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Alexandria
Alexandria (Arabic: الإسكندرية Al Iskandariyya, Coptic: Ⲣⲁⲕⲟⲧⲉ Rakotə, Greek: Αλεξάνδρεια Alexándria, Koine Greek: Ἀλεξάνδρεια ἡ κατ' Αἴγυπτον IPA: [ɑlɛˈksɑndɾiɑ e kɑt ˈɛʝypton] "Alexandria in Egypt", Egyptian Arabic: اسكندريه [eskendeˈrejːæ]) is the second-largest city of Egypt, with a population of 4.1 million, extending about 32 km (20 mi) along the coast of the Mediterranean Sea in the north central part of the country; it is also the largest city lying directly on the Mediterranean coast. It is Egypt's largest seaport, serving approximately 80% of Egypt's imports and exports. Alexandria is also an important tourist resort. It is home to the Bibliotheca Alexandrina (the new Library of Alexandria). It is an important industrial centre because of its natural gas and oil pipelines from Suez.
Alexandria was founded around a small pharaonic town c. 331 BC by Alexander the Great. It remained Egypt's capital for nearly a thousand years, until the Muslim conquest of Egypt in AD 641, when a new capital was founded at Fustat (Fustat was later absorbed into Cairo). Alexandria was known because of its Lighthouse of Alexandria (Pharos), one of the Seven Wonders of the Ancient World; its library (the largest library in the ancient world); and the Catacombs of Kom el Shoqafa, one of the Seven Wonders of the Middle Ages. Ongoing maritime archaeology in the harbor of Alexandria, which began in 1994, is revealing details of Alexandria both before the arrival of Alexander, when a city named Rhacotis existed there, and during the Ptolemaic dynasty.
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Gresham College
Coordinates: 51°31′04″N 0°06′35″W / 51.5177°N 0.1098°W / 51.5177; -0.1098
Gresham College is an institution of higher learning located at Barnard's Inn Hall off Holborn in central London, England. It was founded in 1597 under the will of Sir Thomas Gresham and today it hosts over 140 free public lectures every year within the City of London.
Sir Thomas Gresham, founder of the Royal Exchange, left his estate jointly to the City of London Corporation and to the Mercers' Company, which today support the college through the Joint Grand Gresham Committee under the presidency of the Lord Mayor of London. Gresham's will provided for the setting up of the College—in Gresham's mansion in Bishopsgate, on the site now occupied by Tower 42, the former NatWest Tower—and endowed it with the rental income from shops sited around the Royal Exchange, which Gresham had established.
The early success of the College led to the incorporation of the Royal Society in 1663, which pursued its activities at the College in Bishopsgate before moving to its own premises in Crane Court in 1710. The College remained in Gresham's mansion in Bishopsgate until 1768, and moved about London thereafter until the construction in 1842 of its own buildings in Gresham Street EC2. Gresham College did not become part of the University of London on the founding of the University in the 19th century, although a close association between the College and the University persisted for many years. Since 1991, the College has operated at Barnard’s Inn Hall, Holborn EC1.
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Arthur Cayley
Arthur Cayley F.R.S. (16 August 1821 – 26 January 1895) was a British mathematician. He helped found the modern British school of pure mathematics.
As a child, Cayley enjoyed solving complex math problems for amusement. He entered Trinity College, Cambridge, where he excelled in Greek, French, German, and Italian, as well as mathematics. He worked as a lawyer for 14 years.
He postulated the Cayley–Hamilton theorem—that every square matrix is a root of its own characteristic polynomial, and verified it for matrices of order 2 and 3. He was the first to define the concept of a group in the modern way—as a set with a binary operation satisfying certain laws. Formerly, when mathematicians spoke of "groups", they had meant permutation groups.
See also Cayley's theorem.
Arthur Cayley was born in Richmond, London, England, on 16 August 1821. His father, Henry Cayley, was a distant cousin of Sir George Cayley the aeronautics engineer innovator, and descended from an ancient Yorkshire family. He settled in Saint Petersburg, Russia, as a merchant. His mother was Maria Antonia Doughty, daughter of William Doughty. According to some writers she was Russian, but her father's name indicates an English origin. His brother was the linguist Charles Bagot Cayley. Arthur spent his first eight years in Saint Petersburg. In 1829 his parents were settled permanently at Blackheath, near London. Arthur was sent to a private school. He early showed great liking for, and aptitude in, numerical calculation. At age 14 he was sent to King's College School. The school's master observed indications of mathematical genius and advised the father to educate his son not for his own business, as he had intended, but to enter the University of Cambridge.
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Augustus De Morgan
Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. The crater De Morgan on the Moon is named after him.
Augustus De Morgan was born in 1806. His father was Col. Augustus De Morgan, who held various appointments in the service of the East India Company. His mother descended from James Dodson, who computed a table of anti-logarithms, that is, the numbers corresponding to exact logarithms. Augustus De Morgan became blind in one eye a month or two after he was born. The family moved to England when Augustus was seven months old. As his father and grandfather had both been born in India, De Morgan used to say that he was neither English, nor Scottish, nor Irish, but a Briton "unattached", using the technical term applied to an undergraduate of Oxford or Cambridge who is not a member of any one of the Colleges.
When De Morgan was ten years old, his father died. Mrs. De Morgan resided at various places in the southwest of England, and her son received his elementary education at various schools of no great account. His mathematical talents went unnoticed until he was fourteen, when a family-friend discovered him making an elaborate drawing of a figure in Euclid with ruler and compasses. She explained the aim of Euclid to Augustus, and gave him an initiation into demonstration.
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Brahmagupta
Brahmagupta (Sanskrit: ब्रह्मगुप्त;
listen (help·info)) (598–668 CE) was an Indian mathematician and astronomer who wrote many important works on mathematics and astronomy. His best known work is the Brāhmasphuṭasiddhānta (Correctly Established Doctrine of Brahma), written in 628 in Bhinmal. Its 25 chapters contain several unprecedented mathematical results.
Brahmagupta is believed to have been born in 598 AD in Bhinmal city in the state of Rajasthan of Northwest India. In ancient times Bhillamala was the seat of power of the Gurjars. His father was Jisnugupta. He likely lived most of his life in Bhillamala (modern Bhinmal in Rajasthan) during the reign (and possibly under the patronage) of King Vyaghramukha. As a result, Brahmagupta is often referred to as Bhillamalacarya, that is, the teacher from Bhillamala. He was the head of the astronomical observatory at Ujjain, and during his tenure there wrote four texts on mathematics and astronomy: the Cadamekela in 624, the Brahmasphutasiddhanta in 628, the Khandakhadyaka in 665, and the Durkeamynarda in 672. The Brahmasphutasiddhanta (Corrected Treatise of Brahma) is arguably his most famous work. The historian al-Biruni (c. 1050) in his book Tariq al-Hind states that the Abbasid caliph al-Ma'mun had an embassy in India and from India a book was brought to Baghdad which was translated into Arabic as Sindhind. It is generally presumed that Sindhind is none other than Brahmagupta's Brahmasphuta-siddhanta.
