title: Snake River

text: The Snake River has over 20 major tributaries, most of which are in the mountainous regions of the basin. The largest by far is the Clearwater River, which drains in north central Idaho. Many of the rivers that flow into the Snake River Plain from the north sink into the Snake River Aquifer, but still contribute their water to the river. Aside from rivers, the Snake is fed by many significant springs, many of which arise from the aquifer on the west side of the plain.


title: Wanda De Jesus

text: Wanda De Jesus (born August 26, 1958) is an American actress. She was the fourth actress to portray Santana Andrade in NBC's soap opera "Santa Barbara".


title: Wanda De Jesus

text: Aside from her work on soap operas, De Jesus appeared in several TV shows in a guest starring role, and she had a recurring role on the CBS crime series "" as MDPD Detective Adelle Sevilla. On the week of June 14, 2010, she made her debut on "All My Children" as Iris Blanco, the mayor of Pine Valley. In 2010, De Jesus was cast as the head of the police squad on "", but the role – which had already been changed significantly before De Jesus joined – was changed again and reshot with a different actress after two episodes had been filmed.


title: 'Til Tuesday

text: Til Tuesday (often stylized as til tuesday) was an American new wave band formed in Boston, Massachusetts, United States. The band, consisting of Aimee Mann (lead vocals, bass), Robert Holmes (guitar), Joey Pesce (keyboards), and Michael Hausman (drums), was active from 1982 to 1989. They are best known for their 1985 hit single "Voices Carry".


title: 'Til Tuesday

text: 'Til Tuesday first gained fame six months after its formation when it won Boston's WBCN Rock & Roll Rumble in 1983. Their original composition "Love in a Vacuum" (credited to all members of the group) received a fair amount of airplay on the station, and the group was eventually signed to Epic Records.


title: 'Til Tuesday

text: "Love in a Vacuum" was re-recorded for the Epic debut album, 1985's "Voices Carry"; however, the breakthrough song turned out to be the title track. The "Voices Carry" single peaked at number eight on the U.S. "Billboard" Hot 100, and is said to have been inspired by an argument between Mann and Hausman, who had broken off a relationship before the album's release. According to producer Mike Thorne on his Stereo Society web site, "The title track was originally written and sung by Aimee as if to a woman... The record company was predictably unhappy with such lyrics."


title: 'Til Tuesday

text: The band became an early MTV staple with the "Voices Carry" video, which depicts an domineering boyfriend trying to convert Mann to his upper-class lifestyle; she finally lashes out at him during a concert, standing up from her seat in the audience and removing her cap to reveal her signature spiky, rat-tailed hair. The group won that year's MTV Video Music Award for Best New Artist.


title: 'Til Tuesday

text: By the 1986 follow-up "Welcome Home", Mann was beginning to write more of the songs herself and the band was moving away from the slick new wave sound of their debut. But while critical reaction was generally strong, the #26 placing for the lead single, "What About Love", was a commercial disappointment, especially after the top-ten success of "Voices Carry". Even more problematic, the album just barely sneaked into the U.S. top 50, also a letdown after the #19 placing for their debut.


title: 'Til Tuesday

text: After the album's release Pesce left the band and was replaced by Michael Montes. At about the same time, Mann's two-year relationship with singer-songwriter Jules Shear, whom she had been dating since the release of the "Voices Carry" album, came to an end. This breakup somewhat informed the band's final album, 1988's "Everything's Different Now", particularly in the song "J for Jules", though Mann insisted that not every song on the LP was about the relationship. Shear collaborated with Matthew Sweet on the album's title track; it also featured "The Other End (Of the Telescope)", a collaboration between Mann and Elvis Costello on which Costello provides a guest vocal. Holmes and Montes played on every track of the album, but left before its release; for live dates, Til Tuesday was now a duo of Mann and Hausmann, supported by session musicians.


title: 'Til Tuesday

text: While critical praise continued to flow, "Everything's Different Now" album peaked at No. 124 in the U.S., while the lead single "(Believed You Were) Lucky" (co-written with Shear) reached number 95.


title: 'Til Tuesday

text: 'Til Tuesday essentially broke up after the release of "Everything's Different Now". However, for a short time Mann continued to tour under the 'Til Tuesday name with various session players (including guitarists Jon Brion and Clayton Scoble) while legal problems with the band's label Epic prevented her from beginning work on a solo record for several years.


title: Dave Calhoun

text: David L. Calhoun (born April 18, 1957) is an American businessman who has been president and chief executive officer (CEO) of The Boeing Company since January 2020. He was previously Boeing's chairman and was appointed president and CEO after his predecessor Dennis Muilenburg was fired amidst safety concerns regarding the 737 MAX after two fatal crashes that claimed the lives of 346 passengers and crew on board.


title: Dave Calhoun

text: Calhoun was born April 18, 1957 in Philadelphia and grew up in Allentown, Pennsylvania. He attended Parkland High School in South Whitehall Township, Pennsylvania, graduating in 1975. In high school, Calhoun was one of three captains of the varsity basketball team and played golf. He attended Virginia Tech, graduating in 1979 with a degree in accounting.


title: Dave Calhoun

text: After Calhoun graduated from college, he gained employment at General Electric (GE). He decided to work for GE at that time, due to its close proximity to where he lived in Lehigh Valley. He worked at GE for 26 years, overseeing transportation, aircraft engines, reinsurance, lighting and other GE units, before ultimately being appointed as vice chairman of the company and a member of GE's Board of Directors in 2005.


title: Dave Calhoun

text: Calhoun left GE to join privately held global information services firm VNU as CEO in 2006. Under his leadership the company rebranded itself as Nielsen Holdings, returned to the public markets in 2011, and was added to the S&P 500 Index in 2013. In 2014, Calhoun became executive chairman of Nielsen, and also joined The Blackstone Group as a senior managing director and head of portfolio operations and a member of Blackstone's management committee. Blackstone had been one of six private equity firms that backed Nielsen's transformation. During his career, Calhoun was on the board of directors of Caterpillar, Gates Corporation, and Medtronic.


title: Dave Calhoun

text: Since 2009, Calhoun has served as a director at Boeing, and was named lead independent director in 2018. In October 2019, he was named chairman of Boeing in the wake of the Boeing 737 MAX groundings, and on December 23, 2019, he stepped down as chairman in preparation for becoming Boeing's CEO and president, effective January 13, 2020.


title: Dave Calhoun

text: In a March 2020 interview with "The New York Times", Calhoun discussed the 737 MAX's MCAS software, saying the Boeing had made a "fatal mistake" in expecting that pilots could immediately correct the software problems. He went on to explain that "pilots [in Ethiopia and Indonesia] don't have anywhere near the experience that they have here in the U.S."; he unsuccessfully requested to go off the record after being asked whether American pilots would have been able to control the situation, and then replied, "[f]orget it, you can guess the answer."


title: Dave Calhoun

text: In 2020, Boeing had a historically bad year, reporting a $12 billion loss and laying off 30,000 workers. At the same time, Calhoun earned $21.1 million in compensation.


title: Autonomous regions of China

text: The autonomous regions () are the highest-level administrative divisions of China. Like Chinese provinces, an autonomous region has its own local government, but under Chinese law, an autonomous region has more legislative rights, such as the right to "formulate self-government regulations and other separate regulations." An autonomous region is the highest level of minority autonomous entity in China, which has a comparably higher population of a particular minority ethnic group.


title: Autonomous regions of China

text: The autonomous regions are the creations of the People's Republic of China (PRC), as they are not recognized by the Republic of China (ROC) based in Taiwan, which previously ruled Mainland China before the PRC's establishment in 1949.


title: Autonomous regions of China

text: Established in 1947, the Inner Mongolia Autonomous Region became the first autonomous region in the Chinese liberated zone. Xinjiang was made autonomous in 1955 after the PRC's founding, and Guangxi and Ningxia were made autonomous in 1958. Tibet was placed under PRC control in 1951, with some Western observers calling this an annexation, and was named a Chinese Autonomous Region in 1965. The designation of Guangxi and Ningxia as Zhuang and Hui autonomous areas, respectively, was bitterly protested by the local Han Chinese, who made up two-thirds of the population of each region. Although Mongols made up an even smaller percentage of Inner Mongolia than either of these, the ensuing Chinese Civil War gave little opportunity for protest.


title: Autonomous regions of China

text: Autonomous regions in China have no legal right to secede, unlike in the Soviet Union – the "Law of the People's Republic of China on Regional Ethnic Autonomy", written in 1984, states that "each and every ethnic autonomous region is an inseparable part of the People's Republic of China," and that "any form of... separatism... is absolutely prohibited."


title: Autonomous regions of China

text: Note: In the "Third Largest Ethnic Group" column is the ethnic group given in brackets, after the names of the autonomous regions and Han people.


title: Opiate

text: An opiate, in classical pharmacology, is a substance derived from opium. In more modern usage, the term "opioid" is used to designate all substances, both natural and synthetic, that bind to opioid receptors in the brain (including antagonists). Opiates are alkaloid compounds naturally found in the Papaver somniferum plant (opium poppy)"Papaver somniferum".


title: Opiate

text: The psychoactive compounds found in the opium plant include morphine, codeine, and thebaine. Opiates have long been used for a variety of medical conditions with evidence of opiate trade and use for pain relief as early as the eighth century AD. Opiates are considered drugs with moderate to high abuse potential and are listed on various "Substance-Control Schedules" under the Uniform Controlled Substances Act of the United States of America.


title: Opiate

text: In 2014, between 13 and 20 million people used opiates recreationally (0.3% to 0.4% of the global population between the ages of 15 and 65). According to the CDC, from this population, there were 47,000 deaths, with a total of 500,000 deaths from 2000 to 2014. In 2016, the World Health Organization reported that 27 million people suffer from Opioid use disorder. They also reported that in 2015, 450,000 people died as a result of drug use, with between a third and a half of that number being attributed to opioids.


