title: American Bully

text: The American Bully, as it is now known, began development in the 1980s with the majority of the final behavioral and aesthetic product being completed in the 1990s. There is consensus that at least five other breeds were used to attain the more "bully" physical traits desired as well as the more diminutive size of some lines. The American Pit Bull Terrier (APBT) was the foundation (parent breed) used to create the American Bully. The APBT has maintained a characteristic appearance and temperament for over a century. Within that span of time different strains of APBT emerged within the breed, each with different physical attributes. One particular APBT strain was crossbred to create developed a specific, stockier, physique that breeders originally misrepresented as purebred American Pit Bull Terriers. Eventually, enough breeders agreed that these dogs were disparate enough from American Pit Bull Terriers that it warranted them admitting that they were different breed altogether. The bloodline of these mixed breeds were further influenced with further, openly acknowledged breeding to the American Bulldog, English Bulldog, and Olde English Bulldogge in order to fine-tune desired physical characteristics and personality traits.


title: American Bully

text: The breed was first recognized by its breed club, the American Bully Kennel Club (ABKC), in 2004. This registry first acted as a means to document pedigrees and show the breed against its written standard. According to the ABKC the initial desire for this breed was to produce a dog with a lower prey drive and more of the "bully" traits and characteristics than the American Staffordshire Terrier. Mass and heavy bone was prioritized to ensure such a look, and due to this many of the dogs shown today display the wide front for which they were originally bred.


title: American Bully

text: The breed's development and popularity are commonly tied to the growth of hip-hop culture. The American Bully should not be confused with the several other bulldog-type breeds.


title: American Bully

text: Within the ABKC, the four varieties are separated by height without specification of weight. All these varieties are expected to follow the same standard with minor alterations.


title: American Bully

text: All dogs are classified and shown as "Standard" until they reach a year of age, at which point they are separated into the varieties and shown against their own type.


title: American Bully

text: The standard American Bully type is a medium-sized dog with a compact bulky muscular body, heavy bone structure and blocky head. Male dogs must be , while females must be .


title: American Bully

text: The "pocket" type is a smaller variant, with full-grown males under 17 inches (43 cm), but no less than 14 inches (36 cm), at the withers and females under 16 inches (40 cm), but no less than 13 inches (33 cm), at the withers.


title: American Bully

text: An XL type is determined by its adult height, with males between 21 inches (51  cm) and 23 inches (57  cm) at the withers and females between 19 inches (48  cm) and 22 inches (54  cm) at the withers.


title: American Bully

text: The "classic" is a lighter-framed dog than the "standard", but falls within the same height range. These dogs do not display the exaggerated features often found in the other varieties, and arguably display clearer American Pit Bull Terrier/American Staffordshire Terrier lineage.


title: American Bully

text: Outside of the breed standard, dogs shorter or taller than the named variations have been bred. Smaller dogs are sometimes called "Micro", and larger ones are called "XXL", but neither are recognized by the kennel clubs as legitimate varieties.


title: American Bully

text: The American Bully is a highly adaptable and trainable breed. Many dogs, despite acting as lapdogs in the home, do well in sports such as weight pull and flirt pole. Human aggression is discouraged in breed standards, however a level of dog aggression is characteristic of the breed. Breeders have acknowledged that American Bully dogs can be very dangerous if improperly raised or bred. On July 16, 2022, Joanne Robinson of Rotherham, England, was killed in her own home when she was attacked by her pet American Bully XL. The dog, one of a pair, was reputed to weigh 196 lbs. Furthermore, on August 10 of the same year, another attack by the breed against a person was reported. Ian Symes, an experienced and professional dog-walker, was mauled to death while walking through a park in Fareham, Hampshire.


title: American Bully

text: In 2022, six fatal dog attacks in the UK listed American Bully as the breed responsible for the attacks, with victims ranging in ages from 17 months to 62 years old.


title: American Bully

text: Health problems vary within the breed and span the entire spectrum, with some varieties being plagued by problems, and others being well-documented for health and quality. Testing is not as commonplace in the breed as in older breeds, though hip and elbow scoring are the most frequently conducted. Cherry eye, ectropion, and entropion are often seen affecting the eyes, while Brachycephalic Respiratory Syndrome can be seen in the shorter muzzled dogs.


title: The Founder

text: The Founder is a 2016 American biographical drama film directed by John Lee Hancock and written by Robert Siegel. Starring Michael Keaton as businessman Ray Kroc, the film portrays the story of his creation of the McDonald's fast-food restaurant chain, which eventually involved forcing out the company's original founders to take control with conniving ruthlessness. Nick Offerman and John Carroll Lynch co-star as McDonald's founders Richard and Maurice McDonald, alongside Linda Cardellini as Ray Kroc's third wife Joan Smith, and B. J. Novak as McDonald's president and chief executive Harry J. Sonneborn.


title: The Founder

text: The film premiered at Arclight Hollywood on December 7, 2016, and was released theatrically in the United States on January 20, 2017, by the Weinstein Company. It grossed $24 million worldwide and received generally positive reviews from critics, who praised Keaton's performance.


title: The Founder

text: In 1954, Ray Kroc is a traveling milkshake machine salesman. While he and his supportive wife Ethel have saved enough money to live a comfortable life in Arlington Heights, Illinois, he craves more. Ray also observes that many of the drive-in restaurants that he tries to sell to are inefficiently run. After learning that a drive-in in San Bernardino is ordering an unusually large number of milkshake mixers, Ray drives to California to see it. What he finds is McDonald's—a popular walk-up restaurant with fast service, high-quality food, disposable packaging and a family-friendly atmosphere.


title: The Founder

text: Ray meets with the two McDonald brothers, Maurice "Mac" and Richard "Dick" McDonald. They give Ray a tour of the kitchen and he notes the employees' strong work ethic. Ray is astounded with the restaurant, and takes the brothers to dinner. They tell him the origin story of McDonald's and how they came to design their "fast food" system. The next day, Ray suggests that the brothers franchise the restaurant, but they hesitate, pointing out that they already tried, only to have encountered absentee franchisees who were lackadaisical in upholding their system. Ray persists and eventually convinces the brothers to allow him to lead their franchising efforts on the condition that he agrees to a strict contract, which requires all changes be subject to the McDonald brothers' approval.


title: The Founder

text: Initially, Ray begins building a McDonald's restaurant in Des Plaines, Illinois, while attempting to entice wealthy investors (specifically fellow members at his country club) to open franchises, but encounters the same poor management ethic which doomed the original franchise efforts. After encountering a Bible salesman who's Jewish, Ray hits on the idea of franchising to middle-class investors, who have more incentive to be hands-on and are willing to follow the McDonald's formula. This proves successful, and new franchises begin opening across the Midwest, with Ray representing himself as the creator of McDonald's and Fred Turner, a burger cook at the Des Plaines restaurant who caught Ray's eye on the grill, as his associate. During this time, Ray meets Rollie Smith, an upscale restaurant owner in Minnesota who wishes to invest, and Rollie's wife, Joan, to whom Ray is immediately attracted.


title: The Founder

text: Despite his success, Ray begins to encounter financial difficulties as his share of franchise profits are limited due to his contract, which the McDonald brothers decline to renegotiate. Meanwhile, the owners are encountering higher than expected costs, particularly for refrigeration of large amounts of ice cream for milkshakes. Joan suggests a powdered milkshake as a way to avoid these costs, but the brothers consider it degrading to their food quality. Ray is called to the bank as his mortgage is past due, but this is overheard by Harry Sonneborn, a financial consultant, who requests to review Ray's books. Sonneborn realizes that the real profit opportunity is in providing real estate to the franchisees, which will not only provide a revenue stream, but give Ray leverage over his franchisees and the brothers. Ray incorporates a new company, Franchise Realty Corporation, and attracts new investors. This allows him to open new restaurants without the brothers' approval. This upsets the brothers and emboldens Ray: he increasingly defies them by circumventing their authority and providing powdered milkshakes to all franchisees, but not their restaurant. Ray also divorces his wife, Ethel, who gets all his assets except any shares in his business.


title: The Founder

text: Ray renames the company the McDonald's Corporation and demands to be released from his contract and buy the McDonald brothers out, the news of which sends Mac into diabetic shock. Ray visits him in the hospital and offers a blank check to settle their business. The brothers agree to a $2.7 million lump sum payment (equivalent to $26 million in 2020), ownership of their original restaurant in San Bernardino, and a 1% annual royalty, but when the time comes to finalize the agreement, Ray refuses to include the royalty in the settlement and instead offers it as a handshake deal. Afterwards, Dick confronts Ray and asks why he had to take over their business, when he could have easily stolen their idea and recreated it. Ray argues that the true value of McDonald's is the name itself, which expresses all the attributes of Americana.


title: The Founder

text: The McDonald brothers are forced to take their own name off the original restaurant and Ray opens a new McDonald's franchise directly across the street, finally putting the brothers out of business. The film ends in 1970, where Ray prepares a speech in his elaborate mansion with his new wife: Joan, now divorced from Rollie Smith. In the speech, Ray, plagiarizing heavily from a motivational speech he heard earlier in the film, praises himself for his success. He states that he was able to achieve it, not through talent or a strong work ethic, but through persistence.


title: The Founder

text: An epilogue reveals that the McDonald brothers were never paid their royalties, which would have eventually been in the area of $100 million a year, and that every day McDonald's feeds approximately 1% of the Earth's population.


