1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
2.
Golden Age of Islam
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This period is traditionally said to have ended with the collapse of the Abbasid caliphate due to Mongol invasions and the Sack of Baghdad in 1258 AD. A few contemporary scholars place the end of the Islamic Golden Age as late as the end of 15th to 16th centuries, the metaphor of a golden age began to be applied in 19th-century literature about Islamic history, in the context of the western aesthetic fashion known as Orientalism. There is no definition of term, and depending on whether it is used with a focus on cultural or on military achievement. During the early 20th century, the term was used only occasionally, the Muslim government heavily patronized scholars. The money spent on the Translation Movement for some translations is estimated to be equivalent to twice the annual research budget of the United Kingdom’s Medical Research Council. The best scholars and notable translators, such as Hunayn ibn Ishaq, had salaries that are estimated to be the equivalent of professional athletes today, the House of Wisdom was a library established in Abbasid-era Baghdad, Iraq by Caliph al-Mansur. During this period, the Muslims showed a strong interest in assimilating the knowledge of the civilizations that had been conquered. They also excelled in fields, in particular philosophy, science. For a long period of time the personal physicians of the Abbasid Caliphs were often Assyrian Christians, among the most prominent Christian families to serve as physicians to the caliphs were the Bukhtishu dynasty. Throughout the 4th to 7th centuries, Christian scholarly work in the Greek, the House of Wisdom was founded in Baghdad in 825, modelled after the Academy of Gondishapur. It was led by Christian physician Hunayn ibn Ishaq, with the support of Byzantine medicine, many of the most important philosophical and scientific works of the ancient world were translated, including the work of Galen, Hippocrates, Plato, Aristotle, Ptolemy and Archimedes. Many scholars of the House of Wisdom were of Christian background, the use of paper spread from China into Muslim regions in the eighth century, arriving in Al-Andalus on the Iberian peninsula, present-day Spain in the 10th century. It was easier to manufacture than parchment, less likely to crack than papyrus, Islamic paper makers devised assembly-line methods of hand-copying manuscripts to turn out editions far larger than any available in Europe for centuries. It was from countries that the rest of the world learned to make paper from linen. Ibn Rushd and Ibn Sina played a role in saving the works of Aristotle, whose ideas came to dominate the non-religious thought of the Christian. Ibn Sina and other such as al-Kindi and al-Farabi combined Aristotelianism and Neoplatonism with other ideas introduced through Islam. Arabic philosophic literature was translated into Latin and Ladino, contributing to the development of modern European philosophy, during this period, non-Muslims were allowed to flourish relative to treatment of religious minorities in the Christian Byzantine Empire. The Jewish philosopher Moses Maimonides, who lived in Andalusia, is an example, in epistemology, Ibn Tufail wrote the novel Hayy ibn Yaqdhan and in response Ibn al-Nafis wrote the novel Theologus Autodidactus
3.
Greek mathematics
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Greek mathematics, as the term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by culture, Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. The word mathematics itself derives from the ancient Greek μάθημα, meaning subject of instruction, the study of mathematics for its own sake and the use of generalized mathematical theories and proofs is the key difference between Greek mathematics and those of preceding civilizations. The origin of Greek mathematics is not well documented, the earliest advanced civilizations in Greece and in Europe were the Minoan and later Mycenaean civilization, both of which flourished during the 2nd millennium BC. While these civilizations possessed writing and were capable of advanced engineering, including four-story palaces with drainage and beehive tombs, though no direct evidence is available, it is generally thought that the neighboring Babylonian and Egyptian civilizations had an influence on the younger Greek tradition. Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus. Little is known about the life and work of Thales, so little indeed that his date of birth and death are estimated from the eclipse of 585 BC, despite this, it is generally agreed that Thales is the first of the seven wise men of Greece. The two earliest mathematical theorems, Thales theorem and Intercept theorem are attributed to Thales. The former, which states that an angle inscribed in a semicircle is a right angle and it is for this reason that Thales is often hailed as the father of the deductive organization of mathematics and as the first true mathematician. Thales is also thought to be the earliest known man in history to whom specific mathematical discoveries have been attributed, another important figure in the development of Greek mathematics is Pythagoras of Samos. Like Thales, Pythagoras also traveled to Egypt and Babylon, then under the rule of Nebuchadnezzar, Pythagoras established an order called the Pythagoreans, which held knowledge and property in common and hence all of the discoveries by individual Pythagoreans were attributed to the order. And since in antiquity it was customary to give all credit to the master, aristotle for one refused to attribute anything specifically to Pythagoras as an individual and only discussed the work of the Pythagoreans as a group. One of the most important characteristics of the Pythagorean order was that it maintained that the pursuit of philosophical and mathematical studies was a basis for the conduct of life. Indeed, the philosophy and mathematics are said to have been coined by Pythagoras. From this love of knowledge came many achievements and it has been customarily said that the Pythagoreans discovered most of the material in the first two books of Euclids Elements. The reason it is not clear exactly what either Thales or Pythagoras actually did is that almost no documentation has survived. The only evidence comes from traditions recorded in such as Proclus’ commentary on Euclid written centuries later. Some of these works, such as Aristotle’s commentary on the Pythagoreans, are themselves only known from a few surviving fragments
4.
Euclid
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Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the father of geometry. He was active in Alexandria during the reign of Ptolemy I, in the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, Euclid is the anglicized version of the Greek name Εὐκλείδης, which means renowned, glorious. Very few original references to Euclid survive, so little is known about his life, the date, place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is rarely mentioned by name by other Greek mathematicians from Archimedes onward, the few historical references to Euclid were written centuries after he lived by Proclus c.450 AD and Pappus of Alexandria c.320 AD. Proclus introduces Euclid only briefly in his Commentary on the Elements, Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclids Elements, Euclid replied there is no royal road to geometry. This anecdote is questionable since it is similar to a story told about Menaechmus, a detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be completely fictitious, however, this hypothesis is not well accepted by scholars and there is little evidence in its favor. The only reference that historians rely on of Euclid having written the Elements was from Proclus, although best known for its geometric results, the Elements also includes number theory. The geometrical system described in the Elements was long known simply as geometry, today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century. In addition to the Elements, at least five works of Euclid have survived to the present day and they follow the same logical structure as Elements, with definitions and proved propositions. Data deals with the nature and implications of information in geometrical problems. On Divisions of Figures, which only partially in Arabic translation. It is similar to a first-century AD work by Heron of Alexandria, catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J OConnor, phaenomena, a treatise on spherical astronomy, survives in Greek, it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC. Optics is the earliest surviving Greek treatise on perspective, in its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth, Things seen under a greater angle appear greater, proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Other works are attributed to Euclid, but have been lost
5.
Archimedes
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Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the scientists in classical antiquity. He was also one of the first to apply mathematics to physical phenomena, founding hydrostatics and statics and he is credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion. Archimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting the tomb of Archimedes, which was surmounted by a sphere and a cylinder, unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Archimedes was born c.287 BC in the city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years, in The Sand Reckoner, Archimedes gives his fathers name as Phidias, an astronomer about whom nothing is known. Plutarch wrote in his Parallel Lives that Archimedes was related to King Hiero II, a biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he married or had children. During his youth, Archimedes may have studied in Alexandria, Egypt and he referred to Conon of Samos as his friend, while two of his works have introductions addressed to Eratosthenes. Archimedes died c.212 BC during the Second Punic War, according to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, the soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives an account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he considered him a valuable asset and had ordered that he not be harmed. Marcellus called Archimedes a geometrical Briareus, the last words attributed to Archimedes are Do not disturb my circles, a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. This quote is given in Latin as Noli turbare circulos meos. The phrase is given in Katharevousa Greek as μὴ μου τοὺς κύκλους τάραττε
6.
Apollonius of Perga
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Apollonius of Perga was a Greek geometer and astronomer known for his theories on the topic of conic sections. Beginning from the theories of Euclid and Archimedes on the topic and his definitions of the terms ellipse, parabola, and hyperbola are the ones in use today. Apollonius worked on other topics, including astronomy. Most of the work has not survived except in references in other authors. His hypothesis of eccentric orbits to explain the apparently aberrant motion of the planets, for such an important contributor to the field of mathematics, scant biographical information remains. The 6th century Palestinian commentator, Eutocius of Ascalon, on Apollonius’ major work, Conics, states, “Apollonius, the geometrician. Came from Perga in Pamphylia in the times of Ptolemy Euergetes, the ruins of the city yet stand. It was a center of Hellenistic culture, Euergetes, “benefactor, ” identifies Ptolemy III Euergetes, third Greek dynast of Egypt in the diadochi succession. Presumably, his “times” are his regnum, 246-222/221 BC, times are always recorded by ruler or officiating magistrate, so that if Apollonius was born earlier than 246, it would have been the “times” of Euergetes’ father. The identity of Herakleios is uncertain, the approximate times of Apollonius are thus certain, but no exact dates can be given. The figure Specific birth and death years stated by the scholars are only speculative. Eutocius appears to associate Perga with the Ptolemaic dynasty of Egypt, never under Egypt, Perga in 246 BC belonged to the Seleucid Empire, an independent diadochi state ruled by the Seleucid dynasty. Someone designated “of Perga” might well be expected to have lived and worked there, to the contrary, if Apollonius was later identified with Perga, it was not on the basis of his residence. The remaining autobiographical material implies that he lived, studied and wrote in Alexandria, philip was assassinated in 336 BC. Alexander went on to fulfill his plan by conquering the vast Iranian empire, the material is located in the surviving false “Prefaces” of the books of his Conics. These are letters delivered to friends of Apollonius asking them to review the book enclosed with the letter. The Preface to Book I, addressed to one Eudemus, reminds him that Conics was initially requested by a house guest at Alexandria, Naucrates had the first draft of all eight books in his hands by the end of the visit. Apollonius refers to them as being “without a thorough purgation” and he intended to verify and emend the books, releasing each one as it was completed
7.
Indian mathematics
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Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics, important contributions were made by scholars like Aryabhata, Brahmagupta, Mahāvīra, Bhaskara II, Madhava of Sangamagrama, the decimal number system in worldwide use today was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, in addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China and this was followed by a second section consisting of a prose commentary that explained the problem in more detail and provided justification for the solution. In the prose section, the form was not considered so important as the ideas involved, all mathematical works were orally transmitted until approximately 500 BCE, thereafter, they were transmitted both orally and in manuscript form. A later landmark in Indian mathematics was the development of the series expansions for functions by mathematicians of the Kerala school in the 15th century CE. Their remarkable work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series. However, they did not formulate a theory of differentiation and integration. Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley Civilisation have uncovered evidence of the use of practical mathematics. The people of the Indus Valley Civilization manufactured bricks whose dimensions were in the proportion 4,2,1, considered favourable for the stability of a brick structure. They used a system of weights based on the ratios, 1/20, 1/10, 1/5, 1/2,1,2,5,10,20,50,100,200. They mass-produced weights in regular geometrical shapes, which included hexahedra, barrels, cones, the inhabitants of Indus civilisation also tried to standardise measurement of length to a high degree of accuracy. They designed a ruler—the Mohenjo-daro ruler—whose unit of length was divided into ten equal parts, bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this unit of length. The religious texts of the Vedic Period provide evidence for the use of large numbers, by the time of the Yajurvedasaṃhitā-, numbers as high as 1012 were being included in the texts. The solution to partial fraction was known to the Rigvedic People as states in the purush Sukta, With three-fourths Puruṣa went up, the Satapatha Brahmana contains rules for ritual geometric constructions that are similar to the Sulba Sutras. The Śulba Sūtras list rules for the construction of fire altars. Most mathematical problems considered in the Śulba Sūtras spring from a single theological requirement, according to, the Śulba Sūtras contain the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. The diagonal rope of an oblong produces both which the flank and the horizontal <ropes> produce separately and they contain lists of Pythagorean triples, which are particular cases of Diophantine equations
8.
