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Thursday, July 14, 2022

History of lysergic acid diethylamide

The psychedelic drug (or entheogen) lysergic acid diethylamide (LSD) was first synthesized on November 16, 1938, by the Swiss chemist Albert Hofmann in the Sandoz (now Novartis) laboratories in Basel, Switzerland. It was not until five years later on April 19, 1943, that the psychedelic properties were found.

Discovery

Albert Hofmann, born in Switzerland, joined the pharmaceutical-chemical department of Sandoz Laboratories, located in Basel, as a co-worker with professor Arthur Stoll, founder and director of the pharmaceutical department. He began studying the medicinal plant squill and the fungus ergot as part of a program to purify and synthesize active constituents for use as pharmaceuticals. His main contribution was to elucidate the chemical structure of the common nucleus of Scilla glycosides (an active principle of Mediterranean squill). While researching lysergic acid derivatives, Hofmann first synthesized LSD on November 16, 1938. The main intention of the synthesis was to obtain a respiratory and circulatory stimulant (an analeptic). It was set aside for five years, until April 16, 1943, when Hofmann decided to take a second look at it. While re-synthesizing LSD, he accidentally absorbed a small amount of the drug and discovered its powerful effects. He described what he felt as being:

... affected by a remarkable restlessness, combined with a slight dizziness. At home I lay down and sank into a not unpleasant intoxicated-like condition, characterized by an extremely stimulated imagination. In a dreamlike state, with eyes closed (I found the daylight to be unpleasantly glaring), I perceived an uninterrupted stream of fantastic pictures, extraordinary shapes with intense, kaleidoscopic play of colors. After about two hours this condition faded away.

"Bicycle Day"

BICYCLE DAY HOFFMAN 25 TABS.jpg
LSD blotter commemorating Bicycle Day
TypeSecular
CelebrationsConsumption of lysergic acid diethylamide (LSD)
ObservancesHonors the anniversary of the first ever acid trip, undergone by Swiss chemist Dr. Albert Hofmann on April 19, 1943 in Basel, Switzerland.
DateApril 19
Next time19 April 2023
FrequencyAnnual

On April 19, 1943, Hofmann ingested 0.25 milligrams (250 micrograms) of the substance. Less than one hour later, Hofmann experienced sudden and intense changes in perception. He asked his laboratory assistant to escort him home. As was customary in Basel, they made the journey by bicycle. On the way, Hofmann's condition rapidly deteriorated as he struggled with feelings of anxiety, alternating in his beliefs that the next-door neighbor was a malevolent witch, that he was going insane, and that the LSD had poisoned him. When the house doctor arrived, however, he could detect no physical abnormalities, save for a pair of incredibly dilated pupils. Hofmann was reassured, and soon his terror began to give way to a sense of good fortune and enjoyment, as he later wrote:

... Little by little I could begin to enjoy the unprecedented colors and plays of shapes that persisted behind my closed eyes. Kaleidoscopic, fantastic images surged in on me, alternating, variegated, opening and then closing themselves in circles and spirals, exploding in colored fountains, rearranging and hybridizing themselves in constant flux ...

The events of the first LSD trip, now known as "Bicycle Day", after the bicycle ride home, proved to Hofmann that he had indeed made a significant discovery: a psychoactive substance with extraordinary potency, capable of causing significant shifts of consciousness in incredibly low doses. (The term trip was first coined by US Army scientists during the 1950s when they were experimenting with LSD.) Hofmann foresaw the drug as a powerful psychiatric tool; because of its intense and introspective nature, he could not imagine anyone using it recreationally. Bicycle Day is increasingly observed in psychedelic communities as a day to celebrate the discovery of LSD.

The celebration of Bicycle Day originated in DeKalb, Illinois, in 1985, when Thomas B. Roberts, then a professor at Northern Illinois University, invented the name "Bicycle Day" when he founded the first celebration at his home. Several years later, he sent an announcement made by one of his students to friends and Internet lists, thus propagating the idea and the celebration. His original intent was to commemorate Hofmann's original, accidental exposure on April 16, but that date fell midweek and was not a good time for the party, so he chose the 19th to honor Hofmann's first intentional exposure.

Psychiatric use

LSD was introduced as a commercial medication under the trade-name Delysid for various psychiatric uses in 1947.

LSD was brought to the attention of the United States in 1949 by Sandoz Laboratories because they believed LSD might have clinical applications.

Throughout the 1950s, mainstream media reported on research into LSD and its growing use in psychiatry, and undergraduate psychology students taking LSD as part of their education described the effects of the drug. Time magazine published six positive reports on LSD between 1954 and 1959.

LSD was originally perceived as a psychotomimetic capable of producing model psychosis. By the mid-1950s, LSD research was being conducted in major American medical centers, where researchers used LSD as a means of temporarily replicating the effects of mental illness. One of the leading authorities on LSD during the 1950s in the United States was the psychoanalyst Sidney Cohen. Cohen first took the drug on October 12, 1955, and expected to have an unpleasant trip, but was surprised when he experienced "no confused, disoriented delirium." He reported that the "problems and strivings, the worries and frustrations of everyday life vanished; in their place was a majestic, sunlit, heavenly inner quietude." Cohen immediately began his own experiments with LSD with the help of Aldous Huxley whom he had met in 1955. In 1957, with the help of psychologist Betty Eisner, Cohen began experimenting on whether or not LSD might have a helpful effect in facilitating psychotherapy, curing alcoholism, and enhancing creativity. Between 1957 and 1958, they treated 22 patients who had minor personality disorders. LSD was also given to artists in order to track their mental deterioration, but Huxley believed LSD might enhance their creativity. Between 1958 and 1962, psychiatrist Oscar Janiger tested LSD on more than 100 painters, writers, and composers.

In one study in the late 1950s, Dr. Humphry Osmond gave LSD to alcoholics in Alcoholics Anonymous who had failed to quit drinking. After one year, around 50% of the study group had not had a drink—a success rate that has never been duplicated by any other means. Bill Wilson, the founder of Alcoholics Anonymous, participated in medically supervised experiments on the effects of LSD on alcoholism and believed LSD could be used to cure alcoholics.