title: Opiate

text: Opiates belong to the large biosynthetic group of benzylisoquinoline alkaloids, and are so named because they are naturally occurring alkaloids found in the opium poppy. The major psychoactive opiates are morphine, codeine, and thebaine. Papaverine, noscapine, and approximately 24 other alkaloids are also present in opium but have little to no effect on the human central nervous system. Alkaloids that have no effect on the central nervous system are not considered to be opiates. Very small quantities of hydrocodone and hydromorphone are detected in assays of opium on rare occasions; it appears to be produced by the plant under circumstances and by processes that are not understood at this time and may die. Dihydrocodeine, oxymorphol, oxycodone, oxymorphone, metopon, skeuomorphine and possibly other derivatives of morphine and/or hydromorphone also are found in trace amounts in opium.


title: Opiate

text: Despite morphine being the most medically significant opiate, larger quantities of codeine are consumed medically, most of it synthesized from morphine. Codeine has greater and more predictable oral bioavailability, making it easier to titrate the dose. Codeine also has less abuse potential than morphine, and because it is milder, larger doses of codeine are required.


title: Opiate

text: Opiate withdrawal syndrome effects are associated with the abrupt cessation or reduction of prolonged opiate usage.


title: Opiate

text: The manifestation of opiate dependence and abuse relies on a variety of factors, including the opiate's pharmacokinetic properties and the user's predisposition for addiction.


title: Opiate

text: While the full synthesis of opioids from naphthoquinone (Gates synthesis) or other simple organic starting materials is possible, they are tedious and uneconomical processes. Therefore, most of the opiate-type analgesics in use today are either extracted from "Papaver somniferum" or synthesized from those opiates, especially thebaine.


title: Opiate

text: In 2015 researches reported successful biosynthesis of thebaine and hydrocodone using genetically modified yeast. Once scaled for commercial use the process would cut production time from a year to several days and could reduce costs by 90%.


title: Opiate

text: Codeine is a prodrug which is converted to morphine and acts on "μ"-opiate receptors. It is converted to morphine by metabolism of CYP2D6 enzymes. Individuals who have lower CYP2D6 activity may not metabolize codeine efficiently enough to experience its analgesic effects. Conversely, individuals with higher CYP2D6 activity may metabolize the drug too quickly and experience dose-related side effects such as sedation and respiratory depression.


title: Opiate

text: Fentanyl is a synthetic opioid structurally similar to arylpiperidines. It is a strong "μ-r"eceptor agonist that is 80–100 times more potent than morphine, and has a fast onset with a shorter duration of action than morphine. It is metabolized in the liver by CYP3A4 enzymes to the compound norfentanyl.


title: Opiate

text: Heroin, the brand name for diacetylmorphine, is the first of several semi-synthetic opioids to be derived from morphine, a component of natural opium. Although it is derived from, rather than directly found in, natural opium, it is commonly referred to as an opiate. Heroin (diacetylmorphine) is a morphine prodrug; it is metabolized by the liver into morphine after administration. One of the major metabolites of heroin, 6-monoacetylmorphine (6-MAM), is also a morphine prodrug.


title: Opiate

text: Hydromorphone is used as an alternative to morphine. It has a high first-pass metabolism, and is primarily glucuronidated in the liver to hydromorphone-3-glucoronide (H3G). 75% of hydromorphone is renally excreted, with 7% excreted as the parent opiate.


title: Opiate

text: Meperidine is a synthetic opiate part of the arylpiperidine class. It is a strong "μ-r"eceptor agonist with 1/10th the potency of morphine. It is used to treat rigors, and has a half-life of three to four hours. It is hepatically metabolized to the active metabolites of normeperidine, normepiridinic acid, and medperidinic acid. Normeperidine at toxic levels can cause CNS excitation and seizures.


title: Opiate

text: Methadone has a higher bioavailability and half life compared to morphine. It is metabolized to an inactive product by N-demethylation by CYP3A4 enzymes in the liver. It has high person to person variability because of varying levels of CYP3A4 in individuals. It is approved for treatment of moderate to severe pain as well as opioid dependence. Because of its high risk of drug interactions, liver toxicity, and patient variability, patients have to be monitored closely at methadone clinics.


title: Opiate

text: Nicomorphine (Vilan, morphine dinicotinate), Diamorphine (Heroin, morphine diacetate), dipropanoylmorphine (morphine dipropionate), desomorphine (Permonid, di-hydro-desoxy-morphine), methyldesorphine, acetylpropionylmorphine, dibenzoylmorphine, diacetyldihydromorphine, and several others are also derived from morphine.


title: Opiate

text: Morphine is metabolized in the liver to morphine-3-glucuronide (M3G) and morphine-6-glucuronide (M6G), and are excreted by the kidneys. They are also able to cross into the blood-brain barrier into the cerebrospinal fluid. M6G has potent analgesic activity, binds to opioid receptors, and is a main contributor to the therapeutic benefit of morphine. M3G does not act as an analgesic, has a low affinity for opioid receptors, and may possibly antagonize the therapeutic effects of morphine and M6G. Moreover, high doses of morphine, and thus M3G, is associated with neurotoxic side effects such as hyperalgesia, allodynia and myoclonus.


title: Opiate

text: Oxymorphone is a congener of morphine. It is metabolized to 6-hydroxy-oxymorphone and oxymorphone-3-glucuronide, and 40% is excreted as metabolites. 6-hydroxy-oxymorphine is active and exists in a 1:1 ratio with the parent drug. Oxymorphone-3-glucuronide's activity is unknown.


title: Opiate

text: Opiates are mainly used for pain management with the premise that there will be benefits for both pain & function that will outweigh the risks to the patient. Another indication is symptomatic relief of shortage of breath, both in the acute setting (for example, pulmonary edema) and in terminally ill patients.


title: Opiate

text: Opiate dose conversions may be necessary when switching medications given the differing pharmacodynamics between opiates. Generally, morphine is used as the standard for converting between opiates to achieve equivalent analgesic effects. These differences in morphine-equivalents may differ between formulations of the same medication. Calculating total daily dose using morphine milligram equivalents is used to identify patients at risk of overdose.


title: Opiate

text: Common side effects associated to opioid use include: sedation, nausea, dizziness, vomiting, constipation, physical dependence, tolerance, and respiratory depression. Of these the most common are constipation & nausea and there are no development of tolerance to these side effects. This is why stool softeners or laxatives (polyethylene glycol, docusate, and senna) are often prescribed with opioids.


title: Opiate

text: Less common side effects include: delayed gastric emptying, hyperalgesia, immunologic and hormonal dysfunction, muscle rigidity, and myoclonus.


title: Opiate

text: Opiate use for pain is widely accepted in the healthcare system. However, long-term treatment for chronic pain is highly controversial as there is a high risk of addiction associated with its use leading to abuse and diversion to others even when taken properly. Those addicted to opiates will prioritize acquiring these drugs over other activities in their lives, negatively impacting their professional and personal relationships.


title: Opiate

text: Chronic opioid use can lead to tolerance. This often causes the patient to need higher and/or more frequent doses of the drug to get the desired effects.


title: Opiate

text: Concentration-dependence adverse effects may vary based on the user's genetic polymorphisms which can alter drug metabolism. Cytochrome P450 (notably CYP2D6) is responsible for the metabolism of various opiates to active metabolites and variations in CYP450 activity lead to varying serum drug levels.


title: Opiate

text: Pain is an unpleasant sensory and emotional experience associated with actual or potential tissue damage. It is an essential defensive function where pain works as an alarm to avoid or limit tissue damage. Opiates act upon opioid receptors that are coupled to inhibitor G protein coupled receptors (GPCR). These receptors fall into 3 classes: μ (mu), δ (delta), and κ (kappa) receptors.


title: Opiate

text: More than 70% of opioid receptors are μ receptors, predominantly located on the central terminals of nociceptors in the dorsal horn of the spinal cord. The remaining 30% of opioid receptors are located post-synaptically on dendrites of second-order spinothalamic neurons & interneurons.


title: Opiate

text: When an opiate binds as an agonist to the GPCR, there will be a signaling cascade resulting in the inhibition of adenylate cyclase and calcium ion channels with the stimulation of potassium ion channels. The net effect of these changes is a reduced intracellular cAMP and hyperpolarization of the neuronal cell reducing neurotransmitter release. Through this pathway, when opiates bind to and activate the mu receptor, there is a decrease transmission of pain signalling. This pathway targeted for the analgesia properties that opiates are known and used for. Other clinically important roles of mu are its involvement in respiratory and cardiovascular functions, gastrointestinal peristalsis, feeding, and mood. These other pathways are important because they explain the side effects of opiate use like respiratory depression at high doses, constipation with chronic use, and addicting properties.


title: Opiate

text: Statistically, middle-aged patients with substance use history and psychiatric comorbidities are seen with higher mortality risks such as suicide.


title: Isolation tank

text: An isolation tank, sensory deprivation tank, float tank, float pod, float cabin, flotation tank, or sensory attenuation tank is a water filled, pitch-black, light-proof, soundproof environment heated to the same temperature as the skin, developed in 1954 by John C. Lilly.


title: Isolation tank

text: Flotation tanks are widely advertised as a form of alternative medicine but beneficial health effects are unproven.


title: Isolation tank

text: The tank is filled with 10 inches of water which contains enough dissolved Epsom salt to create a specific gravity of approximately 1.25–1.26. This environment allows an individual to float effortlessly on the surface of the water.


title: Isolation tank

text: The isolation tank was developed in 1954 by John C. Lilly, a medical practitioner and neuropsychiatrist. During his training in psychoanalysis at the US National Institute of Mental Health (NIMH), Lilly experimented with sensory deprivation.


title: Isolation tank

text: Widespread commercial interest and use of the isolation tank did not occur until 1972, when Glenn Perry, a computer systems programmer, began selling the first commercial tanks after he attended a five-day workshop by Lilly.


title: Isolation tank

text: In 1981, there were about $4 million in sales and rentals in the industry, and expectations were that the market would grow, spurred by the film "Altered States" (a film starring William Hurt as a psychopathologist experimenting with a flotation tank), which came out in 1980. According to one source in the industry, in the 1980s the rise of AIDS and the fear of shared water reduced demand for flotation centers. By 2013, flotation was more popular in Europe than the US, but had undergone some growth in the area around San Francisco; at that time a low-end tank cost about $10,000, and an hour-long flotation session cost about $70.