title: The Founder

text: The screenplay for "The Founder" was written by Robert Siegel, based on Ray Kroc's autobiography and an unauthorized biography. According to early reports, the film was to be developed in the same vein as "There Will Be Blood" and "The Social Network". According to "Deadline Hollywood", it was ranked the 13th-best unproduced script of 2014. In December 2014, John Lee Hancock was signed to direct the picture.


title: The Founder

text: In February 2015, Michael Keaton was signed to the role of Ray Kroc. Laura Dern joined the film on May 11, 2015 to play Kroc's wife Ethel Fleming, whom Kroc divorced in 1961. The next day, it was announced that Nick Offerman joined the film, set to play Richard "Dick" McDonald. On May 28, 2015, it was announced that B. J. Novak joined the film as Kroc's financial consultant, Harry J. Sonneborn. On June 9, 2015, it was reported that Linda Cardellini had joined the film, and on June 26, 2015, it was announced that John Carroll Lynch and Patrick Wilson had also been cast.


title: The Founder

text: Principal photography for the film began in Newnan, Georgia on June 1, 2015. Production designer Michael Corenblith had previously worked on films such as "Apollo 13", "Saving Mr. Banks" and "The Blind Side" in which attention to historic detail was important. Corenblith worked from archival photos, training films, materials provided by the McDonald family, blueprints obtained from eBay and research at the oldest McDonald's restaurant in Downey, California. The McDonald brothers' original octagonal San Bernardino restaurant was built in Newnan in the parking lot of the Coweta County administration building.


title: The Founder

text: After a month of searching for a suitable location, an old-style McDonald's building set with the "golden arches" was constructed in a church parking lot in seven working days in Douglasville, Georgia. The set included a working kitchen with period-accurate kitchen equipment that was brought up to current code. Rearrangement of exterior features such as parking lot striping allowed that set to serve as each franchise location portrayed in the film. Both interior and exterior portions of the restaurant were modular, allowing countertops or entire wall sized glass panes to be removed to make room for cameras and other equipment.


title: The Founder

text: The J. Mack Robinson College of Business Administration Building in downtown Atlanta, which houses a Bank of America branch, served as the Illinois First Federal Savings & Loan association building. Some interior sets were built on soundstages at EUE Screen Gems Studios in Atlanta. Atlanta's East Lake Golf Club served as Rolling Green Country Club in the film.


title: The Founder

text: On March 2, 2015, The Weinstein Company paid $7 million for the film's distribution rights. On March 26, 2015, the studio set the film for a November 25, 2016 release date. In March 2016, the film was moved up to August 5, 2016. On July 13, 2016, the film's release date was delayed until a limited December 16, 2016 date, followed by a wide release on January 20, 2017. The film ultimately opened in the United States at Arclight Hollywood on December 7, 2016, in order to qualify for the 2017 Oscars.


title: The Founder

text: In February 2017, FilmNation Entertainment, one of the film's production companies, sued The Weinstein Company for $15 million. The Weinstein Company released "Gold" on January 27, 2017, a week after "The Founder", which FilmNation claimed was a breach of contract, saying the two companies had an agreement that no Weinstein Company film would be released within a week before or after "The Founder".


title: The Founder

text: "The Founder" grossed $12.8 million in the United-States and Canada and $11.3 million in others territories, for a worldwide total of $24.1 million.


title: The Founder

text: In North America, the film was expected to gross $3 million from 1,115 theaters in its opening weekend. It ended up earning $3.8 million, finishing 9th at the box office. In its second week, the film made $2.6 million, a drop of 23.4%.


title: The Founder

text: On review aggregator Rotten Tomatoes, the film has an approval rating of 81% based on 243 reviews, with a weighted average of 6.90/10. The website's critical consensus reads, ""The Founder" puts Michael Keaton's magnetic performance at the center of a smart, satisfying biopic that traces the rise of one of America's most influential businessmen – and the birth of one of its most far-reaching industries." On Metacritic, the film holds a weighted average score 66 out of 100, based on 47 critics, indicating "generally favorable reviews". Audiences polled by CinemaScore gave the film an average grade of "B+" on an A+ to F scale.


title: The Founder

text: "Rolling Stone" magazine's Peter Travers gave the film three out of four stars, stating that director Hancock and screenwriter Siegel did "strive hard – and mostly succeed – at keeping Hollywood sentiment out of the storytelling... Set more than a half century ago, "The Founder" proves to be a movie for a divisive here and now. Step right up. You might just learn something." RogerEbert.com's Matt Zoller Seitz gave the film three out of four stars, stating that despite the film over-relying on exposition and failing to skillfully incorporate Ray Kroc's personal life into the narrative, "I'd be lying if I said I hadn't thought about "The Founder" constantly since seeing it... It's an ad that becomes a warning before circling around and becoming another, darker kind of advertisement, and one of the most intriguing and surprising things about "The Founder" is that, in the end, it seems vaguely ashamed of itself for letting this happen".


title: Siouxsie Sioux

text: Susan Janet Ballion (born 27 May 1957), known professionally as Siouxsie Sioux, is an English singer, songwriter, musician and record producer. She was the lead singer of the rock band Siouxsie and the Banshees (1976–1996). They released 11 studio albums, and had several UK Top 20 singles including "Hong Kong Garden", "Happy House" and "Peek-a-Boo", plus a US "Billboard" Top 25 hit, "Kiss Them for Me".


title: Siouxsie Sioux

text: Siouxsie also formed a second group, the Creatures (1981–2005). With the Creatures she recorded four studio albums and the hit single "Right Now". After disbanding the Creatures in the mid-2000s, she has continued as a solo artist, using the name Siouxsie, and released the album "Mantaray" to critical acclaim in 2007.


title: Siouxsie Sioux

text: AllMusic named Siouxsie as "one of the most influential British singers of the rock era". Her songs have been covered by Jeff Buckley, Tricky and LCD Soundsystem. Her records have also been praised by artists such as PJ Harvey and TV on the Radio. In 2011, she was awarded for Outstanding Contribution to Music at the Q Awards and in 2012, at the Ivor Novello Awards, she received the Inspiration Award.


title: Siouxsie Sioux

text: Siouxsie was born Susan Janet Ballion on 27 May 1957 at Guy's Hospital in Southwark, England. She is ten years younger than her two siblings. Her brother and sister were born while the family was in the Belgian Congo. Her parents met in that colony and worked there for a few years. Her mother, Betty, was of Scottish and English descent and was a secretary who spoke both French and English. Her father was a bacteriologist who milked venom from snakes, and came from Wallonia, the French-speaking part of Belgium. In the mid-1950s, before Siouxsie's birth, the family moved to England.


title: Siouxsie Sioux

text: The Ballions lived in a suburban district in Chislehurst, Kent. Siouxsie was an isolated child, being unable to invite friends to her house because of her alcoholic, unemployed father. Despite his issues, Siouxsie regarded him as intelligent and well-read, and sympathised with his inability to fit in with a "rigid, middle-class society". During moments of sobriety, her father shared with her his love for books. Siouxsie was aware that her family was different; the Ballions were not involved in the local community and Siouxsie, aware that her family's house differed from the neighbours', would later state that "the suburbs inspired intense hatred."


title: Siouxsie Sioux

text: At the age of nine, she and a friend were sexually assaulted by a stranger. The assault was ignored by both her parents and the police, and was not spoken of in the family. The incident and its later treatment would lead Siouxsie to distrust adults. Years later, she stated: "I grew up having no faith in adults as responsible people. And being the youngest in the family I was isolated – I had no one to confide in. So I invented my own world, my own reality. It was my own way of defending myself – protecting myself from the outside world. The only way I could deal with how to survive was to get some strong armour."


title: Siouxsie Sioux

text: Her father died of alcoholism-related illness when Siouxsie was 14 years old, resulting in a decline in her health. Siouxsie lost a great deal of weight and failed to attend school. After several misdiagnoses, she was operated on and survived a bout of ulcerative colitis. During the weeks of recovery in mid-1972, she watched television in the hospital and saw David Bowie on "Top of the Pops".


title: Siouxsie Sioux

text: At 17, she left school. During this period she began visiting the local gay discos frequented by her sister's friends. She later introduced her own friends to that scene. In November 1975, the Sex Pistols performed at the local art college in Chislehurst. Siouxsie did not attend, but one of her friends told her how they sounded like the Stooges, and that singer Johnny Rotten had threatened students attending the gig. In February 1976, Siouxsie and her friend Steven Severin went to see the Sex Pistols play in London. After chatting with members of the band, Siouxsie and Severin decided to follow them regularly. In the following months, journalist Caroline Coon coined the term "Bromley Contingent" to describe this group of eccentric teenagers devoted to the Sex Pistols.


title: Siouxsie Sioux

text: Siouxsie became well known in the London club scene for her glam, fetish- and bondage-inspired attire, which later became part of punk fashion. She would also heavily influence the later development of gothic fashion with her signature cat-eye makeup, deep red lipstick, spiky dyed-black hair, and black clothing. In early September 1976, the Bromley Contingent followed the Sex Pistols to France, where Siouxsie was beaten up for wearing a cupless bra and a black armband with a swastika on it. She claimed her intent was to shock the bourgeoisie, not to make a political statement. She later wrote the song "Metal Postcard (Mittageisen)" (in memory of the anti-Nazi artist John Heartfield).


title: Siouxsie Sioux

text: Following the DIY ethos and the idea that the people in the audience could be the people on stage, Siouxsie and Severin decided to form a band. When a support slot at the 100 Club Punk Festival (organised by Malcolm McLaren) opened up, they decided to make an attempt at performing, although at that time they did not know how to play any songs. On 20 September 1976, the band improvised 20 minutes of music while Siouxsie sang the "Lord's Prayer".


