1.
Fibonacci number
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The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics, the sequence described in Liber Abaci began with F1 =1. Fibonacci numbers are related to Lucas numbers L n in that they form a complementary pair of Lucas sequences U n = F n and V n = L n. They are intimately connected with the ratio, for example. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is a journal dedicated to their study. The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody, in the Sanskrit tradition of prosody, there was interest in enumerating all patterns of long syllables that are 2 units of duration, and short syllables that are 1 unit of duration. Counting the different patterns of L and S of a given duration results in the Fibonacci numbers, susantha Goonatilake writes that the development of the Fibonacci sequence is attributed in part to Pingala, later being associated with Virahanka, Gopāla, and Hemachandra. He dates Pingala before 450 BC, however, the clearest exposition of the sequence arises in the work of Virahanka, whose own work is lost, but is available in a quotation by Gopala, Variations of two earlier meters. For example, for four, variations of meters of two three being mixed, five happens, in this way, the process should be followed in all mātrā-vṛttas. The sequence is also discussed by Gopala and by the Jain scholar Hemachandra, outside India, the Fibonacci sequence first appears in the book Liber Abaci by Fibonacci. The puzzle that Fibonacci posed was, how many pairs will there be in one year, at the end of the first month, they mate, but there is still only 1 pair. At the end of the month the female produces a new pair. At the end of the month, the original female produces a second pair. At the end of the month, the original female has produced yet another new pair. At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs plus the number of pairs alive last month and this is the nth Fibonacci number. The name Fibonacci sequence was first used by the 19th-century number theorist Édouard Lucas, the most common such problem is that of counting the number of compositions of 1s and 2s that sum to a given total n, there are Fn+1 ways to do this. For example, if n =5, then Fn+1 = F6 =8 counts the eight compositions, 1+1+1+1+1 = 1+1+1+2 = 1+1+2+1 = 1+2+1+1 = 2+1+1+1 = 2+2+1 = 2+1+2 = 1+2+2, all of which sum to 5. The Fibonacci numbers can be found in different ways among the set of strings, or equivalently
2.
Otto Fibonacci
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This article contains a list of minor characters in the American television series Prison Break. The listed characters are those who are played by guest stars, the characters are listed alphabetically by their last name or by the name which appears in the episode credits. Jonathan Krantz, or the Pad Man to fans, is the main antagonist in the series and he is the leader of The Company and holds the rank of General. He appears only fleetingly in his first two seasons, as William Kims shadowy superior and architect of Michael Scofields incarceration at Sona. The character plays a large and important role in season 4, in earlier seasons, he is identified by his tendency to issue orders on notepads, to avoid being recorded, a tendency which he fully abandons by the fourth season. In his appearance in the season 4 premiere, he discovers that James Whistler, after Whistlers betrayal, he orders drastic measures to be taken against The Companys enemies, dispatching the ruthless assassin Wyatt to kill anyone and everyone necessary to achieve those ends. When it is reported that Agent Don Self did an identity search on the General, he seems to be very shocked, in Safe and Sound, it is revealed to the viewers that the Generals name is Jonathan Krantz, as seen from the images of him. Learning that the General has one of the Scylla Cards, Michaels team plan an attack against his limo in The Price, due to the betrayal of team member Roland Glenn, the General is alerted and the attack is not successful. Now knowing that Scylla is in danger, the General orders it to be moved and he is tricked in the next episode into thinking that Wyatt has killed the brothers, but decides to go ahead with moving Scylla anyway. He plays a role in Selfless, where he comes face to face with the Scylla team for the first time. When he goes to confront Michael in the bunker, he is forced to hand over his card by Michaels team at gunpoint. Thinking it is useless without the other cards, hes shocked when Michael unlocks Scylla with copies of the five cards. Under duress and the threat of his daughter Lisas life, he allows the Scylla team to escape, in the following episodes, the General is put under immense pressure as he must direct efforts to retrieve Scylla while dealing with dissent amongst his own ranks. After learning in Just Business that Gretchen Morgan and Self has Scylla, he sends his men to retrieve it and his men fail, but capture Michael instead. Discovering that he is in a condition, he is taken back to Company HQ. The General orders the doctors to look after Michael and keep him stable, the General has a somewhat reduced role in these final episodes, often having only a couple of scenes per episode. This is in due to the introduction of a new prominent antagonist. A former employee of The Company, Christina turns rogue and acquires Scylla to set her own plan in motion
3.
Camposanto Monumentale
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The Campo Santo, also known as Camposanto Monumentale or Camposanto Vecchio, is a historical edifice at the northern edge of the Cathedral Square in Pisa, Italy. A legend claims that bodies buried in that ground will rot in just 24 hours, the burial ground lies over the ruins of the old baptistery of the church of Santa Reparata, the church that once stood where the cathedral now stands. The term monumental serves to differentiate it from the urban cemetery in Pisa. The building was the fourth and last one to be raised in the Cathedral Square and it dates from a century after the bringing of the soil from Golgotha, and was erected over the earlier burial ground. The construction of huge, oblong Gothic cloister was begun in 1278 by the architect Giovanni di Simone. He died in 1284 when Pisa suffered a defeat in the battle of Meloria against the Genoans. The cemetery was completed in 1464. It seems that the building was not meant to be a cemetery, but a church called Santissima Trinità. However we know that the part was the western one. The outer wall is composed of 43 blind arches, the one on the right is crowned by a gracious Gothic tabernacle. It contains the Virgin Mary with Child, surrounded by four saints and it is the work from the second half of the 14th century by a follower of Giovanni Pisano. This was the entrance door. Most of the tombs are under the arcades, although a few are on the central lawn, the inner court is surrounded by elaborate round arches with slender mullions and plurilobed tracery. In the Aulla chapel we can see also the original incense lamp that Galileo Galilei used for calculation of pendular movements and this lamp is the one Galileo saw inside the cathedral, now replaced by a larger more elaborate one. The last chapel was Dal Pozzo, commissioned by archbishop of Pisa Carlo Antonio Dal Pozzo in 1594, it has a dedicated to St. Jerome. Also in the Dal Pozzo chapel sometimes a Mass is celebrated, the sarcophagi were initially all around the cathedral, often attached to the building itself. That until the cemetery was built, then they were collected in the all over the meadow. Carlo Lasinio, in the years he was the curator of the Campo Santo, nowadays the sarcophagi are inside the galleries, near the walls
4.
Pisa
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Pisa is a city in Tuscany, Central Italy, straddling the Arno just before it empties into the Tyrrhenian Sea. It is the city of the Province of Pisa. Although Pisa is known worldwide for its tower, the city of over 90,834 residents contains more than 20 other historic churches, several medieval palaces. Much of the architecture was financed from its history as one of the Italian maritime republics. The origin of the name, Pisa, is a mystery, while the origin of the city had remained unknown for centuries, the Pelasgi, the Greeks, the Etruscans, and the Ligurians had variously been proposed as founders of the city. Archaeological remains from the 5th century BC confirmed the existence of a city at the sea, trading with Greeks, the presence of an Etruscan necropolis, discovered during excavations in the Arena Garibaldi in 1991, confirmed its Etruscan origins. Ancient Roman authors referred to Pisa as an old city, strabo referred Pisas origins to the mythical Nestor, king of Pylos, after the fall of Troy. Virgil, in his Aeneid, states that Pisa was already a center by the times described. The Virgilian commentator Servius wrote that the Teuti, or Pelops, the maritime role of Pisa should have been already prominent if the ancient authorities ascribed to it the invention of the naval ram. Pisa took advantage of being the port along the western coast from Genoa to Ostia. Pisa served as a base for Roman naval expeditions against Ligurians, Gauls, in 180 BC, it became a Roman colony under Roman law, as Portus Pisanus. In 89 BC, Portus Pisanus became a municipium, Emperor Augustus fortified the colony into an important port and changed the name in Colonia Iulia obsequens. It is supposed that Pisa was founded on the shore, however, due to the alluvial sediments from the Arno and the Serchio, whose mouth lies about 11 kilometres north of the Arnos, the shore moved west. Strabo states that the city was 4.0 kilometres away from the coast, currently, it is located 9.7 kilometres from the coast. However it was a city, with ships sailing up the Arno. In the 90s AD, a complex was built in the city. During the later years of the Roman Empire, Pisa did not decline as much as the cities of Italy, probably thanks to the complexity of its river system. After Charlemagne had defeated the Lombards under the command of Desiderius in 774, Pisa went through a crisis, politically it became part of the duchy of Lucca
5.
Mathematician
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A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems. Mathematics is concerned with numbers, data, quantity, structure, space, models, one of the earliest known mathematicians was Thales of Miletus, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, the number of known mathematicians grew when Pythagoras of Samos established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was All is number. It was the Pythagoreans who coined the term mathematics, and with whom the study of mathematics for its own sake begins, the first woman mathematician recorded by history was Hypatia of Alexandria. She succeeded her father as Librarian at the Great Library and wrote works on applied mathematics. Because of a dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked. Science and mathematics in the Islamic world during the Middle Ages followed various models and it was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. As these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences, an example of a translator and mathematician who benefited from this type of support was al-Khawarizmi. A notable feature of many working under Muslim rule in medieval times is that they were often polymaths. Examples include the work on optics, maths and astronomy of Ibn al-Haytham, the Renaissance brought an increased emphasis on mathematics and science to Europe. As time passed, many gravitated towards universities. Moving into the 19th century, the objective of universities all across Europe evolved from teaching the “regurgitation of knowledge” to “encourag productive thinking. ”Thus, seminars, overall, science became the focus of universities in the 19th and 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge. ”Mathematicians usually cover a breadth of topics within mathematics in their undergraduate education, and then proceed to specialize in topics of their own choice at the graduate level. In some universities, a qualifying exam serves to test both the breadth and depth of an understanding of mathematics, the students, who pass, are permitted to work on a doctoral dissertation. Mathematicians involved with solving problems with applications in life are called applied mathematicians. Applied mathematicians are mathematical scientists who, with their knowledge and professional methodology. With professional focus on a variety of problems, theoretical systems
6.