Aryabhata
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Aryabhata or Aryabhata I was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the Āryabhaṭīya and the Arya-siddhanta, furthermore, in most instances Aryabhatta would not fit the metre either. Aryabhata mentions in the Aryabhatiya that it was composed 3,600 years into the Kali Yuga and this corresponds to 499 CE, and implies that he was born in 476. Aryabhata called himself a native of Kusumapura or Pataliputra, Bhāskara I describes Aryabhata as āśmakīya, one belonging to the Aśmaka country. During the Buddhas time, a branch of the Aśmaka people settled in the region between the Narmada and Godavari rivers in central India. It has been claimed that the aśmaka where Aryabhata originated may be the present day Kodungallur which was the capital city of Thiruvanchikkulam of ancient Kerala. This is based on the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr, however, K. Chandra Hari has argued for the Kerala hypothesis on the basis of astronomical evidence. Aryabhata mentions Lanka on several occasions in the Aryabhatiya, but his Lanka is an abstraction and it is fairly certain that, at some point, he went to Kusumapura for advanced studies and lived there for some time. Both Hindu and Buddhist tradition, as well as Bhāskara I, identify Kusumapura as Pāṭaliputra, Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar. Aryabhata is the author of treatises on mathematics and astronomy. His major work, Aryabhatiya, a compendium of mathematics and astronomy, was referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry and it also contains continued fractions, quadratic equations, sums-of-power series, and a table of sines. This work appears to be based on the older Surya Siddhanta and uses the midnight-day reckoning, a third text, which may have survived in the Arabic translation, is Al ntf or Al-nanf. It claims that it is a translation by Aryabhata, but the Sanskrit name of work is not known. Probably dating from the 9th century, it is mentioned by the Persian scholar and chronicler of India, direct details of Aryabhatas work are known only from the Aryabhatiya. The name Aryabhatiya is due to later commentators, Aryabhata himself may not have given it a name. His disciple Bhaskara I calls it Ashmakatantra and it is also occasionally referred to as Arya-shatas-aShTa, because there are 108 verses in the text. It is written in the terse style typical of sutra literature
9.
Brahmagupta
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Brahmagupta was an Indian mathematician and astronomer. He is the author of two works on mathematics and astronomy, the Brāhmasphuṭasiddhānta, a theoretical treatise, and the Khaṇḍakhādyaka. According to his commentators, Brahmagupta was a native of Bhinmal, Brahmagupta was the first to give rules to compute with zero. The texts composed by Brahmagupta were composed in verse in Sanskrit. As no proofs are given, it is not known how Brahmaguptas results were derived, Brahmagupta was born in 598 CE according to his own statement. He lived in Bhillamala during the reign of the Chapa dynasty ruler Vyagrahamukha and he was the son of Jishnugupta. He was a Shaivite by religion, even though most scholars assume that Brahmagupta was born in Bhillamala, there is no conclusive evidence for it. However, he lived and worked there for a part of his life. Prithudaka Svamin, a commentator, called him Bhillamalacharya, the teacher from Bhillamala. Sociologist G. S. Ghurye believed that he might have been from the Multan region or the Abu region and it was also a center of learning for mathematics and astronomy. Brahmagupta became an astronomer of the Brahmapaksha school, in the year 628, at an age of 30, he composed Brāhmasphuṭasiddhānta which is believed to be a revised version of the received siddhanta of the Brahmapaksha school. Scholars state that he has incorported a great deal of originality to his revision, the book consists of 24 chapters with 1008 verses in the ārya meter. Later, Brahmagupta moved to Ujjain, which was also a centre for astronomy. At the mature age of 67, he composed his next well known work Khanda-khādyaka and he is believed to have died in Ujjain. Brahmagupta had a plethora of criticism directed towards the work of rival astronomers, the division was primarily about the application of mathematics to the physical world, rather than about the mathematics itself. In Brahmaguptas case, the disagreements stemmed largely from the choice of astronomical parameters, the historian of science George Sarton called him one of the greatest scientists of his race and the greatest of his time. Brahmaguptas mathematical advances were carried on to further extent by Bhāskara II, a descendant in Ujjain. Prithudaka Svamin wrote commentaries on both of his works, rendering difficult verses into simpler language and adding illustrations, lalla and Bhattotpala in the 8th and 9th centuries wrote commentaries on the Khanda-khadyaka
10.
Place-value system
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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
11.
Decimal fractions
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This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation, the decimal numeral system has ten as its base, which, in decimal, is written 10, as is the base in every positional numeral system. It is the base most widely used by modern civilizations. Decimal fractions have terminating decimal representations and other fractions have repeating decimal representations, Decimal notation is the writing of numbers in a base-ten numeral system. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, Roman numerals have symbols for the decimal powers and secondary symbols for half these values. Brahmi numerals have symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100, Chinese numerals have symbols for 1–9, and additional symbols for powers of ten, which in modern usage reach 1072. Positional decimal systems include a zero and use symbols for the ten values to represent any number, positional notation uses positions for each power of ten, units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier multiplied with that position has a value ten times that of the position to its right. There were at least two independent sources of positional decimal systems in ancient civilization, the Chinese counting rod system. Ten is the number which is the count of fingers and thumbs on both hands, the English word digit as well as its translation in many languages is also the anatomical term for fingers and toes. In English, decimal means tenth, decimate means reduce by a tenth, however, the symbols used in different areas are not identical, for instance, Western Arabic numerals differ from the forms used by other Arab cultures. A decimal fraction is a fraction the denominator of which is a power of ten. g, Decimal fractions 8/10, 1489/100, 24/100000, and 58900/10000 are expressed in decimal notation as 0.8,14.89,0.00024,5.8900 respectively. In English-speaking, some Latin American and many Asian countries, a period or raised period is used as the separator, in many other countries, particularly in Europe. The integer part, or integral part of a number is the part to the left of the decimal separator. The part from the separator to the right is the fractional part. It is usual for a number that consists only of a fractional part to have a leading zero in its notation. Any rational number with a denominator whose only prime factors are 2 and/or 5 may be expressed as a decimal fraction and has a finite decimal expansion. 1/2 =0.5 1/20 =0.05 1/5 =0.2 1/50 =0.02 1/4 =0.25 1/40 =0.025 1/25 =0.04 1/8 =0.125 1/125 =0.008 1/10 =0
12.
Algebra
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Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols, as such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine, abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are unknown or allowed to take on many values. For example, in x +2 =5 the letter x is unknown, in E = mc2, the letters E and m are variables, and the letter c is a constant, the speed of light in a vacuum. Algebra gives methods for solving equations and expressing formulas that are easier than the older method of writing everything out in words. The word algebra is used in certain specialized ways. A special kind of object in abstract algebra is called an algebra. A mathematician who does research in algebra is called an algebraist, the word algebra comes from the Arabic الجبر from the title of the book Ilm al-jabr wal-muḳābala by Persian mathematician and astronomer al-Khwarizmi. The word entered the English language during the century, from either Spanish, Italian. It originally referred to the procedure of setting broken or dislocated bones. The mathematical meaning was first recorded in the sixteenth century, the word algebra has several related meanings in mathematics, as a single word or with qualifiers. As a single word without an article, algebra names a broad part of mathematics, as a single word with an article or in plural, an algebra or algebras denotes a specific mathematical structure, whose precise definition depends on the author. Usually the structure has an addition, multiplication, and a scalar multiplication, when some authors use the term algebra, they make a subset of the following additional assumptions, associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word refers to a generalization of the above concept. With a qualifier, there is the distinction, Without an article, it means a part of algebra, such as linear algebra, elementary algebra. With an article, it means an instance of some abstract structure, like a Lie algebra, sometimes both meanings exist for the same qualifier, as in the sentence, Commutative algebra is the study of commutative rings, which are commutative algebras over the integers
13.
Al-Khwarizmi
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Muḥammad ibn Mūsā al-Khwārizmī, formerly Latinized as Algoritmi, was a Persian mathematician, astronomer, and geographer during the Abbasid Caliphate, a scholar in the House of Wisdom in Baghdad. In the 12th century, Latin translations of his work on the Indian numerals introduced the decimal number system to the Western world. Al-Khwārizmīs The Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and he is often considered one of the fathers of algebra. He revised Ptolemys Geography and wrote on astronomy and astrology, some words reflect the importance of al-Khwārizmīs contributions to mathematics. Algebra is derived from al-jabr, one of the two operations he used to solve quadratic equations, algorism and algorithm stem from Algoritmi, the Latin form of his name. His name is also the origin of guarismo and of algarismo, few details of al-Khwārizmīs life are known with certainty. He was born in a Persian family and Ibn al-Nadim gives his birthplace as Khwarezm in Greater Khorasan, muhammad ibn Jarir al-Tabari gives his name as Muḥammad ibn Musá al-Khwārizmiyy al-Majūsiyy al-Quṭrubbaliyy. The epithet al-Qutrubbulli could indicate he might instead have come from Qutrubbul and this would not be worth mentioning if a series of errors concerning the personality of al-Khwārizmī, occasionally even the origins of his knowledge, had not been made. Recently, G. J. Toomer. with naive confidence constructed an entire fantasy on the error which cannot be denied the merit of amusing the reader. Regarding al-Khwārizmīs religion, Toomer writes, Another epithet given to him by al-Ṭabarī, al-Majūsī, Ibn al-Nadīms Kitāb al-Fihrist includes a short biography on al-Khwārizmī together with a list of the books he wrote. Al-Khwārizmī accomplished most of his work in the period between 813 and 833, douglas Morton Dunlop suggests that it may have been possible that Muḥammad ibn Mūsā al-Khwārizmī was in fact the same person as Muḥammad ibn Mūsā ibn Shākir, the eldest of the three Banū Mūsā. Al-Khwārizmīs contributions to mathematics, geography, astronomy, and cartography established the basis for innovation in algebra, on the Calculation with Hindu Numerals written about 825, was principally responsible for spreading the Hindu–Arabic numeral system throughout the Middle East and Europe. It was translated into Latin as Algoritmi de numero Indorum, al-Khwārizmī, rendered as Algoritmi, led to the term algorithm. Some of his work was based on Persian and Babylonian astronomy, Indian numbers, al-Khwārizmī systematized and corrected Ptolemys data for Africa and the Middle East. Another major book was Kitab surat al-ard, presenting the coordinates of places based on those in the Geography of Ptolemy but with improved values for the Mediterranean Sea, Asia and he also wrote on mechanical devices like the astrolabe and sundial. He assisted a project to determine the circumference of the Earth and in making a map for al-Mamun. When, in the 12th century, his works spread to Europe through Latin translations, the Compendious Book on Calculation by Completion and Balancing is a mathematical book written approximately 830 CE. The term algebra is derived from the name of one of the operations with equations described in this book
14.
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
15.
Trigonometry
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Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies, Trigonometry is also the foundation of surveying. Trigonometry is most simply associated with planar right-angle triangles, thus the majority of applications relate to right-angle triangles. One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, Trigonometry on surfaces of negative curvature is part of hyperbolic geometry. Trigonometry basics are often taught in schools, either as a course or as a part of a precalculus course. Sumerian astronomers studied angle measure, using a division of circles into 360 degrees, the ancient Nubians used a similar method. In 140 BC, Hipparchus gave the first tables of chords, analogous to modern tables of sine values, in the 2nd century AD, the Greco-Egyptian astronomer Ptolemy printed detailed trigonometric tables in Book 1, chapter 11 of his Almagest. Ptolemy used chord length to define his trigonometric functions, a difference from the sine convention we use today. The modern sine convention is first attested in the Surya Siddhanta and these Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, at about the same time, Chinese mathematicians developed trigonometry independently, although it was not a major field of study for them. At the same time, another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond, Trigonometry was still so little known in 16th-century northern Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explain its basic concepts. Driven by the demands of navigation and the growing need for maps of large geographic areas. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595, gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry, the works of the Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series. Also in the 18th century, Brook Taylor defined the general Taylor series, if one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees, they are complementary angles, the shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, if the length of one of the sides is known, the other two are determined. Sin A = opposite hypotenuse = a c, Cosine function, defined as the ratio of the adjacent leg to the hypotenuse
16.