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Diophantus
Diophantus of Alexandria (Ancient Greek: Διόφαντος ὁ Ἀλεξανδρεύς. b. between A.D. 200 and 214, d. between 284 and 298 at age 84), sometimes called "the father of algebra", was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations, many of which are now lost. In studying Arithmetica, Pierre de Fermat concluded that a certain equation considered by Diophantus had no solutions, and noted without elaboration that he had found "a truly marvelous proof of this proposition," now referred to as Fermat's Last Theorem. This led to tremendous advances in number theory, and the study of Diophantine equations ("Diophantine geometry") and of Diophantine approximations remain important areas of mathematical research. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. In modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought. Diophantus also made advances in mathematical notation.
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Fibonacci
Leonardo Pisano Bigollo (c. 1170 – c. 1250) also known as Leonardo of Pisa, Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or, most commonly, simply Fibonacci, was an Italian mathematician, considered by some "the most talented western mathematician of the Middle Ages."
Fibonacci is best known to the modern world for the spreading of the Hindu–Arabic numeral system in Europe, primarily through the publication in 1202 of his Liber Abaci (Book of Calculation), and for a number sequence named the Fibonacci numbers after him, which he did not discover but used as an example in the Liber Abaci.
Leonardo Fibonacci was born around 1170 to Guglielmo Bonacci, a wealthy Italian merchant. Guglielmo directed a trading post (by some accounts he was the consultant for Pisa) in Bugia, a port east of Algiers in the Almohad dynasty's sultanate in North Africa (now Bejaia, Algeria). As a young boy, Leonardo traveled with him to help; it was there he learned about the Hindu–Arabic numeral system.
Recognizing that arithmetic with Hindu–Arabic numerals is simpler and more efficient than with Roman numerals, Fibonacci traveled throughout the Mediterranean world to study under the leading Arab mathematicians of the time. Leonardo returned from his travels around 1200. In 1202, at age 32, he published what he had learned in Liber Abaci (Book of Abacus or Book of Calculation), and thereby popularized Hindu–Arabic numerals in Europe.
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Florian Cajori
Florian Cajori (St Aignan, Switzerland, February 28, 1859 – August 14 or 15, 1930,Berkeley, United States) was one of the most celebrated historians of mathematics in his day.
Florian Cajori emigrated to the United States at the age of sixteen. He received a Ph.D. at Tulane University, where he taught for a few years before being driven north by his health. He taught at Colorado College, where he founded the Colorado College Scientific Society. Even today his History of Mathematical Notations (1928–29) has been described as "unsurpassed." In 1918, he was appointed to a specially created chair in history of mathematics at the University of California, Berkeley. He remained in Berkeley, California until his death in 1930.
His last work was revising Motte's Translation of Newton's Principia, vol.1 The Motion of Bodies, but he died before it was finished. The work was finished by R.T.Crawford of Berkeley, California.
The Cajori crater on the Moon was named in his honour.
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François Viète
François Viète (Latin: Franciscus Vieta; 1540 – 23 February 1603), Seigneur de la Bigotière, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations. He was a lawyer by trade, and served as a privy councillor to both Henry III and Henry IV.
Viete was born at Fontenay-le-Comte, Vendée. His grandfather was a merchant from La Rochelle. His father, Etienne Viète, was an attorney in Fontenay-le-Comte and a notary in Le Busseau. His mother was the aunt of Barnabé Brisson, a magistrate and the first president of parliament during the ascendancy of the Catholic League of France.
Vieta went to a Franciscan school and in 1558 studied law at Poitiers, graduating as a Bachelor of Law in 1559. A year later, he began his career as an attorney in his native town. From the outset, he was entrusted with some major cases, including the settlement of rent in Poitou for the widow of King Francis I of France and looking after the interests of Mary, Queen of Scots.
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Gabriel Cramer
Gabriel Cramer (31 July 1704 – 4 January 1752) was a Swiss mathematician, born in Geneva. He showed promise in mathematics from an early age. At 18 he received his doctorate and at 20 he was co-chair of mathematics. In 1728 he proposed a solution to the St. Petersburg Paradox that came very close to the concept of expected utility theory given ten years later by Daniel Bernoulli. He published his best-known work in his forties. This was his treatise on algebraic curves, "Introduction à l'analyse des lignes courbes algébriques", published in 1750. It contains the earliest demonstration that a curve of the n-th degree is determined by
on it, in general position. He edited the works of the two elder Bernoullis; and wrote on the physical cause of the spheroidal shape of the planets and the motion of their apsides (1730), and on Newton's treatment of cubic curves (1746). He was professor at Geneva, and died at Bagnols-sur-Cèze. He did extensive travel throughout Europe in the late 1730s which greatly influenced his works in mathematics.
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George Peacock
George Peacock (9 April 1791 – 8 November 1858) was an English mathematician.
Peacock was born on 9 April 1791 at Thornton Hall, Denton, near Darlington, County Durham. His father, the Rev. Thomas Peacock, was a clergyman of the Church of England, incumbent and for 50 years curate of the parish of Denton, where he also kept a school. In early life Peacock did not show any precocity of genius, and was more remarkable for daring feats of climbing than for any special attachment to study. He received his elementary education from his father, and at 17 years of age, was sent to Richmond, to a school taught by a graduate of Cambridge University to receive instruction preparatory to entering that university. At this school he distinguished himself greatly both in classics and in the rather elementary mathematics then required for entrance at Cambridge. In 1809 he became a student of Trinity College, Cambridge.
In 1812 Peacock took the rank of Second Wrangler, and the second Smith's prize, the senior wrangler being John Herschel. Two years later he became a candidate for a fellowship in his college and won it immediately, partly by means of his extensive and accurate knowledge of the classics. A fellowship then meant about pounds 200 a year, tenable for seven years provided the Fellow did not marry meanwhile, and capable of being extended after the seven years provided the Fellow took clerical Orders.
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Greeks
The Greeks, also known as the Hellenes (Greek: Ἕλληνες, [ˈelines]), are a nation and ethnic group native to Greece, Cyprus and other regions. They also form a significant diaspora, with Greek communities established around the world.
Greek colonies and communities have been historically established in most corners of the Mediterranean, but Greeks have always been centered around the Aegean Sea, where the Greek language has been spoken since antiquity. Until the early 20th century, Greeks were uniformly distributed between the Greek peninsula, the western coast of Asia Minor, Pontus, Egypt, Cyprus and Constantinople; many of these regions coincided to a large extent with the borders of the Byzantine Empire of the late 11th century and the Eastern Mediterranean areas of the ancient Greek colonization.
In the aftermath of the Greco-Turkish War (1919–1922), a large-scale population exchange between Greece and Turkey transferred and confined Christians from Turkey, except Constantinople (effectively ethnic Greeks) into the borders of the modern Greek state and Cyprus. Other ethnic Greek populations can be found from southern Italy to the Caucasus and in diaspora communities in a number of other countries. Today, most Greeks are officially registered as members of the Greek Orthodox Church.