Liber Abaci
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Liber Abaci is a historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci. Liber Abaci was among the first Western books to describe Hindu–Arabic numbers traditionally described as Arabic Numerals, by addressing the applications of both commercial tradesmen and mathematicians, it contributed to convincing the public of the superiority of the Hindu–Arabic numeral system. The title of Liber Abaci means The Book of Calculation, the second version of Liber Abaci was dedicated to Michael Scot in 1227 CE. No versions of the original 1202 CE book have been found, the first section introduces the Hindu–Arabic numeral system, including methods for converting between different representation systems. The second section presents examples from commerce, such as conversions of currency and measurements, another example in this chapter, describing the growth of a population of rabbits, was the origin of the Fibonacci sequence for which the author is most famous today. The fourth section derives approximations, both numerical and geometrical, of irrational numbers such as square roots, the book also includes proofs in Euclidean geometry. Fibonaccis method of solving algebraic equations shows the influence of the early 10th-century Egyptian mathematician Abū Kāmil Shujāʿ ibn Aslam, there are three key differences between Fibonaccis notation and modern fraction notation. We generally write a fraction to the right of the number to which it is added. Fibonacci instead would write the same fraction to the left, i. e.132. That is, b a d c = a c + b c d, the notation was read from right to left. For example, 29/30 could be written as 124235 and this can be viewed as a form of mixed radix notation, and was very convenient for dealing with traditional systems of weights, measures, and currency. Sigler also points out an instance where Fibonacci uses composite fractions in which all denominators are 10, Fibonacci sometimes wrote several fractions next to each other, representing a sum of the given fractions. For instance, 1/3+1/4 = 7/12, so a notation like 14132 would represent the number that would now more commonly be written as the mixed number 2712, or simply the improper fraction 3112. Notation of this form can be distinguished from sequences of numerators and denominators sharing a fraction bar by the break in the bar. If all numerators are 1 in a written in this form, and all denominators are different from each other. This notation was also combined with the composite fraction notation. The complexity of this notation allows numbers to be written in different ways. In the Liber Abaci, Fibonacci says the following introducing the Modus Indorum or the method of the Indians, today known as Hindu–Arabic numerals or traditionally, just Arabic numerals
7.
Italians
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Italians are a nation and ethnic group native to Italy who share a common culture, ancestry and speak the Italian language as a native tongue. The majority of Italian nationals are speakers of Standard Italian. Italians have greatly influenced and contributed to the arts and music, science, technology, cuisine, sports, fashion, jurisprudence, banking, Italian people are generally known for their localism and their attention to clothing and family values. The term Italian is at least 3,000 years old and has a history that goes back to pre-Roman Italy. According to one of the common explanations, the term Italia, from Latin, Italia, was borrowed through Greek from the Oscan Víteliú. The bull was a symbol of the southern Italic tribes and was often depicted goring the Roman wolf as a defiant symbol of free Italy during the Social War. Greek historian Dionysius of Halicarnassus states this account together with the legend that Italy was named after Italus, mentioned also by Aristotle and Thucydides. The Etruscan civilization reached its peak about the 7th century BC, but by 509 BC, when the Romans overthrew their Etruscan monarchs, its control in Italy was on the wane. By 350 BC, after a series of wars between Greeks and Etruscans, the Latins, with Rome as their capital, gained the ascendancy by 272 BC, and they managed to unite the entire Italian peninsula. This period of unification was followed by one of conquest in the Mediterranean, in the course of the century-long struggle against Carthage, the Romans conquered Sicily, Sardinia and Corsica. Finally, in 146 BC, at the conclusion of the Third Punic War, with Carthage completely destroyed and its inhabitants enslaved, octavian, the final victor, was accorded the title of Augustus by the Senate and thereby became the first Roman emperor. After two centuries of rule, in the 3rd century AD, Rome was threatened by internal discord and menaced by Germanic and Asian invaders. Emperor Diocletians administrative division of the empire into two parts in 285 provided only temporary relief, it became permanent in 395, in 313, Emperor Constantine accepted Christianity, and churches thereafter rose throughout the empire. However, he moved his capital from Rome to Constantinople. The last Western emperor, Romulus Augustulus, was deposed in 476 by a Germanic foederati general in Italy and his defeat marked the end of the western part of the Roman Empire. During most of the period from the fall of Rome until the Kingdom of Italy was established in 1861, Odoacer ruled well for 13 years after gaining control of Italy in 476. Then he was attacked and defeated by Theodoric, the king of another Germanic tribe, Theodoric and Odoacer ruled jointly until 493, when Theodoric murdered Odoacer. Theodoric continued to rule Italy with an army of Ostrogoths and a government that was mostly Italian, after the death of Theodoric in 526, the kingdom began to grow weak
8.
Republic of Pisa
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The Republic of Pisa was a de facto independent state centered on the Tuscan city of Pisa during the late 10th and 11th centuries. It rose to become a powerhouse, a commercial center whose merchants dominated Mediterranean and Italian trade for a century before being surpassed and superseded by the Republic of Genoa. The power of Pisa as a mighty maritime nation began to grow, during the High Middle Ages the city grew into a very important commercial and naval center and controlled a significant Mediterranean merchant fleet and navy. It expanded its influence through the sack of Reggio di Calabria in the south of Italy in 1005, Pisa was in continuous conflict with the Saracens, whose bases were in the Italian astersa, for control of the Mediterranean. In alliance with Genoa, Sardinia was captured in 1016 with the defeat of the Saracen leader Mujāhid al-‘Āmirī and this victory gave Pisa supremacy in the Tyrrhenian Sea. When the Pisans subsequently ousted the Genoese from Sardinia, a new conflict, between 1030 and 1035 Pisa went on to successfully defeat several rival towns in the Emirate of Sicily and conquer Carthage in North Africa. In 1051-1052 Admiral Jacopo Ciurini conquered Corsica, provoking more resentment from the Genoese. In 1063, the Pisans approached the Norman Roger I of Sicily, Roger declined due to other commitments. With no land support, the Pisan attack against Palermo failed, in 1060 Pisa engaged in its first battle against Genoa and the Pisan victory helped to consolidate its position in the Mediterranean. This was simply a confirmation of the present situation, because at the time the marquis of Tuscany had already excluded from power. Pisa sacked the Zirid city of Mahdia in 1088, four years later, Pisan and Genoese ships helped Alfonso VI of Castile force El Cid out of Valencia. In 1092 Pope Urban II awarded Pisa supremacy over Corsica and Sardinia, a Pisan fleet of 120 ships participated in the First Crusade and the Pisans were instrumental in the siege of Jerusalem in 1099. On their way to the Holy Land the Pisan ships did not miss the opportunity to sack several Byzantine islands, the Pisan crusaders were led by their archbishop, Dagobert, the future Latin Patriarch of Jerusalem. Pisa and the maritime republics took advantage of the crusade to establish trading posts and colonies in the eastern coastal regions of Syria, Lebanon. In particular the Pisans founded colonies in Antioch, Acre, Jaffa, Tripoli, Tyre, in all these cities the Pisans were granted privileges and immunity from taxation, but had to contribute to their defence in case of attack. In the 12th century the Pisan quarter in the part of Constantinople had grown to 1,000 people. For some years of that century Pisa was the most prominent merchant and military ally of the Byzantine Empire, Pisa, as an international power, was destroyed forever by the crushing defeat of its navy in the Battle of Meloria against Genoa in 1284. In this battle, most of the Pisan galleys were destroyed, in 1290, an assault by Genoese ships against the Porto Pisano sealed the fate of the independent Pisan state
9.
Middle Ages
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In the history of Europe, the Middle Ages or Medieval Period lasted from the 5th to the 15th century. It began with the fall of the Western Roman Empire and merged into the Renaissance, the Middle Ages is the middle period of the three traditional divisions of Western history, classical antiquity, the medieval period, and the modern period. The medieval period is subdivided into the Early, High. Population decline, counterurbanisation, invasion, and movement of peoples, the large-scale movements of the Migration Period, including various Germanic peoples, formed new kingdoms in what remained of the Western Roman Empire. In the seventh century, North Africa and the Middle East—once part of the Byzantine Empire—came under the rule of the Umayyad Caliphate, although there were substantial changes in society and political structures, the break with classical antiquity was not complete. The still-sizeable Byzantine Empire survived in the east and remained a major power, the empires law code, the Corpus Juris Civilis or Code of Justinian, was rediscovered in Northern Italy in 1070 and became widely admired later in the Middle Ages. In the West, most kingdoms incorporated the few extant Roman institutions, monasteries were founded as campaigns to Christianise pagan Europe continued. The Franks, under the Carolingian dynasty, briefly established the Carolingian Empire during the later 8th, the Crusades, first preached in 1095, were military attempts by Western European Christians to regain control of the Holy Land from Muslims. Kings became the heads of centralised nation states, reducing crime and violence, intellectual life was marked by scholasticism, a philosophy that emphasised joining faith to reason, and by the founding of universities. Controversy, heresy, and the Western Schism within the Catholic Church paralleled the conflict, civil strife. Cultural and technological developments transformed European society, concluding the Late Middle Ages, the Middle Ages is one of the three major periods in the most enduring scheme for analysing European history, classical civilisation, or Antiquity, the Middle Ages, and the Modern Period. Medieval writers divided history into periods such as the Six Ages or the Four Empires, when referring to their own times, they spoke of them as being modern. In the 1330s, the humanist and poet Petrarch referred to pre-Christian times as antiqua, leonardo Bruni was the first historian to use tripartite periodisation in his History of the Florentine People. Bruni and later argued that Italy had recovered since Petrarchs time. The Middle Ages first appears in Latin in 1469 as media tempestas or middle season, in early usage, there were many variants, including medium aevum, or middle age, first recorded in 1604, and media saecula, or middle ages, first recorded in 1625. The alternative term medieval derives from medium aevum, tripartite periodisation became standard after the German 17th-century historian Christoph Cellarius divided history into three periods, Ancient, Medieval, and Modern. The most commonly given starting point for the Middle Ages is 476, for Europe as a whole,1500 is often considered to be the end of the Middle Ages, but there is no universally agreed upon end date. English historians often use the Battle of Bosworth Field in 1485 to mark the end of the period
10.