title: Isolation tank

text: Flotation has been widely advertised as a form of alternative medicine that has a number of health benefits, but the claims are often exaggerated and poorly evidenced. Despite the lack of scientific support, people have sought treatment from flotation for many conditions including muscle tension, chronic pain, hypertension, and rheumatoid arthritis to PMS.


title: Isolation tank

text: A 2005 meta-analysis of clinical trials that had been conducted at that time, found that the trials were generally small and highly prone to error, but given that limitation, use of isolation tanks, (called "flotation REST" or "restricted environmental stimulation therapy" in the literature) appeared to be useful for stress management.


title: Isolation tank

text: Scientific research with float tank therapy generally uses the term "flotation-REST" (reduced environmental stimulation therapy) to refer to the technique. This term is preferred over "sensory deprivation" due to the fact that 1) the float tank experience actually enhanced sensory input from the body (e.g., cardiorespiratory sensations) and can also be conceptualized as a form of sensory enhancement and 2) the term "sensory deprivation" carries negative connotations of torture and hallucinations that have likely impeded legitimate research in this field.


title: Tales of the City (2019 miniseries)

text: Armistead Maupin's Tales of the City is an American drama streaming television miniseries that premiered June 7, 2019, on Netflix, based on the "Tales of the City" novels by Armistead Maupin. Laura Linney, Paul Gross, Olympia Dukakis, and Barbara Garrick reprise their roles from previous television adaptations of Maupin's books: the original "Tales of the City" in 1993, and the sequels "More Tales of the City" (1998) and "Further Tales of the City" (2001). The series was Dukakis's final television role before her death.


title: Tales of the City (2019 miniseries)

text: The show won the GLAAD Media Award for Outstanding Limited Series at the 31st GLAAD Media Awards, the third time the series has won the award following the original "Tales of the City" at the 1995 awards ceremony and sequel "More Tales of the City" at the 1999 ceremony.


title: Tales of the City (2019 miniseries)

text: Mary Ann Singleton returns to 28 Barbary Lane in San Francisco after a 23-year absence for the 90th birthday of her former landlady, Anna Madrigal. Mary Ann is happily reunited with Michael Tolliver and Anna, but things are more complicated with her ex-husband, Brian Hawkins, and Shawna, the daughter she left behind to pursue a broadcasting career. Shawna follows a mysterious attraction to a new girl in town who is making a documentary about 28 Barbary Lane. The relationship between transgender man Jake Rodriguez and his lesbian girlfriend Margot Park faces challenges as Jake explores his newfound attraction to men. Michael struggles with the option to stop using condoms now that he is in a relationship with Ben Marshall, who is on PrEP. Anna begins receiving mysterious letters threatening to expose a secret from her past.


title: Tales of the City (2019 miniseries)

text: Netflix announced June 28, 2017, the development of a revival of the "Tales of the City" television miniseries, based on the novels by Armistead Maupin. A subsequent April 23, 2018, announcement included a production order for ten episodes, with the series to be written by Lauren Morelli, who would serve as showrunner and executive producer, alongside an all-LGBTQ writer's room. Other executive producers would include Maupin, Alan Poul, Laura Linney, Andrew Stearn, Liza Chasin, Tim Bevan, and Eric Fellner. Michael Cunningham would serve as consulting producer and Poul was also expected to direct. Production companies involved with the series include Working Title Television and NBCUniversal International. An announcement on April 9, 2019, revealed the series would be released on June 7, 2019.


title: Tales of the City (2019 miniseries)

text: Alongside the initial series development announcement, it was reported that Laura Linney and Olympia Dukakis would reprise their roles as Mary Ann Singleton and Anna Madrigal, respectively. Concurrent with the news of the series order, it was confirmed that Linney, Dukakis, and Barbara Garrick would be reprising their roles, and that Elliot Page would be joining the main cast of the production in a new role.


title: Tales of the City (2019 miniseries)

text: In October 2018, it was announced that Paul Gross would reprise his role of Brian Hawkins and that Murray Bartlett, Charlie Barnett, Josiah Victoria Garcia, and May Hong had also joined the main cast. Additionally, it was further announced that Jen Richards, Daniela Vega, Michelle Buteau, Ashley Park, Christopher Larkin, Caldwell Tidicue, Matthew Risch, Michael Park, Dickie Hearts, Benjamin Thys, Samantha Soule, Juan Castano, Zosia Mamet, and Victor Garber had been cast in recurring roles. Molly Ringwald later joined the cast in a recurring capacity.


title: Tales of the City (2019 miniseries)

text: Principal photography for the series began by July 2018 in New York City. Filming of interior scenes of the series were shot in New York while exterior scenes were set to be filmed in San Francisco, California. On October 5, 2018, the production was shooting in the Nodine Hill section of Yonkers, New York. On October 24, 2018, filming took place at Mission Dolores Park in San Francisco. By January 2019, filming for the series had reportedly concluded.


title: Tales of the City (2019 miniseries)

text: On Rotten Tomatoes, the series received an approval of 83%, and a 7.50/10 average rating from 42 reviews. The critics' consensus states, "Like a pleasant visit to a place you used to live, "Tales of the City" provides ample nostalgic comforts and continues the series' mission of celebrating the diversity of San Francisco on its own terms." Metacritic scored it 63 out of 100 based on 15 critics, signifying "generally favorable reviews".


title: Methods of computing square roots

text: Methods of computing square roots are numerical analysis algorithms for approximating the principal, or non-negative, square root (usually denoted formula_1, formula_2, or formula_3) of a real number. Arithmetically, it means given formula_4, a procedure for finding a number which when multiplied by itself, yields formula_4; algebraically, it means a procedure for finding the non-negative root of the equation formula_6; geometrically, it means given two line segments, a procedure for constructing their geometric mean.


title: Methods of computing square roots

text: Every real number except zero has two square roots. The principal square root of most numbers is an irrational number with an infinite decimal expansion. As a result, the decimal expansion of any such square root can only be computed to some finite-precision approximation. However, even if we are taking the square root of a perfect square integer, so that the result does have an exact finite representation, the procedure used to compute it may only return a series of increasingly accurate approximations.


title: Methods of computing square roots

text: The continued fraction representation of a real number can be used instead of its decimal or binary expansion and this representation has the property that the square root of any rational number (which is not already a perfect square) has a periodic, repeating expansion, similar to how rational numbers have repeating expansions in the decimal notation system.


title: Methods of computing square roots

text: The most common analytical methods are iterative and consist of two steps: finding a suitable starting value, followed by iterative refinement until some termination criterion is met. The starting value can be any number, but fewer iterations will be required the closer it is to the final result. The most familiar such method, most suited for programmatic calculation, is Newton's method, which is based on a property of the derivative in the calculus. A few methods like paper-and-pencil synthetic division and series expansion, do not require a starting value. In some applications, an integer square root is required, which is the square root rounded or truncated to the nearest integer (a modified procedure may be employed in this case).


title: Methods of computing square roots

text: The method employed depends on what the result is to be used for (i.e. how accurate it has to be), how much effort one is willing to put into the procedure, and what tools are at hand. The methods may be roughly classified as those suitable for mental calculation, those usually requiring at least paper and pencil, and those which are implemented as programs to be executed on a digital electronic computer or other computing device. Algorithms may take into account convergence (how many iterations are required to achieve a specified precision), computational complexity of individual operations (i.e. division) or iterations, and error propagation (the accuracy of the final result).


title: Methods of computing square roots

text: Procedures for finding square roots (particularly the square root of 2) have been known since at least the period of ancient Babylon in the 17th century BCE. Heron's method from first century Egypt was the first ascertainable algorithm for computing square root. Modern analytic methods began to be developed after introduction of the Arabic numeral system to western Europe in the early Renaissance. Today, nearly all computing devices have a fast and accurate square root function, either as a programming language construct, a compiler intrinsic or library function, or as a hardware operator, based on one of the described procedures.


title: Methods of computing square roots

text: Many iterative square root algorithms require an initial seed value. The seed must be a non-zero positive number; it should be between 1 and formula_4, the number whose square root is desired, because the square root must be in that range. If the seed is far away from the root, the algorithm will require more iterations. If one initializes with formula_8 (or formula_4), then approximately formula_10 iterations will be wasted just getting the order of magnitude of the root. It is therefore useful to have a rough estimate, which may have limited accuracy but is easy to calculate. In general, the better the initial estimate, the faster the convergence. For Newton's method (also called Babylonian or Heron's method), a seed somewhat larger than the root will converge slightly faster than a seed somewhat smaller than the root.


title: Methods of computing square roots

text: In general, an estimate is pursuant to an arbitrary interval known to contain the root (such as formula_11). The estimate is a specific value of a functional approximation to formula_12 over the interval. Obtaining a better estimate involves either obtaining tighter bounds on the interval, or finding a better functional approximation to formula_13. The latter usually means using a higher order polynomial in the approximation, though not all approximations are polynomial. Common methods of estimating include scalar, linear, hyperbolic and logarithmic. A decimal base is usually used for mental or paper-and-pencil estimating. A binary base is more suitable for computer estimates. In estimating, the exponent and mantissa are usually treated separately, as the number would be expressed in scientific notation.


title: Methods of computing square roots

text: Typically the number formula_4 is expressed in scientific notation as formula_15 where formula_16 and "n" is an integer, and the range of possible square roots is formula_17 where formula_18.


title: Methods of computing square roots

text: Scalar methods divide the range into intervals, and the estimate in each interval is represented by a single scalar number. If the range is considered as a single interval, the arithmetic mean (5.5) or geometric mean (formula_19) times formula_20 are plausible estimates. The absolute and relative error for these will differ. In general, a single scalar will be very inaccurate. Better estimates divide the range into two or more intervals, but scalar estimates have inherently low accuracy.


title: Methods of computing square roots

text: This estimate has maximum absolute error of formula_23 at a = 100, and maximum relative error of 100% at a = 1.


title: Methods of computing square roots

text: For example, for formula_24 factored as formula_25, the estimate is formula_26. formula_27, an absolute error of 246 and relative error of almost 70%.