History of algebra
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As a branch of mathematics, algebra emerged at the end of the 16th century in Europe, with the work of François Viète. Algebra can essentially be considered as doing computations similar to those of arithmetic, however, until the 19th century, algebra consisted essentially of the theory of equations. For example, the theorem of algebra belongs to the theory of equations and is not, nowadays. This article describes the history of the theory of equations, called here algebra, the treatise provided for the systematic solution of linear and quadratic equations. According to one history, t is not certain just what the terms al-jabr and muqabalah mean, Arabic influence in Spain long after the time of al-Khwarizmi is found in Don Quixote, where the word algebrista is used for a bone-setter, that is, a restorer. Algebra did not always make use of the symbolism that is now ubiquitous in mathematics, instead, the stages in the development of symbolic algebra are approximately as follows, Rhetorical algebra, in which equations are written in full sentences. For example, the form of x +1 =2 is The thing plus one equals two or possibly The thing plus 1 equals 2. Rhetorical algebra was first developed by the ancient Babylonians and remained dominant up to the 16th century, syncopated algebra, in which some symbolism is used, but which does not contain all of the characteristics of symbolic algebra. For instance, there may be a restriction that subtraction may be used once within one side of an equation. Syncopated algebraic expression first appeared in Diophantus Arithmetica, followed by Brahmaguptas Brahma Sphuta Siddhanta, symbolic algebra, in which full symbolism is used. Early steps toward this can be seen in the work of several Islamic mathematicians such as Ibn al-Banna and al-Qalasadi, later, René Descartes introduced the modern notation and showed that the problems occurring in geometry can be expressed and solved in terms of algebra. Equally important as the use or lack of symbolism in algebra was the degree of the equations that were addressed, X2 + p x = q x 2 = p x + q x 2 + q = p x where p and q are positive. This trichotomy comes about because quadratic equations of the form x 2 + p x + q =0, for instance, an equation of the form x 2 = A was solved by finding the side of a square of area A. In addition to the three stages of expressing algebraic ideas, there were four stages in the development of algebra that occurred alongside the changes in expression. These four stages were as follows, Geometric stage, where the concepts of algebra are largely geometric and this dates back to the Babylonians and continued with the Greeks, and was later revived by Omar Khayyám. Static equation-solving stage, where the objective is to find numbers satisfying certain relationships, the move away from geometric algebra dates back to Diophantus and Brahmagupta, but algebra didnt decisively move to the static equation-solving stage until Al-Khwarizmis Al-Jabr. Dynamic function stage, where motion is an underlying idea, the idea of a function began emerging with Sharaf al-Dīn al-Tūsī, but algebra did not decisively move to the dynamic function stage until Gottfried Leibniz. Abstract stage, where mathematical structure plays a central role, abstract algebra is largely a product of the 19th and 20th centuries
17.
Arabic language
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Arabic is a Central Semitic language that was first spoken in Iron Age northwestern Arabia and is now the lingua franca of the Arab world. Arabic is also the language of 1.7 billion Muslims. It is one of six languages of the United Nations. The modern written language is derived from the language of the Quran and it is widely taught in schools and universities, and is used to varying degrees in workplaces, government, and the media. The two formal varieties are grouped together as Literary Arabic, which is the language of 26 states. Modern Standard Arabic largely follows the standards of Quranic Arabic. Much of the new vocabulary is used to denote concepts that have arisen in the post-Quranic era, Arabic has influenced many languages around the globe throughout its history. During the Middle Ages, Literary Arabic was a vehicle of culture in Europe, especially in science, mathematics. As a result, many European languages have borrowed many words from it. Many words of Arabic origin are found in ancient languages like Latin. Balkan languages, including Greek, have acquired a significant number of Arabic words through contact with Ottoman Turkish. Arabic has also borrowed words from languages including Greek and Persian in medieval times. Arabic is a Central Semitic language, closely related to the Northwest Semitic languages, the Ancient South Arabian languages, the Semitic languages changed a great deal between Proto-Semitic and the establishment of the Central Semitic languages, particularly in grammar. Innovations of the Central Semitic languages—all maintained in Arabic—include, The conversion of the suffix-conjugated stative formation into a past tense, the conversion of the prefix-conjugated preterite-tense formation into a present tense. The elimination of other prefix-conjugated mood/aspect forms in favor of new moods formed by endings attached to the prefix-conjugation forms, the development of an internal passive. These features are evidence of descent from a hypothetical ancestor. In the southwest, various Central Semitic languages both belonging to and outside of the Ancient South Arabian family were spoken and it is also believed that the ancestors of the Modern South Arabian languages were also spoken in southern Arabia at this time. To the north, in the oases of northern Hijaz, Dadanitic and Taymanitic held some prestige as inscriptional languages, in Najd and parts of western Arabia, a language known to scholars as Thamudic C is attested
18.
Islamic golden age
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This period is traditionally said to have ended with the collapse of the Abbasid caliphate due to Mongol invasions and the Sack of Baghdad in 1258 AD. A few contemporary scholars place the end of the Islamic Golden Age as late as the end of 15th to 16th centuries, the metaphor of a golden age began to be applied in 19th-century literature about Islamic history, in the context of the western aesthetic fashion known as Orientalism. There is no definition of term, and depending on whether it is used with a focus on cultural or on military achievement. During the early 20th century, the term was used only occasionally, the Muslim government heavily patronized scholars. The money spent on the Translation Movement for some translations is estimated to be equivalent to twice the annual research budget of the United Kingdom’s Medical Research Council. The best scholars and notable translators, such as Hunayn ibn Ishaq, had salaries that are estimated to be the equivalent of professional athletes today, the House of Wisdom was a library established in Abbasid-era Baghdad, Iraq by Caliph al-Mansur. During this period, the Muslims showed a strong interest in assimilating the knowledge of the civilizations that had been conquered. They also excelled in fields, in particular philosophy, science. For a long period of time the personal physicians of the Abbasid Caliphs were often Assyrian Christians, among the most prominent Christian families to serve as physicians to the caliphs were the Bukhtishu dynasty. Throughout the 4th to 7th centuries, Christian scholarly work in the Greek, the House of Wisdom was founded in Baghdad in 825, modelled after the Academy of Gondishapur. It was led by Christian physician Hunayn ibn Ishaq, with the support of Byzantine medicine, many of the most important philosophical and scientific works of the ancient world were translated, including the work of Galen, Hippocrates, Plato, Aristotle, Ptolemy and Archimedes. Many scholars of the House of Wisdom were of Christian background, the use of paper spread from China into Muslim regions in the eighth century, arriving in Al-Andalus on the Iberian peninsula, present-day Spain in the 10th century. It was easier to manufacture than parchment, less likely to crack than papyrus, Islamic paper makers devised assembly-line methods of hand-copying manuscripts to turn out editions far larger than any available in Europe for centuries. It was from countries that the rest of the world learned to make paper from linen. Ibn Rushd and Ibn Sina played a role in saving the works of Aristotle, whose ideas came to dominate the non-religious thought of the Christian. Ibn Sina and other such as al-Kindi and al-Farabi combined Aristotelianism and Neoplatonism with other ideas introduced through Islam. Arabic philosophic literature was translated into Latin and Ladino, contributing to the development of modern European philosophy, during this period, non-Muslims were allowed to flourish relative to treatment of religious minorities in the Christian Byzantine Empire. The Jewish philosopher Moses Maimonides, who lived in Andalusia, is an example, in epistemology, Ibn Tufail wrote the novel Hayy ibn Yaqdhan and in response Ibn al-Nafis wrote the novel Theologus Autodidactus
19.
Muhammad ibn Musa al-Khwarizmi
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Muḥammad ibn Mūsā al-Khwārizmī, formerly Latinized as Algoritmi, was a Persian mathematician, astronomer, and geographer during the Abbasid Caliphate, a scholar in the House of Wisdom in Baghdad. In the 12th century, Latin translations of his work on the Indian numerals introduced the decimal number system to the Western world. Al-Khwārizmīs The Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and he is often considered one of the fathers of algebra. He revised Ptolemys Geography and wrote on astronomy and astrology, some words reflect the importance of al-Khwārizmīs contributions to mathematics. Algebra is derived from al-jabr, one of the two operations he used to solve quadratic equations, algorism and algorithm stem from Algoritmi, the Latin form of his name. His name is also the origin of guarismo and of algarismo, few details of al-Khwārizmīs life are known with certainty. He was born in a Persian family and Ibn al-Nadim gives his birthplace as Khwarezm in Greater Khorasan, muhammad ibn Jarir al-Tabari gives his name as Muḥammad ibn Musá al-Khwārizmiyy al-Majūsiyy al-Quṭrubbaliyy. The epithet al-Qutrubbulli could indicate he might instead have come from Qutrubbul and this would not be worth mentioning if a series of errors concerning the personality of al-Khwārizmī, occasionally even the origins of his knowledge, had not been made. Recently, G. J. Toomer. with naive confidence constructed an entire fantasy on the error which cannot be denied the merit of amusing the reader. Regarding al-Khwārizmīs religion, Toomer writes, Another epithet given to him by al-Ṭabarī, al-Majūsī, Ibn al-Nadīms Kitāb al-Fihrist includes a short biography on al-Khwārizmī together with a list of the books he wrote. Al-Khwārizmī accomplished most of his work in the period between 813 and 833, douglas Morton Dunlop suggests that it may have been possible that Muḥammad ibn Mūsā al-Khwārizmī was in fact the same person as Muḥammad ibn Mūsā ibn Shākir, the eldest of the three Banū Mūsā. Al-Khwārizmīs contributions to mathematics, geography, astronomy, and cartography established the basis for innovation in algebra, on the Calculation with Hindu Numerals written about 825, was principally responsible for spreading the Hindu–Arabic numeral system throughout the Middle East and Europe. It was translated into Latin as Algoritmi de numero Indorum, al-Khwārizmī, rendered as Algoritmi, led to the term algorithm. Some of his work was based on Persian and Babylonian astronomy, Indian numbers, al-Khwārizmī systematized and corrected Ptolemys data for Africa and the Middle East. Another major book was Kitab surat al-ard, presenting the coordinates of places based on those in the Geography of Ptolemy but with improved values for the Mediterranean Sea, Asia and he also wrote on mechanical devices like the astrolabe and sundial. He assisted a project to determine the circumference of the Earth and in making a map for al-Mamun. When, in the 12th century, his works spread to Europe through Latin translations, the Compendious Book on Calculation by Completion and Balancing is a mathematical book written approximately 830 CE. The term algebra is derived from the name of one of the operations with equations described in this book
20.
House of Wisdom
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The House of Wisdom was a major intellectual center during the Islamic Golden Age. The House of Wisdom was founded by Caliph Harun al-Rashid and culminated in prominence under his son al-Mamun who is credited with its formal institution, Al-Mamun is also credited with bringing many well-known scholars to share information, ideas, and culture in the House of Wisdom. Based in Baghdad from the 9th to 13th centuries, beside Muslim scholars, besides translating books into Arabic and preserving them, scholars associated with the House of Wisdom also made many remarkable original contributions to diverse fields. Drawing primarily on Greek, but also Syriac, Indian and Persian texts, the scholars accumulated a collection of world knowledge. By the middle of the century, the House of Wisdom had the largest selection of books in the world. It was destroyed in the sack of the city following the Mongol Siege of Baghdad, throughout the 4th to 7th centuries, scholarly work in the Greek and Syriac languages was either newly initiated, or carried on from the Hellenistic period. Through the Umayyad era, founded by Caliph Muawiyah I, during the reign of Caliph Al-Mamun and he then formed a library that were referred by the name of Bayt al-Hikma. These were fundamental elements that directly to the flourishing of scholarship in the Arab world. In 750, the Abbasid dynasty replaced the Umayyad as the dynasty of the Islamic Empire, and, in 762. Baghdads location and cosmopolitan population made the location for a stable commercial and intellectual center. For this purpose, al-Mansur founded a library, modeled after the Sassanian Imperial Library. He also invited delegations of scholars from India and other places to share their knowledge of mathematics, in the Abbasid Empire, many foreign works were translated into Arabic from Greek, Chinese, Sanskrit, Persian and Syriac. The Translation Movement gained great momentum during the reign of caliph al-Rashid, originally the texts concerned mainly medicine, mathematics and astronomy, but, other disciplines, especially philosophy, soon followed. Al-Rashids library, direct predecessor to the House of Wisdom, was known as Bayt al-Hikma or, as the historian Al-Qifti called it. Under the sponsorship of caliph al-Mamun, economic support of the House of Wisdom, moreover, Abbasid society itself came to understand and appreciate the value of knowledge, and support also came from merchants and the military. It was easy for scholars and translators to make a living, Wisdom was so valuable that books and ancient texts were sometimes preferred as war booty instead of other riches. Indeed, Ptolemys Almagest was claimed as a condition for peace after a war between the Abbasids and the Byzantine Empire, the House of Wisdom was much more than an academic center removed from the broader society. Its experts served several functions in Baghdad, scholars from the Bayt al-Hikma usually doubled as engineers and architects in major construction projects
21.