Guglielmo Libri Carucci dalla Sommaja
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Guglielmo Libri Carucci dalla Sommaja was an Italian count and mathematician, who became known for his love and subsequent theft of ancient and precious manuscripts. He was sentenced in France to 10 years in jail in absentia, some of the works were returned when he died. The letter was written by Descartes to Father Marin Mersenne who had been overseeing the publication of Descartess Meditations on First Philosophy and he was born on born on New Years Day, January 1,1803 in Florence, Italy. He entered the University of Pisa in 1816, starting to study law and he graduated in 1820, his first works being praised by Babbage, Cauchy, and Gauss. In 1823, at the age of 20, he was appointed Professor of Mathematical Physics at Pisa, but did not relish teaching and the following year went on sabbatical leave, traveling to Paris. There, he became friends with many of the most prominent French mathematicians of the day, including Laplace, Poisson, Ampère, Fourier and Arago. Upon his return to Italy, he involved in politics. Faced with arrest and prosecution, he fled to France In 1833 and he was elected to the Academy and given the Légion dhonneur. Although his friendship with Arago helped him obtain some these prestigious posts, eventually their relationship went sour, between 1838 and 1841 Count Libri wrote and published a four volume History of the Mathematical Sciences in Italy from the Renaissance of literature to the 17th Century. In 1841, Libri obtained an appointment as Chief Inspector of French Libraries through his friendship with influential French Chief of Police François Guizot, in 1842, he stole the Ashburnham Pentateuch at the Library of Tours. He did not, occasionally, hesitate to mutilate certain manuscripts, five volumes of the Peiresc fund, in 1848, as France was involved in a liberal revolution and the government fell, a warrant was issued for Libris arrest. However he received a tip-off and fled to London, shipping 18 large trunks of books and manuscripts, about 30,000 items, before doing so. On June 22,1850, he was, however, found guilty of theft by a French Court and his friend, the archaeologist and writer Prosper Mérimée, argued in his favor and was prosecuted for this. Although Libri had arrived in England with nothing but his books and manuscripts, he led a good life and his money came from selling his books. Two large sales held in 1861 reputedly netted him over a million francs, in 1868, when his health started to deteriorate, Libri returned to Florence and died in Fiesole, Italy on September 28,1869. In 1888, the 5th Earl of Ashburnham sold a part of the documents stolen in France to the French national library including the Ashburnham Pentateuch. In 2010, one of the items, a letter from Descartes to Father Marin Mersenne concerning the publication of “Meditations on First Philosophy”, was discovered in the library of Haverford College. The college has returned the letter to the Institut de France on June 8,2010, andrea Del Centina, Alessandra Fiocca, Guglielmo Libri, matematico e storico della matematica
11.
Consul
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Consul was the title of one of the chief magistrates of the Roman Republic, and subsequently a somewhat significant title under the Roman Empire. The title was used in other city states and also revived in modern states. The relating adjective is consular, from the consularis, in modern terminology, a consul is a type of diplomat. The American Heritage Dictionary defines consul as an appointed by a government to reside in a foreign country. Throughout most of southern France, a consul was an equivalent to the échevins of the north. The most prominent were those of Bordeaux and Toulouse, which came to be known as jurats and capitouls, the capitouls of Toulouse were granted transmittable nobility. In many other towns the first consul, was the equivalent of a mayor today, assisted by a variable number of secondary consuls. His main task was to levy and collect tax, the Dukes of Gaeta often used also the title of consul in its Greek form Hypatos. The city-state of Genoa, unlike ancient Rome, bestowed the title of consul on various state officials, among these were Genoese officials stationed in various Mediterranean ports, whose role included helping Genoese merchants and sailors in difficulties with the local authorities. This institution, with its name, was emulated by other powers and is reflected in the modern usage of the word. In reality, the first consul, Bonaparte, dominated his two colleagues and held power, soon making himself consul for life and eventually, in 1804. Chief magistrate, an office held for four months by one of the consuls. As noted above, Bologna already had consuls at some parts of its Medieval history, while many cities had a double-headed chief magistracy, often another title was used, such as Duumvir or native styles such as Meddix, but consul was used in some. It was not uncommon for an organization under Roman private law to copy the terminology of state, the founding statute, or contract, of such an organisation was called lex, law. The people elected each year were patricians, members of the upper class. org, see each present country
12.
Almohad Caliphate
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The Almohad Caliphate was a Moroccan Berber Muslim movement founded in the 12th century. The Almohad movement was founded by Ibn Tumart among the Berber Masmuda tribes of southern Morocco, around 1120, the Almohads first established a Berber state in Tinmel in the Atlas Mountains. They succeeded in overthrowing the ruling Almoravid dynasty governing Morocco by 1147 and they then extended their power over all of the Maghreb by 1159. Al-Andalus followed the fate of North Africa and all Islamic Iberia was under Almohad rule by 1172, nearly all of the Moorish dominions in Iberia were lost soon after, with the great Moorish cities of Cordova and Seville falling to the Christians in 1236 and 1248 respectively. The Almohads continued to rule in Africa until the loss of territory through the revolt of tribes and districts enabled the rise of their most effective enemies. The Almohad movement originated with Ibn Tumart, a member of the Masmuda, at the time, Morocco, and much of the rest of North Africa and Spain, was under the rule of the Almoravids, a Sanhaja Berber dynasty. Early in his life, Ibn Tumart went to Spain to pursue his studies, in Baghdad, Ibn Tumart attached himself to the theological school of al-Ashari, and came under the influence of the teacher al-Ghazali. He soon developed his own system, combining the doctrines of various masters, Ibn Tumarts main principle was a strict unitarianism, which denied the independent existence of the attributes of God as being incompatible with His unity, and therefore a polytheistic idea. Ibn Tumart represented a revolt against what he perceived as anthropomorphism in Muslim orthodoxy and his followers would become known as the al-Muwahhidun, meaning those who affirm the unity of God. After his return to the Maghreb c.1117, Ibn Tumart spent some time in various Ifriqiyan cities, preaching and agitating, heading riotous attacks on wine-shops and on other manifestations of laxity. He laid the blame for the latitude on the dynasty of the Almoravids. His antics and fiery preaching led fed-up authorities to him along from town to town. After being expelled from Bejaia, Ibn Tumart set up camp in Mellala, in the outskirts of the city, where he received his first disciples - notably, al-Bashir and Abd al-Mumin. In 1120, Ibn Tumart and his band of followers proceeded to Morocco, stopping first in Fez. He even went so far as to assault the sister of the Almoravid emir Ali ibn Yusuf, in the streets of Fez, because she was going about unveiled, after the manner of Berber women. After the debate, the scholars concluded that Ibn Tumarts views were blasphemous and the man dangerous, but the emir decided merely to expel him from the city. Ibn Tumart took refuge among his own people, the Hargha, in his village of Igiliz. He retreated to a cave, and lived out an ascetic lifestyle, coming out only to preach his program of puritan reform
13.