title: Siouxsie Sioux

text: For critic Jon Savage, Siouxsie was "unlike any female singer before or since, commanding yet aloof, entirely modern". Viv Albertine from the Slits said: "Siouxsie just appeared fully made, fully in control, utterly confident. It totally blew me away. There she was doing something that I dared to dream but she took it and did it and it wiped the rest of the festival for me, that was it. I can't even remember everything else about it except that one performance."


title: Siouxsie Sioux

text: One of Siouxsie's first public appearances was with the Sex Pistols on Bill Grundy's television show, on Thames Television in December 1976. Standing next to the band, Siouxsie made fun of the presenter when he asked her how she was doing. She responded: "I've always wanted to meet you, Bill." Grundy, who claimed he was drunk, suggested a meeting after the show, which provoked guitarist Steve Jones to respond with a series of expletives, never heard before on early-evening television. This episode created a media furore on the front covers of several tabloids, including the "Daily Mirror", which published the headline "Siouxsie's a Punk Shocker". The event had a major impact on the Sex Pistols' subsequent career, and they became a household name overnight.


title: Siouxsie Sioux

text: Aware of the press surrounding both herself and the Sex Pistols, Siouxsie began to distance herself from the scene and stopped seeing Sex Pistols after the 15 December 1976 gig at Notre Dame Hall. From then, she focused her energy on her own band, Siouxsie and the Banshees.


title: Siouxsie Sioux

text: In February 1977, Siouxsie began touring in England as Siouxsie and the Banshees. One year later, their first single, "Hong Kong Garden", reached number 7 in the UK Singles Chart; With its oriental-inflected xylophone motif, "Melody Maker" deemed it "a glorious debut [...] All the elements come together with remarkable effect. The song is strident and powerful with tantalising oriental guitar riffs plus words and vocals that are the result of anger, disdain and isolation. No-one will be singled out because everyone is part and parcel of the whole. It might even be a hit".


title: Siouxsie Sioux

text: Their debut album, "The Scream", was one of the first post-punk records released. It received 5-star reviews in "Sounds" and "Record Mirror". The latter said that the record "points to the future, real music for the new age". The music was different from the single; it was angular, dark and jagged. "The Scream" was later hailed by "NME" as one of the best debut albums of all time along with Patti Smith's "Horses". "Join Hands" followed in 1979 with war as the lyrical theme.


title: Siouxsie Sioux

text: The 1980 album "Kaleidoscope" marked a change in musical direction with the arrival of John McGeoch, considered "one of the most innovative and influential guitarists" by "The Guardian", and drummer Budgie, the latter of whom would continue to perform and record with Siouxsie throughout her career until 2004. The hit single "Happy House" was qualified as "great Pop" with "liquid guitar" and other songs like "Red Light" were layered with electronic sounds. "Kaleidoscope" widened Siouxsie's audience, reaching the top 5 in the UK charts. "Juju" followed in 1981, reaching number 7; the singles "Spellbound" and "Arabian Knights" were described as "pop marvels" by "The Guardian". During recording sessions for "Juju", Siouxsie and Budgie formed a percussion-oriented duo called the Creatures, characterized by a stripped-down sound focused on vocals and drums; their first record, the EP "Wild Things", was a commercial success.


title: Siouxsie Sioux

text: In 1982, the Siouxsie and the Banshees' album "A Kiss in the Dreamhouse" was widely acclaimed by critics. Richard Cook of "NME" depicted it as "a feat of imagination scarcely ever recorded". The single "Slowdive" was "a violin-colored dance beat number". They included strings for the first time on several songs. However, the recording sessions took their toll, and McGeoch was forced to quit the band.


title: Siouxsie Sioux

text: In 1983, Siouxsie went to Hawaii to record the Creatures' first album, "Feast", which included the hit single "Miss the Girl". It was her first incursion into exotica, incorporating sounds of waves, local Hawaiian choirs and local percussions. Later that year, Siouxsie and Budgie released "Right Now", a song from Mel Tormé's repertoire that the Creatures re-orchestrated with brass arrangements; "Right Now" soon became a top 20 hit single in the UK. Then, with the Banshees (including guitarist Robert Smith of the Cure), she covered the Beatles' "Dear Prudence", which reached number 3 on the UK Singles Chart. Two albums followed with Smith: "Nocturne", recorded live in London in 1983, and 1984's "Hyæna". In 1985, the single "Cities in Dust" was recorded with sequencers; it climbed to number 21 in the UK charts. "Entertainment Weekly" noted that it was the first of a handful of Alternative rock radio hits in the U.S. 1986's "Tinderbox" and the 1987 covers album "Through the Looking Glass" both reached the top 15 in the UK.


title: Siouxsie Sioux

text: In 1988, the single "Peek-a-Boo" marked a musical departure from her previous work, anticipating hip hop-inspired rock with the use of samples. The song was praised by "NME" as "oriental marching band hip hop with farting horns and catchy accordion" and hailed by "Melody Maker" as "a brightly unexpected mixture of black steel and pop disturbance". The "Peepshow" album received a five star review in "Q" magazine. The ballad "The Last Beat of My Heart" issued as a single, saw her exploring new ground with accordion and strings.


title: Siouxsie Sioux

text: Siouxsie and Budgie then went to Andalusia in Spain to record the second Creatures album, "Boomerang". The songs took a different direction from previous Creatures works, with backing music ranging from flamenco to jazz and blues styles. It featured brass on most of the songs. The first single was "Standing There". "NME" hailed "Boomerang" as "a rich and unsettling landscape of exotica". Anton Corbijn visited the group during the recording near Jerez de la Frontera, and Siouxsie convinced him to take photographs in color, unlike his prior work which was in black-and-white: the photos used for the promotion showed Siouxsie and Budgie in fields surrounded with sunflowers. In 1990, she toured for the first time with the Creatures, in Europe and North America.


title: Siouxsie Sioux

text: On 1991's dance-oriented "Kiss Them for Me" single, Siouxsie and the Banshees used South Asian instrumentation, which had become popular in the UK club scene with the growth of bhangra. Indian tabla player Talvin Singh (who was later Björk's percussionist on her 1993 "Debut" album) took part in the session and provided vocals for the bridge. With "Kiss Them for Me", the Banshees scored a hit on the US "Billboard" Hot 100 peaking at number 23. After the release of "Superstition" which received enthusiastic reviews, the group co-headlined the first Lollapalooza tour, further increasing their American following.


title: Siouxsie Sioux

text: In 1992, film director Tim Burton requested that she write a song for "Batman Returns", and the Banshees composed the single "Face to Face".


title: Siouxsie Sioux

text: In the mid-1990s, Siouxsie started to do one-off collaborations with other artists. Suede invited her to a benefit concert for the Red Hot Organization. With guitarist Bernard Butler, she performed a version of Lou Reed's "Caroline Says". "Spin" reviewed it as "haughty and stately". Morrissey, ex-lead singer of the Smiths, recorded a duet with Siouxsie in 1994. They both sang on the single "Interlude", a track that was initially performed by Timi Yuro, a female torch singer of the 1960s. "Interlude" was released under the name of "Morrissey and Siouxsie".


title: Siouxsie Sioux

text: The last Banshees studio album, "The Rapture", was released in 1995; it was written partly in the Toulouse area of France, where she had recently moved. After the accompanying tour, the Banshees announced their split during a press conference called "20 Minutes into 20 Years". The Creatures de facto became her only band. At the same time, she released the song "The Lighthouse" on French producer Hector Zazou's album "Chansons des mers froides" (which translates to "Songs from the Cold Seas"), with jazz trumpetist Mark Isham. Siouxsie and Zazou adapted the poem "Flannan Isle" by English poet Wilfred Wilson Gibson.


title: Siouxsie Sioux

text: Her first live performance in three years was in February 1998 when former Velvet Underground member John Cale invited her to a festival called "With a Little Help From My Friends" at the Paradiso in Amsterdam. The concert was shown on Dutch national television and featured an unreleased Creatures composition, "Murdering Mouth", sung as a duet with Cale. The collaboration between the two artists worked so well that they decided to tour the US from June until August, performing "Murdering Mouth", and Cale's "Gun" together as the encores of a Creatures and Cale double bill.


title: Siouxsie Sioux

text: The following year, Siouxsie and Budgie released "Anima Animus", the first Creatures album since the split of the Banshees. It included the singles "2nd Floor" and "Prettiest Thing". The material diverged from their former work, with a more urban sound blending art rock and electronica. "Anima Animus" was described by "The Times" as "hypnotic and inventive". Also in 1999, Siouxsie collaborated with Marc Almond on the track "Threat of Love".


title: Siouxsie Sioux

text: In 2002, she was rated as one of the 10 best female rock artists by "Q". That same year, Universal released "The Best of Siouxsie and the Banshees" as the first reissue of her back catalogue.


title: Siouxsie Sioux

text: In 2003, Siouxsie and Budgie released the last Creatures album, "Hái!", which was in part recorded in Japan, collaborating with taiko player Leonard Eto (previously of the Kodo Drummers). Peter Wratts wrote in "Time Out": "Her voice is the dominant instrument here, snaking and curling around the bouncing drumming backdrop, elegiac and inhuman as she chants, purrs and whispers her way around the album". He termed the record a "spine-tingling achievement". "Hái!" was preceded by the single "Godzilla!". That year, Siouxsie was featured on the track "Cish Cash" by Basement Jaxx, from their album "Kish Kash", which later won Best Electronic/Dance Album at the "Grammy Awards".


