]] of given [[frequency]].

In the [[United States]], the [[Telecommunications Act of 1996]] makes some prank calls a [[felony]] with penalties of up to two years in prison, and possible fines (depending on severity). However, such penalties are rarely carried out. As an example, the [[Chicago]] [[shock jock]] [[Mancow Muller|Erich "Mancow" Muller]], after being criticized for the extensive use of prank calls on his radio show, broadcasted the sarcastic remark: "Reality check for you people: Chicago's the murder capital of America. The police don't care if you get a prank call."

If [[Fourier analysis]] is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as complex valued functions of the form

:<math> f ( t ) = z e^{i\omega t} \,</math>

where ω represents the [[angular frequency]] and the complex number ''z'' encodes the phase and amplitude as explained above.

Moreover, to make a prank call that falls afoul of the Telecommunications Act, {{UnitedStatesCode|47|223}} (a)(1), the call must be done with the intent to "annoy, abuse, threaten, or harass". Arguably then, if the intent of the call is to amuse, confuse, or simply to engage the call's recipient, there is no violation of the Telecommunications Act.

In [[electrical engineering]], the Fourier transform is used to analyze varying [[voltage]]s and [[current (electricity)|currents]]. 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 [[impedance]]. (Electrical engineers and some physicists use the letter ''j'' for the imaginary unit since ''i'' is typically reserved for varying currents and may come into conflict with ''i''.) This use is also extended into [[digital signal processing]] and [[digital image processing]], which utilize digital versions of Fourier analysis (and [[Wavelet]] analysis) to transmit, [[Data compression|compress]], restore, and otherwise process [[digital]] [[Sound|audio]] signals, still images, and [[video]] signals.

=== Frequency (spectral) ___domain electromagnetism ===

[[Maxwell's equations]] are (usually) written in terms of real vector function's of space and time. When fourier transformed to functions of space and frequency the fields become complex, as in signal analysis. The majority of electromagnetic calculations are done in the frequency ___domain.

==See also==

==== Sign convention ====

* [[Celebrity prank call]]

In such calculations, engineers tend to use <math> e^{j(\omega t - k x)} \,</math> to describe a plane wave, for compatibility with signal analysis, while physicists tend to use <math> e^{i(k x - \omega t)} \,</math> for compatibility with quantum mechanics and other scattering calculations. It is therefore sometimes said that j = –i.

* ''[[The Simpsons]]'' (specifically, [[Moe's Tavern]])

* [[Tube Bar prank calls]]

===Improper integrals===

* [[Obscene phone call]]s

In applied fields, the use of complex analysis is often used to compute certain real-valued [[improper integral]]s, by means of complex-valued functions. Several methods exist to do this, see [[methods of contour integration]].

* [[Phone Losers of America]]

* [[Dead Ringers (comedy)]]

===Quantum mechanics===

* [[The Happy Telephone]]

The complex number field is also of utmost importance in [[quantum mechanics]]

* [[Davin & Devyn]]

since the underlying theory is built on (infinite dimensional) [[Hilbert space]]s over '''C'''. The more limited original formulations of [[Erwin Schrödinger|Schrödinger]] and [[Werner Heisenberg|Heisenberg]] are also in terms of complex numbers.

* [[Touch-Tone Terrorists]]

* [[The Jerky Boys]]

===Relativity===

* [[Rickey Smiley]]

In [[special relativity|special]] and [[general relativity]], some formulae for the metric on [[spacetime]] become simpler if one takes the time variable to be imaginary.

===Applied mathematics===

In [[differential equations]], it is common to

first find all complex roots ''r'' of the [[characteristic equation]] of a

[[linear differential equation]] and then attempt to solve the system

in terms of base functions of the form ''f''(''t'') = ''e''^''rt''.

===Fluid dynamics===

In [[fluid dynamics]], complex functions are used to describe [[potential flow in 2d]].

===Fractals===

Certain [[fractal]]s are plotted in the complex plane e.g. [[Mandelbrot set]] and [[Julia set]].

==History==

The earliest fleeting reference to [[square root]]s of [[negative numbers]] perhaps occurred in the work of the [[Greece|Greek]] [[Hellenistic mathematics|mathematician]] and inventor [[Hero of Alexandria|Heron of Alexandria]] in the [[1st century]] [[common era|CE]], when he considered the volume of an impossible [[frustum]] of a [[pyramid]] <ref>http://people.bath.ac.uk/aab20/complexnumbers.html</ref>, though negative numbers were not conceived in the [[Hellenistic civilization|Hellenistic world]].