North Africa
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North Africa or Northern Africa is the northernmost region of Africa. The United Nationss definition of Northern Africa is, Algeria, Egypt, Libya, Morocco, Sudan, Tunisia, the countries of Algeria, Morocco, Tunisia, and Libya are often collectively referred to as the Maghreb, which is the Arabic word for sunset. Egypt lies to the northeast and encompasses part of West Asia, while Sudan is situated on the edge of the Sahel, Egypt is a transcontinental country because of the Sinai Peninsula, which geographically lies in Western Asia. North Africa also includes a number of Spanish possessions, the Canary Islands and Madeira in the North Atlantic Ocean northwest of the African mainland are included in considerations of the region. From 3500 BC, following the abrupt desertification of the Sahara due to changes in the Earths orbit. The Islamic influence in the area is significant, and North Africa is a major part of the Muslim world. Some researchers have postulated that North Africa rather than East Africa served as the point for the modern humans who first trekked out of the continent in the Out of Africa migration. The Atlas Mountains extend across much of Morocco, northern Algeria and Tunisia, are part of the mountain system that also runs through much of Southern Europe. They recede to the south and east, becoming a steppe landscape before meeting the Sahara desert, the sediments of the Sahara overlie an ancient plateau of crystalline rock, some of which is more than four billion years old. Sheltered valleys in the Atlas Mountains, the Nile Valley and Delta, a wide variety of valuable crops including cereals, rice and cotton, and woods such as cedar and cork, are grown. Typical Mediterranean crops, such as olives, figs, dates and citrus fruits, the Nile Valley is particularly fertile, and most of the population in Egypt and Sudan live close to the river. Elsewhere, irrigation is essential to improve yields on the desert margins. The inhabitants of Saharan Africa are generally divided in a manner corresponding to the principal geographic regions of North Africa, the Maghreb, the Nile valley. The edge of the Sahel, to the south of Egypt has mainly been inhabited by Nubians, Ancient Egyptians record extensive contact in their Western desert with people that appear to have been Berber or proto-Berber, as well as Nubians from the south. They have contributed to the Arabized Berber populations, the official language or one of the official languages in all of the countries in North Africa is Arabic. The people of the Maghreb and the Sahara regions speak Berber languages and several varieties of Arabic, the Arabic and Berber languages are distantly related, both being members of the Afroasiatic language family. The Tuareg Berber languages are more conservative than those of the coastal cities. Over the years, Berbers have been influenced by contact with cultures, Greeks, Phoenicians, Egyptians, Romans, Vandals, Arabs, Europeans
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Algeria
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Algeria, officially the Peoples Democratic Republic of Algeria, is a sovereign state in North Africa on the Mediterranean coast. Its capital and most populous city is Algiers, located in the far north. With an area of 2,381,741 square kilometres, Algeria is the tenth-largest country in the world, the country is a semi-presidential republic consisting of 48 provinces and 1,541 communes. Abdelaziz Bouteflika has been President since 1999, Berbers are the indigenous inhabitants of Algeria. Algeria is a regional and middle power, the North African country supplies large amounts of natural gas to Europe, and energy exports are the backbone of the economy. According to OPEC Algeria has the 16th largest oil reserves in the world, Sonatrach, the national oil company, is the largest company in Africa. Algeria has one of the largest militaries in Africa and the largest defence budget on the continent, most of Algerias weapons are imported from Russia, with whom they are a close ally. Algeria is a member of the African Union, the Arab League, OPEC, the countrys name derives from the city of Algiers. The citys name in turn derives from the Arabic al-Jazāir, a form of the older Jazāir Banī Mazghanna. In the region of Ain Hanech, early remnants of hominid occupation in North Africa were found, neanderthal tool makers produced hand axes in the Levalloisian and Mousterian styles similar to those in the Levant. Algeria was the site of the highest state of development of Middle Paleolithic Flake tool techniques, tools of this era, starting about 30,000 BC, are called Aterian. The earliest blade industries in North Africa are called Iberomaurusian and this industry appears to have spread throughout the coastal regions of the Maghreb between 15,000 and 10,000 BC. Neolithic civilization developed in the Saharan and Mediterranean Maghreb perhaps as early as 11,000 BC or as late as between 6000 and 2000 BC and this life, richly depicted in the Tassili nAjjer paintings, predominated in Algeria until the classical period. The amalgam of peoples of North Africa coalesced eventually into a native population that came to be called Berbers. These settlements served as market towns as well as anchorages, as Carthaginian power grew, its impact on the indigenous population increased dramatically. Berber civilization was already at a stage in which agriculture, manufacturing, trade, by the early 4th century BC, Berbers formed the single largest element of the Carthaginian army. In the Revolt of the Mercenaries, Berber soldiers rebelled from 241 to 238 BC after being unpaid following the defeat of Carthage in the First Punic War. They succeeded in obtaining control of much of Carthages North African territory, the Carthaginian state declined because of successive defeats by the Romans in the Punic Wars
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Mediterranean Sea
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The sea is sometimes considered a part of the Atlantic Ocean, although it is usually identified as a separate body of water. The name Mediterranean is derived from the Latin mediterraneus, meaning inland or in the middle of land and it covers an approximate area of 2.5 million km2, but its connection to the Atlantic is only 14 km wide. The Strait of Gibraltar is a strait that connects the Atlantic Ocean to the Mediterranean Sea and separates Gibraltar. In oceanography, it is called the Eurafrican Mediterranean Sea or the European Mediterranean Sea to distinguish it from mediterranean seas elsewhere. The Mediterranean Sea has a depth of 1,500 m. The sea is bordered on the north by Europe, the east by Asia and it is located between latitudes 30° and 46° N and longitudes 6° W and 36° E. Its west-east length, from the Strait of Gibraltar to the Gulf of Iskenderun, the seas average north-south length, from Croatia’s southern shore to Libya, is approximately 800 km. The Mediterranean Sea, including the Sea of Marmara, has an area of approximately 2,510,000 square km. The sea was an important route for merchants and travelers of ancient times that allowed for trade, the history of the Mediterranean region is crucial to understanding the origins and development of many modern societies. In addition, the Gaza Strip and the British Overseas Territories of Gibraltar and Akrotiri, the term Mediterranean derives from the Latin word mediterraneus, meaning amid the earth or between land, as it is between the continents of Africa, Asia and Europe. The Ancient Greek name Mesogeios, is similarly from μέσο, between + γη, land, earth) and it can be compared with the Ancient Greek name Mesopotamia, meaning between rivers. The Mediterranean Sea has historically had several names, for example, the Carthaginians called it the Syrian Sea and latter Romans commonly called it Mare Nostrum, and occasionally Mare Internum. Another name was the Sea of the Philistines, from the people inhabiting a large portion of its shores near the Israelites, the sea is also called the Great Sea in the General Prologue by Geoffrey Chaucer. In Ottoman Turkish, it has also been called Bahr-i Sefid, in Modern Hebrew, it has been called HaYam HaTikhon, the Middle Sea, reflecting the Seas name in ancient Greek, Latin, and modern languages in both Europe and the Middle East. Similarly, in Modern Arabic, it is known as al-Baḥr al-Mutawassiṭ, in Turkish, it is known as Akdeniz, the White Sea since among Turks the white colour represents the west. Several ancient civilisations were located around the Mediterranean shores, and were influenced by their proximity to the sea. It provided routes for trade, colonisation, and war, as well as food for numerous communities throughout the ages, due to the shared climate, geology, and access to the sea, cultures centered on the Mediterranean tended to have some extent of intertwined culture and history. Two of the most notable Mediterranean civilisations in classical antiquity were the Greek city states, later, when Augustus founded the Roman Empire, the Romans referred to the Mediterranean as Mare Nostrum
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Frederick II, Holy Roman Emperor
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Frederick II was a Holy Roman Emperor and King of Sicily in the Middle Ages, a member of the House of Hohenstaufen. His political and cultural ambitions, based in Sicily and stretching through Italy to Germany, however, his enemies, especially the popes, prevailed, and his dynasty collapsed soon after his death. As such, he was King of Germany, of Italy, at the age of three, he was crowned King of Sicily as a co-ruler with his mother, Constance of Hauteville, the daughter of Roger II of Sicily. His other royal title was King of Jerusalem by virtue of marriage, Pope Gregory IX went so far as to call him an Antichrist. Speaking six languages, Frederick was a patron of science. He played a role in promoting literature through the Sicilian School of poetry. His Sicilian royal court in Palermo, from around 1220 to his death, saw the first use of a form of an Italo-Romance language. The poetry that emanated from the school had a significant influence on literature and he was also the first king who explicitly outlawed trials by ordeal as they were considered irrational. After his death, his line died out and the House of Hohenstaufen came to an end. Born in Iesi, near Ancona, Italy, Frederick was the son of the emperor Henry VI and he was known as the puer Apuliae. Some chronicles say that his mother, the forty-year-old Constance, gave birth to him in a square in order to forestall any doubt about his origin. In 1196 at Frankfurt am Main the infant Frederick was elected King of the Germans and his rights in Germany were disputed by Henrys brother Philip of Swabia and Otto of Brunswick. At the death of his father in 1197, Frederick was in Italy travelling towards Germany when the bad news reached his guardian, Conrad of Spoleto. Frederick was hastily brought back to his mother Constance in Palermo, Sicily, Constance of Sicily was in her own right queen of Sicily, and she established herself as regent. Upon Constances death in 1198, Pope Innocent III succeeded as Fredericks guardian, Fredericks tutor during this period was Cencio, who would become Pope Honorius III. However, Markward of Annweiler, with the support of Henrys brother, Philip of Swabia, reclaimed the regency for himself, in 1200, with the help of Genoese ships, he landed in Sicily and one year later seized the young Frederick. He thus ruled Sicily until 1202, when he was succeeded by another German captain, William of Capparone, Frederick was subsequently under tutor Walter of Palearia, until, in 1208, he was declared of age. His first task was to reassert his power over Sicily and southern Italy, Otto of Brunswick had been crowned Holy Roman Emperor by Pope Innocent III in 1209
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National Central Library (Florence)
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The library was founded in 1714 when scholar Antonio Magliabechi bequeathed his entire collection of books, encompassing approximately 30,000 volumes, to the city of Florence. By 1743, it was required that a copy of work published in Tuscany be submitted to the library. Originally known as the Magliabechiana, the library was opened to the public in 1747 and its holdings were combined with those of the Biblioteca Palatina in 1861, and by 1885, the library had been renamed as the National Central Library of Florence, or the BNCF. Since 1870, the library has collected copies of all Italian publications, since 1935, the collections have been housed in a building designed by Cesare Bazzani and V. Mazzei, located along the Arno River in the quarter of Santa Croce. Before this, they were found in various rooms belonging to the Uffizi Gallery, the National Library System, located in the BNCF, is responsible for the automation of library services and the indexing of national holdings. Unfortunately, a flood of the Arno River in 1966 damaged nearly one-third of the librarys holdings, most notably its periodicals and Palatine. The Restoration Center was subsequently established and may be credited with saving many of these priceless artifacts, however, much work remains to be done and some items are forever lost
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Positional notation