In the United Kingdom the use of LSD was pioneered by Dr. Ronald A. Sandison in 1952, at Powick Hospital, Worcestershire. A special LSD unit was set up in 1958. After Sandison left the hospital in 1964, medical superintendent Arthur Spencer took over and continued the clinical use of the drug until it was withdrawn in 1965. In all, 683 patients were treated with LSD in 13,785 separate sessions at Powick, but Spencer was the last member of the medical staff to use it.

From the late 1940s through the mid-1970s, extensive research and testing was conducted on LSD. During a 15-year period beginning in 1950, research on LSD and other hallucinogens generated over 1,000 scientific papers, several dozen books, and six international conferences. Overall, LSD was prescribed as treatment to over 40,000 patients. Film star Cary Grant was one of many men during the 1950s and 1960s who were given LSD in concert with psychotherapy. Many psychiatrists began taking the drug recreationally and sharing it with friends. Dr. Leary's experiments (see Timothy Leary below) spread LSD usage to a much wider segment of the general populace.

Sandoz halted LSD production in August 1965 after growing governmental protests at its proliferation among the general populace. The National Institute of Mental Health in the United States distributed LSD on a limited basis for scientific research. Scientific study of LSD largely ceased by about 1980 as research funding declined, and governments became wary of permitting such research, fearing that the results of the research might encourage illicit LSD use. By the end of the 20th century, there were few authorized researchers left, and their efforts were mostly directed towards establishing approved protocols for further work with LSD in easing the suffering of the dying and with drug addicts and alcoholics.

A 2014 study showed evidence that LSD can have therapeutic benefits in treating anxiety associated with life-threatening diseases. Rick Doblin, an American drug researcher, described the work as "a proof of concept" that he hoped would "break these substances out of the mold of the counterculture and bring them back to the lab as part of a psychedelic renaissance."

Eight subjects received a full 200-microgram dose of LSD while four others received one-tenth as much. Participants then took part in two LSD-assisted therapy sessions two to three weeks apart. Subjects who took the full dose experienced reductions in anxiety averaging 20 per cent while those given the low dose reported becoming more anxious.

When subjects taking the low dose were switched to the full dose they too showed reduced anxiety, with the positive effects lasting for up to a year. The effects of the drug itself lasted for up to 10 hours with participants talking to Dr Gasser throughout the experience.

"These results indicate that when administered safely in a methodologically rigorous medically supervised psychotherapeutic setting, LSD can reduce anxiety," the study concludes, "suggesting that larger controlled studies are warranted."

Resistance and prohibition

LSD blotter
 

By the mid-1960s the backlash against the use of LSD and its perceived corrosive effects on cultural values resulted in governmental action to restrict the availability of the drug by making use of it illegal. LSD was declared a "Schedule I" substance, legally designating that the drug has a "high potential for abuse" and is without any "currently accepted medical use in treatment." LSD was removed from legal circulation. The United States Drug Enforcement Administration claimed:

Although the initial observations on the benefits of LSD were highly optimistic, empirical data developed subsequently proved less promising ... Its use in scientific research has been extensive and its use has been widespread. Although the study of LSD and other hallucinogens increased the awareness of how chemicals could affect the mind, its use in psychotherapy largely has been debunked. It produces no aphrodisiac effects, does not increase creativity, has no lasting positive effect in treating alcoholics or criminals, does not produce a 'model psychosis', and does not generate immediate personality change.

However, drug studies have confirmed that the powerful hallucinogenic effects of this drug can produce profound adverse reactions, such as acute panic reactions, psychotic crises, and "flashbacks", especially in users ill-equipped to deal with such trauma.

The governors of Nevada and California each signed bills into law on May 30, 1966, that make them the first two American states to outlaw the manufacture, sale, and possession of the drug. The law went into effect immediately in Nevada, and on October 6, 1966, in California. Other U.S. states and many other countries soon followed with similar bans.

Influential individuals

Aldous Huxley

Renowned British intellectual Aldous Huxley was one of the most important figures in the early history of LSD. He was a figure of high repute in the world of letters and had become internationally famous through his novels Crome Yellow, Antic Hay and his dystopian novel Brave New World. His experiments with psychedelic drugs (initially mescaline) and his descriptions of them in his writings did much to spread awareness of psychedelic drugs to the general public and arguably helped to glamorize their recreational use, although Huxley himself treated them very seriously.

Huxley was introduced to psychedelic drugs in 1953 by a friend, psychiatrist Humphry Osmond. Osmond had become interested in hallucinogens and their relationship to mental illness in the 1940s. During the 1950s, he completed extensive studies of a number of drugs, including mescaline and LSD. As noted above, Osmond had some remarkable success in treating alcoholics with LSD.

In May 1953 Osmond gave Huxley his first dose of mescaline at the Huxley home. In 1954 Huxley recorded his experiences in the landmark book The Doors of Perception; the title was drawn from a quotation by British artist and poet William Blake. Huxley tried LSD for the first time in 1955, obtained from "Captain" Al Hubbard.

Alfred Hubbard

Alfred Matthew Hubbard is reputed to have introduced more than 6,000 people to LSD, including scientists, politicians, intelligence officials, diplomats, and church figures. He became known as the original "Captain Trips", travelling about with a leather case containing pharmaceutically pure LSD, mescaline, and psilocybin. He became a 'freelance' apostle for LSD in the early 1950s after supposedly receiving an angelic vision telling him that something important to the future of mankind would soon be coming. When he read about LSD the next year, he immediately sought and acquired LSD, which he tried for himself in 1951.

Although he had no medical training, Hubbard collaborated on running psychedelic sessions with LSD with Ross McLean at Vancouver's Hollywood Hospital, with psychiatrists Abram Hoffer and Humphry Osmond; with Myron Stolaroff at the International Federation for Advanced Study in Menlo Park, California; and with Willis Harman at the Stanford Research Institute (SRI). At various times over the next 20 years, Hubbard also reportedly worked for the Canadian Special Services, the U.S. Justice Department and the U.S. Bureau of Alcohol, Tobacco & Firearms. It is also rumored that he was involved with the CIA's MK-ULTRA project. How his government positions actually interacted with his work with LSD is unknown.

Harold A. Abramson

In 1955, Time magazine reported:

"In Manhattan, Psychiatrist Harold A. Abramson of the Cold Spring Harbor Biological Laboratory has developed a technique of serving dinner to a group of subjects, topping off the meal with a liqueur glass containing 40 micrograms of LSD."