Baghdad
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Baghdad is the capital of the Republic of Iraq. The population of Baghdad, as of 2016, is approximately 8,765,000 making it the largest city in Iraq, the second largest city in the Arab world, and the second largest city in Western Asia. Located along the Tigris River, the city was founded in the 8th century, within a short time of its inception, Baghdad evolved into a significant cultural, commercial, and intellectual center for the Islamic world. This, in addition to housing several key institutions, garnered the city a worldwide reputation as the Centre of Learning. Throughout the High Middle Ages, Baghdad was considered to be the largest city in the world with a population of 1,200,000 -3,000,000 people. The city was destroyed at the hands of the Mongol Empire in 1258, resulting in a decline that would linger through many centuries due to frequent plagues. With the recognition of Iraq as an independent state in 1938, in contemporary times, the city has often faced severe infrastructural damage, most recently due to the 2003 invasion of Iraq, and the subsequent Iraq War that lasted until December 2011. In recent years, the city has been subjected to insurgency attacks. As of 2012, Baghdad was listed as one of the least hospitable places in the world to live, the site where the city of Baghdad developed has been populated for millennia. By the 8th century AD, several villages had developed there, including a Persian hamlet called Baghdad, the name is of Indo-European origin and a Middle Persian compound of Bagh god and dād given by, translating to Bestowed by God or Gods gift. In Old Persian the first element can be traced to boghu and is related to Slavic bog god, a similar term in Middle Persian is the name Mithradāt, known in English by its Hellenistic form Mithridates, meaning gift of Mithra. There are a number of locations in the wider region whose names are compounds of the word bagh, including Baghlan. The name of the town Baghdati in Georgia shares the same etymological origins, when the Abbasid caliph, al-Mansur, founded a completely new city for his capital, he chose the name Madinat al-Salaam or City of Peace. This was the name on coins, weights, and other official usage. By the 11th century, Baghdad became almost the exclusive name for the world-renowned metropolis, after the fall of the Umayyads, the first Muslim dynasty, the victorious Abbasid rulers wanted their own capital whence they could rule. They chose a site north of the Sassanid capital of Ctesiphon, on 30 July 762, the caliph Al-Mansur commissioned the construction of the city, mansur believed that Baghdad was the perfect city to be the capital of the Islamic empire under the Abbasids. Mansur loved the site so much he is quoted saying, This is indeed the city that I am to found, where I am to live, and where my descendants will reign afterward. The citys growth was helped by its excellent location, based on at least two factors, it had control over strategic and trading routes along the Tigris, the abundance of water in a dry climate
22.
Greek people
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The Greeks or Hellenes are an ethnic group native to Greece, Cyprus, southern Albania, Turkey, Sicily, Egypt and, to a lesser extent, other countries surrounding the Mediterranean Sea. They also form a significant diaspora, with Greek communities established around the world, many of these regions coincided to a large extent with the borders of the Byzantine Empire of the late 11th century and the Eastern Mediterranean areas of ancient Greek colonization. The cultural centers of the Greeks have included Athens, Thessalonica, Alexandria, Smyrna, most ethnic Greeks live nowadays within the borders of the modern Greek state and Cyprus. The Greek genocide and population exchange between Greece and Turkey nearly ended the three millennia-old Greek presence in Asia Minor, other longstanding Greek populations can be found from southern Italy to the Caucasus and southern Russia and Ukraine and in the Greek diaspora communities in a number of other countries. Today, most Greeks are officially registered as members of the Greek Orthodox Church, the Greeks speak the Greek language, which forms its own unique branch within the Indo-European family of languages, the Hellenic. They are part of a group of ethnicities, described by Anthony D. Smith as an archetypal diaspora people. Both migrations occur at incisive periods, the Mycenaean at the transition to the Late Bronze Age, the Mycenaeans quickly penetrated the Aegean Sea and, by the 15th century BC, had reached Rhodes, Crete, Cyprus and the shores of Asia Minor. Around 1200 BC, the Dorians, another Greek-speaking people, followed from Epirus, the Dorian invasion was followed by a poorly attested period of migrations, appropriately called the Greek Dark Ages, but by 800 BC the landscape of Archaic and Classical Greece was discernible. The Greeks of classical antiquity idealized their Mycenaean ancestors and the Mycenaean period as an era of heroes, closeness of the gods. The Homeric Epics were especially and generally accepted as part of the Greek past, as part of the Mycenaean heritage that survived, the names of the gods and goddesses of Mycenaean Greece became major figures of the Olympian Pantheon of later antiquity. The ethnogenesis of the Greek nation is linked to the development of Pan-Hellenism in the 8th century BC, the works of Homer and Hesiod were written in the 8th century BC, becoming the basis of the national religion, ethos, history and mythology. The Oracle of Apollo at Delphi was established in this period, the classical period of Greek civilization covers a time spanning from the early 5th century BC to the death of Alexander the Great, in 323 BC. It is so named because it set the standards by which Greek civilization would be judged in later eras, the Peloponnesian War, the large scale civil war between the two most powerful Greek city-states Athens and Sparta and their allies, left both greatly weakened. Many Greeks settled in Hellenistic cities like Alexandria, Antioch and Seleucia, two thousand years later, there are still communities in Pakistan and Afghanistan, like the Kalash, who claim to be descended from Greek settlers. The Hellenistic civilization was the period of Greek civilization, the beginnings of which are usually placed at Alexanders death. This Hellenistic age, so called because it saw the partial Hellenization of many non-Greek cultures and this age saw the Greeks move towards larger cities and a reduction in the importance of the city-state. These larger cities were parts of the still larger Kingdoms of the Diadochi, Greeks, however, remained aware of their past, chiefly through the study of the works of Homer and the classical authors. An important factor in maintaining Greek identity was contact with barbarian peoples and this led to a strong desire among Greeks to organize the transmission of the Hellenic paideia to the next generation
23.
Diophantus
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Diophantus of Alexandria, sometimes called the father of algebra, was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica, many of which are now lost. These texts deal with solving algebraic equations and this led to tremendous advances in number theory, and the study of Diophantine equations and of Diophantine approximations remain important areas of mathematical research. Diophantus coined the term παρισότης to refer to an approximate equality and this term was rendered as adaequalitas in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves. Diophantus was the first Greek mathematician who recognized fractions as numbers, thus he allowed positive rational numbers for the coefficients, in modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought. Diophantus also made advances in mathematical notation, little is known about the life of Diophantus. He lived in Alexandria, Egypt, probably from between AD200 and 214 to 284 or 298, much of our knowledge of the life of Diophantus is derived from a 5th-century Greek anthology of number games and puzzles created by Metrodorus. One of the states, Here lies Diophantus, the wonder behold. Alas, the child of master and sage After attaining half the measure of his fathers life chill fate took him. After consoling his fate by the science of numbers for four years and this puzzle implies that Diophantus age x can be expressed as x = x/6 + x/12 + x/7 +5 + x/2 +4 which gives x a value of 84 years. However, the accuracy of the information cannot be independently confirmed, the Arithmetica is the major work of Diophantus and the most prominent work on algebra in Greek mathematics. It is a collection of problems giving numerical solutions of both determinate and indeterminate equations, of the original thirteen books of which Arithmetica consisted only six have survived, though there are some who believe that four Arab books discovered in 1968 are also by Diophantus. Some Diophantine problems from Arithmetica have been found in Arabic sources and it should be mentioned here that Diophantus never used general methods in his solutions. Hermann Hankel, renowned German mathematician made the following remark regarding Diophantus, “Our author not the slightest trace of a general, comprehensive method is discernible, each problem calls for some special method which refuses to work even for the most closely related problems. The portion of the Greek Arithmetica that survived, however, was, like all ancient Greek texts transmitted to the modern world, copied by. In addition, some portion of the Arithmetica probably survived in the Arab tradition. ”Arithmetica was first translated from Greek into Latin by Bombelli in 1570, however, Bombelli borrowed many of the problems for his own book Algebra. The editio princeps of Arithmetica was published in 1575 by Xylander, the best known Latin translation of Arithmetica was made by Bachet in 1621 and became the first Latin edition that was widely available. Pierre de Fermat owned a copy, studied it, and made notes in the margins. I have a marvelous proof of this proposition which this margin is too narrow to contain. ”Fermats proof was never found
24.
Positive number
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In mathematics, the concept of sign originates from the property of every non-zero real number of being positive or negative. Zero itself is signless, although in some contexts it makes sense to consider a signed zero, along with its application to real numbers, change of sign is used throughout mathematics and physics to denote the additive inverse, even for quantities which are not real numbers. Also, the sign can indicate aspects of mathematical objects that resemble positivity and negativity. A real number is said to be if its value is greater than zero. The attribute of being positive or negative is called the sign of the number, zero itself is not considered to have a sign. Also, signs are not defined for complex numbers, although the argument generalizes it in some sense, in common numeral notation, the sign of a number is often denoted by placing a plus sign or a minus sign before the number. For example, +3 denotes positive three, and −3 denotes negative three, when no plus or minus sign is given, the default interpretation is that a number is positive. Because of this notation, as well as the definition of numbers through subtraction. In this context, it makes sense to write − = +3, any non-zero number can be changed to a positive one using the absolute value function. For example, the value of −3 and the absolute value of 3 are both equal to 3. In symbols, this would be written |−3| =3 and |3| =3, the number zero is neither positive nor negative, and therefore has no sign. In arithmetic, +0 and −0 both denote the same number 0, which is the inverse of itself. Note that this definition is culturally determined, in France and Belgium,0 is said to be both positive and negative. The positive resp. negative numbers without zero are said to be strictly positive resp, in some contexts, such as signed number representations in computing, it makes sense to consider signed versions of zero, with positive zero and negative zero being different numbers. One also sees +0 and −0 in calculus and mathematical analysis when evaluating one-sided limits and this notation refers to the behaviour of a function as the input variable approaches 0 from positive or negative values respectively, these behaviours are not necessarily the same. Because zero is positive nor negative, the following phrases are sometimes used to refer to the sign of an unknown number. A number is negative if it is less than zero, a number is non-negative if it is greater than or equal to zero. A number is non-positive if it is less than or equal to zero, thus a non-negative number is either positive or zero, while a non-positive number is either negative or zero
25.
Rational numbers
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In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. The set of all numbers, often referred to as the rationals, is usually denoted by a boldface Q, it was thus denoted in 1895 by Giuseppe Peano after quoziente. The decimal expansion of a rational number always either terminates after a number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number and these statements hold true not just for base 10, but also for any other integer base. A real number that is not rational is called irrational, irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating, since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be defined as equivalence classes of pairs of integers such that q ≠0, for the equivalence relation defined by ~ if. In abstract algebra, the numbers together with certain operations of addition and multiplication form the archetypical field of characteristic zero. As such, it is characterized as having no proper subfield or, alternatively, finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the numbers form a dense subset of the real numbers. The real numbers can be constructed from the numbers by completion, using Cauchy sequences, Dedekind cuts. The term rational in reference to the set Q refers to the fact that a number represents a ratio of two integers. In mathematics, rational is often used as a noun abbreviating rational number, the adjective rational sometimes means that the coefficients are rational numbers. However, a curve is not a curve defined over the rationals. Any integer n can be expressed as the rational number n/1, a b = c d if and only if a d = b c. Where both denominators are positive, a b < c d if and only if a d < b c. If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations, − a − b = a b, two fractions are added as follows, a b + c d = a d + b c b d
26.