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Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the symbol for the different orders of magnitude. This greatly simplified arithmetic, leading to the spread of the notation across the world. With the use of a point, the notation can be extended to include fractions. The Hindu–Arabic numeral system, base-10, is the most commonly used system in the world today for most calculations, today, the base-10 system, which is likely motivated by counting with the ten fingers, is ubiquitous. Other bases have been used in the past however, and some continue to be used today, for example, the Babylonian numeral system, credited as the first positional numeral system, was base-60, but it lacked a real 0 value. Zero was indicated by a space between sexagesimal numerals, by 300 BC, a punctuation symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges, the Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number, thus numbers like 2 and 120,3 and 180,4 and 240, looked the same because the larger numbers lacked a final sexagesimal placeholder. Counting rods and most abacuses have been used to represent numbers in a numeral system. This approach required no memorization of tables and could produce practical results quickly, for four centuries there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the continues to be used in Japan. After the French Revolution, the new French government promoted the extension of the decimal system, some of those pro-decimal efforts—such as decimal time and the decimal calendar—were unsuccessful. Other French pro-decimal efforts—currency decimalisation and the metrication of weights and measures—spread widely out of France to almost the whole world. According to Joseph Needham and Lam Lay Yong, decimal fractions were first developed and used by the Chinese in the 1st century BC, the written Chinese decimal fractions were non-positional. However, counting rod fractions were positional, the Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them. A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century. A key argument against the system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing 100 into 5100
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Numeral system
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A numeral system is a writing system for expressing numbers, that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols 11 to be interpreted as the symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases. The number the numeral represents is called its value, ideally, a numeral system will, Represent a useful set of numbers Give every number represented a unique representation Reflect the algebraic and arithmetic structure of the numbers. For example, the decimal representation of whole numbers gives every nonzero whole number a unique representation as a finite sequence of digits. Etc. all of which have the same meaning except for some scientific, such systems are, however, not the topic of this article. The most commonly used system of numerals is the Hindu–Arabic numeral system, two Indian mathematicians are credited with developing it. Aryabhata of Kusumapura developed the notation in the 5th century. The numeral system and the concept, developed by the Hindus in India, slowly spread to other surrounding countries due to their commercial. The Arabs adopted and modified it, even today, the Arabs call the numerals which they use Rakam Al-Hind or the Hindu numeral system. The Arabs translated Hindu texts on numerology and spread them to the world due to their trade links with them. The Western world modified them and called them the Arabic numerals, hence the current western numeral system is the modified version of the Hindu numeral system developed in India. It also exhibits a great similarity to the Sanskrit–Devanagari notation, which is used in India. The simplest numeral system is the numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol / is chosen, for example, then the seven would be represented by ///////. Tally marks represent one such system still in common use, the unary system is only useful for small numbers, although it plays an important role in theoretical computer science. Elias gamma coding, which is used in data compression. The unary notation can be abbreviated by introducing different symbols for new values. The ancient Egyptian numeral system was of type, and the Roman numeral system was a modification of this idea
20.
Bookkeeping
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Bookkeeping is the recording of financial transactions, and is part of the process of accounting in business. Transactions include purchases, sales, receipts, and payments by a person or an organization/corporation. Bookkeeping is usually performed by a bookkeeper, a bookkeeper is a person who records the day-to-day financial transactions of a business. He or she is responsible for writing the daybooks, which contain records of purchases, sales, receipts. The bookkeeper brings the books to the trial stage, an accountant may prepare the income statement and balance sheet using the trial balance. The origin of book-keeping is lost in obscurity, but recent researches would appear to show that some method of keeping accounts has existed from the remotest times. Babylonian records have been found dating back as far as 2600 B. C. written with a stylus on small slabs of clay, the term waste book was used in colonial America referring to bookkeeping. The purpose was to document daily transactions including receipts and expenditures and this was recorded in chronological order, and the purpose was for temporary use only. The daily transactions would then be recorded in a daybook or account ledger in order to balance the accounts, the name waste book comes from the fact that once the waste books data were transferred to the actual journal, the waste book could be discarded. The bookkeeping process primarily records the effects of transactions. The difference between a manual and any electronic accounting system results from the formers latency between the recording of a transaction and its posting in the relevant account. In the normal course of business, a document is produced each time a transaction occurs, Sales and purchases usually have invoices or receipts. Deposit slips are produced when lodgements are made to a bank account, checks are written to pay money out of the account. Bookkeeping first involves recording the details of all of these documents into multi-column journals. For example, all sales are recorded in the sales journal. Each column in a journal normally corresponds to an account, in the single entry system, each transaction is recorded only once. Most individuals who balance their check-book each month are using such a system, after a certain period, typically a month, each column in each journal is totalled to give a summary for that period. Using the rules of double-entry, these summaries are then transferred to their respective accounts in the ledger
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Abacus
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The exact origin of the abacus is still unknown. Today, abaci are often constructed as a frame with beads sliding on wires. The use of the word abacus dates before 1387 AD, when a Middle English work borrowed the word from Latin to describe a sandboard abacus, the Latin word came from Greek ἄβαξ abax which means something without base, and improperly, any piece of rectangular board or plank. Alternatively, without reference to ancient texts on etymology, it has suggested that it means a square tablet strewn with dust. Whereas the table strewn with dust definition is popular, there are those that do not place credence in this at all, Greek ἄβαξ itself is probably a borrowing of a Northwest Semitic, perhaps Phoenician, word akin to Hebrew ʾābāq, dust. The preferred plural of abacus is a subject of disagreement, with both abacuses and abaci in use, the user of an abacus is called an abacist. The period 2700–2300 BC saw the first appearance of the Sumerian abacus, some scholars point to a character from the Babylonian cuneiform which may have been derived from a representation of the abacus. Archaeologists have found ancient disks of various sizes that are thought to have used as counters. However, wall depictions of this instrument have not been discovered, during the Achaemenid Empire, around 600 BC the Persians first began to use the abacus. The earliest archaeological evidence for the use of the Greek abacus dates to the 5th century BC, also Demosthenes talked of the need to use pebbles for calculations too difficult for your head. The Greek abacus was a table of wood or marble, pre-set with small counters in wood or metal for mathematical calculations and this Greek abacus saw use in Achaemenid Persia, the Etruscan civilization, Ancient Rome and, until the French Revolution, the Western Christian world. A tablet found on the Greek island Salamis in 1846 AD, dates back to 300 BC and it is a slab of white marble 149 cm long,75 cm wide, and 4.5 cm thick, on which are 5 groups of markings. Below these lines is a space with a horizontal crack dividing it. Also from this frame the Darius Vase was unearthed in 1851. It was covered with pictures including a holding a wax tablet in one hand while manipulating counters on a table with the other. The earliest known documentation of the Chinese abacus dates to the 2nd century BC. The Chinese abacus, known as the suanpan, is typically 20 cm tall and it usually has more than seven rods. There are two beads on each rod in the deck and five beads each in the bottom for both decimal and hexadecimal computation
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Accounting
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Accounting or accountancy is the measurement, processing and communication of financial information about economic entities such as businesses and corporations. The modern field was established by the Italian mathematician Luca Pacioli in 1494, practitioners of accounting are known as accountants. The terms accounting and financial reporting are often used as synonyms, Accounting can be divided into several fields including financial accounting, management accounting, external auditing, and tax accounting. Accounting information systems are designed to support accounting functions and related activities, Accounting is facilitated by accounting organizations such as standard-setters, accounting firms and professional bodies. Financial statements are audited by accounting firms, and are prepared in accordance with generally accepted accounting principles. GAAP is set by various standard-setting organizations such as the Financial Accounting Standards Board in the United States, as of 2012, all major economies have plans to converge towards or adopt the International Financial Reporting Standards. The history of accounting is thousands of old and can be traced to ancient civilizations. By the time of the Emperor Augustus, the Roman government had access to detailed financial information, double-entry bookkeeping developed in medieval Europe, and accounting split into financial accounting and management accounting with the development of joint-stock companies. The first work on a double-entry bookkeeping system was published in Italy, both the words accounting and accountancy were in use in Great Britain by the mid-1800s, and are derived from the words accompting and accountantship used in the 18th century. In Middle English the verb to account had the form accounten, which was derived from the Old French word aconter, which is in turn related to the Vulgar Latin word computare, meaning to reckon. The base of computare is putare, which meant to prune, to purify, to correct an account, hence, to count or calculate. The word accountant is derived from the French word compter, which is derived from the Italian. Accountancy refers to the occupation or profession of an accountant, particularly in British English, Accounting has several subfields or subject areas, including financial accounting, management accounting, auditing, taxation and accounting information systems. Financial accounting focuses on the reporting of a financial information to external users of the information, such as investors. It calculates and records business transactions and prepares financial statements for the users in accordance with generally accepted accounting principles. GAAP, in turn, arises from the agreement between accounting theory and practice, and change over time to meet the needs of decision-makers. This branch of accounting is also studied as part of the exams for qualifying as an actuary. It is interesting to note that two professionals, accountants and actuaries, have created a culture of being archrivals
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Irrational numbers
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In mathematics, the irrational numbers are all the real numbers, which are not rational numbers, the latter being the numbers constructed from ratios of integers. Irrational numbers may also be dealt with via non-terminating continued fractions, for example, the decimal representation of the number π starts with 3.14159265358979, but no finite number of digits can represent π exactly, nor does it repeat. Mathematicians do not generally take terminating or repeating to be the definition of the concept of rational number, as a consequence of Cantors proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational. The first proof of the existence of numbers is usually attributed to a Pythagorean. The then-current Pythagorean method would have claimed that there must be sufficiently small. However, Hippasus, in the 5th century BC, was able to deduce that there was in no common unit of measure. His reasoning is as follows, Start with a right triangle with side lengths of integers a, b. The ratio of the hypotenuse to a leg is represented by c, b, assume a, b, and c are in the smallest possible terms. By the Pythagorean theorem, c2 = a2+b2 = b2+b2 = 2b2, since c2 = 2b2, c2 is divisible by 2, and therefore even. Since c2 is even, c must be even, since c is even, dividing c by 2 yields an integer. Squaring both sides of c = 2y yields c2 =2, or c2 = 4y2, substituting 4y2 for c2 in the first equation gives us 4y2= 2b2. Dividing by 2 yields 2y2 = b2, since y is an integer, and 2y2 = b2, b2 is divisible by 2, and therefore even. Since b2 is even, b must be even and we have just show that both b and c must be even. Hence they have a factor of 2. However this contradicts the assumption that they have no common factors and this contradiction proves that c and b cannot both be integers, and thus the existence of a number that cannot be expressed as a ratio of two integers. Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible. ”Another legend states that Hippasus was merely exiled for this revelation, the discovery of incommensurable ratios was indicative of another problem facing the Greeks, the relation of the discrete to the continuous. Brought into light by Zeno of Elea, who questioned the conception that quantities are discrete and composed of a number of units of a given size. ”However Zeno found that in fact “ in general are not discrete collections of units. That in fact, these divisions of quantity must necessarily be infinite, for example, consider a line segment, this segment can be split in half, that half split in half, the half of the half in half, and so on
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Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
25.