This mention in America's most popular newsweekly is noteworthy because Abramson was not a psychiatrist or even a psychologist, but was an allergist who was a key participant in the CIA MK-ULTRA mind-control program.

R. Gordon Wasson

In 1957, R. Gordon Wasson, the vice president of J.P. Morgan, published an article in Life magazine extolling the virtues of magic mushrooms. This prompted Albert Hofmann to isolate psilocybin in 1958 for distribution by Sandoz with its product LSD in the U.S., further raising interest in LSD in the mass media. Following Wasson's report, Timothy Leary visited Mexico to experience the mushrooms.

Timothy Leary

DEA agents Howard Safir (left) and Don Strange (right) with Leary in custody (1972)

Dr. Timothy Leary, a lecturer in psychology at Harvard University, was the most prominent pro-LSD researcher. Leary claimed that using LSD with the right dosage, set (one's emotional mindset at time of ingestion), and setting, preferably with the guidance of professionals, could alter behavior in dramatic and beneficial ways. Leary began conducting experiments with psilocybin in 1960 on himself and a number of Harvard graduate students after trying hallucinogenic mushrooms used in Native American religious rituals while visiting Mexico. His group began conducting experiments on state prisoners, where they claimed a 90% success rate preventing repeat offenses.

Later reexamination of Leary's data reveals his results to be skewed, whether intentionally or not; the percent of men in the study who ended up back in prison later in life was approximately 2% lower than the usual rate. Leary was later introduced to LSD, and he then incorporated that drug into his research as his mental catalyst of choice. Leary claimed that his experiments produced no murders, suicides, psychotic breaks, or bad trips. Almost all of Leary's participants reported profound mystical experiences which they felt had a tremendously positive effect on their lives. While it is true that Leary's experiments did not lead to any murders, he willfully chose to ignore the bad trips which occurred, as well as the attempted suicide of a woman the day after she was given mescaline by Leary.

By 1962, the Harvard faculty's disapproval with Leary's experiments reached critical mass. Leary was informed that the CIA was monitoring his research (see Government experiments below). Many of the other faculty members had harbored reservations about Leary's research, and parents began complaining to the university about Leary's distribution of hallucinogenic drugs to their children. Further, many undergraduate students who were not part of Leary's research program heard of the profound experiences other students had undergone and began taking LSD for recreational purposes, which was not illegal at the time . Leary described LSD as a potent aphrodisiac in an interview with Playboy magazine. Leary left the university for an extended amount of time during the spring semester, thus failing to fulfill his duties as professor. Leary and another Harvard psychologist, Richard Alpert, were dismissed from the university in 1963.

In 1964, they published The Psychedelic Experience: A Manual Based on the Tibetan Book of the Dead, which argued that the psychedelic experience paralleled the death/rebirth experience described in the Bardo Thodol (Tibetan Book of the Dead). Leary and Alpert, unfazed by their dismissals, relocated first to Mexico, but were expelled from the country by the Mexican government. They then set up at a large private mansion owned by William Hitchcock, named after the small town in New York State where it is located, Millbrook, where they continued their experiments. Their research lost its controlled scientific character as the experiments transformed into LSD parties. Leary later wrote, "We saw ourselves as anthropologists from the twenty-first century inhabiting a time module set somewhere in the Dark Ages of the 1960s. On this space colony, we were attempting to create a new paganism and a new dedication to life as art."

A judge who expressed dislike for Leary's books sentenced him to 30 years in prison for possession of half a marijuana cigarette in violation of the Marihuana Tax Act of 1937. However, this decision was reversed in the 1969 U.S. Supreme Court case Timothy Leary v. United States (395 U.S. 6) on the grounds that the Act required self-incrimination, thus violating the Fifth Amendment of the U.S. Constitution. Publicity surrounding the case further cemented Leary's growing reputation as a counter cultural guru. Around this time, President Richard Nixon described Leary as "the most dangerous man in America." Repeated FBI raids instigated the end of the Millbrook experiment. Leary refocused his efforts towards countering the tremendous amount of anti-LSD propaganda then being issued by the United States government, popularizing the slogan "Turn on, tune in, drop out." Many experts blame Leary and his activism for the near-total suppression of psychedelic research over the next 35 years.

Owsley Stanley

Historically, LSD was distributed not for profit, but because those who made and distributed it truly believed that the psychedelic experience could be beneficial for humanity. A limited number of chemists, probably fewer than a dozen, are believed to have manufactured nearly all of the illicit LSD available in the United States. The best known of these is undoubtedly Augustus Owsley Stanley III, usually known simply as Owsley or Bear. The former chemistry student set up a private LSD lab in the mid-60s in San Francisco and supplied the LSD consumed at the famous Acid Test parties held by Ken Kesey and his Merry Pranksters, as well as the Human Be-In in San Francisco in January 1967 and the Monterey International Pop Festival in June 1967. He also had close social connections the Grateful Dead, Jefferson Airplane, and Big Brother and The Holding Company, regularly supplying them with LSD and working as their live sound engineer, creating many tapes of these groups in concert. Owsley's LSD activities—immortalized by Steely Dan in their song "Kid Charlemagne"—ended with his arrest at the end of 1967, but some other manufacturers most likely operated continuously for 30 years or more. Announcing Owsley's first bust in 1966, The San Francisco Chronicle's headline "LSD Millionaire Arrested" inspired the rare Grateful Dead song "Alice D. Millionaire".

Owsley associated with other early LSD producers, Tim Scully and Nicholas Sand.

Ken Kesey

Ken Kesey was born in 1935 in La Junta, Colorado to dairy farmers Frederick A. Kesey and Ginevra Smith. In 1946, the family moved to Springfield, Oregon. A champion wrestler in both high school and college, he graduated from Springfield High School in 1953.

Kesey attended the University of Oregon's School of Journalism, where he received a degree in speech and communication in 1957, where he was also a brother of Beta Theta Pi. He was awarded a Woodrow Wilson National Fellowship in 1958 to enroll in the creative writing program at Stanford University, which he did the following year. While at Stanford, he studied under Wallace Stegner and began the manuscript that would become One Flew Over the Cuckoo's Nest.