Complex numbers became more prominent in the [[16th century]], when closed formulas for the roots of [[Cube (arithmetic)|cubic]] and [[quartic]] [[polynomial]]s were discovered by Italian mathematicians (see [[Niccolo Fontana Tartaglia]], [[Gerolamo Cardano]]). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. For example, Tartaglia's cubic formula gives the following solution to the equation <math>x^3-x=0</math>:

:<math>\frac{1}{\sqrt{3}}\left(\sqrt{-1}^{1/3}+\frac{1}{\sqrt{-1}^{1/3}}\right).</math>

At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation <math>z^3=i</math> has solutions &minus;''i'', <math>\frac{\sqrt{3}}{2}+\frac{1}{2}i</math> and <math>\frac{-\sqrt{3}}{2}+\frac{1}{2}i</math>. Substituting these in turn for <math>\sqrt{-1}^{1/3}</math> into the cubic formula and simplifying, one gets 0, 1 and &minus;1 as the solutions of <math>x^3-x=0.</math>

This was doubly unsettling since not even negative numbers were considered to be on firm ground at the time. The term "imaginary" for these quantities was coined by [[René Descartes]] in [[1637]] and was meant to be derogatory (see [[imaginary number]] for a discussion of the "reality" of complex numbers). A further source of confusion was that the equation <math>\sqrt{-1}^2=\sqrt{-1}\sqrt{-1}=-1</math> seemed to be capriciously inconsistent with the algebraic identity <math>\sqrt{a}\sqrt{b}=\sqrt{ab}</math>, which is valid for positive real numbers ''a'' and ''b'', and which was also used in complex number calculations with one of ''a'', ''b'' positive and the other negative. The incorrect use of this identity (and the related identity <math>\frac{1}{\sqrt{a}}=\sqrt{\frac{1}{a}}</math>) in the case when both ''a'' and ''b'' are negative even bedeviled [[Euler]]. This difficulty eventually led to the convention of using the special symbol ''i'' in place of <math>\sqrt{-1}</math> to guard against this mistake.

The [[18th century]] saw the labors of [[Abraham de Moivre]] and [[Leonhard Euler]]. To De Moivre is due (1730) the well-known formula which bears his name, [[de Moivre's formula]]:

:<math>(\cos \theta + i\sin \theta)^{n} = \cos n \theta + i\sin n \theta \,</math>

and to Euler (1748) [[Euler's formula]] of [[complex analysis]]:

:<math>\cos \theta + i\sin \theta = e ^{i\theta }. \,</math>

The existence of complex numbers was not completely accepted until the geometrical interpretation (see below) had been described by [[Caspar Wessel]] in [[1799]]; it was rediscovered several years later and popularized by [[Carl Friedrich Gauss]], and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in [[John Wallis|Wallis's]] ''De Algebra tractatus''.

Wessel's memoir appeared in the Proceedings of the [[Copenhagen Academy]] for 1799, and is exceedingly clear and complete, even in comparison with modern works. He also considers the sphere, and gives a [[quaternion]] theory from which he develops a complete spherical trigonometry. In 1804 the Abbé Buée independently came upon the same idea which Wallis had suggested, that <math>\pm\sqrt{-1}</math> should represent a unit line, and its negative, perpendicular to the real axis. [[Buée]]'s paper was not published until 1806, in which year [[Jean-Robert Argand]] also issued a pamphlet on the same subject. It is to Argand's essay that the scientific foundation for the graphic representation of complex numbers is now generally referred. Nevertheless, in 1831 Gauss found the theory quite unknown, and in 1832 published his chief memoir on the subject, thus bringing it prominently before the mathematical world. Mention should also be made of an excellent little treatise by [[Mourey]] (1828), in which the foundations for the theory of directional numbers are scientifically laid. The general acceptance of the theory is not a little due to the labors of [[Augustin Louis Cauchy]] and [[Niels Henrik Abel]], and especially the latter, who was the first to boldly use complex numbers with a success that is well known.

The common terms used in the theory are chiefly due to the founders. Argand called <math>\cos \phi + i\sin \phi</math> the ''direction factor'', and <math>r = \sqrt{a^2+b^2}</math> the ''modulus''; Cauchy (1828) called <math>\cos \phi + i\sin \phi</math> the ''reduced form'' (l'expression réduite); Gauss used ''i'' for <math>\sqrt{-1}</math>, introduced the term ''complex number'' for <math>a+bi</math>, and called <math>a^2+b^2</math> the ''norm''.