Golden ratio
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In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship, expressed algebraically, for quantities a and b with a > b >0, a + b a = a b = def φ, where the Greek letter phi represents the golden ratio. Its value is, φ =1 +52 =1.6180339887 …, A001622 The golden ratio is also called the golden mean or golden section. Other names include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, the golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts. The golden ratio has also used to analyze the proportions of natural objects as well as man-made systems such as financial markets. Two quantities a and b are said to be in the golden ratio φ if a + b a = a b = φ, one method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ, a + b a =1 + b a =1 +1 φ, multiplying by φ gives φ +1 = φ2 which can be rearranged to φ2 − φ −1 =0. First, the line segment A B ¯ is about doubled and then the semicircle with the radius A S ¯ around the point S is drawn, now the semicircle is drawn with the radius A B ¯ around the point B. The arising intersection point E corresponds 2 φ, next up, the perpendicular on the line segment A E ¯ from the point D will be establish. The subsequent parallel F S ¯ to the line segment C M ¯, produces, as it were and it is well recognizable, this triangle and the triangle M S C are similar to each other. The hypotenuse F S ¯ has due to the cathetuses S D ¯ =1 and D F ¯ =2 according the Pythagorean theorem, finally, the circle arc is drawn with the radius 5 around the point F. The golden ratio has been claimed to have held a fascination for at least 2,400 years. But the fascination with the Golden Ratio is not confined just to mathematicians, biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry, the division of a line into extreme and mean ratio is important in the geometry of regular pentagrams and pentagons. Euclid explains a construction for cutting a line in extreme and mean ratio, throughout the Elements, several propositions and their proofs employ the golden ratio. The golden ratio is explored in Luca Paciolis book De divina proportione, since the 20th century, the golden ratio has been represented by the Greek letter φ or less commonly by τ. Timeline according to Priya Hemenway, Phidias made the Parthenon statues that seem to embody the golden ratio, plato, in his Timaeus, describes five possible regular solids, some of which are related to the golden ratio
26.
Piazza dei Miracoli
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Considered sacred by the Catholic Church, its owner, the square is dominated by four great religious edifices, the Pisa Cathedral, the Pisa Baptistry, the Campanile, and the Camposanto Monumentale. Partly paved and partly grassed, the Piazza dei Miracoli is also the site of the Ospedale Nuovo di Santo Spirito, which houses the Sinopias Museum, the square is sometimes called the Campo dei Miracoli. In 1987, the square was declared a UNESCO World Heritage Site. The heart of the Piazza del Duomo is the Duomo, the cathedral of the Archdiocese of Pisa. The cathedral has two aisles on either side of the nave, the transept consists of three aisles. The church is also as the Primatial, the archbishop of Pisa being a Primate since 1092. Its construction began in 1064 by the architect Buscheto and it set the model for the distinctive Pisan Romanesque style of architecture. The mosaics of the interior, as well as the pointed arches, the façade, of grey marble and white stone set with discs of coloured marble, was built by a master named Rainaldo, as indicated by an inscription above the middle door, Rainaldus prudens operator. The massive bronze doors were made in the workshops of Giambologna. The original central door was of bronze, made around 1180 by Bonanno Pisano, however, worshippers have never used the façade doors to enter, instead entering by way of the Porta di San Ranieri, in front of the Leaning Tower, built around 1180 by Bonanno Pisano. Above the doors are four rows of galleries with, on top, statues of Madonna with Child and, on the corners. Also in the façade is found the tomb of Buscheto and an inscription about the foundation of the Cathedral, the interior is faced with black and white marble and has a gilded ceiling and a frescoed dome. It was largely redecorated after a fire in 1595, which destroyed most of the Renaissance art works, fortunately, the impressive mosaic of Christ in Majesty, in the apse, flanked by the Blessed Virgin and St. John the Evangelist, survived the fire. It evokes the mosaics in the church of Monreale, Sicily, although it is said that the mosaic was done by Cimabue, only the head of St. John was done by the artist in 1302, his last work, since he died in Pisa the same year. The cupola, at the intersection of the nave and transept, was decorated by Riminaldi showing the assumption of the Blessed Virgin. Galileo is believed to have formulated his theory about the movement of a pendulum by watching the swinging of the lamp hanging from the ceiling of the nave. That lamp, smaller and simpler than the present one, is now kept in the Camposanto, the granite Corinthian columns between the nave and the aisle came originally from the mosque of Palermo, captured by the Pisans in 1063. The coffer ceiling of the nave was replaced after the fire of 1595, the present gold-decorated ceiling carries the coat of arms of the Medici
27.
The Fibonaccis
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The Fibonaccis were an American art rock band formed in 1981 in Los Angeles. The band consisted of songwriters John Dentino and Ron Stringer, Magie Song, Joe Berardi, the Fibonaccis were formed out of the Los Angeles art punk scene which included bands such as Wall of Voodoo, Oingo Boingo and Sparks. Lyrically, the band regularly explored dark and esoteric subject matter ranging from serial killers to UFOs, presented in a satirical, the Fibonaccis released their debut EP in 1982, following up with a 12 single/EP, Tumor/Psycho/Slow Beautiful Sex, the next year. In 1984, the group filmed a music video for an unreleased cover of Jimi Hendrixs Purple Haze. Throughout their career, The Fibonaccis regularly contributed their music to independent film soundtracks, in 1986, the band collaborated with composer Richard Band on the score for the horror-comedy TerrorVision, recording five tracks including the movies theme song. Their song Sergio Leone was used for the credits of 1982s Android. The Fibonaccis appeared onscreen as the band Sexy Holiday in the 1987 comedy Valet Girls, lipsynching to Slow Beautiful Sex, in 1987, the band released their sole studio LP, Civilization and Its Discotheques on the Blue Yonder Sounds label. In explaining the reason for the LPs delay, the said that various hassles and difficulties with record companies had plagued a more timely release. Their frustration over the recording, added with a lack of media recognition. In 1992, Restless Records released a 26-track retrospective of the work called Repressed - The Best of the Fibonaccis. To celebrate the release of the album, The Fibonaccis performed a reunion show in Los Angeles on November 19,1992. Following The Fibonaccis disbandment, John Dentino continued to compose music independently and has recently been working on independent documentary films. Joe Berardi went on to perform and tour with Wall of Voodoos Stan Ridgway, Magie Song acted in a number of independent films in the early 1990s, including Gregg Arakis The Living End and Stephen Sayadians Dr. Caligari, and currently works as an acupuncturist in Los Angeles. Ron Stringer most recently served as editor and critic for the LA Weekly. Tom Corey died from an aneurysm in late 2001. Currently, the entire Fibonaccis discography is out of print, with Repressed being their only released on compact disc. In 2006, John Dentino created a website for the band. The Ordinary Women -2,234, rice Song -2,41 Tumor 1
28.