At Stanford in 1959, Kesey volunteered to take part in a CIA-financed study named Project MKULTRA at the Menlo Park Veterans Hospital. The project studied the effects on the patients of psychoactive drugs, particularly LSD, psilocybin, mescaline, cocaine, AMT, and DMT. Kesey wrote many detailed accounts of his experiences with these drugs, both during the Project MKULTRA study and in the years of private experimentation that followed. Kesey's role as a medical guinea pig inspired him to write the book One Flew Over the Cuckoo's Nest in 1962. The success of the book, as well as the sale of his residence at Stanford, allowed him to move to La Honda, California in the mountains west of Stanford University. He frequently entertained friends and many others with parties he called "Acid Tests" involving music (such as Kesey's favorite band, The Warlocks, later known as the Grateful Dead), black lights, fluorescent paint, strobes and other "psychedelic" effects, and, of course, LSD. These parties were noted in some of Allen Ginsberg's poems and are also described in the books The Electric Kool-Aid Acid Test by Tom Wolfe, Hell's Angels: The Strange and Terrible Saga of the Outlaw Motorcycle Gangs by Hunter S. Thompson, and Freewheelin Frank, Secretary of the Hell's Angels by Frank Reynolds. Ken Kesey was also said to have experimented with LSD with Ringo Starr in 1965 and that he influenced the setup for future performances with The Beatles in the UK.

In the summer of 1964, Kesey's Merry Pranksters customized a bus named "Furthur" and set out on a tour to propagate LSD use.

Sidney Cohen

Sidney Cohen was a Los Angeles-based psychiatrist. His work primarily focused on the effects of psychedelics, primarily LSD. Cohen published 13 books in his life, all of them being based around drugs and substance abuse. He began working on LSD in the 1950s. One of his earlier works is a video of an experiment that shows Cohen interviewing a woman before and after administering her LSD. In the later part of the 1960s he worked as a director for the National Institute of Mental Health in their Division of Narcotic Addiction and Drug Abuse. He has been open about having taken LSD many times himself, but was always opposed to the growing use of LSD amongst members of the counterculture movement. Cohen thought LSD was only safe if used under medical supervision and that the average person was not equipped with the ability to safely handle the drug. Through his work he had become known as one of the leading experts in LSD research.

William Leonard Pickard

William Leonard Pickard earned a scholarship to Princeton University but dropped out after one term, instead preferring to hang out at Greenwich Village jazz clubs in New York City. In 1971, he got a job as a research manager at the University of California, Berkeley in the Department of Bacteriology and Immunology, a job he held until 1974.

In December 1988, a neighbor reported a strange chemical odor coming from an architectural shop at a Mountain View, California industrial park. Federal agents arrived to find 200,000 doses of LSD and William Pickard inside. Pickard was charged with manufacturing LSD and served five years in prison.

By 1994, Pickard had enrolled at the John F. Kennedy School of Government at Harvard University. His studies focused on drug abuse in the former Soviet Union, where he theorized that the booming black market and many unemployed chemists could lead to a flood of the drug market.

In 2000, Pickard was arrested for manufacturing LSD in Kansas and was serving two life sentences at United States Penitentiary, Tucson. On July 27, 2020, Pickard was granted Compassionate Release from federal prison after serving 17 years of his sentence.

Secret government research

The U.S. Central Intelligence Agency (CIA) became interested in LSD when they read reports alleging that American prisoners during the Korean War were being brainwashed with the use of some sort of drug or "lie serum". LSD was the original centerpiece of the top secret MKULTRA project, an ambitious undertaking conducted from the 1950s through the 70s designed to explore the possibilities of pharmaceutical mind control. Hundreds of participants, including CIA agents, government employees, military personnel, prostitutes, members of the general public, and mental patients were given LSD, many without their knowledge or consent. The experiments often involved severe psychological torture. To guard against outward reactions, doctors conducted experiments in clinics and laboratories where subjects were monitored by EEG machines and had their words recorded. Some studies investigated whether drugs, stress or specific environmental conditions could be used to break prisoners or to induce confessions.

The CIA also created The Society for the Investigation of Human Ecology, which was a CIA funding front which provided grants to social scientists and medical researchers investigating questions of interest related to the MKULTRA program. Between 1960 and 1963, the CIA gave $856,782 worth of grants to different organizations. The researchers eventually concluded that LSD's effects were too varied and uncontrollable to make it of any practical use as a truth drug, and the project moved on to other substances. It would be decades before the U.S. government admitted the existence of the project and offered apologies to the families of those who were forced to participate in the experiments. During this time period, the use of LSD for psychochemical warfare was under consideration and testing, among other substances. Looking to replicate the effects of nerve gas created by the Germans during World War II without the toxicity, LSD was sought for use under the pretense that it could induce hysteria and psychoses, or at least an inability to fight without wholesale destruction of the enemy and their properties. Thousands of tests on willing research subjects took place at the Edgewood Arsenal in Maryland, with the ultimate conclusion being that LSD was too unpredictable and uncontrollable for any tactical use.

Recreational use

From 1960 to 1980

Estimated number of first-time LSD users has fluctuated between 200,000 and 1,000,000.
Estimated annual numbers of first-time LSD use in the United States among persons aged 12 or older: 1967–2008

LSD began to be used recreationally in certain (primarily medical) circles. Mainly academics and medical professionals, who became acquainted with LSD in their work, began using it themselves and sharing it with friends and associates. Among the first to do so was British psychiatrist Humphry Osmond.

Psychedelic subculture goes mainstream

LSD historian Jay Stevens, author of the 1987 book Storming Heaven: LSD and the American Dream, has said that in the early days of its recreational use, LSD users (who were at that time mostly academics and medical professionals) fell into two broadly delineated groups. The first group, which was essentially conservative and exemplified by Aldous Huxley, felt that LSD was too powerful and too dangerous to allow its immediate and widespread introduction, and that its use ought to be restricted to the 'elite' members of society—artists, writers, scientists—who could mediate its gradual distribution throughout society. The second and more radical group, typified by Richard Alpert and Timothy Leary, felt that LSD had the power to revolutionize society and that it should be spread as widely as possible and be available to all.