title: Methods of computing square roots

text: A better estimate, and the standard method used, is a linear approximation to the function formula_28 over a small arc. If, as above, powers of the base are factored out of the number formula_4 and the interval reduced to formula_30, a secant line spanning the arc, or a tangent line somewhere along the arc may be used as the approximation, but a least-squares regression line intersecting the arc will be more accurate.


title: Methods of computing square roots

text: A least-squares regression line minimizes the average difference between the estimate and the value of the function. Its equation is formula_31. Reordering, formula_32. Rounding the coefficients for ease of computation,


title: Methods of computing square roots

text: That is the best estimate "on average" that can be achieved with a single piece linear approximation of the function y=x in the interval formula_30. It has a maximum absolute error of 1.2 at a=100, and maximum relative error of 30% at S=1 and 10.


title: Methods of computing square roots

text: To divide by 10, subtract one from the exponent of formula_35, or figuratively move the decimal point one digit to the left. For this formulation, any additive constant 1 plus a small increment will make a satisfactory estimate so remembering the exact number isn't a burden. The approximation (rounded or not) using a single line spanning the range formula_30 is less than one significant digit of precision; the relative error is greater than 1/2, so less than 2 bits of information are provided. The accuracy is severely limited because the range is two orders of magnitude, quite large for this kind of estimation.


title: Methods of computing square roots

text: A much better estimate can be obtained by a piece-wise linear approximation: multiple line segments, each approximating some subarc of the original. The more line segments used, the better the approximation. The most common way is to use tangent lines; the critical choices are how to divide the arc and where to place the tangent points. An efficacious way to divide the arc from y=1 to y=100 is geometrically: for two intervals, the bounds of the intervals are the square root of the bounds of the original interval, 1*100, i.e. [1,] and [,100]. For three intervals, the bounds are the cube roots of 100: [1,], [,()], and [(),100], etc. For two intervals, = 10, a very convenient number. Tangent lines are easy to derive, and are located at x = and x = . Their equations are: formula_37 and formula_38. Inverting, the square roots are: formula_39 and formula_40. Thus for formula_41:


title: Methods of computing square roots

text: The maximum absolute errors occur at the high points of the intervals, at a=10 and 100, and are 0.54 and 1.7 respectively. The maximum relative errors are at the endpoints of the intervals, at a=1, 10 and 100, and are 17% in both cases. 17% or 0.17 is larger than 1/10, so the method yields less than a decimal digit of accuracy.


title: Methods of computing square roots

text: In some cases, hyperbolic estimates may be efficacious, because a hyperbola is also a convex curve and may lie along an arc of Y = x better than a line. Hyperbolic estimates are more computationally complex, because they necessarily require a floating division. A near-optimal hyperbolic approximation to x on the interval formula_30 is y=190/(10-x)-20. Transposing, the square root is x = -190/(y+20)+10. Thus for formula_44:


title: Methods of computing square roots

text: The floating division need be accurate to only one decimal digit, because the estimate overall is only that accurate, and can be done mentally. A hyperbolic estimate is better on average than scalar or linear estimates. It has maximum absolute error of 1.58 at 100 and maximum relative error of 16.0% at 10. For the worst case at a=10, the estimate is 3.67. If one starts with 10 and applies Newton-Raphson iterations straight away, two iterations will be required, yielding 3.66, before the accuracy of the hyperbolic estimate is exceeded. For a more typical case like 75, the hyperbolic estimate is 8.00, and 5 Newton-Raphson iterations starting at 75 would be required to obtain a more accurate result.


title: Methods of computing square roots

text: A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater than or equal to 25 but less than 36, begins with a 5. Similarly for numbers between other squares. This method will yield a correct first digit, but it is not accurate to one digit: the first digit of the square root of 35 for example, is 5, but the square root of 35 is almost 6.


title: Methods of computing square roots

text: A better way is to the divide the range into intervals half way between the squares. So any number between 25 and half way to 36, which is 30.5, estimate 5; any number greater than 30.5 up to 36, estimate 6. The procedure only requires a little arithmetic to find a boundary number in the middle of two products from the multiplication table. Here is a reference table of those boundaries:


title: Methods of computing square roots

text: The method implicitly yields one significant digit of accuracy, since it rounds to the best first digit.


title: Methods of computing square roots

text: The method can be extended 3 significant digits in most cases, by interpolating between the nearest squares bounding the operand. If formula_48, then formula_49 is approximately k plus a fraction, the difference between and "k" divided by the difference between the two squares:


title: Methods of computing square roots

text: Example: find the square root of 75. , so is 75 and is 0. From the multiplication tables, the square root of the mantissa must be 8 point "something" because 8 × 8 is 64, but 9 × 9 is 81, too big, so is 8; "something" is the decimal representation of . The fraction is 75 - "k" = 11, the numerator, and 81 - "k" = 17, the denominator. 11/17 is a little less than 12/18, which is 2/3s or .67, so guess .66 (it's ok to guess here, the error is very small). So the estimate is . to three significant digits is 8.66, so the estimate is good to 3 significant digits. Not all such estimates using this method will be so accurate, but they will be close.


title: Methods of computing square roots

text: When working in the binary numeral system (as computers do internally), by expressing formula_4 as formula_54 where formula_55, the square root formula_56 can be estimated as


title: Methods of computing square roots

text: which is the least-squares regression line to 3 significant digit coefficients. formula_49 has maximum absolute error of 0.0408 at formula_35=2, and maximum relative error of 3.0% at formula_35=1. A computationally convenient rounded estimate (because the coefficients are powers of 2) is:


title: Methods of computing square roots

text: which has maximum absolute error of 0.086 at 2 and maximum relative error of 6.1% at formula_35=0.5 and formula_35=2.0.


title: Methods of computing square roots

text: For formula_64, the binary approximation gives formula_65. formula_66, so the estimate has an absolute error of 19 and relative error of 5.3%. The relative error is a little less than 1/2, so the estimate is good to 4+ bits.


title: Methods of computing square roots

text: An estimate for formula_35 good to 8 bits can be obtained by table lookup on the high 8 bits of formula_35, remembering that the high bit is implicit in most floating point representations, and the bottom bit of the 8 should be rounded. The table is 256 bytes of precomputed 8-bit square root values. For example, for the index 11101101 representing 1.8515625, the entry is 10101110 representing 1.359375, the square root of 1.8515625 to 8 bit precision (2+ decimal digits).


title: Methods of computing square roots

text: Perhaps the first algorithm used for approximating formula_1 is known as the Babylonian method, despite there being no direct evidence beyond informed conjecture that the eponymous Babylonian mathematicians employed this method. The method is also known as Heron's method, after the first-century Greek mathematician Hero of Alexandria who gave the first explicit description of the method in his AD 60 work "Metrica". The basic idea is that if is an overestimate to the square root of a non-negative real number then will be an underestimate, or vice versa, and so the average of these two numbers may reasonably be expected to provide a better approximation (though the formal proof of that assertion depends on the inequality of arithmetic and geometric means that shows this average is always an overestimate of the square root, as noted in the article on square roots, thus assuring convergence). This is equivalent to using Newton's method to solve formula_70.


title: Methods of computing square roots

text: More precisely, if is our initial guess of formula_1 and is the error in our estimate such that , then we can expand the binomial


title: Methods of computing square roots

text: Since the computed error was not exact, this becomes our next best guess. The process of updating is iterated until desired accuracy is obtained. This is a quadratically convergent algorithm, which means that the number of correct digits of the approximation roughly doubles with each iteration. It proceeds as follows:


title: Methods of computing square roots

text: This algorithm works equally well in the -adic numbers, but cannot be used to identify real square roots with -adic square roots; one can, for example, construct a sequence of rational numbers by this method that converges to +3 in the reals, but to −3 in the 2-adics.


title: Methods of computing square roots

text: To calculate , where = 125348, to six significant figures, use the rough estimation method above to get


title: Methods of computing square roots

text: Suppose that "x" > 0 and "S" > 0. Then for any natural number "n", "x" > 0. Let the relative error in "x" be defined by


title: Methods of computing square roots

text: If using the rough estimate above with the Babylonian method, then the least accurate cases in ascending order are as follows:


title: Methods of computing square roots

text: Rounding errors will slow the convergence. It is recommended to keep at least one extra digit beyond the desired accuracy of the being calculated to minimize round off error.


title: Methods of computing square roots

text: This method for finding an approximation to a square root was described in an ancient South Asian manuscript from Pakistan, called the Bakhshali manuscript. It is equivalent to two iterations of the Babylonian method beginning with "x". Thus, the algorithm is quartically convergent, which means that the number of correct digits of the approximation roughly quadruples with each iteration. The original presentation, using modern notation, is as follows: To calculate formula_1, let formula_94 be the initial approximation to formula_4. Then, successively iterate as:


title: Methods of computing square roots

text: This can be used to construct a rational approximation to the square root by beginning with an integer. If formula_97 is an integer chosen so formula_98 is close to formula_4, and formula_100 is the difference whose absolute value is minimized, then the first iteration can be written as:


title: Methods of computing square roots

text: The Bakhshali method can be generalized to the computation of an arbitrary root, including fractional roots.


title: Methods of computing square roots

text: Using the same example as given with the Babylonian method, let formula_102 Then, the first iteration gives


title: Methods of computing square roots

text: This is a method to find each digit of the square root in a sequence. This method is based on the binomial theorem and basically an inverse algorithm solving (x+y)² = x²+2xy+y². It is slower than the Babylonian method, but it has several advantages:


title: Methods of computing square roots

text: Napier's bones include an aid for the execution of this algorithm. The shifting "n"th root algorithm is a generalization of this method.


title: Methods of computing square roots

text: First, consider the case of finding the square root of a number "Z", that is the square of a two-digit number "XY", where "X" is the tens digit and "Y" is the units digit. Specifically:


title: Methods of computing square roots

text: Now using the digit-by-digit algorithm, we first determine the value of "X". "X" is the largest digit such that X is less or equal to "Z" from which we removed the two rightmost digits.


title: Methods of computing square roots

text: In the next iteration, we pair the digits, multiply "X" by 2, and place it in the tenth's place while we try to figure out what the value of "Y" is.


title: Methods of computing square roots

text: Since this is a simple case where the answer is a perfect square root "XY", the algorithm stops here.