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Hero of Alexandria
Hero (or Heron) of Alexandria (Greek: Ἥρων ὁ Ἀλεξανδρεύς) (c. 10–70 AD) was an ancient Greek mathematician and engineer who was active in his native city of Alexandria, Roman Egypt. He is considered the greatest experimenter of antiquity and his work is representative of the Hellenistic scientific tradition.
Hero published a well recognized description of a steam-powered device called an aeolipile (hence sometimes called a "Hero engine"). Among his most famous inventions was a windwheel, constituting the earliest instance of wind harnessing on land. He is said to have been a follower of the Atomists. Some of his ideas were derived from the works of Ctesibius.
Much of Hero's original writings and designs have been lost, but some of his works were preserved in Arab manuscripts.
A number of references mention dates around 150 BC, but these are inconsistent with the dates of his publications and inventions. This may be due to a misinterpretation of the phrase "first century" or because Hero was a common name.
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Josiah Willard Gibbs
Josiah Willard Gibbs (February 11, 1839 – April 28, 1903) was an American scientist who made important theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in transforming physical chemistry into a rigorous deductive science. Together with James Clerk Maxwell and Ludwig Boltzmann, he created statistical mechanics (a term which he coined), explaining the laws of thermodynamics in terms of the statistical properties of large ensembles of particles. As a mathematician, he invented modern vector calculus (independently of Oliver Heaviside).
In 1863, Yale University awarded Gibbs the first American doctorate in engineering. After a three-year sojourn in Europe, Gibbs spent the rest of his career at Yale, where he was professor of mathematical physics. Working in relative isolation, he became the earliest theoretical scientist in the United States to earn an international reputation and was praised by Albert Einstein as "the greatest mind in American history." In 1901 Gibbs received what was then considered the highest honor awarded by the international scientific community, the Copley Medal of the Royal Society of London, "for his contributions to mathematical physics."
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Leopold Kronecker
Leopold Kronecker (December 7, 1823 – December 29, 1891) was a German mathematician who worked on number theory and algebra. He criticized Cantor's work on set theory, and was quoted by Weber (1893) as having said, "God made integers; all else is the work of man". Kronecker was a student and lifelong friend of Ernst Kummer.
Leopold Kronecker was born on 7 December 1823 in Liegnitz, Prussia (now Legnica, Poland) in a wealthy Jewish family. His parents, Isidor and Johanna (née Prausnitzer), took care of their children's education and provided them private tutoring at home - Leopold's younger brother Hugo Kronecker would also follow a scientific path later becoming a notable physiologist. Kronecker then went to the Liegnitz Gymnasium where he was interested in a wide range of topics including science, history and philosophy, while also practicing gymnastics and swimming. At the gymnasium he was taught by Ernst Kummer, who noticed and encouraged the boy's interest for mathematics.
In 1841 Kronecker became a student at the University of Berlin where his interest did not immediately focus on mathematics, but rather spread over several subjects including astronomy and philosophy. He spent the summer of 1843 at the University of Bonn to study astronomy and 1843-44 at the University of Breslau following his former teacher Kummer. Back in Berlin, Kronecker studied mathematics with Gustav Lejeune Dirichlet and in 1845 defended his dissertation in algebraic number theory written under Dirichlet's supervision.
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Paolo Ruffini
Paolo Ruffini (September 22, 1765 – May 10, 1822) was an Italian mathematician and philosopher.
By 1788 he had earned university degrees in philosophy, medicine/surgery, and mathematics. Among his work was an incomplete proof (Abel–Ruffini theorem) that quintic (and higher-order) equations cannot be solved by radicals (1799), and Ruffini's rule which is a quick method for polynomial division. Ruffini also made contributions to group theory in addition to probability and quadrature of the circle.
He practiced as both a professor of mathematics (University of Modena) and a doctor including scientific work on typhus.
Ruffini’s 1799 work marked a major development for group theory. Ruffini developed Joseph Louis Lagrange's work on permutation theory, following 29 years after Lagrange’s "Réflexions sur la théorie algébrique des equations" (1770–1771) which was largely ignored until Ruffini who established strong connections between permutations and the solvability of algebraic equations. Ruffini was the first to controversially assert the unsolvability by radicals of algebraic equations higher than quartics. This angered many members of the community such as Malfatti (1731–1807). Work in this area was later carried on by those such as Gauss and Galois who succeeded in such a proof.
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Plato
Plato ( /ˈpleɪtoʊ/; Greek: Πλάτων, Plátōn, "broad"; 424/423 BC – 348/347 BC) was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the foundations of Western philosophy and science. In the words of A. N. Whitehead:
The safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato. I do not mean the systematic scheme of thought which scholars have doubtfully extracted from his writings. I allude to the wealth of general ideas scattered through them.
Plato's sophistication as a writer is evident in his Socratic dialogues; thirty-six dialogues and thirteen letters have been ascribed to him. Plato's writings have been published in several fashions; this has led to several conventions regarding the naming and referencing of Plato's texts. Plato's dialogues have been used to teach a range of subjects, including philosophy, logic, ethics, rhetoric, and mathematics.
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René Descartes
René Descartes (French pronunciation: [ʁəne dekaʁt]; Latinized form: Renatus Cartesius; adjectival form: "Cartesian"; 31 March 1596 – 11 February 1650) was a French philosopher, mathematician, and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day. In particular, his Meditations on First Philosophy continues to be a standard text at most university philosophy departments. Descartes' influence in mathematics is equally apparent; the Cartesian coordinate system — allowing algebraic equations to be expressed as geometric shapes in a two-dimensional coordinate system — was named after him. He is credited as the father of analytical geometry, the bridge between algebra and geometry, crucial to the discovery of infinitesimal calculus and analysis. Descartes was also one of the key figures in the Scientific Revolution and has been described as an example of genius.
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Richard Dedekind
Julius Wilhelm Richard Dedekind (October 6, 1831 – February 12, 1916) was a German mathematician who did important work in abstract algebra (particularly ring theory), algebraic number theory and the foundations of the real numbers.
Dedekind's father was Julius Levin Ulrich Dedekind, an administrator at Collegium Carolinum in Braunschweig. Dedekind had three older siblings. As an adult, he never employed the names Julius Wilhelm. He was born, lived most of his life, and died in Braunschweig (often called "Brunswick" in English).
He first attended the Collegium Carolinum in 1848 before moving to the University of Göttingen in 1850. There, Dedekind studied number theory under Moritz Stern. Gauss was still teaching, although mostly at an elementary level, and Dedekind became his last student. Dedekind received his doctorate in 1852, for a thesis titled Über die Theorie der Eulerschen Integrale ("On the Theory of Eulerian integrals[disambiguation needed
]"). This thesis did not display the talent evident in Dedekind's subsequent publications.