During the 1960s, this second 'group' of casual LSD users evolved and expanded into a subculture that extolled the mystical and religious symbolism often engendered by the drug's powerful effects, and advocated its use as a method of raising consciousness. The personalities associated with the subculture included spiritual gurus such as Dr. Timothy Leary and psychedelic rock musicians such as the Grateful Dead, Jimi Hendrix, Pink Floyd, Jefferson Airplane and the Beatles, and soon attracted a great deal of publicity, generating further interest in LSD.

The popularization of LSD outside of the medical world was hastened when individuals such as author Ken Kesey participated in drug trials and liked what they saw. Tom Wolfe wrote a widely read account of the early days of LSD's entrance into the non-academic world in his book The Electric Kool Aid Acid Test, which documented the cross-country, acid-fueled voyage of Kesey and the Merry Pranksters on the psychedelic bus "Furthur" and the Pranksters' later 'Acid Test' LSD parties.

In 1965, Sandoz laboratories stopped its still legal shipments of LSD to the United States for research and psychiatric use, after a request from the U.S. government concerned about its use. By April 1966, LSD use had become so widespread that Time magazine warned about its dangers.

In December 1966, the exploitation film Hallucination Generation was released. This was followed by the films The Trip in 1967 and Psych-Out in 1968.

Musicians and LSD

On March 27, 1965, Beatles members John Lennon and George Harrison (and their wives) were dosed with LSD without their permission by their dentist, Dr. John Riley. John Lennon mentioned the incident in his famous 1970 Rolling Stone interview, but the name of the dentist was revealed only in 2006. On August 24, 1965, Lennon, Harrison and Ringo Starr took their second trip on LSD. Actor Peter Fonda repeatedly said "I know what it's like to be dead" to John Lennon during an LSD trip. John Lennon wrote "Lucy in the Sky with Diamonds", a fanciful song which many assumed referred to LSD, although he always denied the connection as coincidence. The songs "She Said She Said" and "Tomorrow Never Knows" from the Beatles' Revolver album explicitly reference LSD trips, and many lines of "Tomorrow Never Knows" were borrowed from Timothy Leary's book The Psychedelic Experience. Around the same time, bands such as Pink Floyd, Jefferson Airplane, and The Grateful Dead helped give birth to a genre known as "psychedelic rock" or acid rock. In 1965, The Pretty Things released an album called Get the Picture? which included a track titled "L.S.D."

LSD became a headline item in early 1967, and the Beatles admitted to having been under the influence of LSD. Earlier in the year, British tabloid News of the World ran a sensational three-week series on 'drug parties' hosted by rock group The Moody Blues and attended by leading stars including Donovan, The Who's Pete Townshend and Cream drummer Ginger Baker. Largely as a result of collusion between News of the World journalists and the London Drug Squad, many pop stars including Donovan and Rolling Stones members Mick Jagger and Keith Richards were arrested for drug possession, although none of the arrests involved LSD.

The FBI suggested in now declassified documents that the Grateful Dead were responsible for introducing LSD to the U.S. The Grateful Dead were the "house band" at Ken Kesey and the Merry Pranksters' Acid Tests. These free-form parties introduced many people on the West Coast to LSD for the first time, as documented in Tom Wolfe's The Electric Kool-Aid Acid Test and Phil Lesh's Searching for the Sound. Acid historian Jesse Jarnow describes how Grateful Dead concerts served as the United States' primary distribution network for LSD in the second half of the twentieth century.

In 1992, Mike Dirnt of Green Day wrote the famous "Longview" bass line while under the influence of LSD. In an interview, Green Day lead singer and guitarist Billie Joe Armstrong recalled that he arrived at their house and saw Mike sitting on the floor with highly dilated pupils, holding his bass guitar. Mike looked up at Billie and exclaimed, "Listen to this!"

LSD in Australia

LSD was evidently in limited recreational use in Australia in the early 1960s, but is believed to have been initially restricted to those with connections to the scientific and the medical communities. LSD overdose was suggested as a possible cause of the January 2, 1962 deaths of CSIRO scientists Dr. Gilbert Bogle and his lover Dr. Margaret Chandler, but is very unlikely as there are no known cases of a LSD fatal overdose and other more likely causes of death have been suggested. Large quantities of LSD began to appear in Australia around 1968, and soon permeated the music scene and youth culture in general, especially in the capital cities. The major source of supply during this period is believed to have been American servicemen visiting Australia (mainly Sydney) from Vietnam on 'rest and recreation' (R&R) leave, although the growing connections between American and Australian organized crime in the late 1960s may also have facilitated its importation. Recreational LSD use among young people was on a par with that in other countries in Australia by the early 1970s and continued until late in the decade. LSD is not believed to have been manufactured locally in a significant quantity (if at all) and most if not all supplies were sourced from overseas.

Production of LSD

During the 1960s and early 1970s, the drug culture adopted LSD as the psychedelic drug of choice, particularly amongst the hippie community. However, LSD dramatically decreased in popularity in the mid-1970s. This decline was due to negative publicity centred on side-effects of LSD use, its criminalization, and the increasing effectiveness of drug law enforcement efforts, rather than new medical information. The last country to produce LSD legally (until 1975) was Czechoslovakia; during the 1960s, high-quality LSD was imported from the communist country to California, a fact appreciated by Leary in The Politics of Ecstasy.

Victor James Kapur had the first known home grown UK 'acid lab'. Up to then, all LSD had been imported from the U.S. or was remnant produce of Sandoz before it stopped producing LSD. In 1967, Kapur was caught distributing 19 grams of crystalline LSD and subsequently the police raided both of his laboratories. One was in the back room of Kapur's chemist shop and another, larger one, was in a garage he rented from a friend of his brother-in-law.

A second group was busted in 1969. A lab in Kent, and a flat in London were raided simultaneously and quantities of equipment and LSD seized along with the two men who had been making the LSD, Quentin Theobald and Peter Simmons.

The availability of LSD had been drastically reduced by the late 1970s due to a combination of governmental controls and law enforcement. The supply of constituent chemicals including lysergic acid, which was used for production of LSD in the 1960s, and ergotamine tartrate, which was used for production in the 1970s, were placed under tight surveillance and government funding for LSD research was almost eliminated. These efforts were augmented by a series of major busts in England and Europe. One of the most famous was "Operation Julie" in Britain in 1978, named after the first name of the female drug squad officer involved; it broke up one of the largest LSD manufacturing and distribution operations in the world at that time, headed by chemist Richard Kemp. The group targeted by the Julie task force were reputed to have had links to the mysterious The Brotherhood of Eternal Love and to Ronald Stark.