title: Methods of computing square roots

text: The same idea can be extended to any arbitrary square root computation next. Suppose we are able to find the square root of "N" by expressing it as a sum of "n" positive numbers such that


title: Methods of computing square roots

text: This expression allows us to find the square root by sequentially guessing the values of formula_109s. Suppose that the numbers formula_110 have already been guessed, then the m-th term of the right-hand-side of above summation is given by formula_111 where formula_112 is the approximate square root found so far. Now each new guess formula_113 should satisfy the recursion


title: Methods of computing square roots

text: such that formula_115 for all formula_116 with initialization formula_117 When formula_118 the exact square root has been found; if not, then the sum of formula_109s gives a suitable approximation of the square root, with formula_120 being the approximation error.


title: Methods of computing square roots

text: where formula_122 are place holders and the coefficients formula_123. At any m-th stage of the square root calculation, the approximate root found so far, formula_124 and the summation term formula_125 are given by


title: Methods of computing square roots

text: Here since the place value of formula_125 is an even power of 10, we only need to work with the pair of most significant digits of the remaining term formula_129 at any m-th stage. The section below codifies this procedure.


title: Methods of computing square roots

text: It is obvious that a similar method can be used to compute the square root in number systems other than the decimal number system. For instance, finding the digit-by-digit square root in the binary number system is quite efficient since the value of formula_109 is searched from a smaller set of binary digits {0,1}. This makes the computation faster since at each stage the value of formula_125 is either formula_132 for formula_133 or formula_134 for formula_135. The fact that we have only two possible options for formula_113 also makes the process of deciding the value of formula_113 at m-th stage of calculation easier. This is because we only need to check if formula_138 for formula_139 If this condition is satisfied, then we take formula_135; if not then formula_141 Also, the fact that multiplication by 2 is done by left bit-shifts helps in the computation.


title: Methods of computing square roots

text: Write the original number in decimal form. The numbers are written similar to the long division algorithm, and, as in long division, the root will be written on the line above. Now separate the digits into pairs, starting from the decimal point and going both left and right. The decimal point of the root will be above the decimal point of the square. One digit of the root will appear above each pair of digits of the square.


title: Methods of computing square roots

text: This section uses the formalism from the digit-by-digit calculation section above, with the slight variation that we let formula_144, with each formula_145 or formula_133.


title: Methods of computing square roots

text: We iterate all formula_147, from formula_148 down to formula_149, and build up an approximate solution formula_150, the sum of all formula_109 for which we have determined the value.


title: Methods of computing square roots

text: To determine if formula_113 equals formula_147 or formula_154, we let formula_155. If formula_156 (i.e. the square of our approximate solution including formula_147 does not exceed the target square) then formula_145, otherwise formula_133 and formula_160.


title: Methods of computing square roots

text: To avoid squaring formula_161 in each step, we store the difference formula_162 and incrementally update it by setting formula_163 with formula_164.


title: Methods of computing square roots

text: As an extra optimization, we store formula_168 and formula_169, the two terms of formula_125 in case that formula_113 is nonzero, in separate variables formula_172, formula_173:


title: Methods of computing square roots

text: Faster algorithms, in binary and decimal or any other base, can be realized by using lookup tables—in effect trading more storage space for reduced run time.


title: Methods of computing square roots

text: Pocket calculators typically implement good routines to compute the exponential function and the natural logarithm, and then compute the square root of "S" using the identity found using the properties of logarithms (formula_182) and exponentials (formula_183):


title: Methods of computing square roots

text: The denominator in the fraction corresponds to the "n"th root. In the case above the denominator is 2, hence the equation specifies that the square root is to be found. The same identity is used when computing square roots with logarithm tables or slide rules.


title: Methods of computing square roots

text: This method is applicable for finding the square root of formula_185 and converges best for formula_186.


title: Methods of computing square roots

text: This, however, is no real limitation for a computer based calculation, as in base 2 floating point and fixed point representations, it is trivial to multiply formula_187 by an integer power of 4, and therefore formula_1 by the corresponding power of 2, by changing the exponent or by shifting, respectively. Therefore, formula_187 can be moved to the range formula_190. Moreover, the following method does not employ general divisions, but only additions, subtractions, multiplications, and divisions by powers of two, which are again trivial to implement. A disadvantage of the method is that numerical errors accumulate, in contrast to single variable iterative methods such as the Babylonian one.


title: Methods of computing square roots

text: and therefore the convergence of formula_198 to the desired result formula_1 is ensured by the convergence of formula_197 to 0, which in turn follows from formula_205.


title: Methods of computing square roots

text: This method was developed around 1950 by M. V. Wilkes, D. J. Wheeler and S. Gill for use on EDSAC, one of the first electronic computers. The method was later generalized, allowing the computation of non-square roots.


title: Methods of computing square roots

text: The following are iterative methods for finding the reciprocal square root of "S" which is formula_206. Once it has been found, find formula_1 by simple multiplication: formula_208. These iterations involve only multiplication, and not division. They are therefore faster than the Babylonian method. However, they are not stable. If the initial value is not close to the reciprocal square root, the iterations will diverge away from it rather than converge to it. It can therefore be advantageous to perform an iteration of the Babylonian method on a rough estimate before starting to apply these methods.


title: Methods of computing square roots

text: Goldschmidt's algorithm finds formula_1 faster than Newton-Raphson iteration on a computer with a fused multiply–add instruction and either a pipelined floating point unit or two independent floating-point units.


title: Methods of computing square roots

text: until formula_227 is sufficiently close to 1, or a fixed number of iterations. The iterations converge to


title: Methods of computing square roots

text: Note that it is possible to omit either formula_230 and formula_231 from the computation, and if both are desired then formula_232 may be used at the end rather than computing it through in each iteration.


title: Methods of computing square roots

text: If "N" is an approximation to formula_1, a better approximation can be found by using the Taylor series of the square root function:


title: Methods of computing square roots

text: As an iterative method, the order of convergence is equal to the number of terms used. With two terms, it is identical to the Babylonian method. With three terms, each iteration takes almost as many operations as the Bakhshali approximation, but converges more slowly. Therefore, this is not a particularly efficient way of calculation. To maximize the rate of convergence, choose "N" so that formula_244 is as small as possible.


title: Methods of computing square roots

text: Quadratic irrationals (numbers of the form formula_245, where "a", "b" and "c" are integers), and in particular, square roots of integers, have periodic continued fractions. Sometimes what is desired is finding not the numerical value of a square root, but rather its continued fraction expansion, and hence its rational approximation. Let "S" be the positive number for which we are required to find the square root. Then assuming "a" to be a number that serves as an initial guess and "r" to be the remainder term, we can write formula_246 Since we have formula_247, we can express the square root of "S" as


title: Methods of computing square roots

text: The first step to evaluating such a fraction to obtain a root is to do numerical substitutions for the root of the number desired, and number of denominators selected. For example, in canonical form, formula_252 is 1 and for , formula_35 is 1, so the numerical continued fraction for 3 denominators is:


title: Methods of computing square roots

text: Step 2 is to reduce the continued fraction from the bottom up, one denominator at a time, to yield a rational fraction whose numerator and denominator are integers. The reduction proceeds thus (taking the first three denominators):


title: Methods of computing square roots

text: Finally (step 3), divide the numerator by the denominator of the rational fraction to obtain the approximate value of the root:


title: Methods of computing square roots

text: The actual value of is 1.41 to three significant digits. The relative error is 0.17%, so the rational fraction is good to almost three digits of precision. Taking more denominators gives successively better approximations: four denominators yields the fraction formula_259, good to almost 4 digits of precision, etc.


title: Methods of computing square roots

text: The following are examples of square roots, their simple continued fractions, and their first terms called "convergents" up to and including denominator 99:


title: Methods of computing square roots

text: In general, the larger the denominator of a rational fraction, the better the approximation. It can also be shown that truncating a continued fraction yields a rational fraction that is the best approximation to the root of any fraction with denominator less than or equal to the denominator of that fraction e.g., no fraction with a denominator less than or equal to 70 is as good an approximation to as 99/70.


title: Methods of computing square roots

text: the Lucas sequence of the first kind "U"("P","Q") is defined by the recurrence relations:formula_260and the characteristic equation of it is:formula_261it has the discriminant formula_262 and the roots:formula_263all that yield the following positive value:formula_264so when we want formula_265, we can choose formula_266 and formula_267, and then calculate formula_268 using formula_269 and formula_270for large value of formula_271.


title: Methods of computing square roots

text: The most effective way to calculate formula_269 and formula_270is:formula_274Summary:formula_275then when formula_276:formula_277


title: Methods of computing square roots

text: A number is represented in a floating point format as formula_278 which is also called scientific notation. Its square root is formula_279 and similar formulae would apply for cube roots and logarithms. On the face of it, this is no improvement in simplicity, but suppose that only an approximation is required: then just formula_280 is good to an order of magnitude. Next, recognise that some powers, , will be odd, thus for 3141.59 = 3.14159 rather than deal with fractional powers of the base, multiply the mantissa by the base and subtract one from the power to make it even. The adjusted representation will become the equivalent of 31.4159 so that the square root will be .


title: Methods of computing square roots

text: If the integer part of the adjusted mantissa is taken, there can only be the values 1 to 99, and that could be used as an index into a table of 99 pre-computed square roots to complete the estimate. A computer using base sixteen would require a larger table, but one using base two would require only three entries: the possible bits of the integer part of the adjusted mantissa are 01 (the power being even so there was no shift, remembering that a normalised floating point number always has a non-zero high-order digit) or if the power was odd, 10 or 11, these being the first "two" bits of the original mantissa. Thus, 6.25 = 110.01 in binary, normalised to 1.1001 × 2 an even power so the paired bits of the mantissa are 01, while .625 = 0.101 in binary normalises to 1.01 × 2 an odd power so the adjustment is to 10.1 × 2 and the paired bits are 10. Notice that the low order bit of the power is echoed in the high order bit of the pairwise mantissa. An even power has its low-order bit zero and the adjusted mantissa will start with 0, whereas for an odd power that bit is one and the adjusted mantissa will start with 1. Thus, when the power is halved, it is as if its low order bit is shifted out to become the first bit of the pairwise mantissa.