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Zhu Shijie
Zhu Shijie (Chinese: 朱世杰; pinyin: Zhū Shìjié; Wade–Giles: Chu Shih-chieh, fl thirteenth century, 1270 - 1330), courtesy name Hanqing (汉卿), pseudonym Songting (松庭), was one of the greatest Chinese mathematicians living during the Yuan Dynasty.
Zhu was born close to today's Beijing. Two of his mathematical works have survived. Introduction to Computational Studies (算学启蒙Suanxue qimeng), and Jade Mirror of the Four Unknowns.
Suanxue qimeng written in 1299, is an elementary textbook on mathematics in three volumes, 20 chapters and 259 problems. This book also showed how to measure different two-dimensional shapes and three-dimensional solids. The Introduction had an important influence on the development of mathematics in Japan. The book was once lost in China until Qing dynasty mathematician Luo Shilin bought a Korean printed edtion, and re published in Yangzhou, since then this book was reprinted several times.
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Why can't you just forget about algebra
It's all about you now
And all your talk of logic and formula
Could never help you now
Not anymore
'Cause you were always
On the run
From the darkness
In your heart
So you wear it
On the outside
Of your chest
I have taken the liberty
To tell your ghosts to go
Brighten with sunlight and simpathy
They promise not to show
For a while
'Cause you were always
The little boy
Who couldn't keep it
To yourself
So your heart
Is on the outside
Of your chest
At the speed of love
You moved inside my home
And nothing is alright
If you are still alone
And your heart
Is waiting that
The sound of you and everyone
Is still you're always on the wrong
From the poison in your love
And your heart
Is on the outside
Why can't you just
Forget about algebra
It's all about you now
And all your talk
Of logic and formula
Could never help you now
Not anymore
Cause you were always
On the run
From the darkness
In your heart
So you wear it
On the outside
Of your chest
I have taken
The liberty
To tell your ghost to go
Bribing them with
Sunlight and sympathy
They promised not to show
For a while
Cause you were always
The little boy
Who couldn't keep it
To yourself
So your heart
Is on the outside
Of your chest
At the speed of light
You moved inside my home
Nothing is alright
If you are still alone
And your heart
Is greater than
The sum of you
And everyone
But still you're always
On the run
From the poison
In your lungs
And your heart
Is on the outside
Of your chest
Kept back by the deeper rumblin's in my breast
Bad vibes puttin' mad pressure on my chest
Fly time is a rhythm clockin' exocet
Mad lines from a stranger you've already met
Tribal is the plate I'm eatin' from, you guessed
I'm full yet I cannot fit inside my vest
'Where's he at? Is the question I refuse to get
Here I'm at, right here, in your headphone set, ah
You'll try to diss take a crack at this not algebra or calculus
You'll try to diss take a crack at this not algebra or calculus
So we'll take a crack at this now
Two truths make a lie, first you die then fly
Cross the sky jambalaya with your chocolate pie
Suck it deep, don't let it interrupt your sleep
You'll get fleas and disease when you're countin' your sheep
If you please, don't please 'em run 'em off the ranch
Let them grow like leaves then rip 'em off the branch
Fly time is a rhythm clockin' do-si-do
B-boys rocking with the freak-freak, so now you know
You'll try to diss take a crack at this not algebra or calculus
You'll try to diss take a crack at this not algebra or calculus
So we'll take a crack at this now
It's the P H A R M A K O N
The remedy, yes the cure from the poisonous pen
C O M M U N I C A T I O N
What you'll find inside my rhyme is enemy and friend
Going N to the E the B the U.L.A.
Burning all the competition with the rhymes that I say
H the double O the l the I the G.A.N
Now the soul is what will signify the hooligan
You'll try to diss take a crack at this not algebra or calculus
You'll try to diss take a crack at this not algebra or calculus
You'll try to diss take a crack at this not algebra or calculus
You'll try to diss take a crack at this not algebra or calculus
We'll take a crack at this now
You'll try to diss take a crack at this not algebra or calculus
You'll try to diss take a crack at this not algebra or calculus
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Algebra: Introduction to Algebra
- Order: Reorder
- Duration: 11:22
- Updated: 22 Jan 2013
[NOTE: This video is only the first 11 minutes of the full lesson, please visit my website to purchase complete version] This lesson consists of providing you with a Self-Tutorial on what is algebra, what are variables, constants, coefficients, terms, and expressions. The tutor explains the use of proper notation, how to combine like terms, find the negation of an algebraic expression, how to evaluate an expression (by hand and by using your calculator), and finally, there is a VERY detailed section on how to translate English phrases into algebraic expressions.
http://wn.com/Algebra_Introduction_to_Algebra
[NOTE: This video is only the first 11 minutes of the full lesson, please visit my website to purchase complete version] This lesson consists of providing you with a Self-Tutorial on what is algebra, what are variables, constants, coefficients, terms, and expressions. The tutor explains the use of proper notation, how to combine like terms, find the negation of an algebraic expression, how to evaluate an expression (by hand and by using your calculator), and finally, there is a VERY detailed section on how to translate English phrases into algebraic expressions.
College Algebra - Lecture 1 - Numbers
- Order: Reorder
- Duration: 79:35
- Updated: 23 Jan 2013
College Algebra with Professor Richard Delaware - UMKC VSI - Lecture 1 - Numbers. This lecture discusses about set of objects,Natural numbers,Real numbers and how to find distance between two points.
http://wn.com/College_Algebra_Lecture_1_Numbers
College Algebra with Professor Richard Delaware - UMKC VSI - Lecture 1 - Numbers. This lecture discusses about set of objects,Natural numbers,Real numbers and how to find distance between two points.
- published: 04 May 2009
- views: 216765
- author: UMKC
Algebra: Linear Equations 1
- Order: Reorder
- Duration: 7:27
- Updated: 24 Jan 2013
Learn more: www.khanacademy.org equations of the form AX=B
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Learn more: www.khanacademy.org equations of the form AX=B
- published: 19 Nov 2006
- views: 606224
- author: khanacademy
Algebra -- At This Time
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- Duration: 5:12
- Updated: 24 Jan 2013
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The Beauty of Algebra
- Order: Reorder
- Duration: 10:07
- Updated: 23 Jan 2013
Learn more: www.khanacademy.org Why the abstraction of mathematics is so fundamental
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Learn more: www.khanacademy.org Why the abstraction of mathematics is so fundamental
- published: 09 Feb 2012
- views: 740470
- author: khanacademy
Lec 1 | MIT 18.06 Linear Algebra, Spring 2005
- Order: Reorder
- Duration: 39:49
- Updated: 24 Jan 2013
Lecture 1: The Geometry of Linear Equations. View the complete course at: ocw.mit.edu License: Creative Commons BY-NC-SA More information at ocw.mit.edu More courses at ocw.mit.edu
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Lecture 1: The Geometry of Linear Equations. View the complete course at: ocw.mit.edu License: Creative Commons BY-NC-SA More information at ocw.mit.edu More courses at ocw.mit.edu
- published: 07 May 2009
- views: 858585
- author: MIT
Algebra Help - The Quadratic Formula - YourTeacher.com
- Order: Reorder
- Duration: 3:39
- Updated: 23 Jan 2013
For a complete lesson on the quadratic formula, go to www.yourteacher.com. Get the exact algebra help you need with 1000+ online math lessons featuring a personal math teacher inside every lesson! In this lesson, students learn to solve quadratic equations in the form ax^2 + bx + c = 0 using the quadratic formula, which states that x = -b plus or minus the square root of b^2 -- 4ac over 2a. For example, to solve the equation x^2 -- 3x -- 8 = 0, since a = 1, b = -3, and c = -8, the quadratic formula states that x = 3 plus or minus the square root of (-3)^2 -- 4(1)(-8) over 2(1). Simplifying on the right side of the equation, x = 3 plus or minus the square root of 9 + 32 over 2, or x = 3 plus or minus the square root of 41 over 2, which is the final answer. Note that students also learn to derive the quadratic formula using completing the square in the first example problem in this lesson.