title: Siouxsie Sioux

text: 2004 was a pivotal year for the singer. She toured for the first time as a solo act combining Banshees and Creatures songs. A live DVD called "Dreamshow" was recorded at the last London concert, in which she and her musicians were accompanied by a 16-piece orchestra, the Millennia Ensemble. Released in August 2005, this DVD reached the number 1 position in the UK music DVD charts.


title: Siouxsie Sioux

text: Her first solo album, "Mantaray", was released in September 2007. Pitchfork wrote, "She really "is" pop", before finishing the review by declaring, "It's a success". "Mojo" stated: "a thirst for sonic adventure radiates from each track". "Mantaray" included three singles: "Into a Swan", "Here Comes That Day" and "About to Happen". In 2008, Siouxsie performed vocals for the track "Careless Love" on "The Edge of Love" soundtrack by composer Angelo Badalamenti, a frequent collaborator with director David Lynch. She performed another Badalamenti number, "Who Will Take My Dreams Away", at the annual edition of the World Soundtrack Awards. After a year of touring, the singer played the last show of her tour in London in September 2008. A live DVD of this performance, "", was released in 2009.


title: Siouxsie Sioux

text: In June 2013, after a hiatus of five years, Siouxsie played two nights at the Royal Festival Hall in London during Yoko Ono's Meltdown festival. She performed 1980's "Kaleidoscope" album live in its entirety, along with other works from her back catalogue, and her performance was praised by the press. She also appeared at Ono's "Double Fantasy" concert, to sing the final song, "Walking on Thin Ice".


title: Siouxsie Sioux

text: In October 2014, she and fellow Banshee Steven Severin compiled a CD titled "It's a Wonderfull Life" for the November 2014 issue of "Mojo" magazine, in which she appeared on the cover. The disc included 15 tracks that inspired the Banshees.


title: Siouxsie Sioux

text: "Love Crime", her first song in eight years, was featured in the finale of the TV series "Hannibal", broadcast in August 2015. The song was then included on both CD and vinyl editions of "Hannibal Season 3 – Volume 2" and was digitally released as download only.


title: Siouxsie Sioux

text: On 15 December 2022, it was announced that Siouxsie will be one of the headliners of the Latitude festival 2023 in Suffolk: she will be headlining the BBC Stage on Sunday 23 July 2023. It will be her first live performance in a decade.


title: Siouxsie Sioux

text: Journalist Paul Morley noted that Siouxsie's songs topics dealt with "mental illness, medical terrors, surreal diseases, depraved urges, sinister intensity, unearthly energy, sexual abuse, childhood disturbances, sordid mysteries, unbearable nervous anxiety, fairytale fears, urban discontent and the bleak dignity of solitude". Many of her songs are about damage; her childhood marked her profoundly. She said, "Damaged lives, damaged souls, damaged relationships. Most of the damage I sing about first happened when I was younger and I am still feeding off it and working it out. Early experiences are what create a lifetime of damage. The songs you write can help you fix the damage. And just the environment you are in is so important and can waste potential and corrupt something. For me, there was neglect. An alcoholic father who is not there because the most important thing for him is just to get alcohol and your mother is trying to compensate for the non-existent second parent so she's never there because she's working all the time and when she is around she's stressed out. Being isolated and not having anyone to connect with, there was just no physical touching back then".


title: Siouxsie Sioux

text: Siouxsie has been praised by artists of many genres. She had a strong impact on two trip-hop acts. Tricky covered 1983's proto trip-hop "Tattoo", to open his second album "Nearly God", and Massive Attack sampled "Metal Postcard (Mittageisen)" on their song "Superpredators (Metal Postcard)" for the soundtrack to the film "The Jackal."


title: Siouxsie Sioux

text: Other acts have covered Siouxsie's songs. Jeff Buckley performed "Killing Time" several times, he first recorded it during a radio session for WFMU in 1992. LCD Soundsystem recorded a cover of "Slowdive" for the B-side of "Disco Infiltrator", their version was also released on "Introns". Santigold based her track "My Superman", on the music of "Red Light". In 2003, the Beta Band sampled "Painted Bird" and changed the title to "Liquid Bird" on their "Heroes to Zeros" album. Red Hot Chili Peppers performed "Christine" at the V2001 festival and introduced it to their British audience as "your national anthem". "Christine" was also revisited by Simple Minds. Indie folk group DeVotchKa covered "The Last Beat of My Heart" at the suggestion of Arcade Fire singer Win Butler in 2007. The Weeknd sampled "Happy House" on the title track of his first mixtape "House of Balloons" in 2011.


title: Siouxsie Sioux

text: Morrissey said that "Siouxsie and the Banshees were excellent. They were one of the great groups of the late 70s, early 80s". In 1994, discussing modern bands, he also stated: "None of them are as good as Siouxsie and the Banshees at full pelt. That's not dusty nostalgia, that's fact". Another member of the Smiths, Johnny Marr, said: "Really my generation was all about a guy called John McGeoch, from Siouxsie and the Banshees. He was a great player". Marr hailed McGeoch for his work on Siouxsie's single "Spellbound". Marr qualified it as "clever" with "really good picky thing going on which is very un-rock'n'roll". Radiohead also cited McGeoch-era Siouxsie records when mentioning the recording of "There There". Their singer, Thom Yorke, said: "The band that really changed my life was R.E.M. and Siouxsie and the Banshees ...". "My favourite show I ever saw then was Siouxsie and she was absolutely amazing. ... She's totally in command of the whole audience". Yorke added that she "made an especially big impression in concert, she was really sexy but absolutely terrifying." Sonic Youth singer and guitarist Thurston Moore named "Hong Kong Garden" as one of his 25 all-time favourite songs.


title: Siouxsie Sioux

text: Siouxsie has influenced other bands ranging from contemporaries Joy Division, U2, and the Cure, to later acts like the Jesus and Mary Chain, Jane's Addiction and TV on the Radio. Joy Division co-founder Peter Hook said that "The Scream" inspired them for the "really unusual way of playing" of the guitarist and the drummer and cited the Banshees as "one of our big influences". U2 frontman Bono named her as an influence in the band's 2006 autobiography "U2 by U2". He was inspired by her way of singing. "I still think that I sing like Siouxsie from The Banshees on the first two U2 albums". With his band, he selected "Christine" for a compilation made for "Mojo"<nowiki>'s readers.</nowiki> U2 guitarist the Edge also was the presenter of an award given to Siouxsie at a "Mojo" ceremony in 2005. The Cure's Robert Smith related what the "Join Hands" tour brought him musically: "When we supported The Banshees in 1979, we suddenly became aware of how limited our palette was. I felt constrained, so when the opportunity arose to play with them I jumped at it and juggled the two bands for a while. It taught me a lot – they had fantastic rhythm sections and this made me think, ‘Why can’t I have this?’." For Smith's record "The Head on the Door" in 1985, he stated: "It reminds me of the "Kaleidoscope" album, the idea of having lots of different sounding things, different colours". Dave Navarro of Jane's Addiction once made a parallel between his band and the Banshees: "There are so many similar threads: melody, use of sound, attitude, sex appeal. I always saw Jane's Addiction as the masculine Siouxsie and the Banshees". Dave Sitek of TV on the Radio hailed the poppiest Siouxsie songs, citing their arrangements: "I've always tried to make a song that begins like "Kiss Them for Me". I think songs like "I Was a Lover" or "Wash the Day Away" came from that element of surprise mode where all of a sudden this giant drum comes in and you're like, what the fuck?! That record was the first one where I was like, okay, even my friends're going to fall for this. I feel like that transition into that record was a relief for me. Really beautiful music was always considered too weird by the normal kids and that was the first example where I thought, we've got them, they're hooked! I watched people dance to that song, people who had never heard of any of the music that I listened to, they heard that music in a club and went crazy". Dave Gahan of Depeche Mode said about her: "She always sounds exciting. She sings with a lot of sex–that's what I like". Mark Lanegan stated that he would like to collaborate with her: "In my wildest dreams I would love to sing with Siouxsie". She is also revered by Damon Albarn and Dave Grohl. Omar Rodriguez-Lopez of the Mars Volta mentioned his liking for "the very textural side of Siouxsie".