Irrational number
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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
27.
Cubic equation
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In algebra, a cubic function is a function of the form f = a x 3 + b x 2 + c x + d, where a is nonzero. Setting f =0 produces an equation of the form. The solutions of this equation are called roots of the polynomial f, If all of the coefficients a, b, c, and d of the cubic equation are real numbers then there will be at least one real root. All of the roots of the equation can be found algebraically. The roots can also be found trigonometrically, alternatively, numerical approximations of the roots can be found using root-finding algorithms like Newtons method. The coefficients do not need to be complex numbers, much of what is covered below is valid for coefficients of any field with characteristic 0 or greater than 3. The solutions of the cubic equation do not necessarily belong to the field as the coefficients. For example, some cubic equations with rational coefficients have roots that are complex numbers. Cubic equations were known to the ancient Babylonians, Greeks, Chinese, Indians, Babylonian cuneiform tablets have been found with tables for calculating cubes and cube roots. The Babylonians could have used the tables to solve cubic equations, the problem of doubling the cube involves the simplest and oldest studied cubic equation, and one for which the ancient Egyptians did not believe a solution existed. Methods for solving cubic equations appear in The Nine Chapters on the Mathematical Art, in the 3rd century, the Greek mathematician Diophantus found integer or rational solutions for some bivariate cubic equations. In the 11th century, the Persian poet-mathematician, Omar Khayyám, in an early paper, he discovered that a cubic equation can have more than one solution and stated that it cannot be solved using compass and straightedge constructions. He also found a geometric solution, in the 12th century, the Indian mathematician Bhaskara II attempted the solution of cubic equations without general success. However, he gave one example of an equation, x3 + 12x = 6x2 +35. He used what would later be known as the Ruffini-Horner method to approximate the root of a cubic equation. He also developed the concepts of a function and the maxima and minima of curves in order to solve cubic equations which may not have positive solutions. He understood the importance of the discriminant of the equation to find algebraic solutions to certain types of cubic equations. Leonardo de Pisa, also known as Fibonacci, was able to approximate the positive solution to the cubic equation x3 + 2x2 + 10x =20
28.
Parabola
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A parabola is a two-dimensional, mirror-symmetrical curve, which is approximately U-shaped when oriented as shown in the diagram below, but which can be in any orientation in its plane. It fits any of several different mathematical descriptions which can all be proved to define curves of exactly the same shape. One description of a parabola involves a point and a line, the focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus, a parabola is a graph of a quadratic function, y = x2, for example. The line perpendicular to the directrix and passing through the focus is called the axis of symmetry, the point on the parabola that intersects the axis of symmetry is called the vertex, and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the focal length, the latus rectum is the chord of the parabola which is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola — that is, conversely, light that originates from a point source at the focus is reflected into a parallel beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other forms of energy and this reflective property is the basis of many practical uses of parabolas. The parabola has many important applications, from an antenna or parabolic microphone to automobile headlight reflectors to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas, the earliest known work on conic sections was by Menaechmus in the fourth century BC. He discovered a way to solve the problem of doubling the cube using parabolas, the name parabola is due to Apollonius who discovered many properties of conic sections. It means application, referring to application of concept, that has a connection with this curve. The focus–directrix property of the parabola and other conics is due to Pappus, Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity. The idea that a reflector could produce an image was already well known before the invention of the reflecting telescope. Designs were proposed in the early to mid seventeenth century by many mathematicians including René Descartes, Marin Mersenne, when Isaac Newton built the first reflecting telescope in 1668, he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror. Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes, solving for y yields y =14 f x 2. The length of the chord through the focus is called latus rectum, one half of it semi latus rectum
29.
Circle
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A circle is a simple closed shape in Euclidean geometry. The distance between any of the points and the centre is called the radius, a circle is a simple closed curve which divides the plane into two regions, an interior and an exterior. Annulus, the object, the region bounded by two concentric circles. Arc, any connected part of the circle, centre, the point equidistant from the points on the circle. Chord, a segment whose endpoints lie on the circle. Circumference, the length of one circuit along the circle, or the distance around the circle and it is a special case of a chord, namely the longest chord, and it is twice the radius. Disc, the region of the bounded by a circle. Lens, the intersection of two discs, passant, a coplanar straight line that does not touch the circle. Radius, a line segment joining the centre of the circle to any point on the circle itself, or the length of such a segment, sector, a region bounded by two radii and an arc lying between the radii. Segment, a region, not containing the centre, bounded by a chord, secant, an extended chord, a coplanar straight line cutting the circle at two points. Semicircle, an arc that extends from one of a diameters endpoints to the other, in non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a half-disc. A half-disc is a case of a segment, namely the largest one. Tangent, a straight line that touches the circle at a single point. The word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, the origins of the words circus and circuit are closely related. The circle has been known since before the beginning of recorded history, natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, in mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Some highlights in the history of the circle are,1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as a value of π.300 BCE – Book 3 of Euclids Elements deals with the properties of circles
30.
Iran
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Iran, also known as Persia, officially the Islamic Republic of Iran, is a sovereign state in Western Asia. Comprising a land area of 1,648,195 km2, it is the second-largest country in the Middle East, with 82.8 million inhabitants, Iran is the worlds 17th-most-populous country. It is the country with both a Caspian Sea and an Indian Ocean coastline. The countrys central location in Eurasia and Western Asia, and its proximity to the Strait of Hormuz, Tehran is the countrys capital and largest city, as well as its leading economic and cultural center. Iran is the site of to one of the worlds oldest civilizations, the area was first unified by the Iranian Medes in 625 BC, who became the dominant cultural and political power in the region. The empire collapsed in 330 BC following the conquests of Alexander the Great, under the Sassanid Dynasty, Iran again became one of the leading powers in the world for the next four centuries. Beginning in 633 AD, Arabs conquered Iran and largely displaced the indigenous faiths of Manichaeism and Zoroastrianism by Islam, Iran became a major contributor to the Islamic Golden Age that followed, producing many influential scientists, scholars, artists, and thinkers. During the 18th century, Iran reached its greatest territorial extent since the Sassanid Empire, through the late 18th and 19th centuries, a series of conflicts with Russia led to significant territorial losses and the erosion of sovereignty. Popular unrest culminated in the Persian Constitutional Revolution of 1906, which established a monarchy and the countrys first legislative body. Following a coup instigated by the U. K. Growing dissent against foreign influence and political repression led to the 1979 Revolution, Irans rich cultural legacy is reflected in part by its 21 UNESCO World Heritage Sites, the third-largest number in Asia and 11th-largest in the world. Iran is a member of the UN, ECO, NAM, OIC. Its political system is based on the 1979 Constitution which combines elements of a democracy with a theocracy governed by Islamic jurists under the concept of a Supreme Leadership. A multicultural country comprising numerous ethnic and linguistic groups, most inhabitants are Shia Muslims, the largest ethnic groups in Iran are the Persians, Azeris, Kurds and Lurs. Historically, Iran has been referred to as Persia by the West, due mainly to the writings of Greek historians who called Iran Persis, meaning land of the Persians. As the most extensive interactions the Ancient Greeks had with any outsider was with the Persians, however, Persis was originally referred to a region settled by Persians in the west shore of Lake Urmia, in the 9th century BC. The settlement was then shifted to the end of the Zagros Mountains. In 1935, Reza Shah requested the international community to refer to the country by its native name, opposition to the name change led to the reversal of the decision, and Professor Ehsan Yarshater, editor of Encyclopædia Iranica, propagated a move to use Persia and Iran interchangeably
31.
Conic section
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In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse, the circle is a special case of the ellipse, and is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, the conic sections of the Euclidean plane have various distinguishing properties. Many of these have used as the basis for a definition of the conic sections. The type of conic is determined by the value of the eccentricity, in analytic geometry, a conic may be defined as a plane algebraic curve of degree 2, that is, as the set of points whose coordinates satisfy a quadratic equation in two variables. This equation may be written in form, and some geometric properties can be studied as algebraic conditions. In the Euclidean plane, the conic sections appear to be different from one another. By extending the geometry to a projective plane this apparent difference vanishes, further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically. The conic sections have been studied for thousands of years and have provided a source of interesting. A conic is the curve obtained as the intersection of a plane, called the cutting plane and we shall assume that the cone is a right circular cone for the purpose of easy description, but this is not required, any double cone with some circular cross-section will suffice. Planes that pass through the vertex of the cone will intersect the cone in a point and these are called degenerate conics and some authors do not consider them to be conics at all. Unless otherwise stated, we assume that conic refers to a non-degenerate conic. There are three types of conics, the ellipse, parabola, and hyperbola, the circle is a special kind of ellipse, although historically it had been considered as a fourth type. The circle and the ellipse arise when the intersection of the cone and plane is a closed curve, if the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola, in this case, the plane will intersect both halves of the cone, producing two separate unbounded curves. A property that the conic sections share is often presented as the following definition, a conic section is the locus of all points P whose distance to a fixed point F is a constant multiple of the distance from P to a fixed line L. For 0 < e <1 we obtain an ellipse, for e =1 a parabola, a circle is a limiting case and is not defined by a focus and directrix, in the plane. The eccentricity of a circle is defined to be zero and its focus is the center of the circle, an ellipse and a hyperbola each have two foci and distinct directrices for each of them
32.
Zero of a function
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In other words, a zero of a function is an input value that produces an output of zero. A root of a polynomial is a zero of the polynomial function. If the function maps real numbers to real numbers, its zeroes are the x-coordinates of the points where its graph meets the x-axis, an alternative name for such a point in this context is an x-intercept. Every equation in the unknown x may be rewritten as f =0 by regrouping all terms in the left-hand side and it follows that the solutions of such an equation are exactly the zeros of the function f. Every real polynomial of odd degree has an odd number of roots, likewise. Consequently, real odd polynomials must have at least one real root, the fundamental theorem of algebra states that every polynomial of degree n has n complex roots, counted with their multiplicities. The non-real roots of polynomials with real coefficients come in conjugate pairs, vietas formulas relate the coefficients of a polynomial to sums and products of its roots. Computing roots of functions, for polynomial functions, frequently requires the use of specialised or approximation techniques. However, some functions, including all those of degree no greater than 4. In topology and other areas of mathematics, the set of a real-valued function f, X → R is the subset f −1 of X. Zero sets are important in many areas of mathematics. One area of importance is algebraic geometry, where the first definition of an algebraic variety is through zero-sets. For instance, for each set S of polynomials in k, one defines the zero-locus Z to be the set of points in An on which the functions in S simultaneously vanish, that is to say Z =. Then a subset V of An is called an algebraic set if V = Z for some S. These affine algebraic sets are the building blocks of algebraic geometry. Zero Pole Fundamental theorem of algebra Newtons method Sendovs conjecture Mardens theorem Vanish at infinity Zero crossing Weisstein, Eric W. Root
33.
Tus, Iran
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Tus, also spelled as Tous, Toos or Tūs, is an ancient city in Razavi Khorasan Province in Iran near Mashhad. To the ancient Greeks, it was known as Susia and it was captured by Alexander the Great in 330 BCE. It was also known as Tusa, Tus was taken by the Umayyad caliph Abd al-Malik and remained under Umayyad control until 747, when a subordinate of Abu Muslim Khorasani defeated the Umayyad governor during the Abbasid Revolution. In 809, the Abbasid Caliph Harun al-Rashid fell ill and died in Tus and his grave is located in the region. Tus was almost entirely destroyed by the Mongol conquests between 1220 and 1259, the most famous person who has emerged from that area was the theologian, jurist, philosopher and mystic al-Ghazali. Another is the poet Ferdowsi, author of the Persian epic Shahnameh, whose mausoleum, built in 1934 in time for the millennium of his birth, dominates the town. Al-Tusi – a descriptor used for individuals associated with Tus Tus citadel Media related to Tus, Iran at Wikimedia Commons Livius. org, Susia
34.