title: Methods of computing square roots

text: A table with only three entries could be enlarged by incorporating additional bits of the mantissa. However, with computers, rather than calculate an interpolation into a table, it is often better to find some simpler calculation giving equivalent results. Everything now depends on the exact details of the format of the representation, plus what operations are available to access and manipulate the parts of the number. For example, Fortran offers an codice_1 function to obtain the power. Effort expended in devising a good initial approximation is to be recouped by thereby avoiding the additional iterations of the refinement process that would have been needed for a poor approximation. Since these are few (one iteration requires a divide, an add, and a halving) the constraint is severe.


title: Methods of computing square roots

text: Many computers follow the IEEE (or sufficiently similar) representation, and a very rapid approximation to the square root can be obtained for starting Newton's method. The technique that follows is based on the fact that the floating point format (in base two) approximates the base-2 logarithm. That is formula_281


title: Methods of computing square roots

text: So for a 32-bit single precision floating point number in IEEE format (where notably, the power has a bias of 127 added for the represented form) you can get the approximate logarithm by interpreting its binary representation as a 32-bit integer, scaling it by formula_282, and removing a bias of 127, i.e.


title: Methods of computing square roots

text: For example, 1.0 is represented by a hexadecimal number 0x3F800000, which would represent formula_284 if taken as an integer. Using the formula above you get formula_285, as expected from formula_286. In a similar fashion you get 0.5 from 1.5 (0x3FC00000).


title: Methods of computing square roots

text: To get the square root, divide the logarithm by 2 and convert the value back. The following program demonstrates the idea. Note that the exponent's lowest bit is intentionally allowed to propagate into the mantissa. One way to justify the steps in this program is to assume formula_287 is the exponent bias and formula_166 is the number of explicitly stored bits in the mantissa and then show that


title: Methods of computing square roots

text: The three mathematical operations forming the core of the above function can be expressed in a single line. An additional adjustment can be added to reduce the maximum relative error. So, the three operations, not including the cast, can be rewritten as


title: Methods of computing square roots

text: where "a" is a bias for adjusting the approximation errors. For example, with "a" = 0 the results are accurate for even powers of 2 (e.g. 1.0), but for other numbers the results will be slightly too big (e.g. 1.5 for 2.0 instead of 1.414... with 6% error). With "a" = −0x4B0D2, the maximum relative error is minimized to ±3.5%.


title: Methods of computing square roots

text: If the approximation is to be used for an initial guess for Newton's method to the equation formula_209, then the reciprocal form shown in the following section is preferred.


title: Methods of computing square roots

text: A variant of the above routine is included below, which can be used to compute the reciprocal of the square root, i.e., formula_291 instead, was written by Greg Walsh. The integer-shift approximation produced a relative error of less than 4%, and the error dropped further to 0.15% with one iteration of Newton's method on the following line. In computer graphics it is a very efficient way to normalize a vector.


title: Methods of computing square roots

text: Some VLSI hardware implements inverse square root using a second degree polynomial estimation followed by a Goldschmidt iteration.


title: Methods of computing square roots

text: is the modulus of "S". The principal square root of a complex number is defined to be the root with the non-negative real part.


title: John Adams Morgan

text: John Adams Morgan (born September 17, 1930) is an American sailor and Olympic champion and the founder and chairman of Morgan Joseph. His father, Henry Sturgis Morgan, was the co-founder of Morgan Stanley and his great-grandfather was J. P. Morgan, founder of J.P. Morgan & Co.


title: John Adams Morgan

text: John Adams Morgan was born on September 17, 1930, in Oyster Bay on Long Island to Henry Sturgis Morgan and Catherine Frances Lovering Adams. His mother was the daughter of Frances Lovering and Charles Francis Adams III, the U.S. Secretary of the Navy under President Herbert Hoover, and a direct descendant of U.S. Presidents John Adams and John Quincy Adams. John attended the Groton School, graduating in 1949. He then attended Yale University, graduating with a Bachelor of Arts in 1953.


title: John Adams Morgan

text: He competed at the 1952 Summer Olympics in Helsinki, where he won a gold medal in the "6 metre class" with the boat "Llanoria".


title: John Adams Morgan

text: From 1956 to 1966, he was a partner in Dominick & Dominick. From 1966 to 1982, he worked at Smith Barney, serving as a senior vice president in charge of the corporate finance department, and as vice chairman of Smith Barney in charge of the firm's merger and acquisition activities, a member of the executive committee and a director of Smith Barney International Inc.


title: John Adams Morgan

text: In 1982, Morgan, the great-grandson of J. P. Morgan, established a retail and brokerage firm known as Morgan Lewis Githens & Ahn, Inc. In 1985, it organized a leveraged buyout with the Olin Corporation, an industrial chemical concern based in Stamford, Connecticut, of Olin's Ecusta cigarette paper business. In 1987, the firm assisted with the acquisition of Service America Corp. from Alleco Inc., formerly Allegheny Beverage Corporation, for $450 million in cash and securities.


title: John Adams Morgan

text: In 2001, the firm and Morgan's broker-dealer license, was bought by the newly established MLGA Holdings. Morgan, along with Fred Joseph (1937–2009), the former president and chief executive officer of the investment bank Drexel Burnham Lambert during the 1980s, co-founded the new entity, which became known as Morgan Joseph LLC in 2002, sought to create a high-yield business for mid-size companies and take advantage of investment bankers who were laid off during the technology stock bubble of 2000. After the new firm was established, Morgan served as chairman of the board of directors of Morgan Lewis.


title: John Adams Morgan

text: In December 2010, Morgan Joseph LLC merged with Tri-Artisan Partners LLC to form Morgan Joseph TriArtisan Group, Inc. In April 2011, Apollo Global Management invested in Morgan Joseph TriArtisan, and registered as a brokerage firm to find clients and deals for its buyout and hedge funds.


title: John Adams Morgan

text: Since 1969, Morgan has served as a director of Upham & Co., Inc. From 1989 until January 1998, he was a director of TriMas Corporation until it was acquired by Metaldyne Corporation. He then served as a director of Metaldyne from 1984 until its recapitalization in November 2000. As of 2001, he was a director of Furnishings International Inc. and a trustee of the Provident Loan Society of New York. He is also a director of the Morgan Library & Museum.


title: John Adams Morgan

text: Morgan has been married five times. In 1953, he married his first wife, Elizabeth Robbins Choate (1933–1998), the daughter of Robert Burnett Choate and the sister of Robert B. Choate Jr. Before their divorce in 1957, they had John Adams Morgan Jr.


title: John Adams Morgan

text: His second marriage was in 1962, to Tania Goss, an alumna of the Ethel Walker School and Vassar College who was the daughter of Natalie Holbrook and Chauncey Porter Goss (d. 1964) of Middlebury, Connecticut. Before their divorce, they had Chauncey Goss Morgan, who was the chief financial officer of Senet Inc.


title: John Adams Morgan

text: In the late 1990s, he met his fourth wife, Sonja Tremont (b. 1963), now known as one of the stars of the Bravo television show, "The Real Housewives of New York City". Morgan and Tremont were said to have met at San Pietro, an Italian restaurant in New York City on Madison Avenue where she was a hostess. After running into each other again in Aspen, they had dinner together and later that night, he proposed to her. They were married a few months later in 1998. Before their divorce 7 years later in 2006, they had one daughter together, Quincy Adams Morgan.


title: John Adams Morgan

text: Morgan owns Caritas Island, a 3.5 acre private island compound off the coast of Stamford, Connecticut with a 26-room, 14,000 square foot home originally built in 1906. In 2011, Morgan listed the island for sale for $18.9 million.


title: Jagdpanther

text: The Jagdpanther (German: "hunting Panther"), Sd.Kfz. 173, was a tank destroyer ("Jagdpanzer", a self-propelled anti-tank gun) built by Germany during World War II.


title: Jagdpanther

text: The "Jagdpanther" combined the 8.8 cm Pak 43 anti-tank gun, similar to the main gun of the Tiger II, and the armor and suspension of the Panther chassis.


title: Jagdpanther

text: It entered service in 1944 and served on the Eastern and Western Fronts. During the last stages of the war, limited German production resulted in small production numbers, shortage of spare parts, and shortened crew training periods of younger operators.


title: Jagdpanther

text: The Jagdpanther was preceded by two attempts at mounting an 8.8 cm gun as a self-propelled anti-tank weapon; "Ferdinand" - also known as "Panzerjäger Tiger" (P) - using the ninety-one leftover Porsche-built VK 45.01 (P) chassis from the Tiger tank competition it lost to Henschel in 1942, and the "Nashorn" on the Geschützwagen III/IV (which used a combination of the Panzer III and Panzer IV components) chassis. "Ferdinand" proved to be too heavy, and "Nashorn" lightly armored and under-powered.


title: Jagdpanther

text: A heavy tank destroyer design based on the 8.8 cm Pak 43 gun and the Panther tank chassis was ordered in late 1942. The full-size model by Daimler-Benz was demonstrated in October 1943 before Hitler. MIAG-built prototypes followed in October/November 1943.


title: Jagdpanther

text: Production started in January 1944; in February, Hitler specified the simpler "Jagdpanther" name instead of its original ""8.8 cm Pak 43/3 auf Fahrgestell Panther"".


title: Jagdpanther

text: To accommodate the heavier-calibre gun, as on previous "Jagdpanzer"-style unturreted tank destroyers, the glacis plate and sloped hull sides of the "Jagdpanther" were extended up into an integral fixed casemate as part of the main hull to provide a roomy interior. The "Jagdpanther" had side plates armour thickness increased (to 50 mm) to offset the slightly reduced angle of the side plate to provide enough interior space. Lower frontal hull plate was reduced to while upper hull frontal plate was kept at . The chassis armour changes were also introduced on the main Panther tank assembly lines with the Panther Ausf. G in spring 1944.