title: Siouxsie Sioux

text: Siouxsie has been hailed by many female singers. When asked if there was any figure who connected with her when she was just a listener, PJ Harvey replied: "It's hard to beat Siouxsie Sioux, in terms of live performance. She is so exciting to watch, so full of energy and human raw quality". Harvey also selected in her top 10 favourite albums of 1999, the "Anima Animus" album by Siouxsie's second band the Creatures. Sinéad O'Connor said that when she started, Siouxsie was one of her main influences. Tracey Thorn of Everything But the Girl wrote in her autobiography that Siouxsie was one of her heroines. Thorn paid homage to Siouxsie in the lyrics of her 2007 song "Hands Up to the Ceiling". Elizabeth Fraser of Cocteau Twins used to have a Siouxsie tattoo on her arm, and mentioned her liking for "Metal Postcard" to the members of Massive Attack in 1998. Sharleen Spiteri of Texas grew up listening to tracks such as "Hong Kong Garden" and was hooked by the Asian vibe present in the song; she stated that Texas' single "In Our Lifetime" was "our tribute to" "Hong Kong Garden". Garbage singer Shirley Manson cited her as an influence: "I learned how to sing listening to "The Scream" and "Kaleidoscope"". Manson also declared that Siouxsie embodied everything she wanted to be as a young woman. Manson would later write the foreword to "Siouxsie & The Banshees: The Authorised Biography". Beth Ditto, of Gossip cited her as one of their influences for their 2009 album "Music for Men". Ana Matronic of Scissor Sisters named Siouxsie as a source of inspiration and the Banshees as her favourite band. Siouxsie was also hailed by Romy Madley Croft of the XX, Kim Deal of the Pixies and the Breeders, and also by Josephine Wiggs of the Breeders, and namechecked by Karin Dreijer Andersson of the Knife. Kate Jackson of the Long Blondes said that Siouxsie was a part of her musical background, thanks to her "sharp lyrics" on Creatures' tracks like "So Unreal". Rachel Goswell mentioned her as a major influence: "From a singing point of view, I was inspired by Siouxsie Sioux, who I just adored. She's amazing. I've never seen anyone else quite like her"; her band Slowdive was named on a suggestion of Goswell, inspired by the Banshees' single of the same name. Lush were initially named "the Baby Machines", which the band culled from the lyrics of "Arabian Knights". Courtney Love of Hole wrote Siouxsie in her favourite records' list when she was a teenager in her diary. While talking about the band Savages, Love also stated: "They are amazing. ... It's kind of very Siouxsie Sioux". Kim Gordon of Sonic Youth said: "Initially I was really inspired by [...] Siouxsie, Patti Smith". FKA twigs named her as a main influence: "Every bit of music that I made sounded like a pastiche of Siouxsie [...] but through that I discovered myself". Róisín Murphy named Siouxsie when asked who were her biggest influences.


title: Siouxsie Sioux

text: Other female artists who have stated their admiration for Siouxsie's work include Charli XCX, Hayley Williams of Paramore, Dolores O'Riordan of the Cranberries, Jennifer Charles of Elysian Fields, Ebony Bones, Toni Halliday of Curve, Kathleen Hanna of Bikini Kill, Chan Marshall a.k.a. Cat Power, Gillian Gilbert of New Order, Alison Goldfrapp, Sarah Cracknell of Saint Etienne, Florence Welch of Florence + the Machine, Chelsea Wolfe, Brody Dalle of the Distillers, Dee Dee of Dum Dum Girls, Joan As Police Woman, Lou Doillon, Emel Mathlouthi, Girlpool, Liz Phair, Billie Ray Martin, An Pierlé, Lauren Mayberry of Chvrches, Meshell Ndegeocello, St. Vincent, Anohni, Jehnny Beth of Savages, Wolf Alice, and Jenny Lee Lindberg of Warpaint. Santigold said: "I keep a Rolodex of the women that vocally inspire me. There aren't that many, but she's definitely one of them. I remember one of the first times I heard "Red Light" it was at a party, and I remember going up to the DJ and being like, "Who's this?". It was that good. I kind of stopped and was like ... wow. There's not a tremendous amount of women who are bold and forward thinking as artists. I feel like her music, at the time especially, was pretty unique in the way that it sort of matched her style. The freedom of experimenting with this dark place that doesn't have a place often in modern music".


title: Siouxsie Sioux

text: In an interview with "The Sunday Times" in August 2007, she announced that she and Budgie had divorced. In an interview with "The Independent", she said: "I've never particularly said I'm hetero or I'm a lesbian. I know there are people who are definitely one way, but not really me. I suppose if I am attracted to men then they usually have more feminine qualities."


title: Complex number

text: In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation formula_1; every complex number can be expressed in the form formula_2, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number formula_3, is called the ', and is called the '. The set of complex numbers is denoted by either of the symbols formula_4 or . Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.


title: Complex number

text: Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation


title: Complex number

text: has no real solution, since the square of a real number cannot be negative, but has the two nonreal complex solutions formula_6 and formula_7.


title: Complex number

text: Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule formula_8 combined with the associative, commutative, and distributive laws. Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field that has the real numbers as a subfield. The complex numbers also form a real vector space of dimension two, with as a standard basis.


title: Complex number

text: This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely expressing in terms of complex numbers some geometric properties and constructions. For example, the real numbers form the real line which is identified to the horizontal axis of the complex plane. The complex numbers of absolute value one form the unit circle. The addition of a complex number is a translation in the complex plane, and the multiplication by a complex number is a similarity centered at the origin. The complex conjugation is the reflection symmetry with respect to the real axis. The complex absolute value is a Euclidean norm.


title: Complex number

text: In summary, the complex numbers form a rich structure that is simultaneously an algebraically closed field, a commutative algebra over the reals, and a Euclidean vector space of dimension two.


title: Complex number

text: A complex number is a number of the form , where and are real numbers, and is an indeterminate satisfying . For example, is a complex number.


title: Complex number

text: This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate , for which the relation is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials. The relation induces the equalities and which hold for all integers ; these allow the reduction of any polynomial that results from the addition and multiplication of complex numbers to a linear polynomial in , again of the form with real coefficients


title: Complex number

text: The real number is called the "real part" of the complex number ; the real number is called its "imaginary part". To emphasize, the imaginary part does not include a factor ; that is, the imaginary part is , not .


title: Complex number

text: Formally, the complex numbers are defined as the quotient ring of the polynomial ring in the indeterminate , by the ideal generated by the polynomial (see below).


title: Complex number

text: A real number can be regarded as a complex number , whose imaginary part is 0. A purely imaginary number is a complex number , whose real part is zero. As with polynomials, it is common to write for and for . Moreover, when the imaginary part is negative, that is, , it is common to write instead of ; for example, for , can be written instead of .


title: Complex number

text: Since the multiplication of the indeterminate and a real is commutative in polynomials with real coefficients, the polynomial may be written as This is often expedient for imaginary parts denoted by expressions, for example, when is a radical.


title: Complex number

text: The real part of a complex number is denoted by , formula_9, or formula_10; the imaginary part of a complex number is denoted by , formula_11, or formula_12 For example,


title: Complex number

text: In some disciplines, particularly in electromagnetism and electrical engineering, is used instead of as is frequently used to represent electric current. In these cases, complex numbers are written as , or .


title: Complex number

text: A complex number can thus be identified with an ordered pair formula_15 of real numbers, which in turn may be interpreted as coordinates of a point in a two-dimensional space. The most immediate space is the Euclidean plane with suitable coordinates, which is then called "complex plane" or "Argand diagram," named after Jean-Robert Argand. Another prominent space on which the coordinates may be projected is the two-dimensional surface of a sphere, which is then called Riemann sphere.


title: Complex number

text: The definition of the complex numbers involving two arbitrary real values immediately suggests the use of Cartesian coordinates in the complex plane. The horizontal ("real") axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical ("imaginary") axis, with increasing values upwards.


title: Complex number

text: A charted number may be viewed either as the point or as a position vector from the origin to this point. The coordinate values of a complex number can hence be expressed in its "Cartesian", "rectangular", or "algebraic" form.


title: Complex number

text: Notably, the operations of addition and multiplication take on a very natural geometric character, when complex numbers are viewed as position vectors: addition corresponds to vector addition, while multiplication (see below) corresponds to multiplying their magnitudes and adding the angles they make with the real axis. Viewed in this way, the multiplication of a complex number by corresponds to rotating the position vector counterclockwise by a quarter turn (90°) about the origin—a fact which can be expressed algebraically as follows:


title: Complex number

text: An alternative option for coordinates in the complex plane is the polar coordinate system that uses the distance of the point from the origin (), and the angle subtended between the positive real axis and the line segment in a counterclockwise sense. This leads to the polar form


title: Complex number

text: If is a real number (that is, if ), then . That is, the absolute value of a real number equals its absolute value as a complex number.


title: Complex number

text: By Pythagoras' theorem, the absolute value of a complex number is the distance to the origin of the point representing the complex number in the complex plane.


title: Complex number

text: The "argument" of (in many applications referred to as the "phase" ) is the angle of the radius with the positive real axis, and is written as . As with the modulus, the argument can be found from the rectangular form —by applying the inverse tangent to the quotient of imaginary-by-real parts. By using a half-angle identity, a single branch of the arctan suffices to cover the range of the -function, and avoids a more subtle case-by-case analysis


title: Complex number

text: Normally, as given above, the principal value in the interval is chosen. If the arg value is negative, values in the range or can be obtained by adding . The value of is expressed in radians in this article. It can increase by any integer multiple of and still give the same angle, viewed as subtended by the rays of the positive real axis and from the origin through . Hence, the arg function is sometimes considered as multivalued. The polar angle for the complex number 0 is indeterminate, but arbitrary choice of the polar angle 0 is common.


title: Complex number

text: Together, and give another way of representing complex numbers, the "polar form", as the combination of modulus and argument fully specify the position of a point on the plane. Recovering the original rectangular co-ordinates from the polar form is done by the formula called "trigonometric form"


title: Complex number

text: In angle notation, often used in electronics to represent a phasor with amplitude and phase , it is written as


title: Complex number

text: When visualizing complex functions, both a complex input and output are needed. Because each complex number is represented in two dimensions, visually graphing a complex function would require the perception of a four dimensional space, which is possible only in projections. Because of this, other ways of visualizing complex functions have been designed.