http://wn.com/Algebra_Help_The_Quadratic_Formula_YourTeachercom
For a complete lesson on the quadratic formula, go to www.yourteacher.com. Get the exact algebra help you need with 1000+ online math lessons featuring a personal math teacher inside every lesson! In this lesson, students learn to solve quadratic equations in the form ax^2 + bx + c = 0 using the quadratic formula, which states that x = -b plus or minus the square root of b^2 -- 4ac over 2a. For example, to solve the equation x^2 -- 3x -- 8 = 0, since a = 1, b = -3, and c = -8, the quadratic formula states that x = 3 plus or minus the square root of (-3)^2 -- 4(1)(-8) over 2(1). Simplifying on the right side of the equation, x = 3 plus or minus the square root of 9 + 32 over 2, or x = 3 plus or minus the square root of 41 over 2, which is the final answer. Note that students also learn to derive the quadratic formula using completing the square in the first example problem in this lesson.
- published: 22 Oct 2007
- views: 400181
- author: yourteachermathhelp
Algebra: Linear Equations 4
- Order: Reorder
- Duration: 7:38
- Updated: 18 Jan 2013
Learn more: www.khanacademy.org Solving linear equations with variable expressions in the denominators of fractions
http://wn.com/Algebra_Linear_Equations_4
Learn more: www.khanacademy.org Solving linear equations with variable expressions in the denominators of fractions
- published: 19 Nov 2006
- views: 632550
- author: khanacademy
Anthony David "4Evermore" feat. Algebra & Phonte Official Video
- Order: Reorder
- Duration: 4:03
- Updated: 23 Jan 2013
Anthony David "4Evermore" feat. Algebra & Phonte Official Video
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Anthony David "4Evermore" feat. Algebra & Phonte Official Video
- published: 14 Feb 2011
- views: 927022
- author: KOCHRECORDS
Linear Algebra: Introduction to Vectors
- Order: Reorder
- Duration: 16:31
- Updated: 21 Jan 2013
Learn more: www.khanacademy.org Linear Algebra: Introduction to Vectors
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Learn more: www.khanacademy.org Linear Algebra: Introduction to Vectors
- published: 08 Oct 2009
- views: 238659
- author: khanacademy
Origins of Algebra
- Order: Reorder
- Duration: 7:18
- Updated: 23 Jan 2013
Learn more: www.khanacademy.org Where did the word "Algebra" and its underlying ideas come from
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Learn more: www.khanacademy.org Where did the word "Algebra" and its underlying ideas come from
- published: 09 Feb 2012
- views: 227735
- author: khanacademy
ALGEBRA BLESSETT: Run And Hide Video
- Order: Reorder
- Duration: 4:12
- Updated: 17 Jan 2013
Video for Algebra Blessett's New single Run And Hide
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Video for Algebra Blessett's New single Run And Hide
- published: 14 May 2008
- views: 98780
- author: 123ally123
Lec 2 | MIT 18.06 Linear Algebra, Spring 2005
- Order: Reorder
- Duration: 47:42
- Updated: 23 Jan 2013
Lecture 2: Elimination with Matrices. View the complete course at: ocw.mit.edu License: Creative Commons BY-NC-SA More information at ocw.mit.edu More courses at ocw.mit.edu
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Lecture 2: Elimination with Matrices. View the complete course at: ocw.mit.edu License: Creative Commons BY-NC-SA More information at ocw.mit.edu More courses at ocw.mit.edu
- published: 07 May 2009
- views: 235696
- author: MIT
Good Will Hunting Scene (Math Problem)
- Order: Reorder
- Duration: 2:39
- Updated: 24 Jan 2013
Janitor solves math problem that took 2 years to solve in one day.
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Janitor solves math problem that took 2 years to solve in one day.
- published: 15 Sep 2008
- views: 602353
- author: samyeargin
Algebra Touch for the iPad
- Order: Reorder
- Duration: 1:05
- Updated: 18 Jan 2013
Available on the App Store: itunes.apple.com
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Available on the App Store: itunes.apple.com
- published: 07 Aug 2010
- views: 28048
- author: algebratouch
Linear Algebra: Vector Examples
- Order: Reorder
- Duration: 25:33
- Updated: 21 Jan 2013
Learn more: www.khanacademy.org Visually understanding basic vector operations
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Learn more: www.khanacademy.org Visually understanding basic vector operations
- published: 08 Oct 2009
- views: 129934
- author: khanacademy
Leo and Satan - Algebra Aversion
- Order: Reorder
- Duration: 2:19
- Updated: 24 Jan 2013
T-SHIRTS sharkrobot.com
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Lec 16 | MIT 18.06 Linear Algebra, Spring 2005
- Order: Reorder
- Duration: 48:05
- Updated: 17 Jan 2013
Lecture 16: Projection Matrices and Least Squares. View the complete course at: ocw.mit.edu License: Creative Commons BY-NC-SA More information at ocw.mit.edu More courses at ocw.mit.edu
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Lecture 16: Projection Matrices and Least Squares. View the complete course at: ocw.mit.edu License: Creative Commons BY-NC-SA More information at ocw.mit.edu More courses at ocw.mit.edu
- published: 07 May 2009
- views: 51824
- author: MIT
Lec 30 | MIT 18.06 Linear Algebra, Spring 2005
- Order: Reorder
- Duration: 49:27
- Updated: 16 Jan 2013
Lecture 30: Linear Transformations and Their Matrices. View the complete course at: ocw.mit.edu License: Creative Commons BY-NC-SA More information at ocw.mit.edu More courses at ocw.mit.edu
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Lecture 30: Linear Transformations and Their Matrices. View the complete course at: ocw.mit.edu License: Creative Commons BY-NC-SA More information at ocw.mit.edu More courses at ocw.mit.edu
- published: 07 May 2009
- views: 54741
- author: MIT
Algebra: Slope
- Order: Reorder
- Duration: 8:27
- Updated: 17 Jan 2013
Learn more: www.khanacademy.org Figuring out the slope of a line
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Learn more: www.khanacademy.org Figuring out the slope of a line
- published: 25 Nov 2006
- views: 331484
- author: khanacademy
Linear Algebra: Introduction to Eigenvalues and Eigenvectors
- Order: Reorder
- Duration: 7:43
- Updated: 20 Jan 2013
Learn more: www.khanacademy.org What eigenvectors and eigenvalues are and why they are interesting
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Learn more: www.khanacademy.org What eigenvectors and eigenvalues are and why they are interesting
- published: 13 Nov 2009
- views: 178766
- author: khanacademy
Lec 34 | MIT 18.06 Linear Algebra, Spring 2005
- Order: Reorder
- Duration: 43:26
- Updated: 09 Jan 2013
Lecture 34: Final Course Review. View the complete course at: ocw.mit.edu License: Creative Commons BY-NC-SA More information at ocw.mit.edu More courses at ocw.mit.edu
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Lecture 34: Final Course Review. View the complete course at: ocw.mit.edu License: Creative Commons BY-NC-SA More information at ocw.mit.edu More courses at ocw.mit.edu
- published: 07 May 2009
- views: 29192
- author: MIT
Lec 3 | MIT 18.06 Linear Algebra, Spring 2005
- Order: Reorder
- Duration: 46:49
- Updated: 24 Jan 2013