Mathematical induction
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Mathematical induction is a mathematical proof technique used to prove a given statement about any well-ordered set. Most commonly, it is used to establish statements for the set of all natural numbers, mathematical induction is a form of direct proof, usually done in two steps. When trying to prove a statement for a set of natural numbers. The second step, known as the step, is to prove that, if the statement is assumed to be true for any one natural number. Having proved these two steps, the rule of inference establishes the statement to be true for all natural numbers, in common terminology, using the stated approach is referred to as using the Principle of mathematical induction. Mathematical induction in this sense is closely related to recursion. Mathematical induction, in form, is the foundation of all correctness proofs for computer programs. Although its name may suggest otherwise, mathematical induction should not be misconstrued as a form of inductive reasoning, mathematical induction is an inference rule used in proofs. In mathematics, proofs including those using mathematical induction are examples of deductive reasoning, in 370 BC, Platos Parmenides may have contained an early example of an implicit inductive proof. The earliest implicit traces of mathematical induction may be found in Euclids proof that the number of primes is infinite, none of these ancient mathematicians, however, explicitly stated the inductive hypothesis. Another similar case was that of Francesco Maurolico in his Arithmeticorum libri duo, the first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique. Another Frenchman, Fermat, made use of a related principle. The inductive hypothesis was also employed by the Swiss Jakob Bernoulli, the modern rigorous and systematic treatment of the principle came only in the 19th century, with George Boole, Augustus de Morgan, Charles Sanders Peirce, Giuseppe Peano, and Richard Dedekind. The simplest and most common form of mathematical induction infers that a statement involving a number n holds for all values of n. The proof consists of two steps, The basis, prove that the statement holds for the first natural number n, usually, n =0 or n =1, rarely, n = –1. The inductive step, prove that, if the statement holds for some number n. The hypothesis in the step that the statement holds for some n is called the induction hypothesis. To perform the step, one assumes the induction hypothesis
35.
Blaise Pascal
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Blaise Pascal was a French mathematician, physicist, inventor, writer and Christian philosopher. He was a prodigy who was educated by his father. Pascal also wrote in defence of the scientific method, in 1642, while still a teenager, he started some pioneering work on calculating machines. After three years of effort and 50 prototypes, he built 20 finished machines over the following 10 years, following Galileo Galilei and Torricelli, in 1647, he rebutted Aristotles followers who insisted that nature abhors a vacuum. Pascals results caused many disputes before being accepted, in 1646, he and his sister Jacqueline identified with the religious movement within Catholicism known by its detractors as Jansenism. Following a religious experience in late 1654, he began writing works on philosophy. His two most famous works date from this period, the Lettres provinciales and the Pensées, the set in the conflict between Jansenists and Jesuits. In that year, he wrote an important treatise on the arithmetical triangle. Between 1658 and 1659 he wrote on the cycloid and its use in calculating the volume of solids, Pascal had poor health, especially after the age of 18, and he died just two months after his 39th birthday. Pascal was born in Clermont-Ferrand, which is in Frances Auvergne region and he lost his mother, Antoinette Begon, at the age of three. His father, Étienne Pascal, who also had an interest in science and mathematics, was a local judge, Pascal had two sisters, the younger Jacqueline and the elder Gilberte. In 1631, five years after the death of his wife, the newly arrived family soon hired Louise Delfault, a maid who eventually became an instrumental member of the family. Étienne, who never remarried, decided that he alone would educate his children, for they all showed extraordinary intellectual ability, the young Pascal showed an amazing aptitude for mathematics and science. Particularly of interest to Pascal was a work of Desargues on conic sections and it states that if a hexagon is inscribed in a circle then the three intersection points of opposite sides lie on a line. Pascals work was so precocious that Descartes was convinced that Pascals father had written it, in France at that time offices and positions could be—and were—bought and sold. In 1631 Étienne sold his position as president of the Cour des Aides for 65,665 livres. The money was invested in a government bond which provided, if not a lavish, then certainly a comfortable income which allowed the Pascal family to move to, but in 1638 Richelieu, desperate for money to carry on the Thirty Years War, defaulted on the governments bonds. Suddenly Étienne Pascals worth had dropped from nearly 66,000 livres to less than 7,300 and it was only when Jacqueline performed well in a childrens play with Richelieu in attendance that Étienne was pardoned
36.
Mathematical proof
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In mathematics, a proof is an inferential argument for a mathematical statement. In the argument, other previously established statements, such as theorems, in principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies, Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning. A proof must demonstrate that a statement is true, rather than enumerate many confirmatory cases. An unproved proposition that is believed to be true is known as a conjecture, Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to examination of current and historical mathematical practice, quasi-empiricism in mathematics. The philosophy of mathematics is concerned with the role of language and logic in proofs, the word proof comes from the Latin probare meaning to test. Related modern words are the English probe, probation, and probability, the Spanish probar, Italian provare, the early use of probity was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof. It is likely that the idea of demonstrating a conclusion first arose in connection with geometry, the development of mathematical proof is primarily the product of ancient Greek mathematics, and one of the greatest achievements thereof. Thales proved some theorems in geometry, eudoxus and Theaetetus formulated theorems but did not prove them. Aristotle said definitions should describe the concept being defined in terms of other concepts already known and his book, the Elements, was read by anyone who was considered educated in the West until the middle of the 20th century. Further advances took place in medieval Islamic mathematics, while earlier Greek proofs were largely geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers as he considered multiplication, division and he used this method to provide a proof of the existence of irrational numbers. An inductive proof for arithmetic sequences was introduced in the Al-Fakhri by Al-Karaji, alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate. Modern proof theory treats proofs as inductively defined data structures, there is no longer an assumption that axioms are true in any sense, this allows for parallel mathematical theories built on alternate sets of axioms
37.
Binomial theorem
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In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. For example,4 = x 4 +4 x 3 y +6 x 2 y 2 +4 x y 3 + y 4, the coefficient a in the term of a xb yc is known as the binomial coefficient or. These coefficients for varying n and b can be arranged to form Pascals triangle and these numbers also arise in combinatorics, where gives the number of different combinations of b elements that can be chosen from an n-element set. Special cases of the theorem were known from ancient times. Greek mathematician Euclid mentioned the case of the binomial theorem for exponent 2. There is evidence that the theorem for cubes was known by the 6th century in India. Binomial coefficients, as combinatorial quantities expressing the number of ways of selecting k objects out of n without replacement, were of interest to the ancient Hindus. The earliest known reference to this problem is the Chandaḥśāstra by the Hindu lyricist Pingala. The commentator Halayudha from the 10th century A. D. explains this method using what is now known as Pascals triangle. By the 6th century A. D. the Hindu mathematicians probably knew how to express this as a quotient n. k. the binomial theorem as such can be found in the work of 11th-century Persian mathematician Al-Karaji, who described the triangular pattern of the binomial coefficients. He also provided a proof of both the binomial theorem and Pascals triangle, using a primitive form of mathematical induction. The Persian poet and mathematician Omar Khayyam was probably familiar with the formula to higher orders, the binomial expansions of small degrees were known in the 13th century mathematical works of Yang Hui and also Chu Shih-Chieh. Yang Hui attributes the method to a much earlier 11th century text of Jia Xian, in 1544, Michael Stifel introduced the term binomial coefficient and showed how to use them to express n in terms of n −1, via Pascals triangle. Blaise Pascal studied the eponymous triangle comprehensively in the treatise Traité du triangle arithmétique, however, the pattern of numbers was already known to the European mathematicians of the late Renaissance, including Stifel, Niccolò Fontana Tartaglia, and Simon Stevin. Isaac Newton is generally credited with the binomial theorem, valid for any rational exponent. This formula is also referred to as the formula or the binomial identity. Using summation notation, it can be written as n = ∑ k =0 n x n − k y k = ∑ k =0 n x k y n − k. A simple variant of the formula is obtained by substituting 1 for y
38.
Pascal's triangle
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In mathematics, Pascals triangle is a triangular array of the binomial coefficients. In the Western world, it is named after French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, the rows of Pascals triangle are conventionally enumerated starting with row n =0 at the top. The entries in each row are numbered from the beginning with k =0 and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the manner, In row 0. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the number in the first row is 1. The entry in the nth row and kth column of Pascals triangle is denoted, for example, the unique nonzero entry in the topmost row is =1. With this notation, the construction of the previous paragraph may be written as follows, = +, for any integer n. This recurrence for the coefficients is known as Pascals rule. Pascals triangle has higher dimensional generalizations, the three-dimensional version is called Pascals pyramid or Pascals tetrahedron, while the general versions are called Pascals simplices. The pattern of numbers that forms Pascals triangle was known well before Pascals time, centuries before, discussion of the numbers had arisen in the context of Indian studies of combinatorics and of binomial numbers and Greeks study of figurate numbers. From later commentary, it appears that the coefficients and the additive formula for generating them. Halayudha also explained obscure references to Meru-prastaara, the Staircase of Mount Meru, in approximately 850, the Jain mathematician Mahāvīra gave a different formula for the binomial coefficients, using multiplication, equivalent to the modern formula = n. r. At around the time, it was discussed in Persia by the Persian mathematician. It was later repeated by the Persian poet-astronomer-mathematician Omar Khayyám, thus the triangle is referred to as the Khayyam triangle in Iran. Several theorems related to the triangle were known, including the binomial theorem, Khayyam used a method of finding nth roots based on the binomial expansion, and therefore on the binomial coefficients. Pascals triangle was known in China in the early 11th century through the work of the Chinese mathematician Jia Xian, in the 13th century, Yang Hui presented the triangle and hence it is still called Yang Huis triangle in China. In the west, the binomial coefficients were calculated by Gersonides in the early 14th century, petrus Apianus published the full triangle on the frontispiece of his book on business calculations in 1527
39.
Latin
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Latin is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from the Etruscan and Greek alphabets, Latin was originally spoken in Latium, in the Italian Peninsula. Through the power of the Roman Republic, it became the dominant language, Vulgar Latin developed into the Romance languages, such as Italian, Portuguese, Spanish, French, and Romanian. Latin, Italian and French have contributed many words to the English language, Latin and Ancient Greek roots are used in theology, biology, and medicine. By the late Roman Republic, Old Latin had been standardised into Classical Latin, Vulgar Latin was the colloquial form spoken during the same time and attested in inscriptions and the works of comic playwrights like Plautus and Terence. Late Latin is the language from the 3rd century. Later, Early Modern Latin and Modern Latin evolved, Latin was used as the language of international communication, scholarship, and science until well into the 18th century, when it began to be supplanted by vernaculars. Ecclesiastical Latin remains the language of the Holy See and the Roman Rite of the Catholic Church. Today, many students, scholars and members of the Catholic clergy speak Latin fluently and it is taught in primary, secondary and postsecondary educational institutions around the world. The language has been passed down through various forms, some inscriptions have been published in an internationally agreed, monumental, multivolume series, the Corpus Inscriptionum Latinarum. Authors and publishers vary, but the format is about the same, volumes detailing inscriptions with a critical apparatus stating the provenance, the reading and interpretation of these inscriptions is the subject matter of the field of epigraphy. The works of several hundred ancient authors who wrote in Latin have survived in whole or in part and they are in part the subject matter of the field of classics. The Cat in the Hat, and a book of fairy tales, additional resources include phrasebooks and resources for rendering everyday phrases and concepts into Latin, such as Meissners Latin Phrasebook. The Latin influence in English has been significant at all stages of its insular development. From the 16th to the 18th centuries, English writers cobbled together huge numbers of new words from Latin and Greek words, dubbed inkhorn terms, as if they had spilled from a pot of ink. Many of these words were used once by the author and then forgotten, many of the most common polysyllabic English words are of Latin origin through the medium of Old French. Romance words make respectively 59%, 20% and 14% of English, German and those figures can rise dramatically when only non-compound and non-derived words are included. Accordingly, Romance words make roughly 35% of the vocabulary of Dutch, Roman engineering had the same effect on scientific terminology as a whole
40.