Modern times

LSD made a comeback in the 1980s accompanying the advent of recreational MDMA use, first in the punk and gothic subcultures through dance clubs, then in the 1990s through the acid house scene and rave subculture. LSD use and availability declined sharply following a raid of a large scale LSD lab in 2000 (see LSD in the United States). The lab was run by William Leonard Pickard (who served 17 years of a two lifetime sentence in US federal prison in Tucson, AZ) and Clyde Apperson (now serving 30 years in prison). Gordon Todd Skinner, who owned the property the large scale lab had been operating on, came to the DEA looking to work as an informant. He and his then-girlfriend Krystle Cole were intimately involved in the case, but were not charged in the bust. The lab was allegedly producing a kilogram of LSD every five weeks, and the U.S. government contends that LSD supply dropped by 90% following the bust. In the decade after the bust, LSD availability and use has gradually risen. Since the late 1980s, there has also been a revival of hallucinogen research more broadly, which, in recent years, has included preclinical and clinical studies involving LSD and other compounds such as members of the 2C family compounds and psilocybin. In particular, a study released in 2012 highlighted the extraordinary effectiveness of LSD in treating alcoholism.

In November 2015, Rolling Stone magazine reported on an increasing number of young professionals, particularly in the San Francisco area, who were using "microdosing" (around 10 micrograms) of LSD in an effort to "work through technical problems and become more innovative." In 2018, the book How to Change Your Mind: What the New Science of Psychedelics Teaches Us About Consciousness, Dying, Addiction, Depression, and Transcendence by Michael Pollan became a No. 1 New York Times best-seller. In 2020, Oregon became the first U.S. state to decriminalize possession of small amounts of LSD.

Algebra

From Wikipedia, the free encyclopedia

The quadratic formula expresses the solution of the equation ax2 + bx + c = 0, where a is not zero, in terms of its coefficients a, b and c.

Algebra (from Arabic الجبر (al-jabr) 'reunion of broken parts, bonesetting') is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.

Elementary algebra deals with the manipulation of variables as if they were numbers (see the image), and is therefore essential in all applications of mathematics. Abstract algebra is the name given in education to the study of algebraic structures such as groups, rings, and fields. Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra and some not, such as Galois theory.

The word algebra is not only used for naming an area of mathematics and some subareas; it is also used for naming some sorts of algebraic structures, such as an algebra over a field, commonly called an algebra. Sometimes, the same phrase is used for a subarea and its main algebraic structures; for example, Boolean algebra and a Boolean algebra. A mathematician specialized in algebra is called an algebraist.

Etymology

The word algebra comes from the title of a book by Muhammad ibn Musa al-Khwarizmi.

The word algebra comes from the Arabic: الجبر, romanizedal-jabr, lit.'reunion of broken parts, bonesetting' from the title of the early 9th century book cIlm al-jabr wa l-muqābala "The Science of Restoring and Balancing" by the Persian mathematician and astronomer al-Khwarizmi. In his work, the term al-jabr referred to the operation of moving a term from one side of an equation to the other, المقابلة al-muqābala "balancing" referred to adding equal terms to both sides. Shortened to just algeber or algebra in Latin, the word eventually entered the English language during the 15th century, from either Spanish, Italian, or Medieval Latin. It originally referred to the surgical procedure of setting broken or dislocated bones. The mathematical meaning was first recorded (in English) in the 16th century.

Different meanings of "algebra"

The word "algebra" has several related meanings in mathematics, as a single word or with qualifiers.

Algebra as a branch of mathematics

Algebra began with computations similar to those of arithmetic, with letters standing for numbers. This allowed proofs of properties that are true no matter which numbers are involved. For example, in the quadratic equation

can be any numbers whatsoever (except that cannot be ), and the quadratic formula can be used to quickly and easily find the values of the unknown quantity which satisfy the equation. That is to say, to find all the solutions of the equation.

Historically, and in current teaching, the study of algebra starts with the solving of equations, such as the quadratic equation above. Then more general questions, such as "does an equation have a solution?", "how many solutions does an equation have?", "what can be said about the nature of the solutions?" are considered. These questions led extending algebra to non-numerical objects, such as permutations, vectors, matrices, and polynomials. The structural properties of these non-numerical objects were then formalized into algebraic structures such as groups, rings, and fields.

Before the 16th century, mathematics was divided into only two subfields, arithmetic and geometry. Even though some methods, which had been developed much earlier, may be considered nowadays as algebra, the emergence of algebra and, soon thereafter, of infinitesimal calculus as subfields of mathematics only dates from the 16th or 17th century. From the second half of the 19th century on, many new fields of mathematics appeared, most of which made use of both arithmetic and geometry, and almost all of which used algebra.

Today, algebra has grown considerably and includes many branches of mathematics, as can be seen in the Mathematics Subject Classification where none of the first level areas (two digit entries) are called algebra. Today algebra includes section 08-General algebraic systems, 12-Field theory and polynomials, 13-Commutative algebra, 15-Linear and multilinear algebra; matrix theory, 16-Associative rings and algebras, 17-Nonassociative rings and algebras, 18-Category theory; homological algebra, 19-K-theory and 20-Group theory. Algebra is also used extensively in 11-Number theory and 14-Algebraic geometry.

History

Early history of algebra

The roots of algebra can be traced to the ancient Babylonians, who developed an advanced arithmetical system with which they were able to do calculations in an algorithmic fashion. The Babylonians developed formulas to calculate solutions for problems typically solved today by using linear equations, quadratic equations, and indeterminate linear equations. By contrast, most Egyptians of this era, as well as Greek and Chinese mathematics in the 1st millennium BC, usually solved such equations by geometric methods, such as those described in the Rhind Mathematical Papyrus, Euclid's Elements, and The Nine Chapters on the Mathematical Art. The geometric work of the Greeks, typified in the Elements, provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations, although this would not be realized until mathematics developed in medieval Islam.