title: Complex number

text: In domain coloring the output dimensions are represented by color and brightness, respectively. Each point in the complex plane as domain is "ornated", typically with "color" representing the argument of the complex number, and "brightness" representing the magnitude. Dark spots mark moduli near zero, brighter spots are farther away from the origin, the gradation may be discontinuous, but is assumed as monotonous. The colors often vary in steps of for to from red, yellow, green, cyan, blue, to magenta. These plots are called color wheel graphs. This provides a simple way to visualize the functions without losing information. The picture shows zeros for and poles at formula_26


title: Complex number

text: The solution in radicals (without trigonometric functions) of a general cubic equation, when all three of its roots are real numbers, contains the square roots of negative numbers, a situation that cannot be rectified by factoring aided by the rational root test, if the cubic is irreducible; this is the so-called "casus irreducibilis" ("irreducible case"). This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545 in his "Ars Magna", though his understanding was rudimentary; moreover he later dismissed complex numbers as "subtle as they are useless". Cardano did use imaginary numbers, but described using them as “mental torture.” This was prior to the use of the graphical complex plane. Cardano and other Italian mathematicians, notably Scipione del Ferro, in the 1500s created an algorithm for solving cubic equations which generally had one real solution and two solutions containing an imaginary number. Since they ignored the answers with the imaginary numbers, Cardano found them useless.


title: Complex number

text: Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field, where any polynomial equation has a root.


title: Complex number

text: Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician Rafael Bombelli. A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions.


title: Complex number

text: The earliest fleeting reference to square roots of negative numbers can perhaps be said to occur in the work of the Greek mathematician Hero of Alexandria in the 1st century AD, where in his "Stereometrica" he considered, apparently in error, the volume of an impossible frustum of a pyramid to arrive at the term formula_27 in his calculations, which today would simplify to formula_28. Negative quantities were not conceived of in Hellenistic mathematics and Hero merely replaced it by its positive formula_29


title: Complex number

text: The impetus to study complex numbers as a topic in itself first arose in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (see Niccolò Fontana Tartaglia, Gerolamo Cardano). It was soon realized (but proved much later) that these formulas, even if one was interested only in real solutions, sometimes required the manipulation of square roots of negative numbers. As an example, Tartaglia's formula for a cubic equation of the form + \sqrt[3]{q/2 - \sqrt{(q/2)^2-(p/3)^3}}</math>. When formula_30 is negative (casus irreducibilis), the second cube root should be regarded as the complex conjugate of the first one.}} gives the solution to the equation as


title: Complex number

text: At first glance this looks like nonsense. However, formal calculations with complex numbers show that the equation has three solutions: formula_32 Substituting these in turn for formula_33 in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of . Of course this particular equation can be solved at sight but it does illustrate that when general formulas are used to solve cubic equations with real roots then, as later mathematicians showed rigorously, the use of complex numbers is unavoidable. Rafael Bombelli was the first to address explicitly these seemingly paradoxical solutions of cubic equations and developed the rules for complex arithmetic trying to resolve these issues.


title: Complex number

text: The term "imaginary" for these quantities was coined by René Descartes in 1637, who was at pains to stress their unreal nature


title: Complex number

text: A further source of confusion was that the equation formula_34 seemed to be capriciously inconsistent with the algebraic identity formula_35, which is valid for non-negative real numbers and , and which was also used in complex number calculations with one of , positive and the other negative. The incorrect use of this identity (and the related identity formula_36) in the case when both and are negative even bedeviled Leonhard Euler. This difficulty eventually led to the convention of using the special symbol in place of formula_37 to guard against this mistake. Even so, Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, "Elements of Algebra", he introduces these numbers almost at once and then uses them in a natural way throughout.


title: Complex number

text: In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abraham de Moivre noted that the identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be re-expressed by the following de Moivre's formula:


title: Complex number

text: by formally manipulating complex power series and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities.


title: Complex number

text: The idea of a complex number as a point in the complex plane (above) was first described by Danish–Norwegian mathematician Caspar Wessel in 1799, although it had been anticipated as early as 1685 in Wallis's "A Treatise of Algebra".


title: Complex number

text: Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806 Jean-Robert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of the fundamental theorem of algebra. Carl Friedrich Gauss had earlier published an essentially topological proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1". It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane, largely establishing modern notation and terminology:


title: Complex number

text: If one formerly contemplated this subject from a false point of view and therefore found a mysterious darkness, this is in large part attributable to clumsy terminology. Had one not called +1, -1, formula_37 positive, negative, or imaginary (or even impossible) units, but instead, say, direct, inverse, or lateral units, then there could scarcely have been talk of such darkness.


title: Complex number

text: In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée, Mourey, Warren, Français and his brother, Bellavitis.


title: Complex number

text: The English mathematician G.H. Hardy remarked that Gauss was the first mathematician to use complex numbers in 'a really confident and scientific way' although mathematicians such as Norwegian Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.


title: Complex number

text: Augustin-Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of complex analysis to a high state of completion, commencing around 1825 in Cauchy's case.


title: Complex number

text: The common terms used in the theory are chiefly due to the founders. Argand called the "direction factor", and formula_41 the "modulus"; + \tfrac{b}{\sqrt{a^2 + b^2}} \sqrt{-1} </math>], whose module is unity [1], would represent its direction.]}} Cauchy (1821) called the "reduced form" (l'expression réduite) and apparently introduced the term "argument"; Gauss used for formula_37, introduced the term "complex number" for , and called the "norm". The expression "direction coefficient", often used for , is due to Hankel (1867), and "absolute value," for "modulus," is due to Weierstrass.


title: Complex number

text: Later classical writers on the general theory include Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others. Important work (including a systematization) in complex multivariate calculus has been started at beginning of the 20th century. Important results have been achieved by Wilhelm Wirtinger in 1927.


title: Complex number

text: Complex numbers have a similar definition of equality to real numbers; two complex numbers and are equal if and only if both their real and imaginary parts are equal, that is, if and . Nonzero complex numbers written in polar form are equal if and only if they have the same magnitude and their arguments differ by an integer multiple of .


title: Complex number

text: Unlike the real numbers, there is no natural ordering of the complex numbers. In particular, there is no linear ordering on the complex numbers that is compatible with addition and multiplication. Hence, the complex numbers do not have the structure of an ordered field. One explanation for this is that every non-trivial sum of squares in an ordered field is nonzero, and is a non-trivial sum of squares. Thus, complex numbers are naturally thought of as existing on a two-dimensional plane.


title: Complex number

text: The "complex conjugate" of the complex number is given by . It is denoted by either or . This unary operation on complex numbers cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division.


title: Complex number

text: Geometrically, is the "reflection" of about the real axis. Conjugating twice gives the original complex number


title: Complex number

text: which makes this operation an involution. The reflection leaves both the real part and the magnitude of unchanged, that is


title: Complex number

text: The product of a complex number and its conjugate is known as the "absolute square". It is always a non-negative real number and equals the square of the magnitude of each:


title: Complex number

text: This property can be used to convert a fraction with a complex denominator to an equivalent fraction with a real denominator by expanding both numerator and denominator of the fraction by the conjugate of the given denominator. This process is sometimes called "rationalization" of the denominator (although the denominator in the final expression might be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator.


title: Complex number

text: Conjugation is also employed in inversive geometry, a branch of geometry studying reflections more general than ones about a line. In the network analysis of electrical circuits, the complex conjugate is used in finding the equivalent impedance when the maximum power transfer theorem is looked for.


title: Complex number

text: Two complex numbers formula_50 and formula_51 are most easily added by separately adding their real and imaginary parts. That is to say:


title: Complex number

text: Multiplication of a complex number formula_50 and a real number can be done similarly by multiplying separately and the real and imaginary parts of :


title: Complex number

text: In particular, subtraction can be done by negating the subtrahend (that is multiplying it with ) and adding the result to the minuend:


title: Complex number

text: Using the visualization of complex numbers in the complex plane, addition has the following geometric interpretation: the sum of two complex numbers and , interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices , and the points of the arrows labeled and (provided that they are not on a line). Equivalently, calling these points , , respectively and the fourth point of the parallelogram the triangles and are congruent.


title: Complex number

text: The rules of the distributive property, the commutative properties (of addition and multiplication), and the defining property apply to complex numbers. It follows that


title: Complex number

text: This can be used to express a division of an arbitrary complex number by a non-zero complex number as


title: Complex number

text: Formulas for multiplication, division and exponentiation are simpler in polar form than the corresponding formulas in Cartesian coordinates. Given two complex numbers and , because of the trigonometric identities


title: Complex number

text: In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. For example, multiplying by corresponds to a quarter-turn counter-clockwise, which gives back . The picture at the right illustrates the multiplication of


title: Complex number

text: Since the real and imaginary part of are equal, the argument of that number is 45 degrees, or (in radian). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are arctan(1/3) and arctan(1/2), respectively. Thus, the formula


title: Complex number

text: holds. As the arctan function can be approximated highly efficiently, formulas like this – known as Machin-like formulas – are used for high-precision approximations of .


title: Complex number

text: where is the signum function. This can be seen by squaring formula_66 to obtain . Here formula_70 is called the modulus of , and the square root sign indicates the square root with non-negative real part, called the principal square root; also formula_71 where .


title: Complex number

text: Analytic continuation allows extending this equality for every complex value of , and thus to define the complex exponentiation with base as


title: Complex number

text: This can be proved either by comparing the power series expansion of both members or by applying analytic continuation from the restriction of the equation to real arguments.


title: Complex number

text: formula_80 of the exponential function. For extending this to the complex domain, one can start from Euler's formula. It implies that, if a complex number formula_81 is written in polar form


title: Complex number

text: However, because cosine and sine are periodic functions, the addition of an integer multiple of to does not change . For example, , so both and are possible values for the natural logarithm of .


title: Complex number

text: If formula_88 is not a non-positive real number (a positive or a non-real number), the resulting principal value of the complex logarithm is obtained with . It is an analytic function outside the negative real numbers, but it cannot be prolongated to a function that is continuous at any negative real number formula_89, where the principal value is .