The expression ''direction coefficient'', often used for <math>\cos \phi + i

\sin \phi</math>, is due to Hankel (1867), and ''absolute value,'' for ''modulus,'' is due to Weierstrass.

Following Cauchy and Gauss have come a number of contributors of high rank, of whom the following may be especially mentioned: [[Ernst Kummer|Kummer]] (1844), [[Leopold Kronecker]] (1845), [[Scheffler]] (1845, 1851, 1880), [[Bellavitis]] (1835, 1852), Peacock (1845), and [[Augustus De Morgan|De Morgan]] (1849). [[August Ferdinand Möbius|Möbius]] must also be mentioned for his numerous memoirs on the geometric applications of complex numbers, and [[Johann Peter Gustav Lejeune Dirichlet|Dirichlet]] for the expansion of the theory to include primes, congruences, reciprocity, etc., as in the case of real numbers.

A complex [[ring (mathematics)|ring]] or [[Field (mathematics)|field]] is a set of complex numbers which is [[closed]] under addition, subtraction, and multiplication. [[Carl Friedrich Gauss|Gauss]] studied complex numbers of the form <math>a + bi</math>, where ''a'' and ''b'' are integral, or rational (and ''i'' is one of the two roots of <math>x^2 + 1 = 0</math>). His student, [[Ferdinand Eisenstein]], studied the type <math>a + b\omega</math>, where <math>\omega</math> is a complex root of <math>x^3 - 1 = 0</math>. Other such classes (called [[cyclotomic fields]]) of complex numbers are derived from the [[roots of unity]] <math>x^k - 1 = 0</math> for higher values of <math>k</math>. This generalization is largely due to [[Kummer]], who also invented [[ideal number]]s, which were expressed as geometrical entities by [[Felix Klein]] in 1893. The general theory of fields was created by [[Évariste Galois]], who studied the fields generated by the roots of any polynomial equation

:<math>\ F(x) = 0.</math>

The late writers (from 1884) on the general theory include [[Karl Weierstrass|Weierstrass]], [[Hermann Schwarz|Schwarz]], [[Richard Dedekind]], [[Otto Hölder]], [[Berloty]], [[Henri Poincaré]], [[Eduard Study]], and [[Alexander MacFarlane]].

The formally correct definition using pairs of real numbers was given in the [[19th century]].

== See also ==

* [[Circular motion#Using complex numbers]]

* [[Complex geometry]]

* [[De Moivre's formula]]

* [[Euler's identity]]

* [[Hypercomplex number]]

* [[Leonhard Euler]]

* [[Local field]]

* [[Mandelbrot set]]

* [[Quaternion]]

* [[Riemann sphere]] (extended complex plane)

* [[Split-complex number]]

* [[Imaginary number]]/[[Imaginary unit]]

== Further reading ==

* ''An Imaginary Tale: The Story of <math>\sqrt{-1}</math>'', by Paul J. Nahin; Princeton University Press; ISBN 0-691-02795-1 (hardcover, 1998). A gentle introduction to the history of complex numbers and the beginnings of complex analysis.

* ''Numbers'', by H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel, R. Remmert; Springer; ISBN 0-387-97497-0 (hardcover, 1991). An advanced perspective on the historical development of the concept of number.

* ''The Road to Reality: A Complete Guide to the Laws of the Universe'', by [[Roger Penrose]]; Alfred A. Knopf, 2005; ISBN 0-679-45443-8. Chapters 4-7 in particular deal extensively (and enthusiastically) with complex numbers.

==References==

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== External links ==

* [http://mathforum.org/johnandbetty/ John and Betty's Journey Through Complex Numbers]

* {{MathWorld | urlname=ComplexNumber | title=Complex Number}}

* [http://www.sosmath.com/complex/complex.html SOS Math - Complex Variables]

* [http://www.cut-the-knot.org/arithmetic/algebra/ComplexNumbers.shtml Algebraic Structure of Complex Numbers] from [[cut-the-knot]]

* [http://math.fullerton.edu/mathews/n2003/ComplexNumberOrigin.html A history of complex numbers].

==External links==

[[Category:Complex numbers]]

{{dmoz|/Recreation/Humor/Pranks/Prank_Calls/|Prank Calls}}

[[Category:Elementary mathematics]]

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