By the time of Plato, Greek mathematics had undergone a drastic change. The Greeks created a geometric algebra where terms were represented by sides of geometric objects, usually lines, that had letters associated with them. Diophantus (3rd century AD) was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations, and have led, in number theory, to the modern notion of Diophantine equation.

Earlier traditions discussed above had a direct influence on the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī (c. 780–850). He later wrote The Compendious Book on Calculation by Completion and Balancing, which established algebra as a mathematical discipline that is independent of geometry and arithmetic.

The Hellenistic mathematicians Hero of Alexandria and Diophantus as well as Indian mathematicians such as Brahmagupta, continued the traditions of Egypt and Babylon, though Diophantus' Arithmetica and Brahmagupta's Brāhmasphuṭasiddhānta are on a higher level. For example, the first complete arithmetic solution written in words instead of symbols, including zero and negative solutions, to quadratic equations was described by Brahmagupta in his book Brahmasphutasiddhanta, published in 628 AD. Later, Persian and Arab mathematicians developed algebraic methods to a much higher degree of sophistication. Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, Al-Khwarizmi's contribution was fundamental. He solved linear and quadratic equations without algebraic symbolism, negative numbers or zero, thus he had to distinguish several types of equations.

In the context where algebra is identified with the theory of equations, the Greek mathematician Diophantus has traditionally been known as the "father of algebra" and in the context where it is identified with rules for manipulating and solving equations, Persian mathematician al-Khwarizmi is regarded as "the father of algebra". It is open to debate whether Diophantus or al-Khwarizmi is more entitled to be known, in the general sense, as "the father of algebra". Those who support Diophantus point to the fact that the algebra found in Al-Jabr is slightly more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully rhetorical. Those who support Al-Khwarizmi point to the fact that he introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation) which the term al-jabr originally referred to, and that he gave an exhaustive explanation of solving quadratic equations, supported by geometric proofs while treating algebra as an independent discipline in its own right. His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study". He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems".

Another Persian mathematician Omar Khayyam is credited with identifying the foundations of algebraic geometry and found the general geometric solution of the cubic equation. His book Treatise on Demonstrations of Problems of Algebra (1070), which laid down the principles of algebra, is part of the body of Persian mathematics that was eventually transmitted to Europe. Yet another Persian mathematician, Sharaf al-Dīn al-Tūsī, found algebraic and numerical solutions to various cases of cubic equations. He also developed the concept of a function. The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji, and the Chinese mathematician Zhu Shijie, solved various cases of cubic, quartic, quintic and higher-order polynomial equations using numerical methods. In the 13th century, the solution of a cubic equation by Fibonacci is representative of the beginning of a revival in European algebra. Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī (1412–1486) took "the first steps toward the introduction of algebraic symbolism". He also computed Σn2, Σn3 and used the method of successive approximation to determine square roots.

Modern history of algebra

Italian mathematician Girolamo Cardano published the solutions to the cubic and quartic equations in his 1545 book Ars magna.

François Viète's work on new algebra at the close of the 16th century was an important step towards modern algebra. In 1637, René Descartes published La Géométrie, inventing analytic geometry and introducing modern algebraic notation. Another key event in the further development of algebra was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th century. The idea of a determinant was developed by Japanese mathematician Seki Kōwa in the 17th century, followed independently by Gottfried Leibniz ten years later, for the purpose of solving systems of simultaneous linear equations using matrices. Gabriel Cramer also did some work on matrices and determinants in the 18th century. Permutations were studied by Joseph-Louis Lagrange in his 1770 paper "Réflexions sur la résolution algébrique des équations" devoted to solutions of algebraic equations, in which he introduced Lagrange resolvents. Paolo Ruffini was the first person to develop the theory of permutation groups, and like his predecessors, also in the context of solving algebraic equations.

Abstract algebra was developed in the 19th century, deriving from the interest in solving equations, initially focusing on what is now called Galois theory, and on constructibility issues. George Peacock was the founder of axiomatic thinking in arithmetic and algebra. Augustus De Morgan discovered relation algebra in his Syllabus of a Proposed System of Logic. Josiah Willard Gibbs developed an algebra of vectors in three-dimensional space, and Arthur Cayley developed an algebra of matrices (this is a noncommutative algebra).

Areas of mathematics with the word algebra in their name

Some subareas of algebra have the word algebra in their name; linear algebra is one example. Others do not: group theory, ring theory, and field theory are examples. In this section, we list some areas of mathematics with the word "algebra" in the name.

Many mathematical structures are called algebras:

Elementary algebra

Algebraic expression notation:
  1 – power (exponent)
  2 – coefficient
  3 – term
  4 – operator
  5 – constant term
  x y c – variables/constants

Elementary algebra is the most basic form of algebra. It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. In arithmetic, only numbers and their arithmetical operations (such as +, −, ×, ÷) occur. In algebra, numbers are often represented by symbols called variables (such as a, n, x, y or z). This is useful because:

  • It allows the general formulation of arithmetical laws (such as a + b = b + a for all a and b), and thus is the first step to a systematic exploration of the properties of the real number system.
  • It allows the reference to "unknown" numbers, the formulation of equations and the study of how to solve these. (For instance, "Find a number x such that 3x + 1 = 10" or going a bit further "Find a number x such that ax + b = c". This step leads to the conclusion that it is not the nature of the specific numbers that allow us to solve it, but that of the operations involved.)
  • It allows the formulation of functional relationships. (For instance, "If you sell x tickets, then your profit will be 3x − 10 dollars, or f(x) = 3x − 10, where f is the function, and x is the number to which the function is applied".)

Polynomials

The graph of a polynomial function of degree 3

A polynomial is an expression that is the sum of a finite number of non-zero terms, each term consisting of the product of a constant and a finite number of variables raised to whole number powers. For example, x2 + 2x − 3 is a polynomial in the single variable x. A polynomial expression is an expression that may be rewritten as a polynomial, by using commutativity, associativity and distributivity of addition and multiplication. For example, (x − 1)(x + 3) is a polynomial expression, that, properly speaking, is not a polynomial. A polynomial function is a function that is defined by a polynomial, or, equivalently, by a polynomial expression. The two preceding examples define the same polynomial function.