Lecture 3: Multiplication and Inverse Matrices. View the complete course at: ocw.mit.edu License: Creative Commons BY-NC-SA More information at ocw.mit.edu More courses at ocw.mit.edu
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Lecture 3: Multiplication and Inverse Matrices. View the complete course at: ocw.mit.edu License: Creative Commons BY-NC-SA More information at ocw.mit.edu More courses at ocw.mit.edu
- published: 07 May 2009
- views: 191681
- author: MIT
photo: AP / Manish
02 Feb 2013
NEW DELHI — A crowd of Tibetans came here to India’s capital last week, bearing flags and political banners and a bittersweet admixture of hope and despair. A grim countdown was under way. The number of Tibetans who have set themselves on fire to protest Chinese rule in Tibet had reached 99, one short of an anguished milestone. Enlarge This Image. Freetibet.org, via Associated Press ... Related ... Connect With Us on Twitter ... (Mr ... Mr ... ....(size: 10.7Kb)
photo: AP / SANA
04 Feb 2013
Tweet. London, Feb. 4 (ANI). Syrian President Bashar al-Assad has accused Israel of trying to 'destabilise' his country. Assad's comments came as he claimed that last week's airstrike showed the Jewish state was behind the revolution seeking to overthrow his regime ... Assad insisted however, that his country could still meet 'current threats and aggression' ... (ANI) ... Israel ... ... ) .. ). ... . . . . . .. . . .. ....(size: 3.0Kb)
photo: UN / John Isaac
04 Feb 2013
About 800,000 people died during the 1994 Rwandan genocide. ....(size: 0.5Kb)
photo: US Navy / MCS2 Matthew D. Leistikow
03 Feb 2013
At least 33 people were killed on Sunday after a suicide bomber and two gunmen attacked a police headquarters in Kirkuk, northern Iraq, official sources said, the third major attack in the region in several weeks. By News Wires (text). . A suicide bomber driving a car and gunmen disguised in police uniforms killed at least 33 people in the Iraqi city Kirkuk on Sunday when they tried to storm the police headquarters ... (REUTERS). IRAQ....(size: 8.5Kb)
photo: AP / Jerome Delay
04 Feb 2013
French warplanes have carried out air strikes in Mali's far north as they try to secure the final rebel stronghold of Kidal after a three-week offensive. Thirty jets targeted Islamist militants' training and communication centres around Tessalit - a mountainous area near the Algerian border. French President Francois Hollande has pledged to help rebuild Mali after the rebels who seized its north are beaten ... ....(size: 2.7Kb)
Fresno Bee
04 Feb 2013
By falling in line with other states, California is abandoning its push for all eighth-graders to take algebra ... away from a 15-year policy of expecting eighth-graders to take Algebra I....(size: 1.0Kb)
Sacramento Bee
04 Feb 2013
-- California is abandoning an algebra requirement for eighth graders, a move already adopted by most states ... Algebra I....(size: 0.6Kb)
MSNBC
04 Feb 2013
California is abandoning an ambitious plan that effectively required middle school students to take algebra in the eighth grade. Even though most other states offer algebra in ... But while other states allow students who are ready for advanced mathematics to take algebra in eighth grade - or in some cases, even younger - none but California required it....(size: 2.0Kb)
noodls
04 Feb 2013
Thanks to his comprehensive mathematical understanding and his ability to apply highly sophisticated techniques from modern algebraic geometry and automorphous forms, Prof ... Rapoport, who has been Professor for arithmetic algebraic geometry at the University of Bonn since 2003, first became famous for his studies on Shimura Varieties and proof of the ......(size: 2.8Kb)
Sacramento Bee
04 Feb 2013
The Times of India
04 Feb 2013
PUNE ... The change will be applicable for science and mathematics subjects, board chairman Sarjerao Jadhav told TOI on Monday ... It will be reduced to 100 marks - 40 marks each for the algebra and geometry written examinations and 20 marks for internal assessment ... ....(size: 1.4Kb)
The Tribune San Luis Obispo
04 Feb 2013
Capitol Alert. Dan Walters Daily. Immigration reform is 'good politics' for GOP. 9 minutes ago. State abandons algebra requirement for 8th graders. 14 minutes ago. Capitol Alert. AM Alert. Lawmakers honor civil rights activist Rosa Parks. 9 minutes ago. Calif. farmworker gets $1M for mangled arm. 34 minutes ago ... 1 hour ago. Deals! ... Next Story > State abandons algebra requirement for 8th graders ... ....(size: 14.6Kb)
Stockhouse
04 Feb 2013
DALLAS, Feb ... "Schools are faced with tremendous fiscal challenges ... Voyager Learning summer school solutions include proven, award-winning solutions such as the TimeWarp® Plus reading intervention, Voyager Adventure Series experiential reading and math solution, Vmath Summer Adventure® math intervention, Voyager Online Algebra, and Everyday English Plus for newcomers to the English language ... About Voyager Learning ... ABCD)....(size: 2.7Kb)
The Guardian
04 Feb 2013
It should have been a car crash. But despite some grinding gear changes, the risky format proved to have lots of fuel in the tank. Jonathan Ross visits the Comic Relief-funded Queensland school in Accra. Photograph. Christian Thompson. Comic Relief, on the face of it, should never really have worked ... In 13 Red Nose Days since that first brave venture in 1988, we have raised £660m ... English, French, science, art, algebra. it was all here ... ....(size: 5.8Kb)
The Tribune San Luis Obispo
04 Feb 2013
Capitol Alert. Dan Walters Daily. Immigration reform is 'good politics' for GOP. 9 minutes ago. State abandons algebra requirement for 8th graders. 14 minutes ago. Capitol Alert. AM Alert. Lawmakers honor civil rights activist Rosa Parks. 9 minutes ago. Calif. farmworker gets $1M for mangled arm. 34 minutes ago. Body found after East LA building fire. 1 hour ago. Deals!. View All. ... 2013 ... ....(size: 12.6Kb)
The Tribune San Luis Obispo
04 Feb 2013
2 found shot to death in car in parking structure. 10 minutes ago. 5 Things-Today. 25 minutes ago. Capitol Alert. Dan Walters Daily. Immigration reform is 'good politics' for GOP. 45 minutes ago. State abandons algebra requirement for 8th graders. 50 minutes ago. Capitol Alert. AM Alert. Lawmakers honor civil rights activist Rosa Parks. 45 minutes ago. Deals!. View All. . . ... ....(size: 12.8Kb)
The Tribune San Luis Obispo
04 Feb 2013
Robbery victim says authorities took his marijuana. 13 minutes ago. 2 found shot to death in car in parking structure. 28 minutes ago. 5 Things-Today. 43 minutes ago. Capitol Alert. Dan Walters Daily. Immigration reform is 'good politics' for GOP. 1 hour ago. State abandons algebra requirement for 8th graders. 1 hour ago. Deals!. View All. . . . . . . ... ....(size: 13.0Kb)
The Tribune San Luis Obispo
04 Feb 2013
Robbery victim says authorities took his marijuana. 13 minutes ago. 2 found shot to death in car in parking structure. 28 minutes ago. 5 Things-Today. 43 minutes ago. Capitol Alert. Dan Walters Daily. Immigration reform is 'good politics' for GOP. 1 hour ago. State abandons algebra requirement for 8th graders. 1 hour ago. Deals!. View All. . . . . . . ... ....(size: 13.1Kb)
This article is about the branch of mathematics. For other uses, see Algebra (disambiguation).