Indian numerals
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Indian numerals are the symbols representing numbers in India. These numerals are used in the context of the decimal Hindu–Arabic numeral system. Below is a list of the Indian numerals in their modern Devanagari form, the corresponding Hindu-Arabic equivalents, their Hindi and Sanskrit pronunciation, since Sanskrit is an Indo-European language, it is obvious that the words for numerals closely resemble those of Greek and Latin. The word Shunya for zero was translated into Arabic as صفر sifr, meaning nothing which became the zero in many European languages from Medieval Latin. The five Indian languages that have adapted the Devanagari script to their use also naturally employ the numeral symbols above, of course, for numerals in Tamil language see Tamil numerals. For numerals in Telugu language see Telugu numerals, Tamil and Malayalam scripts also have distinct forms for 10,100,1000 numbers, ௰, ௱, ௲and ൰, ൱, ൲ respectively in tamil and scripts. A decimal place system has been traced back to ca.500 in India, before that epoch, the Brahmi numeral system was in use, that system did not encompass the concept of the place-value of numbers. Instead, Brahmi numerals included additional symbols for the tens, as well as symbols for hundred. The Indian place-system numerals spread to neighboring Persia, where they were picked up by the conquering Arabs, in 662, Severus Sebokht - a Nestorian bishop living in Syria wrote, I will omit all discussion of the science of the Indians. Of their subtle discoveries in astronomy — discoveries that are more ingenious than those of the Greeks, I wish only to say that this computation is done by means of nine signs. But it is in Khmer numerals of modern Cambodia where the first extant material evidence of zero as a numerical figure, as it was from the Arabs that the Europeans learned this system, the Europeans called them Arabic numerals, the Arabs refer to their numerals as Indian numerals. In academic circles they are called the Hindu–Arabic or Indo–Arabic numerals, but what was the net achievement in the field of reckoning, the earliest art practiced by man. An inflexible numeration so crude as to progress well nigh impossible. Man used these devices for thousands of years without contributing an important idea to the system. Even when compared with the growth of ideas during the Dark Ages. When viewed in light, the achievements of the unknown Hindu. Sanskrit Siddham Numbers Karl Menninger, Number Words and Number Symbols - A Cultural History of Numbers ISBN 0-486-27096-3 David Eugene Smith and Louis Charles Karpinski, The Hindu-Arabic Numerals
41.
Decimal
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This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation, the decimal numeral system has ten as its base, which, in decimal, is written 10, as is the base in every positional numeral system. It is the base most widely used by modern civilizations. Decimal fractions have terminating decimal representations and other fractions have repeating decimal representations, Decimal notation is the writing of numbers in a base-ten numeral system. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, Roman numerals have symbols for the decimal powers and secondary symbols for half these values. Brahmi numerals have symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100, Chinese numerals have symbols for 1–9, and additional symbols for powers of ten, which in modern usage reach 1072. Positional decimal systems include a zero and use symbols for the ten values to represent any number, positional notation uses positions for each power of ten, units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier multiplied with that position has a value ten times that of the position to its right. There were at least two independent sources of positional decimal systems in ancient civilization, the Chinese counting rod system. Ten is the number which is the count of fingers and thumbs on both hands, the English word digit as well as its translation in many languages is also the anatomical term for fingers and toes. In English, decimal means tenth, decimate means reduce by a tenth, however, the symbols used in different areas are not identical, for instance, Western Arabic numerals differ from the forms used by other Arab cultures. A decimal fraction is a fraction the denominator of which is a power of ten. g, Decimal fractions 8/10, 1489/100, 24/100000, and 58900/10000 are expressed in decimal notation as 0.8,14.89,0.00024,5.8900 respectively. In English-speaking, some Latin American and many Asian countries, a period or raised period is used as the separator, in many other countries, particularly in Europe. The integer part, or integral part of a number is the part to the left of the decimal separator. The part from the separator to the right is the fractional part. It is usual for a number that consists only of a fractional part to have a leading zero in its notation. Any rational number with a denominator whose only prime factors are 2 and/or 5 may be expressed as a decimal fraction and has a finite decimal expansion. 1/2 =0.5 1/20 =0.05 1/5 =0.2 1/50 =0.02 1/4 =0.25 1/40 =0.025 1/25 =0.04 1/8 =0.125 1/125 =0.008 1/10 =0
42.
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
43.
Western world
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The Western world or the West is a term usually referring to different nations, depending on the context, most often including at least part of Europe. There are many accepted definitions about what they all have in common, the Western world is also known as the Occident. The concept of the Western part of the earth has its roots in Greco-Roman civilization in Europe, before the Cold War era, the traditional Western viewpoint identified Western Civilization with the Western Christian countries and culture. Its political usage was changed by the antagonism during the Cold War in the mid-to-late 20th Century. The term originally had a literal geographic meaning, Western culture was influenced by many older great civilizations of the ancient Near East, such as Phoenicia, Minoan Crete, Sumer, Babylonia, and also Ancient Egypt. It originated in the Mediterranean basin and its vicinity, Greece, over time, their associated empires grew first to the east and west to include the rest of Mediterranean and Black Sea coastal areas, conquering and absorbing. Later, they expanded to the north of the Mediterranean Sea to include Western, Central, numerous times, this expansion was accompanied by Christian missionaries, who attempted to proselytize Christianity. There is debate among some as to whether Latin America is in a category of its own, specifically, Western culture may imply, a Biblical Christian cultural influence in spiritual thinking, customs and either ethic or moral traditions, around the Post-Classical Era and after. European cultural influences concerning artistic, musical, folkloric, ethic and oral traditions, the concept of Western culture is generally linked to the classical definition of the Western world. In this definition, Western culture is the set of literary, scientific, political, artistic, much of this set of traditions and knowledge is collected in the Western canon. The term has come to apply to countries whose history is marked by European immigration or settlement, such as the Americas, and Oceania. The geopolitical divisions in Europe that created a concept of East and West originated in the Roman Empire, Roman Catholic Western and Central Europe, as such, maintained a distinct identity particularly as it began to redevelop during the Renaissance. Even following the Protestant Reformation, Protestant Europe continued to see itself as more tied to Roman Catholic Europe than other parts of the civilized world. Use of the term West as a cultural and geopolitical term developed over the course of the Age of Exploration as Europe spread its culture to other parts of the world. Additionally, closer contacts between the West and Asia and other parts of the world in recent times have continued to cloud the use, herodotus considered the Persian Wars of the early 5th century BC a conflict of Europa versus Asia. The terms West and East were not used by any Greek author to describe that conflict, the Great Schism and the Fourth Crusade confirmed this deviation. The Renaissance in the West emerged partly from currents within the Roman Empire, Ancient Rome was a civilization that grew from a city-state founded on the Italian Peninsula about the 9th century BC to a massive empire straddling the Mediterranean Sea. In its 12-century existence, Roman civilization shifted from a monarchy, to a republic, nonetheless, despite its great legacy, a number of factors led to the eventual decline of the Roman Empire
44.
Linear equation
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A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. A simple example of an equation with only one variable, x, may be written in the form, ax + b =0, where a and b are constants. The constants may be numbers, parameters, or even functions of parameters. Linear equations can have one or more variables. An example of an equation with three variables, x, y, and z, is given by, ax + by + cz + d =0, where a, b, c, and d are constants and a, b. Linear equations occur frequently in most subareas of mathematics and especially in applied mathematics, an equation is linear if the sum of the exponents of the variables of each term is one. Equations with exponents greater than one are non-linear, an example of a non-linear equation of two variables is axy + b =0, where a and b are constants and a ≠0. It has two variables, x and y, and is non-linear because the sum of the exponents of the variables in the first term and this article considers the case of a single equation for which one searches the real solutions. All its content applies for complex solutions and, more generally for linear equations with coefficients, a linear equation in one unknown x may always be rewritten a x = b. If a ≠0, there is a solution x = b a. The origin of the name comes from the fact that the set of solutions of such an equation forms a straight line in the plane. Linear equations can be using the laws of elementary algebra into several different forms. These equations are referred to as the equations of the straight line. In what follows, x, y, t, and θ are variables, in the general form the linear equation is written as, A x + B y = C, where A and B are not both equal to zero. The equation is written so that A ≥0, by convention. The graph of the equation is a line, and every straight line can be represented by an equation in the above form. If A is nonzero, then the x-intercept, that is, if B is nonzero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis, is C/B, and the slope of the line is −A/B. The general form is written as, a x + b y + c =0
45.
Quadratic equation
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If a =0, then the equation is linear, not quadratic. The numbers a, b, and c are the coefficients of the equation, and may be distinguished by calling them, respectively, the coefficient, the linear coefficient. Because the quadratic equation involves only one unknown, it is called univariate, solutions to problems equivalent to the quadratic equation were known as early as 2000 BC. A quadratic equation with real or complex coefficients has two solutions, called roots and these two solutions may or may not be distinct, and they may or may not be real. It may be possible to express a quadratic equation ax2 + bx + c =0 as a product =0. In some cases, it is possible, by inspection, to determine values of p, q, r. If the quadratic equation is written in the form, then the Zero Factor Property states that the quadratic equation is satisfied if px + q =0 or rx + s =0. Solving these two linear equations provides the roots of the quadratic, for most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed. As an example, x2 + 5x +6 factors as, the more general case where a does not equal 1 can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection. Except for special cases such as where b =0 or c =0 and this means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection. The process of completing the square makes use of the identity x 2 +2 h x + h 2 =2. Starting with an equation in standard form, ax2 + bx + c =0 Divide each side by a. Subtract the constant term c/a from both sides, add the square of one-half of b/a, the coefficient of x, to both sides. This completes the square, converting the left side into a perfect square, write the left side as a square and simplify the right side if necessary. Produce two linear equations by equating the square root of the side with the positive and negative square roots of the right side. Completing the square can be used to derive a formula for solving quadratic equations. The mathematical proof will now be briefly summarized and it can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation,2 = b 2 −4 a c 4 a 2. Taking the square root of both sides, and isolating x, gives, x = − b ± b 2 −4 a c 2 a and these result in slightly different forms for the solution, but are otherwise equivalent
46.
Renaissance
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The Renaissance was a period in European history, from the 14th to the 17th century, regarded as the cultural bridge between the Middle Ages and modern history. It started as a movement in Italy in the Late Medieval period and later spread to the rest of Europe. This new thinking became manifest in art, architecture, politics, science, Early examples were the development of perspective in oil painting and the recycled knowledge of how to make concrete. Although the invention of movable type sped the dissemination of ideas from the later 15th century. In politics, the Renaissance contributed to the development of the customs and conventions of diplomacy, the Renaissance began in Florence, in the 14th century. Other major centres were northern Italian city-states such as Venice, Genoa, Milan, Bologna, the word Renaissance, literally meaning Rebirth in French, first appeared in English in the 1830s. The word also occurs in Jules Michelets 1855 work, Histoire de France, the word Renaissance has also been extended to other historical and cultural movements, such as the Carolingian Renaissance and the Renaissance of the 12th century. The Renaissance was a movement that profoundly affected European intellectual life in the early modern period. Renaissance scholars employed the humanist method in study, and searched for realism, however, a subtle shift took place in the way that intellectuals approached religion that was reflected in many other areas of cultural life. In addition, many Greek Christian works, including the Greek New Testament, were back from Byzantium to Western Europe. Political philosophers, most famously Niccolò Machiavelli, sought to describe life as it really was. Others see more competition between artists and polymaths such as Brunelleschi, Ghiberti, Donatello, and Masaccio for artistic commissions as sparking the creativity of the Renaissance. Yet it remains much debated why the Renaissance began in Italy, accordingly, several theories have been put forward to explain its origins. During the Renaissance, money and art went hand in hand, Artists depended entirely on patrons while the patrons needed money to foster artistic talent. Wealth was brought to Italy in the 14th, 15th, and 16th centuries by expanding trade into Asia, silver mining in Tyrol increased the flow of money. Luxuries from the Eastern world, brought home during the Crusades, increased the prosperity of Genoa, unlike with Latin texts, which had been preserved and studied in Western Europe since late antiquity, the study of ancient Greek texts was very limited in medieval Western Europe. One of the greatest achievements of Renaissance scholars was to bring this entire class of Greek cultural works back into Western Europe for the first time since late antiquity, Arab logicians had inherited Greek ideas after they had invaded and conquered Egypt and the Levant. Their translations and commentaries on these ideas worked their way through the Arab West into Spain and Sicily and this work of translation from Islamic culture, though largely unplanned and disorganized, constituted one of the greatest transmissions of ideas in history
47.