Two important and related problems in algebra are the factorization of polynomials, that is, expressing a given polynomial as a product of other polynomials that cannot be factored any further, and the computation of polynomial greatest common divisors. The example polynomial above can be factored as (x − 1)(x + 3). A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable.

Education

It has been suggested that elementary algebra should be taught to students as young as eleven years old, though in recent years it is more common for public lessons to begin at the eighth grade level (≈ 13 y.o. ±) in the United States. However, in some US schools, algebra is started in ninth grade.

Abstract algebra

Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts. Here are the listed fundamental concepts in abstract algebra.

Sets: Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of sets: collections of objects called elements. All collections of the familiar types of numbers are sets. Other examples of sets include the set of all two-by-two matrices, the set of all second-degree polynomials (ax2 + bx + c), the set of all two dimensional vectors of a plane, and the various finite groups such as the cyclic groups, which are the groups of integers modulo n. Set theory is a branch of logic and not technically a branch of algebra.

Binary operations: The notion of addition (+) is generalized to the notion of binary operation (denoted here by ∗). The notion of binary operation is meaningless without the set on which the operation is defined. For two elements a and b in a set S, ab is another element in the set; this condition is called closure. Addition (+), subtraction (−), multiplication (×), and division (÷) can be binary operations when defined on different sets, as are addition and multiplication of matrices, vectors, and polynomials.

Identity elements: The numbers zero and one are generalized to give the notion of an identity element for an operation. Zero is the identity element for addition and one is the identity element for multiplication. For a general binary operator ∗ the identity element e must satisfy ae = a and ea = a, and is necessarily unique, if it exists. This holds for addition as a + 0 = a and 0 + a = a and multiplication a × 1 = a and 1 × a = a. Not all sets and operator combinations have an identity element; for example, the set of positive natural numbers (1, 2, 3, ...) has no identity element for addition.

Inverse elements: The negative numbers give rise to the concept of inverse elements. For addition, the inverse of a is written −a, and for multiplication the inverse is written a−1. A general two-sided inverse element a−1 satisfies the property that aa−1 = e and a−1a = e, where e is the identity element.

Associativity: Addition of integers has a property called associativity. That is, the grouping of the numbers to be added does not affect the sum. For example: (2 + 3) + 4 = 2 + (3 + 4). In general, this becomes (ab) ∗ c = a ∗ (bc). This property is shared by most binary operations, but not subtraction or division or octonion multiplication.

Commutativity: Addition and multiplication of real numbers are both commutative. That is, the order of the numbers does not affect the result. For example: 2 + 3 = 3 + 2. In general, this becomes ab = ba. This property does not hold for all binary operations. For example, matrix multiplication and quaternion multiplication are both non-commutative.

Groups

Combining the above concepts gives one of the most important structures in mathematics: a group. A group is a combination of a set S and a single binary operation ∗, defined in any way you choose, but with the following properties:

  • An identity element e exists, such that for every member a of S, ea and ae are both identical to a.
  • Every element has an inverse: for every member a of S, there exists a member a−1 such that aa−1 and a−1a are both identical to the identity element.
  • The operation is associative: if a, b and c are members of S, then (ab) ∗ c is identical to a ∗ (bc).

If a group is also commutative – that is, for any two members a and b of S, ab is identical to ba – then the group is said to be abelian.

For example, the set of integers under the operation of addition is a group. In this group, the identity element is 0 and the inverse of any element a is its negation, −a. The associativity requirement is met, because for any integers a, b and c, (a + b) + c = a + (b + c)

The non-zero rational numbers form a group under multiplication. Here, the identity element is 1, since 1 × a = a × 1 = a for any rational number a. The inverse of a is 1/a, since a × 1/a = 1.

The integers under the multiplication operation, however, do not form a group. This is because, in general, the multiplicative inverse of an integer is not an integer. For example, 4 is an integer, but its multiplicative inverse is 1/4, which is not an integer.

The theory of groups is studied in group theory. A major result of this theory is the classification of finite simple groups, mostly published between about 1955 and 1983, which separates the finite simple groups into roughly 30 basic types.

Semi-groups, quasi-groups, and monoids are algebraic structures similar to groups, but with less constraints on the operation. They comprise a set and a closed binary operation but do not necessarily satisfy the other conditions. A semi-group has an associative binary operation but might not have an identity element. A monoid is a semi-group which does have an identity but might not have an inverse for every element. A quasi-group satisfies a requirement that any element can be turned into any other by either a unique left-multiplication or right-multiplication; however, the binary operation might not be associative.

All groups are monoids, and all monoids are semi-groups.

Examples
Set Natural numbers N Integers Z Rational numbers Q
Real numbers R
Complex numbers C
Integers modulo 3
Z/3Z = {0, 1, 2}
Operation + × + × + × ÷ + ×
Closed Yes Yes Yes Yes Yes Yes Yes No Yes Yes
Identity 0 1 0 1 0 N/A 1 N/A 0 1
Inverse N/A N/A a N/A a N/A 1/a
(a ≠ 0)
N/A 0, 2, 1, respectively N/A, 1, 2, respectively
Associative Yes Yes Yes Yes Yes No Yes No Yes Yes
Commutative Yes Yes Yes Yes Yes No Yes No Yes Yes
Structure monoid monoid abelian group monoid abelian group quasi-group monoid quasi-group abelian group monoid

Rings and fields

Groups just have one binary operation. To fully explain the behaviour of the different types of numbers, structures with two operators need to be studied. The most important of these are rings and fields.

A ring has two binary operations (+) and (×), with × distributive over +. Under the first operator (+) it forms an abelian group. Under the second operator (×) it is associative, but it does not need to have an identity, or inverse, so division is not required. The additive (+) identity element is written as 0 and the additive inverse of a is written as −a.

Distributivity generalises the distributive law for numbers. For the integers (a + b) × c = a × c + b × c and c × (a + b) = c × a + c × b, and × is said to be distributive over +.

The integers are an example of a ring. The integers have additional properties which make it an integral domain.

A field is a ring with the additional property that all the elements excluding 0 form an abelian group under ×. The multiplicative (×) identity is written as 1 and the multiplicative inverse of a is written as a−1.

The rational numbers, the real numbers and the complex numbers are all examples of fields.

Equality (mathematics)

From Wikipedia, the free encyclopedia https://en.wikipedia.org/wiki/Equality_...