title: Jagdpanther

text: It was armed with the long-barreled 8.8 cm Pak 43/3 L/71 gun, similar to the main gun of the Tiger II ("King Tiger"). The gun was mounted in a central mantlet, giving it a limited traverse of twelve degrees to each side. A single 7.92 mm MG 34 machine gun was in a ball mount on the right side of the front glacis plate. The machine gunner was also the radio operator. The driver sat on the left. The gunner had a rangefinder and a periscope telescopic sight. The periscope – linked to the gun mount – was under an armoured housing on the roof.


title: Jagdpanther

text: The "Jagdpanther" had a good power-to-weight ratio and a powerful main gun, enabling crews to destroy any type of Allied tank. Based on the established Panther "Ausf" G chassis, the vehicle encountered few mechanical problems. It was crewed by five: driver, radio operator/machine gunner, commander, gunner, and a loader.


title: Jagdpanther

text: There were two main variants. The earlier G1 1944 model has a small welded main gun mantlet, one-piece Pak 43/3 gun, a modified Panther A engine deck, and had two vision openings for the driver. The G2 Jagdpanther used a Panther Ausf. G engine deck, a larger gun mantlet bolted externally, and a two-piece KwK 43/4 L/71 gun. Some later G1 models had 'hybrid' G2 features such as the larger G2 mantlet as changes to the design were implemented. Zimmerit was applied to G1s up to September 1944, then was withdrawn to decrease production time. Early "Jagdpanthers" had two vision openings for the driver, whereas late versions had only one. The main gun originally had a monobloc gun barrel, but, May 1944-on, it was gradually replaced by an economical two-part barrel after crews determined barrel wear was uneven.


title: Jagdpanther

text: A total of 413 "Jagdpanthers" were produced from January 1944 by three manufacturers: MIAG in Braunschweig produced 268 from January 1944 until the end of the war, Maschinenfabrik Niedersachsen Hannover (MNH) produced 112 from November 1944, Maschinenbau und Bahnbedarf (MBA) in Potsdam produced 33 vehicles from December 1944. Planned production was between 100 and 200 a month, but the disruption to German manufacturing made this goal unachievable.


title: Jagdpanther

text: The last 'production' Jagdpanthers were produced at the factory by German staff just after the end of World War II under the supervision of the Royal Electrical and Mechanical Engineers (REME) during the American and British occupation of Germany. Nine Panthers and a dozen Jagdpanthers were produced, then shipped to England for evaluation. A complete Panther and a complete Jagdpanther produced this way are displayed at the Bovington Tank Museum, Dorset, with brass plates on them explaining their history.


title: Jagdpanther

text: "Jagdpanther" equipped heavy antitank battalions ("schwere Panzerjäger-Abteilungen") and served mainly on the Eastern Front. In the West, they were encountered in very small numbers late in the Battle of Normandy; the German "654 schwere Panzerjäger-Abteilung" ("654th Heavy Antitank Battalion") deployed about twelve "Jagdpanthers" against the British armed forces. Seven "Jagdpanthers" of the 1. Schwere Panzerjäger Bataillon 559 were involved in the Battle of Geel in September 1944. Later, significant numbers were concentrated in the West for the Ardennes Offensive.


title: Jagdpanther

text: Three surviving "Jagdpanthers" are restored to running condition one each at the "Deutsches Panzermuseum" at Munster and the "Wehrtechnische Studiensammlung" (WTS) at Koblenz. The Weald Foundation in the UK restored one "Jagdpanther" to running condition and has a second undergoing restoration.


title: Quipu

text: Quipu (also spelled khipu) are recording devices fashioned from strings historically used by a number of cultures in the region of Andean South America.


title: Quipu

text: A "quipu" usually consisted of cotton or camelid fiber strings. The Inca people used them for collecting data and keeping records, monitoring tax obligations, collecting census records, calendrical information, and for military organization. The cords stored numeric and other values encoded as knots, often in a base ten positional system. A "quipu" could have only a few or thousands of cords. The configuration of the "quipus" has been "compared to string mops." Archaeological evidence has also shown the use of finely carved wood as a supplemental, and perhaps sturdier, base to which the color-coded cords would be attached. A relatively small number have survived.


title: Quipu

text: Objects that can be identified unambiguously as "quipus" first appear in the archaeological record in the first millennium AD (though debated quipus are much earlier). They subsequently played a key part in the administration of the Kingdom of Cusco and later the Incan Empire, flourishing across the Andes from c. 1100 to 1532 AD. Some have argued that as the region was subsumed under the Spanish Empire, the quipus were actively destroyed, to be replaced by European writing and numeral systems; but the Spanish response to the "quipu" is much more complicated than this. While certainly some "quipu" were identified as idolatrous and destroyed, the Spaniards actually promoted the adaptation of the "quipu" recording system to the needs of the colonial administration; and priests advocated the use of quipus for ecclesiastical purposes. In several modern villages, "quipus" have continued to be important items for the local community. It is unclear as to where and how many intact quipus still exist, as many have been stored away in mausoleums.


title: Quipu

text: Knotted strings unrelated to "quipu" have been used to record information by the ancient Chinese, Tibetans and Japanese.


title: Quipu

text: "Quipu" is the Spanish spelling and the most common spelling in English. "Khipu" (pronounced , plural: "khipukuna") is the word for "knot" in Cusco Quechua. In most Quechua varieties, the term is "kipu".


title: Quipu

text: ""Quipu"" is a Quechua word meaning "knot" or "to knot". The terms ""quipu"" and ""khipu"" are simply spelling variations on the same word. ""Quipu"" is the traditional Spanish spelling, while ""khipu"" reflects the recent Quechuan and Aymaran spelling shift.


title: Quipu

text: Most information recorded on the "quipus" studied to date by researchers consists of numbers in a decimal system, such as "Indian chiefs ascertain[ing] which province had lost more than another and balanc[ing] the losses between them" after the Spanish invasion. In the early years of the Spanish conquest of Peru, Spanish officials often relied on the quipus to settle disputes over local tribute payments or goods production. "Quipucamayocs" (Quechua "khipu kamayuq" "khipu specialist", plural: "khipu kamayuqkuna") could be summoned to court, where their bookkeeping was recognised as valid documentation of past payments.


title: Quipu

text: Some of the knots, as well as other features, such as color, are thought to represent non-numeric information, which has not been deciphered. It is generally thought that the system did not include phonetic symbols analogous to letters of the alphabet. However Gary Urton has suggested that the "quipus" used a binary system which could record phonological or logographic data. His student Manny Medrano has gone further to find "quipus" that decode to match census data.


title: Quipu

text: The lack of a clear link between any indigenous Peruvian languages and the "quipus" has historically led to the supposition that "quipus" are not a glottographic writing system and have no phonetic referent. Frank Salomon at the University of Wisconsin has argued that "quipus" are actually a semasiographic language, a system of representative symbolssuch as music notation or numeralsthat relay information but are not directly related to the speech sounds of a particular language. The Khipu Database Project (KDP), begun by Gary Urton, may have already decoded the first word from a quipu–the name of a village, Puruchuco–which Urton believes was represented by a three-number sequence, similar to a ZIP code. If this conjecture is correct, "quipus" are the only known example of a complex language recorded in a 3-D system.


title: Quipu

text: Most recently, Sabine Hyland claims to have made the first phonetic decipherment of a "quipu," challenging the assumption that "quipus" do not represent information phonetically. After being contacted by local woman Meche Moreyra Orozco, the head of the Association of Collatinos in Lima, Hyland was granted access to the epistolary "quipus" of San Juan de Collata. These "quipus" were exchanged during an 18th century rebellion against the Spanish government. A combination of color, fiber and ply direction leads to a total of 95 combinations in these "quipus", which is within the range of a logosyllabic writing system. Exchanging information about the rebellion through "quipus" would have prevented the Spanish authorities from understanding the messages if they were intercepted, and the Collata "quipus" are non-numeric. With the help of local leaders, who described the "quipu" as "a language of animals", Hyland was able to translate the names of the two "ayllus", or family lineages, who received and sent the "quipu". The translation relied on phonetic references to the animal fibers and colors of the relevant "quipu" cords.


title: Quipu

text: Marcia Ascher and Robert Ascher, after having analyzed several hundred "quipus", have shown that most information on "quipus" is numeric, and these numbers can be read. Each cluster of knots is a digit, and there are three main types of knots: simple overhand knots; "long knots", consisting of an overhand knot with one or more additional turns; and figure-eight knots. In the Aschers’ system, a fourth type of knot—figure-eight knot with an extra twist—is referred to as "EE". A number is represented as a sequence of knot clusters in base 10.


title: Quipu

text: For example, if 4s represents four simple knots, 3L represents a long knot with three turns, E represents a figure-eight knot and X represents a space:


title: Quipu

text: This reading can be confirmed by a fortunate fact: "quipus" regularly contain sums in a systematic way. For instance, a cord may contain the sum of the next "n" cords, and this relationship is repeated throughout the "quipu". Sometimes there are sums of sums as well. Such a relationship would be very improbable if the knots were incorrectly read.


title: Quipu

text: Some data items are not numbers but what Ascher and Ascher call "number labels". They are still composed of digits, but the resulting number seems to be used as a code, much as we use numbers to identify individuals, places, or things. Lacking the context for individual quipus, it is difficult to guess what any given code might mean. Other aspects of a "quipu" could have communicated information as well: color-coding, relative placement of cords, spacing, and the structure of cords and sub-cords.


title: Quipu

text: Some have argued that far more than numeric information is present and that "quipus" are a writing system. This would be an especially important discovery as there is no surviving record of written Quechua predating the Spanish invasion. Possible reasons for this apparent absence of a written language include destruction by the Spanish of all written records, or the successful concealment by the Inca peoples of those records. Making the matter even more complex, the Inca 'kept separate "khipu" for each province, on which a pendant string recorded the number of people belonging to each category.' This creates yet another step in the process of decryption in addition to the Spanish attempts at eradicating the system. Historians Edward Hyams and George Ordish believe quipus were recording devices, similar to musical notation, in that the notes on the page present basic information, and the performer would then bring those details to life.


title: Quipu

text: In 2003, while checking the geometric signs that appear on drawings of Inca dresses from the "First New Chronicle and Good Government", written by Felipe Guaman Poma de Ayala in 1615, William Burns Glynn found a pattern that seems to decipher some words from "quipus" by matching knots to colors of strings.