title: Complex number

text: It seems natural to extend this formula to complex values of , but there are some difficulties resulting from the fact that the complex logarithm is not really a function, but a multivalued function.


title: Complex number

text: It follows that if is as above, and if is another complex number, then the "exponentiation" is the multivalued function


title: Complex number

text: If, in the preceding formula, is an integer, then the sine and the cosine are independent of . Thus, if the exponent is an integer, then is well defined, and the exponentiation formula simplifies to de Moivre's formula:


title: Complex number

text: for . (Here formula_94 is the usual (positive) th root of the positive real number .) Because sine and cosine are periodic, other integer values of do not give other values.


title: Complex number

text: While the th root of a positive real number is chosen to be the "positive" real number satisfying , there is no natural way of distinguishing one particular complex th root of a complex number. Therefore, the th root is a -valued function of . This implies that, contrary to the case of positive real numbers, one has


title: Complex number

text: The set formula_14 of complex numbers is a field. Briefly, this means that the following facts hold: first, any two complex numbers can be added and multiplied to yield another complex number. Second, for any complex number , its additive inverse is also a complex number; and third, every nonzero complex number has a reciprocal complex number. Moreover, these operations satisfy a number of laws, for example the law of commutativity of addition and multiplication for any two complex numbers and :


title: Complex number

text: These two laws and the other requirements on a field can be proven by the formulas given above, using the fact that the real numbers themselves form a field.


title: Complex number

text: Unlike the reals, formula_14 is not an ordered field, that is to say, it is not possible to define a relation that is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, so precludes the existence of an ordering on formula_99


title: Complex number

text: When the underlying field for a mathematical topic or construct is the field of complex numbers, the topic's name is usually modified to reflect that fact. For example: complex analysis, complex matrix, complex polynomial, and complex Lie algebra.


title: Complex number

text: has at least one complex solution "z", provided that at least one of the higher coefficients is nonzero. This is the statement of the "fundamental theorem of algebra", of Carl Friedrich Gauss and Jean le Rond d'Alembert. Because of this fact, formula_14 is called an algebraically closed field. This property does not hold for the field of rational numbers formula_102 (the polynomial does not have a rational root, since √2 is not a rational number) nor the real numbers formula_103 (the polynomial does not have a real root for , since the square of is positive for any real number ).


title: Complex number

text: There are various proofs of this theorem, by either analytic methods such as Liouville's theorem, or topological ones such as the winding number, or a proof combining Galois theory and the fact that any real polynomial of "odd" degree has at least one real root.


title: Complex number

text: Because of this fact, theorems that hold "for any algebraically closed field" apply to formula_99 For example, any non-empty complex square matrix has at least one (complex) eigenvalue.


title: Complex number

text: It can be shown that any field having these properties is isomorphic (as a field) to formula_99 For example, the algebraic closure of the field formula_109 of the -adic number also satisfies these three properties, so these two fields are isomorphic (as fields, but not as topological fields). Also, formula_14 is isomorphic to the field of complex Puiseux series. However, specifying an isomorphism requires the axiom of choice. Another consequence of this algebraic characterization is that formula_14 contains many proper subfields that are isomorphic to formula_14.


title: Complex number

text: The preceding characterization of formula_14 describes only the algebraic aspects of formula_99 That is to say, the properties of nearness and continuity, which matter in areas such as analysis and topology, are not dealt with. The following description of formula_14 as a topological field (that is, a field that is equipped with a topology, which allows the notion of convergence) does take into account the topological properties. formula_14 contains a subset (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions:


title: Complex number

text: Moreover, formula_14 has a nontrivial involutive automorphism (namely the complex conjugation), such that is in for any nonzero in formula_99


title: Complex number

text: Any field with these properties can be endowed with a topology by taking the sets as a base, where ranges over the field and ranges over . With this topology is isomorphic as a "topological" field to formula_99


title: Complex number

text: The only connected locally compact topological fields are formula_103 and formula_99 This gives another characterization of formula_14 as a topological field, since formula_14 can be distinguished from formula_103 because the nonzero complex numbers are connected, while the nonzero real numbers are not.


title: Complex number

text: William Rowan Hamilton introduced the approach to define the set formula_14 of complex numbers as the set formula_127 of of real numbers, in which the following rules for addition and multiplication are imposed:


title: Complex number

text: Though this low-level construction does accurately describe the structure of the complex numbers, the following equivalent definition reveals the algebraic nature of formula_14 more immediately. This characterization relies on the notion of fields and polynomials. A field is a set endowed with addition, subtraction, multiplication and division operations that behave as is familiar from, say, rational numbers. For example, the distributive law


title: Complex number

text: must hold for any three elements , and of a field. The set formula_103 of real numbers does form a field. A polynomial with real coefficients is an expression of the form


title: Complex number

text: where the are real numbers. The usual addition and multiplication of polynomials endows the set formula_133 of all such polynomials with a ring structure. This ring is called the polynomial ring over the real numbers.


title: Complex number

text: The set of complex numbers is defined as the quotient ring formula_134 This extension field contains two square roots of , namely (the cosets of) and , respectively. (The cosets of) and form a basis of formula_135 as a real vector space, which means that each element of the extension field can be uniquely written as a linear combination in these two elements. Equivalently, elements of the extension field can be written as ordered pairs of real numbers. The quotient ring is a field, because is irreducible over formula_136 so the ideal it generates is maximal.


title: Complex number

text: The formulas for addition and multiplication in the ring formula_137 modulo the relation , correspond to the formulas for addition and multiplication of complex numbers defined as ordered pairs. So the two definitions of the field formula_14 are isomorphic (as fields).


title: Complex number

text: Accepting that formula_14 is algebraically closed, since it is an algebraic extension of formula_140 in this approach, formula_14 is therefore the algebraic closure of formula_142


title: Complex number

text: Here the entries and are real numbers. As the sum and product of two such matrices is again of this form, these matrices form a subring of the ring matrices.


title: Complex number

text: is a ring isomorphism from the field of complex numbers to the ring of these matrices. This isomorphism associates the square of the absolute value of a complex number with the determinant of the corresponding matrix, and the conjugate of a complex number with the transpose of the matrix.


title: Complex number

text: The geometric description of the multiplication of complex numbers can also be expressed in terms of rotation matrices by using this correspondence between complex numbers and such matrices. The action of the matrix on a vector corresponds to the multiplication of by . In particular, if the determinant is , there is a real number such that the matrix has the form:


title: Complex number

text: In this case, the action of the matrix on vectors and the multiplication by the complex number formula_146 are both the rotation of the angle .


title: Complex number

text: 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 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.


title: Complex number

text: The notions of convergent series and continuous functions in (real) analysis have natural analogs in complex analysis. A sequence of complex numbers is said to converge if and only if its real and imaginary parts do. This is equivalent to the (ε, δ)-definition of limits, where the absolute value of real numbers is replaced by the one of complex numbers. From a more abstract point of view, formula_147, endowed with the metric


title: Complex number

text: Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the "exponential function" , also written , is defined as the infinite series


title: Complex number

text: The series defining the real trigonometric functions sine and cosine, as well as the hyperbolic functions sinh and cosh, also carry over to complex arguments without change. For the other trigonometric and hyperbolic functions, such as tangent, things are slightly more complicated, as the defining series do not converge for all complex values. Therefore, one must define them either in terms of sine, cosine and exponential, or, equivalently, by using the method of analytic continuation.


title: Complex number

text: for any complex number . It can be shown that any such solution – called complex logarithm of – satisfies


title: Complex number

text: where arg is the argument defined above, and ln the (real) natural logarithm. As arg is a multivalued function, unique only up to a multiple of , log is also multivalued. The principal value of log is often taken by restricting the imaginary part to the interval .


title: Complex number

text: and is multi-valued, except when is an integer. For , for some natural number , this recovers the non-uniqueness of th roots mentioned above.


title: Complex number

text: Complex numbers, unlike real numbers, do not in general satisfy the unmodified power and logarithm identities, particularly when naïvely treated as single-valued functions; see failure of power and logarithm identities. For example, they do not satisfy


title: Complex number

text: Both sides of the equation are multivalued by the definition of complex exponentiation given here, and the values on the left are a subset of those on the right.


title: Complex number

text: A function "f": formula_147 → formula_147 is called holomorphic if it satisfies the Cauchy–Riemann equations. For example, any formula_140-linear map formula_147 → formula_147 can be written in the form


title: Complex number

text: with complex coefficients and . This map is holomorphic if and only if . The second summand formula_163 is real-differentiable, but does not satisfy the Cauchy–Riemann equations.


title: Complex number

text: Complex analysis shows some features not apparent in real analysis. For example, any two holomorphic functions and that agree on an arbitrarily small open subset of formula_147 necessarily agree everywhere. Meromorphic functions, functions that can locally be written as with a holomorphic function , still share some of the features of holomorphic functions. Other functions have essential singularities, such as at .


title: Complex number

text: Complex numbers have applications in many scientific areas, including signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis. Some of these applications are described below.


title: Complex number

text: Three non-collinear points formula_165 in the plane determine the shape of the triangle formula_166. Locating the points in the complex plane, this shape of a triangle may be expressed by complex arithmetic as


title: Complex number

text: The shape formula_168 of a triangle will remain the same, when the complex plane is transformed by translation or dilation (by an affine transformation), corresponding to the intuitive notion of shape, and describing similarity. Thus each triangle formula_166 is in a similarity class of triangles with the same shape.