Algebra (from Arabic al-jebr meaning "reunion of broken parts"[1]) is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of pure mathematics.
Elementary algebra, often part of the curriculum in secondary education, introduces the concept of variables representing numbers. Statements based on these variables are manipulated using the rules of operations that apply to numbers, such as addition. This can be done for a variety of reasons, including equation solving. Algebra is much broader than elementary algebra and studies what happens when different rules of operations are used and when operations are devised for things other than numbers. Addition and multiplication can be generalized and their precise definitions lead to structures such as groups, rings and fields, studied in the area of mathematics called abstract algebra.
Contents |
History[link]
Main articles: History of algebra and Timeline of algebra
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.[2] Diophantus (3rd century AD), sometimes called "the father of algebra", was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations.[3]
While the word algebra comes from the Arabic language (الجبر al-jabr "restoration") and much of its methods from Arabic/Islamic mathematics, its roots can be traced to earlier traditions, which had a direct influence on Muhammad 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.[4]
The roots of algebra can be traced to the ancient Babylonians,[5] 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 mathematicians 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, though this would not be realized until the medieval Muslim mathematicians.[citation needed]
The Hellenistic mathematicians Hero of Alexandria and Diophantus [6] as well as Indian mathematicians such as Brahmagupta continued the traditions of Egypt and Babylon, though Diophantus' Arithmetica and Brahmagupta's Brahmasphutasiddhanta are on a higher level.[7] For example, the first complete arithmetic solution (including zero and negative solutions) to quadratic equations was described by Brahmagupta in his book Brahmasphutasiddhanta. Later, Arabic and Muslim 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 was the first to solve equations using general methods. He solved the linear indeterminate equations, quadratic equations, second order indeterminate equations and equations with multiple variables.[citation needed]
The Greek mathematician Diophantus has traditionally been known as the "father of algebra" but in more recent times there is much debate over whether al-Khwarizmi, who founded the discipline of al-jabr, deserves that title instead.[8] 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.[9] 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,[10] and that he gave an exhaustive explanation of solving quadratic equations,[11] supported by geometric proofs, while treating algebra as an independent discipline in its own right.[12] 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."[13]
The Persian mathematician Omar Khayyam is credited with identifying the foundations of algebraic geometry and found the general geometric solution of the cubic equation. Another Persian mathematician, Sharaf al-Dīn al-Tūsī, found algebraic and numerical solutions to various cases of cubic equations.[14] He also developed the concept of a function.[15] The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji,[16] 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. As the Islamic world was declining, the European world was ascending. And it is here that algebra was further developed.
François Viète’s work at the close of the 16th century marks the start of the classical discipline of 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 Kowa Seki 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 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, initially focusing on what is now called Galois theory, and on constructibility issues.[17] The "modern algebra" has deep nineteenth-century roots in the work, for example, of Richard Dedekind and Leopold Kronecker and profound interconnections with other branches of mathematics such as algebraic number theory and algebraic geometry.[18] 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).[19]
Classification[link]
Algebra may be divided roughly into the following categories:
- Elementary algebra, in which the properties of operations on the real number system are recorded using symbols as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied. This is usually taught at school under the title algebra (or intermediate algebra and college algebra in subsequent years). University-level courses in group theory may also be called elementary algebra.
- Abstract algebra, sometimes also called modern algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated.
- Linear algebra, in which the specific properties of vector spaces are studied (including matrices);
- Universal algebra, in which properties common to all algebraic structures are studied.
- Algebraic number theory, in which the properties of numbers are studied through algebraic systems. Number theory inspired much of the original abstraction in algebra.
- Algebraic geometry applies abstract algebra to the problems of geometry.
- Algebraic combinatorics, in which abstract algebraic methods are used to study combinatorial questions.
In some directions of advanced study, axiomatic algebraic systems such as groups, rings, fields, and algebras over a field are investigated in the presence of a geometric structure (a metric or a topology) which is compatible with the algebraic structure. The list includes a number of areas of functional analysis:
Elementary algebra[link]
Main article: Elementary algebra
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 denoted by symbols (such as a, x, or y). 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 allows 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.")
Abstract algebra[link]
Main articles: Abstract algebra and Algebraic structure
Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts.
Sets: Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of sets: a collection of all objects (called elements) selected by property, specific for the set. 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 in the plane, and the various finite groups such as the cyclic groups which are the group 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 abstracted to give a binary operation, ∗ say. 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, a ∗ b 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 is addition and multiplication of matrices, vectors, and polynomials.
Identity elements: The numbers zero and one are abstracted 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 a ∗ e = a and e ∗ a = a. This holds for addition as a + 0 = a and 0 + a = a and multiplication a × 1 = a and 1 × a = a. Not all set and operator combinations have an identity element; for example, the positive natural numbers (1, 2, 3, ...) have 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 a ∗ a−1 = 1 and a−1 ∗ a = 1 .
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 (a ∗ b) ∗ c = a ∗ (b ∗ c). 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 a ∗ b = b ∗ a. This property does not hold for all binary operations. For example, matrix multiplication and quaternion multiplication are both non-commutative.
Groups[link]
Main article: Group (mathematics)
See also: Group theory and Examples of 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, e ∗ a and a ∗ e are both identical to a.