Surveying
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Surveying or land surveying is the technique, profession, and science of determining the terrestrial or three-dimensional position of points and the distances and angles between them. A land surveying professional is called a land surveyor, Surveyors work with elements of geometry, trigonometry, regression analysis, physics, engineering, metrology, programming languages and the law. Surveying has been an element in the development of the environment since the beginning of recorded history. The planning and execution of most forms of construction require it and it is also used in transport, communications, mapping, and the definition of legal boundaries for land ownership. It is an important tool for research in other scientific disciplines. Basic surveyance has occurred since humans built the first large structures, the prehistoric monument at Stonehenge was set out by prehistoric surveyors using peg and rope geometry. In ancient Egypt, a rope stretcher would use simple geometry to re-establish boundaries after the floods of the Nile River. The almost perfect squareness and north-south orientation of the Great Pyramid of Giza, built c.2700 BC, the Groma instrument originated in Mesopotamia. The mathematician Liu Hui described ways of measuring distant objects in his work Haidao Suanjing or The Sea Island Mathematical Manual, the Romans recognized land surveyors as a profession. They established the basic measurements under which the Roman Empire was divided, Roman surveyors were known as Gromatici. In medieval Europe, beating the bounds maintained the boundaries of a village or parish and this was the practice of gathering a group of residents and walking around the parish or village to establish a communal memory of the boundaries. Young boys were included to ensure the memory lasted as long as possible, in England, William the Conqueror commissioned the Domesday Book in 1086. It recorded the names of all the owners, the area of land they owned, the quality of the land. It did not include maps showing exact locations, abel Foullon described a plane table in 1551, but it is thought that the instrument was in use earlier as his description is of a developed instrument. Gunters chain was introduced in 1620 by English mathematician Edmund Gunter and it enabled plots of land to be accurately surveyed and plotted for legal and commercial purposes. Leonard Digges described a Theodolite that measured horizontal angles in his book A geometric practice named Pantometria, joshua Habermel created a theodolite with a compass and tripod in 1576. Johnathon Sission was the first to incorporate a telescope on a theodolite in 1725, in the 18th century, modern techniques and instruments for surveying began to be used. Jesse Ramsden introduced the first precision theodolite in 1787 and it was an instrument for measuring angles in the horizontal and vertical planes
29.
Measurement
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Measurement is the assignment of a number to a characteristic of an object or event, which can be compared with other objects or events. The scope and application of a measurement is dependent on the context, however, in other fields such as statistics as well as the social and behavioral sciences, measurements can have multiple levels, which would include nominal, ordinal, interval, and ratio scales. Measurement is a cornerstone of trade, science, technology, historically, many measurement systems existed for the varied fields of human existence to facilitate comparisons in these fields. Often these were achieved by local agreements between trading partners or collaborators, since the 18th century, developments progressed towards unifying, widely accepted standards that resulted in the modern International System of Units. This system reduces all physical measurements to a combination of seven base units. The science of measurement is pursued in the field of metrology, the measurement of a property may be categorized by the following criteria, type, magnitude, unit, and uncertainty. They enable unambiguous comparisons between measurements, the type or level of measurement is a taxonomy for the methodological character of a comparison. For example, two states of a property may be compared by ratio, difference, or ordinal preference, the type is commonly not explicitly expressed, but implicit in the definition of a measurement procedure. The magnitude is the value of the characterization, usually obtained with a suitably chosen measuring instrument. A unit assigns a mathematical weighting factor to the magnitude that is derived as a ratio to the property of a used as standard or a natural physical quantity. An uncertainty represents the random and systemic errors of the measurement procedure, errors are evaluated by methodically repeating measurements and considering the accuracy and precision of the measuring instrument. Measurements most commonly use the International System of Units as a comparison framework, the system defines seven fundamental units, kilogram, metre, candela, second, ampere, kelvin, and mole. Instead, the measurement unit can only ever change through increased accuracy in determining the value of the constant it is tied to and this directly influenced the Michelson–Morley experiment, Michelson and Morley cite Peirce, and improve on his method. With the exception of a few fundamental quantum constants, units of measurement are derived from historical agreements, nothing inherent in nature dictates that an inch has to be a certain length, nor that a mile is a better measure of distance than a kilometre. Over the course of history, however, first for convenience and then for necessity. Laws regulating measurement were originally developed to prevent fraud in commerce.9144 metres, in the United States, the National Institute of Standards and Technology, a division of the United States Department of Commerce, regulates commercial measurements. Before SI units were adopted around the world, the British systems of English units and later imperial units were used in Britain, the Commonwealth. The system came to be known as U. S. customary units in the United States and is still in use there and in a few Caribbean countries. S
30.
Area
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Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the surface of a three-dimensional object. It is the analog of the length of a curve or the volume of a solid. The area of a shape can be measured by comparing the shape to squares of a fixed size, in the International System of Units, the standard unit of area is the square metre, which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the area as three such squares. In mathematics, the square is defined to have area one. There are several formulas for the areas of simple shapes such as triangles, rectangles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles, for shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a motivation for the historical development of calculus. For a solid such as a sphere, cone, or cylinder. Formulas for the areas of simple shapes were computed by the ancient Greeks. Area plays an important role in modern mathematics, in addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, in general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers and it can be proved that such a function exists. An approach to defining what is meant by area is through axioms, area can be defined as a function from a collection M of special kind of plane figures to the set of real numbers which satisfies the following properties, For all S in M, a ≥0. If S and T are in M then so are S ∪ T and S ∩ T, if S and T are in M with S ⊆ T then T − S is in M and a = a − a. If a set S is in M and S is congruent to T then T is also in M, every rectangle R is in M. If the rectangle has length h and breadth k then a = hk, let Q be a set enclosed between two step regions S and T
31.
Volume
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Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre, three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, Volumes of a complicated shape can be calculated by integral calculus if a formula exists for the shapes boundary. Where a variance in shape and volume occurs, such as those that exist between different human beings, these can be calculated using techniques such as the Body Volume Index. One-dimensional figures and two-dimensional shapes are assigned zero volume in the three-dimensional space, the volume of a solid can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas, the combined volume of two substances is usually greater than the volume of one of the substances. However, sometimes one substance dissolves in the other and the volume is not additive. In differential geometry, volume is expressed by means of the volume form, in thermodynamics, volume is a fundamental parameter, and is a conjugate variable to pressure. Any unit of length gives a unit of volume, the volume of a cube whose sides have the given length. For example, a cubic centimetre is the volume of a cube whose sides are one centimetre in length, in the International System of Units, the standard unit of volume is the cubic metre. The metric system also includes the litre as a unit of volume, thus 1 litre =3 =1000 cubic centimetres =0.001 cubic metres, so 1 cubic metre =1000 litres. Small amounts of liquid are often measured in millilitres, where 1 millilitre =0.001 litres =1 cubic centimetre. Capacity is defined by the Oxford English Dictionary as the applied to the content of a vessel, and to liquids, grain, or the like. Capacity is not identical in meaning to volume, though closely related, Units of capacity are the SI litre and its derived units, and Imperial units such as gill, pint, gallon, and others. Units of volume are the cubes of units of length, in SI the units of volume and capacity are closely related, one litre is exactly 1 cubic decimetre, the capacity of a cube with a 10 cm side. In other systems the conversion is not trivial, the capacity of a fuel tank is rarely stated in cubic feet, for example. The density of an object is defined as the ratio of the mass to the volume, the inverse of density is specific volume which is defined as volume divided by mass. Specific volume is an important in thermodynamics where the volume of a working fluid is often an important parameter of a system being studied
32.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
33.
Diophantine equation
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In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied. A linear Diophantine equation is an equation between two sums of monomials of degree zero or one, an exponential Diophantine equation is one in which exponents on terms can be unknowns. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations, in more technical language, they define an algebraic curve, algebraic surface, or more general object, and ask about the lattice points on it. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis, the solutions are described by the following theorem, This Diophantine equation has a solution if and only if c is a multiple of the greatest common divisor of a and b. Moreover, if is a solution, then the solutions have the form, where k is an arbitrary integer. Proof, If d is this greatest common divisor, Bézouts identity asserts the existence of integers e and f such that ae + bf = d, If c is a multiple of d, then c = dh for some integer h, and is a solution. On the other hand, for pair of integers x and y. Thus, if the equation has a solution, then c must be a multiple of d. If a = ud and b = vd, then for every solution, we have a + b = ax + by + k = ax + by + k = ax + by, showing that is another solution. Finally, given two solutions such that ax1 + by1 = ax2 + by2 = c, one deduces that u + v =0. As u and v are coprime, Euclids lemma shows that exists a integer k such that x2 − x1 = kv. Therefore, x2 = x1 + kv and y2 = y1 − ku, the system to be solved may thus be rewritten as B = UC. Calling yi the entries of V−1X and di those of D = UC and it follows that the system has a solution if and only if bi, i divides di for i ≤ k and di =0 for i > k. If this condition is fulfilled, the solutions of the system are V. Hermite normal form may also be used for solving systems of linear Diophantine equations, however, Hermite normal form does not directly provide the solutions, to get the solutions from the Hermite normal form, one has to successively solve several linear equations. Nevertheless, Richard Zippel wrote that the Smith normal form is more than is actually needed to solve linear diophantine equations. Instead of reducing the equation to diagonal form, we only need to make it triangular, the Hermite normal form is substantially easier to compute than the Smith normal form. Integer linear programming amounts to finding some integer solutions of systems that include also inequations
34.
Congruum
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In number theory, a congruum is the difference between successive square numbers in an arithmetic progression of three squares. That is, if x2, y2, and z2 are three numbers that are equally spaced apart from each other, then the spacing between them, z2 − y2 = y2 − x2, is called a congruum. The congruum problem is the problem of finding squares in arithmetic progression and it can be formalized as a Diophantine equation, find integers x, y, and z such that y 2 − x 2 = z 2 − y 2. When this equation is satisfied, both sides of the equation equal the congruum, fibonacci solved the congruum problem by finding a parameterized formula for generating all congrua, together with their associated arithmetic progressions. According to this formula, each congruum is four times the area of a Pythagorean triangle, congrua are also closely connected with congruent numbers, every congruum is a congruent number, and every congruent number is a congruum multiplied by the square of a rational number. For instance, the number 96 is a congruum, since it is the difference between each pair of the three squares 4,100, and 196. The first few congrua are,24,96,120,216,240,336,384,480,600,720 …. Fibonacci was already aware that it is impossible for a congruum to itself be a square, geometrically, this means that it is not possible for the pair of legs of a Pythagorean triangle to be the leg and hypotenuse of another Pythagorean triangle. A proof was given by Pierre de Fermat, and the result is now known as Fermats right triangle theorem. Fermat also conjectured, and Leonhard Euler proved, that there is no sequence of four squares in arithmetic progression, the congruum problem may be solved by choosing two distinct positive integers m and n, then the number 4mn is a congruum. The middle square of the arithmetic progression of squares is 2. Additionally, multiplying a congruum by a square number produces another congruum, all solutions arise in one of these two ways. For instance, the congruum 96 can be constructed by these formulas with m =3 and n =1, the congruum itself is four times the area of the same Pythagorean triangle. The example of a progression with the congruum 96 can be obtained in this way from a right triangle with side and hypotenuse lengths 6,8. A congruent number is defined as the area of a triangle with rational sides. Because every congruum can be obtained as the area of a Pythagorean triangle, conversely, every congruent number is a congruum multiplied by the square of a rational number. However, testing whether a number is a congruum is much easier than testing whether a number is congruent, for the congruum problem, the parameterized solution reduces this testing problem to checking a finite set of parameter values. Spiral of Theodorus, formed by triangles whose sides, when squared, form an infinite arithmetic progression Weisstein
35.