title: Complex number

text: The Mandelbrot set is a popular example of a fractal formed on the complex plane. It is defined by plotting every location formula_170 where iterating the sequence formula_171 does not diverge when iterated infinitely. Similarly, Julia sets have the same rules, except where formula_170 remains constant.


title: Complex number

text: Every triangle has a unique Steiner inellipse – an ellipse inside the triangle and tangent to the midpoints of the three sides of the triangle. The foci of a triangle's Steiner inellipse can be found as follows, according to Marden's theorem: Denote the triangle's vertices in the complex plane as , , and . Write the cubic equation formula_173, take its derivative, and equate the (quadratic) derivative to zero. Marden's theorem says that the solutions of this equation are the complex numbers denoting the locations of the two foci of the Steiner inellipse.


title: Complex number

text: As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in formula_147. "A fortiori", the same is true if the equation has rational coefficients. The roots of such equations are called algebraic numbers – they are a principal object of study in algebraic number theory. Compared to formula_175, the algebraic closure of formula_176, which also contains all algebraic numbers, formula_147 has the advantage of being easily understandable in geometric terms. In this way, algebraic methods can be used to study geometric questions and vice versa. With algebraic methods, more specifically applying the machinery of field theory to the number field containing roots of unity, it can be shown that it is not possible to construct a regular nonagon using only compass and straightedge – a purely geometric problem.


title: Complex number

text: Another example is the Gaussian integers; that is, numbers of the form , where and are integers, which can be used to classify sums of squares.


title: Complex number

text: Analytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can be regarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoretic information in complex-valued functions. For example, the Riemann zeta function is related to the distribution of prime numbers.


title: Complex number

text: In applied fields, complex numbers are often used to compute certain real-valued improper integrals, by means of complex-valued functions. Several methods exist to do this; see methods of contour integration.


title: Complex number

text: In differential equations, it is common to first find all complex roots of the characteristic equation of a linear differential equation or equation system and then attempt to solve the system in terms of base functions of the form . Likewise, in difference equations, the complex roots of the characteristic equation of the difference equation system are used, to attempt to solve the system in terms of base functions of the form .


title: Complex number

text: Eigendecomposition is a useful tool for computing matrix powers and matrix exponentials. However, it often requires the use of complex numbers, even if the matrix is real (for example, a rotation matrix).


title: Complex number

text: Complex numbers often generalize concepts originally conceived in the real numbers. For example, the conjugate transpose generalizes the transpose, hermitian matrices generalize symmetric matrices, and unitary matrices generalize orthogonal matrices.


title: Complex number

text: In control theory, systems are often transformed from the time domain to the complex frequency domain using the Laplace transform. The system's zeros and poles are then analyzed in the "complex plane". The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane.


title: Complex number

text: In the root locus method, it is important whether zeros and poles are in the left or right half planes, that is, have real part greater than or less than zero. If a linear, time-invariant (LTI) system has poles that are


title: Complex number

text: Complex numbers are used in signal analysis and other fields for a convenient description for periodically varying signals. For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a sine wave of a given frequency, the absolute value of the corresponding is the amplitude and the argument is the phase.


title: Complex number

text: 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


title: Complex number

text: where ω represents the angular frequency and the complex number "A" encodes the phase and amplitude as explained above.


title: Complex number

text: This use is also extended into digital signal processing and digital image processing, which use digital versions of Fourier analysis (and wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, and video signals.


title: Complex number

text: In electrical engineering, the Fourier transform is used to analyze varying voltages and currents. The treatment of resistors, capacitors, and inductors 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. This approach is called phasor calculus.


title: Complex number

text: In electrical engineering, the imaginary unit is denoted by , to avoid confusion with , which is generally in use to denote electric current, or, more particularly, , which is generally in use to denote instantaneous electric current.


title: Complex number

text: The complex-valued signal is called the analytic representation of the real-valued, measurable signal .


title: Complex number

text: The complex number field is intrinsic to the mathematical formulations of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the Schrödinger equation and Heisenberg's matrix mechanics – make use of complex numbers.


title: Complex number

text: In special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time component of the spacetime continuum to be imaginary. (This approach is no longer standard in classical relativity, but is used in an essential way in quantum field theory.) Complex numbers are essential to spinors, which are a generalization of the tensors used in relativity.


title: Complex number

text: The process of extending the field formula_183 of reals to formula_4 is known as the Cayley–Dickson construction. It can be carried further to higher dimensions, yielding the quaternions formula_185 and octonions formula_186 which (as a real vector space) are of dimension 4 and 8, respectively.


title: Complex number

text: Just as by applying the construction to reals the property of ordering is lost, properties familiar from real and complex numbers vanish with each extension. The quaternions lose commutativity, that is, for some quaternions , and the multiplication of octonions, additionally to not being commutative, fails to be associative: for some octonions .


title: Complex number

text: Reals, complex numbers, quaternions and octonions are all normed division algebras over formula_183. By Hurwitz's theorem they are the only ones; the sedenions, the next step in the Cayley–Dickson construction, fail to have this structure.


title: Complex number

text: The Cayley–Dickson construction is closely related to the regular representation of formula_188 thought of as an formula_183-algebra (an formula_140-vector space with a multiplication), with respect to the basis . This means the following: the formula_183-linear map


title: Complex number

text: for some fixed complex number can be represented by a matrix (once a basis has been chosen). With respect to the basis , this matrix is


title: Complex number

text: that is, the one mentioned in the section on matrix representation of complex numbers above. While this is a linear representation of formula_4 in the 2 × 2 real matrices, it is not the only one. Any matrix


title: Complex number

text: is also isomorphic to the field formula_188 and gives an alternative complex structure on formula_198 This is generalized by the notion of a linear complex structure.


title: Complex number

text: Hypercomplex numbers also generalize formula_199 formula_188 formula_201 and formula_202 For example, this notion contains the split-complex numbers, which are elements of the ring formula_203 (as opposed to formula_204 for complex numbers). In this ring, the equation has four solutions.


title: Complex number

text: The field formula_183 is the completion of formula_206 the field of rational numbers, with respect to the usual absolute value metric. Other choices of metrics on formula_207 lead to the fields formula_208 of -adic numbers (for any prime number ), which are thereby analogous to formula_140. There are no other nontrivial ways of completing formula_207 than formula_183 and formula_212 by Ostrowski's theorem. The algebraic closures formula_213 of formula_208 still carry a norm, but (unlike formula_4) are not complete with respect to it. The completion formula_216 of formula_213 turns out to be algebraically closed. By analogy, the field is called -adic complex numbers.


title: Complex number

text: The fields formula_199 formula_212 and their finite field extensions, including formula_188 are called local fields.


title: Barbie

text: Barbie is a fashion doll manufactured by American toy company Mattel, Inc. and launched on March 9, 1959. American businesswoman Ruth Handler is credited with the creation of the doll using a German doll called Bild Lilli as her inspiration.


title: Barbie

text: Barbie is the figurehead of a brand of Mattel dolls and accessories, including other family members and collectible dolls. Barbie has been an important part of the toy fashion doll market for over six decades and has been the subject of numerous controversies and lawsuits, often involving parodies of the doll and her lifestyle. Mattel has sold over a billion Barbie dolls, making it the company's largest and most profitable line.


title: Barbie

text: The brand has expanded into a long-running multimedia franchise since the late 1980s, including and CGI/computer-animated films, the latter which began in 2001, became originally available on home video formats and broadcast regularly on the Nickelodeon cable channel in the United States from 2002 to 2017. Since 2017, the franchise has been moved over to streaming services.


title: Barbie

text: Barbie and her best male friend Ken have been described as two most popular dolls in the world. Since its launch, "Barbie" has transformed the toy business in affluent communities globally by becoming a vehicle for the sale of related merchandise (accessories, clothes, friends and relatives of Barbie, etc.). Writing for "Journal of Popular Culture" in 1977, Don Richard Cox noted that Barbie has a significant impact on social values by conveying characteristics of female independence, and with her multitude of accessories, an idealized upscale life-style that can be shared with affluent friends.


title: Barbie

text: Sales of Barbie dolls declined sharply from 2014 to 2016. In 2020, Mattel sold $1.35 billion worth of Barbie dolls and accessories and this was their best sales growth in two decades. This is an increase from the $950 million the brand sold during 2017.


title: Barbie

text: Ruth Handler watched her daughter Barbara play with paper dolls, and noticed that she often enjoyed giving them adult roles. At the time, most children's toy dolls were representations of infants. Realizing that there could be a gap in the market, Handler suggested the idea of an adult-bodied doll to her husband Elliot, a co-founder of the Mattel toy company. He was unenthusiastic about the idea, as were Mattel's directors.


title: Barbie

text: During a trip to Europe in 1956 with her children Barbara and Kenneth, Ruth Handler came across a German toy doll called Bild Lilli. The adult-figured doll was exactly what Handler had in mind, so she purchased three of them. She gave one to her daughter and took the others back to Mattel. The Lilli doll was based on a popular character appearing in a comic strip drawn by Reinhard Beuthin for the newspaper "Bild". Lilli was a blonde bombshell, a working girl who knew what she wanted and was not above using men to get it. The Lilli doll was first sold in Germany in 1955, and although it was initially sold to adults, it became popular with children who enjoyed dressing her up in outfits that were available separately.


title: Barbie

text: Upon her return to the United States, Handler redesigned the doll (with help from local inventor-designer Jack Ryan) and the doll was given a new name, "Barbie", after Handler's daughter Barbara. The doll made its debut at the American International Toy Fair in New York City on March 9, 1959. This date is also used as Barbie's official birthday.