title: Quipu

text: The August 12, 2005, edition of the journal "Science" includes a report titled "Khipu Accounting in Ancient Peru" by anthropologist Gary Urton and mathematician Carrie J. Brezine. Their work may represent the first identification of a "quipu" element for a non-numeric concept, a sequence of three figure-eight knots at the start of a "quipu" that seems to be a unique signifier. It could be a toponym for the city of Puruchuco (near Lima), or the name of the "quipu" keeper who made it, or its subject matter, or even a time designator.


title: Quipu

text: Beynon-Davies considers "quipus" as a sign system and develops an interpretation of their physical structure in terms of the concept of a data system.


title: Quipu

text: "Khipu kamayuqkuna" (knot makers/keepers, i.e., the former Inca record keepers) supplied colonial administrators with a variety and quantity of information pertaining to censuses, tribute, ritual and calendrical organization, genealogies, and other such matters from Inca times. Performing a number of statistical tests for "quipu" sample VA 42527, one study led by Alberto Sáez-Rodríguez discovered that the distribution and patterning of S- and Z-knots can organize the information system from a real star map of the Pleiades cluster.


title: Quipu

text: Laura Minelli, a professor of pre-Columbian studies at the University of Bologna, has discovered something which she believed to be a seventeenth-century Jesuit manuscript that describes literary "quipus", titled . This manuscript consists of nine folios with Spanish, Latin, and ciphered Italian texts. Owned by the family of Neapolitan historian Clara Miccinelli, the manuscript also includes a wool "quipu" fragment. Miccinelli believes that the text was written by two Italian Jesuit missionaries, Joan Antonio Cumis and Giovanni Anello Oliva, around 1610–1638, and Blas Valera, a mestizo Jesuit sometime before 1618. Along with the details of reading literary "quipus", the documents also discuss the events and people of the Spanish conquest of Peru. According to Cumis, since so many "quipus" were burned by the Spanish, very few remained for him to analyze. As related in the manuscript, the word Pacha Kamaq, the Inca deity of earth and time, was used many times in these "quipus", where the syllables were represented by symbols formed in the knots. Following the analysis of the use of "Pacha Kamaq", the manuscript offers a list of many words present in "quipus". However, both Bruce Mannheim, the director of the Center for Latin American Studies at the University of Michigan, and Colgate University's Gary Urton, question its origin and authenticity. These documents seem to be inspired freely by a 1751 writing of Raimondo di Sangro, Prince of Sansevero.


title: Quipu

text: "Quipucamayocs" (Quechua "khipu kamayuq", "khipu-authority"), the accountants of Tawantin Suyu, created and deciphered the "quipu" knots. "Quipucamayocs" could carry out basic arithmetic operations, such as addition, subtraction, multiplication, and division. They kept track of mita, a form of taxation. The "quipucamayocs" also tracked the type of labor being performed, maintained a record of economic output, and ran a census that counted everyone from infants to "old blind men over 80". The system was also used to keep track of the calendar. According to Guaman Poma, "quipucamayocs" could "read" the quipus with their eyes closed.


title: Quipu

text: "Quipucamayocs" were from a class of people, "males, fifty to sixty", and were not the only members of Inca society to use quipus. Inca historians used quipus when telling the Spanish about Tawantin Suyu history (whether they only recorded important numbers or actually contained the story itself is unknown). Members of the ruling class were usually taught to read quipus in the Inca equivalent of a university, the "yachay wasi" (literally, "house of teaching"), in the third year of schooling, for the higher classes who would eventually become the bureaucracy.


title: Quipu

text: In 1532, the Spanish Empire's conquest of the Andean region began, with several Spanish conquerors making note of the existence of "quipus" in their written records about the invasion. The earliest known example comes from Hernando Pizarro, the brother of the Spanish military leader Francisco Pizarro, who recorded an encounter that he and his men had in 1533 as they traveled along the royal road from the highlands to the central coast. It was during this journey that they encountered several "quipu" keepers, later relating that these keepers "untied some of the knots which they had in the deposits section [of the khipu], and they [re-]tied them in another section [of the khipu]."


title: Quipu

text: Christian officials of the Third Council of Lima banned and ordered the burning of all Quipus in 1583 because they were used to record offerings to non-Christian gods and were therefore considered idolatrous objects and an obstacle to religious conversion.


title: Quipu

text: The "quipu" system operated as both a method of calculation and social organization, regulating regional governance and land use. While evidence for the latter is still under the critical eye of scholars around the world, the very fact that they are kept to this day without any confirmed level of fluent literacy in the system is testament to its historical 'moral authority.' Today, "khipu" is regarded as a powerful symbol of heritage, only 'unfurled' and handled by 'pairs of [contemporary] dignitaries,' as the system and its 'construction embed' modern 'cultural knowledge.' Ceremonies in which they are 'curated, even though they can no longer be read,' is even further support for the case of societal honor and significance associated with the quipu. Even today, 'the knotted cords must be present and displayed when village officers leave or begin service, and draping the cords over the incoming office holders instantiates the moral and political authority of the past.' These examples are indicative of how the "quipu" system is not only fundamental mathematically or linguistically for the original Inca, but also for the cultural preservation of the original empire's descendants.


title: Quipu

text: Anthropologists and archaeologists carrying out research in Peru have highlighted two known cases where "quipus" have continued to be used by contemporary communities, albeit as ritual items seen as "communal patrimony" rather than as devices for recording information. The khipu system, being the useful method of social management it was for the Inca, is also a link to the Cuzco census, as it was one of the primary methods of population calculation. This also has allowed historians and anthropologists to understand both the census and the "decimal hierarchy" system the Inca used, and that they were actually 'initiated together,' due to the fact that they were 'conceptually so closely linked.'


title: Quipu

text: In 1994, the American cultural anthropologist Frank Salomon conducted a study in the Peruvian village of Tupicocha, where "quipus" are still an important part of the social life of the village. As of 1994, this was the only known village where quipus with a structure similar to pre-Columbian "quipus" were still used for official local government record-keeping and functions, although the villagers did not associate their quipus with Inca artifacts.


title: Quipu

text: The villagers of San Cristóbal de Rapaz (known as Rapacinos), located in the Province of Oyón, keep a "quipu" in an old ceremonial building, the "Kaha Wayi", that is itself surrounded by a walled architectural complex. Also within the complex is a disused communal storehouse, known as the "Pasa Qullqa", which was formerly used to protect and redistribute the local crops, and some Rapacinos believe that the "quipu" was once a record of this process of collecting and redistributing food. The entire complex was important to the villagers, being "the seat of traditional control over land use, and the centre of communication with the deified mountains who control weather".


title: Quipu

text: In 2004, the archaeologist Renata Peeters (of the UCL Institute of Archaeology in London) and the cultural anthropologist Frank Salomon (of the University of Wisconsin) undertook a project to conserve both the "quipus" in Rapaz and the building that it was in, due to their increasingly poor condition.


title: Quipu

text: In 1912 anthropologist Leslie Leland Locke published "The Ancient Quipu, A Peruvian Knot Record," "American Anthropologist," New Series I4 (1912) 325-332. This was the first work to show how the Inca (Inka) Empire and its predecessor societies used the "quipu" (Khipu) for mathematical and accounting records in the decimal system.


title: Quipu

text: The archaeologist Gary Urton noted in his 2003 book "Signs of the Inka Khipu" that he estimated "from my own studies and from the published works of other scholars that there are about 600 extant "quipu" in public and private collections around the world."


title: Quipu

text: According to the Khipu Database Project undertaken by Harvard University professor Gary Urton and his colleague Carrie Brezine, 751 quipus have been reported to exist across the globe. Their whereabouts range from Europe to North and South America. Most are housed in museums outside of their native countries, but some reside in their native locations under the care of the descendants of those who made the knot records. A table of the largest collections is shown below.


title: Quipu

text: While patrimonial "quipu" collections have not been accounted for in this database, their numbers are likely to be unknown. One prominent patrimonial collection held by the Rapazians of Rapaz, Peru, was recently researched by University of Wisconsin–Madison professor, Frank Salomon. The Anthropology/Archaeology department at the University of California at Santa Barbara also holds one "quipu".


title: Quipu

text: "Quipus" are made of fibers, either spun and plied thread such as wool or hair from alpaca, llama, guanaco or vicuña, though are also commonly made of cellulose like cotton. The knotted strings of "quipus" were often made with an "elaborate system of knotted cords, dyed in various colors, the significance of which was known to the magistrates". Fading of color, natural or dyed, cannot be reversed, and may indicate further damage to the fibers. Colors can darken if damaged by dust or by certain dyes and mordants. "Quipus" have been found with adornments, such as animal shells, attached to the cords, and these non-textile materials may require additional preservation measures.


title: Quipu

text: "Quipus" are now preserved using techniques that will minimise their future degradation. Museums, archives and special collections have adopted preservation guidelines from textile practices.


title: Quipu

text: Environmental controls are used to monitor and control temperature, humidity and light exposure of storage areas. As with all textiles, cool, clean, dry and dark environments are most suitable. The heating, ventilating and air conditioning, or HVAC systems, of buildings that house "quipu" knot records are usually automatically regulated. Relative humidity should be 60% or lower, with low temperatures, as high temperatures can damage the fibres and make them brittle. Damp conditions and high humidity can damage protein-rich material. Textiles suffer damage from ultraviolet (UV) light, which can include fading and weakening of the fibrous material. When quipus are on display, their exposure to ambient conditions is usually minimized and closely monitored.


title: Quipu

text: Damage can occur during storage. The more accessible the items are during storage, the greater the chance of early detection. Storing quipus horizontally on boards covered with a neutral pH paper (paper that is neither acid or alkaline) to prevent potential acid transfer is a preservation technique that extends the life of a collection. The fibers can be abraded by rubbing against each other or, for those attached to sticks or rods, by their own weight if held in an upright position. Extensive handling of "quipus" can also increase the risk of further damage.


title: Quipu

text: "Quipus" are also closely monitored for mold, as well as insects and their larvae. As with all textiles, these are major problems. Fumigation may not be recommended for fiber textiles displaying mold or insect infestations, although it is common practice for ridding paper of mold and insects.