Euclid's Elements
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Euclids Elements is a mathematical and geometric treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt circa 300 BC. It is a collection of definitions, postulates, propositions, the books cover Euclidean geometry and the ancient Greek version of elementary number theory. Elements is the second-oldest extant Greek mathematical treatise after Autolycus On the Moving Sphere and it has proven instrumental in the development of logic and modern science. According to Proclus, the element was used to describe a theorem that is all-pervading. The word element in the Greek language is the same as letter and this suggests that theorems in the Elements should be seen as standing in the same relation to geometry as letters to language. Euclids Elements has been referred to as the most successful and influential textbook ever written, for centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclids Elements was required of all students. Not until the 20th century, by which time its content was taught through other school textbooks. Scholars believe that the Elements is largely a collection of theorems proven by other mathematicians, the Elements may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated the use of letters to refer to figures. This manuscript, the Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions, papyrus Oxyrhynchus 29 is a tiny fragment of an even older manuscript, but only contains the statement of one proposition. Although known to, for instance, Cicero, no record exists of the text having been translated into Latin prior to Boethius in the fifth or sixth century. The Arabs received the Elements from the Byzantines around 760, this version was translated into Arabic under Harun al Rashid circa 800, the Byzantine scholar Arethas commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century. Although known in Byzantium, the Elements was lost to Western Europe until about 1120, the first printed edition appeared in 1482, and since then it has been translated into many languages and published in about a thousand different editions. Theons Greek edition was recovered in 1533, in 1570, John Dee provided a widely respected Mathematical Preface, along with copious notes and supplementary material, to the first English edition by Henry Billingsley. Copies of the Greek text still exist, some of which can be found in the Vatican Library, the manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text, ancient texts which refer to the Elements itself, and to other mathematical theories that were current at the time it was written, are also important in this process. Such analyses are conducted by J. L. Heiberg and Sir Thomas Little Heath in their editions of the text, also of importance are the scholia, or annotations to the text. These additions, which distinguished themselves from the main text. The Elements is still considered a masterpiece in the application of logic to mathematics, in historical context, it has proven enormously influential in many areas of science
36.
Fibonacci numbers in popular culture
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The Fibonacci numbers are a sequence of integers, starting with 0,1 and continuing 1,2,3,5,8,13. Each new number being the sum of the previous two, the Fibonacci numbers, and in conjunction the golden ratio, are a popular theme in culture. They have been mentioned in novels, films, television shows, the numbers have also been used in the creation of music, visual art, and architecture. Stock traders frequently look to the Fibonacci retracement when predicting future share prices, the sequence has been used in the design of a building, the Core, at the Eden Project, near St Austell, Cornwall, England. In 21, the first seven numbers in the Fibonacci Sequence are drawn in icing on Bens Birthday cake, the 8th term,21, is left out. Ben and Miles quickly figure it out, along with the golden rectangle and golden spiral, the Fibonacci sequence is mentioned in Darren Aronofskys independent film Pi. They are used to find the name of God, in The Da Vinci Code, the numbers are used to unlock a safe. They are also placed out of order in a message to indicate that the message is also out of order, in Mr. Magoriums Wonder Emporium, Magorium hires accountant Henry Weston after an interview in which he demonstrates knowledge of Fibonacci numbers. In Death Note, L, Change the World, genius boy Near is seen arranging sugar cubes in a Fibonacci sequence. In Nymphomaniac, the character Seligman notes that when Joe loses her virginity, in a strip of Frazz by Jef Mallett, Frazz and a student are discussing her knitted hat. The student says, Mom sewed one sparkly here and here, to which Frazz replies, Fibonacci sequins, of course. John Waskom postulated that stages of development followed the Fibonacci sequence. This theory was developed and published by Norman Rose in two articles. The first article, which laid out the theory, was entitled Design and Development of Wholeness. The second article laid out the applications and implications of the theory to the topic of moral development, the Fibonacci sequence plays a small part in Dan Browns bestselling novel The Da Vinci Code. In Philip K. Dicks novel VALIS, the Fibonacci sequence are used as identification signs by a called the Friends of God. In the collection of poetry alfabet by the Danish poet Inger Christensen and it was briefly included in the television film adaptation of A Wrinkle in Time. The Fibonacci sequence is referenced in the 2001 book The Perfect Spiral by Jason S. Hornsby
37.
Adelard of Bath
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Adelard of Bath was a 12th-century English natural philosopher. He is known as one of the first to introduce the Hindu–Arabic numeral system to Europe and he stands at the convergence of three intellectual schools, the traditional learning of French schools, the Greek culture of Southern Italy, and the Arabic science of the East. Given the period when he was alive, Adelard’s biography is incomplete in places and leaves some aspects open to interpretation, as a result, much of what is ascribed to Adelard is a product of his own testimony. As his name suggests, he was born in the Roman English city of Bath, despite his extensive travels, by the end of his life he had returned to Bath, where he died. Scholars are hesitant to ascribe definitive parents to the philosopher but Fastred and his name is of Anglo-Saxon origin, which would have placed him in the subordinate class, status wise, in 11th-century England. It is believed that he left England toward the end of the 11th century for Tours, likely on the advice of Bishop John de Villula, during his studies in Tours, an anonymous wise man of Tours inspired Adelard with his interest in astronomy to study the science. Adelard later taught for a time at Laon, leaving Laon for travel no later than 1109, after leaving Laon, he travelled to Southern Italy and Sicily no later than 1116. Adelard also travelled throughout the lands of the Crusades, Greece, West Asia, Sicily, Spain, Tarsus, Antioch. The time spent in these areas would help explain his fascination with mathematics, by 1126, Adelard returned to the West with the intention of spreading the knowledge he had gained about Arab astronomy and geometry to the Latin world. One aspect of particular interest with respect to Adelard, his teachings, and this time of remarkable transition marked an opportunity for someone to gain valuable influence over the evolution of human history. Again, given the 11th-century time period that Adelard was alive, in the absence of a printing press and given the weak public literacy rate, books were rare items in medieval Europe—generally held only by royal courts or Catholic monastic communities. Fittingly, Adelard studied with monks at the Benedictine Monastery at Bath Cathedral, among Adelard of Baths original works is a trio of dialogues, written to mimic the Platonic style, or correspondences with his nephew. The earliest of these is De Eodem et Diverso and it is written in the style of a protreptic, or an exhortation to the study of philosophy. The work is modelled on Boethius Consolation of Philosophy, evident in Adelards vocabulary and it is believed to have been written near Tours after he had already travelled, though there is no indication that he had travelled past Southern Italy and Sicily at the time of writing. The work takes the form of a dialogue between Philocosmia, who advocates worldly pleasures, and Philosophia, whose defence of scholarship leads into a summary of the seven liberal arts. Underlining the entire work is the contrast between Philocosmias res, and Philosophias verba, each section of the liberal arts is divided into two parts. The second of this trio, and arguably Adelards most significant contribution, was his Questiones Naturales or Questions on Natural Science. It can be dated between 1107 and 1133 as, in the text, Adelard himself mentions that seven years have passed since his lecturing in schools at Laon and he chooses to present this work as a forum for Arabic learning, referring often to his experiences in Antioch
38.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
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Keith Devlin
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Keith J. Devlin is a British mathematician and popular science writer. Since 1987 he has lived in the United States, Devlin earned a BSc in Mathematics at Kings College London in 1968, and a PhD in Mathematics at the University of Bristol in 1971 under the supervision of Frederick Rowbottom. He is a commentator on National Public Radios Weekend Edition Saturday, as of 2012, he is the author of 34 books and over 80 research articles. Several of his books are aimed at an audience of the general public, the Joy of Sets, Fundamentals of Contemporary Set Theory. Goodbye, Descartes, the End of Logic and the Search for a New Cosmology of the Mind, john Wiley & Sons, Inc.1997. The Language of Mathematics, Making the Invisible Visible, the Math Gene, How Mathematical Thinking Evolved and Why Numbers Are Like Gossip. The Millennium Problems, the Seven Greatest Unsolved Mathematical Puzzles of Our Time, the Math Instinct, Why Youre a Mathematical Genius. The Numbers Behind NUMB3RS, Solving Crime with Mathematics, with coauthor Gary Lorden The Unfinished Game, Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern. The Man of Numbers, Fibonaccis Arithmetic Revolution, Mathematics Education for a New Era, Video Games as a Medium for Learning. Official website including his curriculum vitae Devlins Angle — column at the Mathematical Association of America