- Every element has an inverse: for every member a of S, there exists a member a−1 such that a ∗ a−1 and a−1 ∗ a are both identical to the identity element.
- The operation is associative: if a, b and c are members of S, then (a ∗ b) ∗ c is identical to a ∗ (b ∗ c).
If a group is also commutative—that is, for any two members a and b of S, a ∗ b is identical to b ∗ a—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 nonzero 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 ¼, which is not an integer.
The theory of groups is studied in group theory. A major result in this theory is the classification of finite simple groups, mostly published between about 1955 and 1983, which is thought to classify all of the finite simple groups into roughly 30 basic types.
| Examples | ||||||||||
| Set: | Natural numbers N | Integers Z | Rational numbers Q (also real R and complex C numbers) | Integers modulo 3: Z3 = {0, 1, 2} | ||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Operation | + | × (w/o zero) | + | × (w/o zero) | + | − | × (w/o zero) | ÷ (w/o zero) | + | × (w/o zero) |
| Closed | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | 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 | 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 | quasigroup | abelian group | quasigroup | abelian group | abelian group (Z2) |
Semigroups, quasigroups, and monoids are structures similar to groups, but more general. They comprise a set and a closed binary operation, but do not necessarily satisfy the other conditions. A semigroup has an associative binary operation, but might not have an identity element. A monoid is a semigroup which does have an identity but might not have an inverse for every element. A quasigroup satisfies a requirement that any element can be turned into any other by a unique pre- or post-operation; however the binary operation might not be associative.
All groups are monoids, and all monoids are semigroups.
Rings and fields[link]
Main articles: ring (mathematics) and field (mathematics)
See also: Ring theory, Glossary of ring theory, Field theory (mathematics), and glossary of field theory
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 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, and specifies the order in which the operators should be applied, (called the precedence). 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.
Polynomials[link]
Main article: Polynomial
A polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, and multiplication (where repeated multiplication of the same variable is standardly denoted as exponentiation with a constant nonnegative integer exponent). For example, x2 + 2x − 3 is a polynomial in the single variable x.
An important class of problems in algebra is factorization of polynomials, that is, expressing a given polynomial as a product of other polynomials. 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.
Objects called algebras[link]
The word algebra is also used for various algebraic structures:
- Algebra over a field or more generally Algebra over a ring
- Algebra over a set
- Boolean algebra
- Heyting algebra
- F-algebra and F-coalgebra in category theory
- Relational algebra
- Sigma-algebra
- T-Algebras of monads.
See also[link]
Notes[link]
- ^ "algebra". Online Etymology Dictionary. http://www.etymonline.com/index.php?term=algebra&allowed_in_frame=0.
- ^ (Boyer 1991, "Europe in the Middle Ages" p. 258) "In the arithmetical theorems in Euclid's Elements VII-IX, numbers had been represented by line segments to which letters had been had been attached, and the geometric proofs in al-Khwarizmi's Algebra made use of lettered diagrams; but all coefficients in the equations used in the Algebra are specific numbers, whether represented by numerals or written out in words. The idea of generality is implied in al-Khwarizmi's exposition, but he had no scheme for expressing algebraically the general propositions that are so readily available in geometry."
- ^ Florian Cajori (2010). "A History of Elementary Mathematics – With Hints on Methods of Teaching". p.34. ISBN 1-4460-2221-8
- ^ Roshdi Rashed (November 2009), Al Khwarizmi: The Beginnings of Algebra, Saqi Books, ISBN 0-86356-430-5
- ^ Struik, Dirk J. (1987). A Concise History of Mathematics. New York: Dover Publications.
- ^ Diophantus, Father of Algebra
- ^ History of Algebra
- ^ Carl B. Boyer, A History of Mathematics, Second Edition (Wiley, 1991), pages 178, 181
- ^ Carl B. Boyer, A History of Mathematics, Second Edition (Wiley, 1991), page 228
- ^ (Boyer 1991, "The Arabic Hegemony" p. 229) "It is not certain just what the terms al-jabr and muqabalah mean, but the usual interpretation is similar to that implied in the translation above. The word al-jabr presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word muqabalah is said to refer to "reduction" or "balancing" – that is, the cancellation of like terms on opposite sides of the equation."
- ^ (Boyer 1991, "The Arabic Hegemony" p. 230) "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was al-Khwarizmi's exposition that his readers must have had little difficulty in mastering the solutions."
- ^ Gandz and Saloman (1936), The sources of al-Khwarizmi's algebra, Osiris i, p. 263–277: "In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers".
- ^ Rashed, R.; Armstrong, Angela (1994), The Development of Arabic Mathematics, Springer, pp. 11–2, ISBN 0-7923-2565-6, OCLC 29181926
- ^ O'Connor, John J.; Robertson, Edmund F., "Sharaf al-Din al-Muzaffar al-Tusi", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Tusi_Sharaf.html .
- ^ Victor J. Katz, Bill Barton; Barton, Bill (October 2007), "Stages in the History of Algebra with Implications for Teaching", Educational Studies in Mathematics (Springer Netherlands) 66 (2): 185–201 [192], DOI:10.1007/s10649-006-9023-7
- ^ (Boyer 1991, "The Arabic Hegemony" p. 239) "Abu'l Wefa was a capable algebraist as well as a trigonometer. [...] His successor al-Karkhi evidently used this translation to become an Arabic disciple of Diophantus – but without Diophantine analysis! [...] In particular, to al-Karkhi is attributed the first numerical solution of equations of the form ax2n + bxn = c (only equations with positive roots were considered),"
- ^ "The Origins of Abstract Algebra". University of Hawaii Mathematics Department.
- ^ "The History of Algebra in the Nineteenth and Twentieth Centuries". Mathematical Sciences Research Institute.
- ^ "The Collected Mathematical Papers".Cambridge University Press.
References[link]
- Donald R. Hill, Islamic Science and Engineering (Edinburgh University Press, 1994).
- Ziauddin Sardar, Jerry Ravetz, and Borin Van Loon, Introducing Mathematics (Totem Books, 1999).
- George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics (Penguin Books, 2000).
- John J O'Connor and Edmund F Robertson, History Topics: Algebra Index. In MacTutor History of Mathematics archive (University of St Andrews, 2005).
- I.N. Herstein: Topics in Algebra. ISBN 0-471-02371-X
- R.B.J.T. Allenby: Rings, Fields and Groups. ISBN 0-340-54440-6
- L. Euler: Elements of Algebra, ISBN 978-1-899618-73-6
- Isaac Asimov Realm of Algebra (Houghton Mifflin), 1961
External links[link]
 |
Wikibooks has more on the topic of |
- 4000 Years of Algebra, lecture by Robin Wilson, at Gresham College, October 17, 2007 (available for MP3 and MP4 download, as well as a text file).
- Algebra entry by Vaughan Pratt in the Stanford Encyclopedia of Philosophy
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This article is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License, which means that you can copy and modify it as long as the entire work (including additions) remains under this license.