Ptolemy
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Claudius Ptolemy was a Greek writer, known as a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in the city of Alexandria in the Roman province of Egypt, wrote in Koine Greek, beyond that, few reliable details of his life are known. His birthplace has been given as Ptolemais Hermiou in the Thebaid in a statement by the 14th-century astronomer Theodore Meliteniotes. This is a very late attestation, however, and there is no reason to suppose that he ever lived elsewhere than Alexandria. Ptolemy wrote several treatises, three of which were of importance to later Byzantine, Islamic and European science. The first is the astronomical treatise now known as the Almagest, although it was entitled the Mathematical Treatise. The second is the Geography, which is a discussion of the geographic knowledge of the Greco-Roman world. The third is the treatise in which he attempted to adapt horoscopic astrology to the Aristotelian natural philosophy of his day. This is sometimes known as the Apotelesmatika but more known as the Tetrabiblos from the Greek meaning Four Books or by the Latin Quadripartitum. The name Claudius is a Roman nomen, the fact that Ptolemy bore it indicates he lived under the Roman rule of Egypt with the privileges and political rights of Roman citizenship. It would have suited custom if the first of Ptolemys family to become a citizen took the nomen from a Roman called Claudius who was responsible for granting citizenship, if, as was common, this was the emperor, citizenship would have been granted between AD41 and 68. The astronomer would also have had a praenomen, which remains unknown and it occurs once in Greek mythology, and is of Homeric form. All the kings after him, until Egypt became a Roman province in 30 BC, were also Ptolemies, abu Mashar recorded a belief that a different member of this royal line composed the book on astrology and attributed it to Ptolemy. The correct answer is not known”, Ptolemy wrote in Greek and can be shown to have utilized Babylonian astronomical data. He was a Roman citizen, but most scholars conclude that Ptolemy was ethnically Greek and he was often known in later Arabic sources as the Upper Egyptian, suggesting he may have had origins in southern Egypt. Later Arabic astronomers, geographers and physicists referred to him by his name in Arabic, Ptolemys Almagest is the only surviving comprehensive ancient treatise on astronomy. Ptolemy presented his models in convenient tables, which could be used to compute the future or past position of the planets. The Almagest also contains a catalogue, which is a version of a catalogue created by Hipparchus
48.
Geography (Ptolemy)
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Its translation into Arabic in the 9th century and Latin in 1406 was highly influential on the geographical knowledge and cartographic traditions of the medieval Caliphate and Renaissance Europe. Versions of Ptolemys work in antiquity were probably proper atlases with attached maps, no Greek manuscript of the Geography survives from earlier than the 13th century. In Europe, maps were sometimes made using the coordinates provided by the text. Later scribes and publishers could then copy these new maps, as Athanasius did for the emperor Andronicus II Palaeologus, the three earliest surviving texts with maps are those from Constantinople based on Planudess work. The first Latin translation of texts was made in 1406 or 1407 by Jacobus Angelus in Florence, Italy. It is not thought that his edition had maps, although Manuel Chrysoloras had given Palla Strozzi a Greek copy of Planudess maps in Florence in 1397, the Geography consists of three sections, divided among 8 books. Book I is a treatise on cartography, describing the methods used to assemble, from Book II through the beginning of Book VII, a gazetteer provides longitude and latitude values for the world known to the ancient Romans. The rest of Book VII provides details on three projections to be used for the construction of a map of the world, varying in complexity and fidelity, Book VIII constitutes an atlas of regional maps. The maps include a recapitulation of some of the values given earlier in the work, Maps based on scientific principles had been made in Europe since the time of Eratosthenes in the 3rd century BC. Ptolemy improved the treatment of map projections and he provided instructions on how to create his maps in the first section of the work. The gazetteer section of Ptolemys work provided latitude and longitude coördinates for all the places and his Prime Meridian ran through the Fortunate Isles, the westernmost land recorded, at around the position of El Hierro in the Canary Islands. The maps spanned 180 degrees of longitude from the Fortunate Isles in the Atlantic to China, Ptolemy was aware that Europe knew only about a quarter of the globe. Ptolemys work included a large and less detailed world map and then separate. As early as the 1420s, these maps were complemented by extra-Ptolemaic regional maps depicting. The original treatise by Marinus of Tyre that formed the basis of Ptolemys Geography has been completely lost, a world map based on Ptolemy was displayed in Augustodunum in late Roman times. Pappus, writing at Alexandria in the 4th century, produced a commentary on Ptolemys Geography, for instance, Grant Parker argues that it would be highly implausible for them to have constructed the Bay of Bengal as precisely as they did without the accounts of sailors. Muslim cartographers were using copies of Ptolemys Almagest and Geography by the 9th century, a 1037 copy of these are the earliest extant maps from Islamic lands. Nallino suggests that the work was not based on Ptolemy but on a world map
49.
C.A. Nallino
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Carlo Alfonso Nallino was an Italian orientalist. He was born in Turin, and studied literature in the University of Turin, at the age of 21 he published is first treatise on Arab geography and astronomy. This was followed by a work on Al-Battani which gained him international recognition, from 1896 he taught in the Istituto Universitario Orientale of Naples and then at the University of Palermo. In 1900 he published a book on the Egyptian Arab dialect, amongs his pupils was Taha Husayn, later Minister of the Interiors. Later Nallino became ordinary professor at the University La Sapienza of Rome, where, in 1921, he had founded the Istituto per lOriente, which published the magazine Oriente Moderno. In 1933 he was named member of the Royal Academy of Arab Language in Cairo, in 1938 he travelled for two months in the Arabic Peninsula, but he died shortly afterwards in Rome for a cardiac crisis after publishing only the first volumes of the studies about his trip. Istituto per lOriente Carlo Alfonso Nallino
50.
Syriac language
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Syriac /ˈsɪri. æk/, also known as Syriac Aramaic, is a dialect of Middle Aramaic that was once spoken across much of the Fertile Crescent and Eastern Arabia. Indeed, Syriac literature comprises roughly 90% of the extant Aramaic literature, Old Aramaic was adopted by the Neo-Assyrian Empire when they conquered the various Aramean city-kingdoms to its west. The Achaemenid Empire, which rose after the fall of the Assyrian Empire, also adopted Old Aramaic as its official language, during the course of the third and fourth centuries AD, the inhabitants of the region began to embrace Christianity. Along with Latin and Greek, Syriac became one of the three most important Christian languages in the centuries of the Christian Era. Primarily a Christian medium of expression, Syriac had a cultural and literary influence on the development of Arabic. Syriac remains the language of Syriac Christianity to this day. Syriac is a Middle Aramaic language and, as such, a language of the Northwestern branch of the Semitic family and it is written in the Syriac alphabet, a derivation of the Aramaic alphabet. Syriac was the local accent of Aramaic in Edessa, that evolved under the influence of Church of the East and it has been found as far afield as Hadrians Wall in Ancient Britain, with inscriptions written by Assyrian and Aramean soldiers of the Roman Empire. Modern Syriac/Modern Syriac Aramaic is an occasionally used to refer to the modern Eastern Aramaic languages. In this terminology, Modern Syriac is divided into, Modern Western Syriac Aramaic, note however that these are sometimes excluded from the category of Modern Syriac. The modern varieties are, therefore, not discussed in this article, in 132 BC, the kingdom of Osroene was founded in Edessa and Proto-Syriac evolved in that kingdom. Many Syriac-speakers still look to Edessa as the cradle of their language, there are about eighty extant early Syriac inscriptions, dated to the first three centuries AD. All of these examples of the language are non-Christian. As an official language, Syriac was given a relatively coherent form, style, in the 3rd century, churches in Edessa began to use Syriac as the language of worship. There is evidence that the adoption of Syriac, the language of the Assyrian people, was to effect mission, much literary effort was put into the production of an authoritative translation of the Bible into Syriac, the Peshitta. At the same time, Ephrem the Syrian was producing the most treasured collection of poetry, in 489, many Syriac-speaking Christians living in the eastern reaches of the Roman Empire fled to the Sassanid Empire to escape persecution and growing animosity with Greek-speaking Christians. The Christological differences with the Church of the East led to the bitter Nestorian schism in the Syriac-speaking world, as a result, Syriac developed distinctive western and eastern varieties. Syriac literature is by far the most prodigious of the various Aramaic languages and its corpus covers poetry, prose, theology, liturgy, hymnody, history, philosophy, science, medicine and natural history
51.
Law of sines
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In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of a triangle to the sines of its angles. When the last of these equations is not used, the law is sometimes stated using the reciprocals, the law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. Numerical calculation using this technique may result in an error if an angle is close to 90 degrees. It can also be used when two sides and one of the angles are known. In some such cases, the triangle is not uniquely determined by this data, the law of sines is one of two trigonometric equations commonly applied to find lengths and angles in scalene triangles, with the other being the law of cosines. The law of sines can be generalized to higher dimensions on surfaces with constant curvature, the area T of any triangle can be written as one half of its base times its height. Thus, depending on the selection of the base the area of the triangle can be written as any of, multiplying these by 2/abc gives 2 T a b c = sin A a = sin B b = sin C c. When using the law of sines to find a side of a triangle, in the case shown below they are triangles ABC and AB′C′. Given a general triangle the following conditions would need to be fulfilled for the case to be ambiguous, The only information known about the triangle is the angle A, the side a is shorter than the side c. The side a is longer than the altitude h from angle B, without further information it is impossible to decide which is the triangle being asked for. The following are examples of how to solve a problem using the law of sines, given, side a =20, side c =24, and angle C = 40°. Using the law of sines, we conclude that sin A20 = sin 40 ∘24, note that the potential solution A =147. 61° is excluded because that would necessarily give A + B + C > 180°. The second equality above readily simplifies to Herons formula for the area, the law of sines takes on a similar form in the presence of curvature. In the spherical case, the formula is, sin A sin α = sin B sin β = sin C sin γ. Here, α, β, and γ are the angles at the center of the sphere subtended by the three arcs of the spherical surface triangle a, b, and c, respectively, a, B, and C are the surface angles opposite their respective arcs. See also Spherical law of cosines and Half-side formula, in hyperbolic geometry when the curvature is −1, the law of sines becomes sin A sinh a = sin B sinh b = sin C sinh c. Define a generalized function, depending also on a real parameter K. The law of sines in constant curvature K reads as sin A sin K a = sin B sin K b = sin C sin K c
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History of trigonometry
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Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics and Babylonian mathematics. Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy, in Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata. During the Middle Ages, the study of continued in Islamic mathematics. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics, the term trigonometry was derived from Greek τρίγωνον trigōnon, triangle and μέτρον metron, measure. Our modern word sine is derived from the Latin word sinus, the Arabic term is in origin a corruption of Sanskrit jīvā, or chord. Sanskrit jīvā in learned usage was a synonym of jyā chord, Sanskrit jīvā was loaned into Arabic as jiba. Particularly Fibonaccis sinus rectus arcus proved influential in establishing the term sinus, the words minute and second are derived from the Latin phrases partes minutae primae and partes minutae secundae. These roughly translate to first small parts and second small parts, the ancient Egyptians and Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries. However, as pre-Hellenic societies lacked the concept of an angle measure, based on one interpretation of the Plimpton 322 cuneiform tablet, some have even asserted that the ancient Babylonians had a table of secants. There is, however, much debate as to whether it is a table of Pythagorean triples, the Egyptians, on the other hand, used a primitive form of trigonometry for building pyramids in the 2nd millennium BC. Ahmes solution to the problem is the ratio of half the side of the base of the pyramid to its height, in other words, the quantity he found for the seked is the cotangent of the angle to the base of the pyramid and its face. Ancient Greek and Hellenistic mathematicians made use of the chord, given a circle and an arc on the circle, the chord is the line that subtends the arc. A chords perpendicular bisector passes through the center of the circle and bisects the angle. One half of the chord is the sine of one half the bisected angle, that is, c h o r d θ =2 sin θ2. Due to this relationship, a number of identities and theorems that are known today were also known to Hellenistic mathematicians. For instance, propositions twelve and thirteen of book two of the Elements are the laws of cosines for obtuse and acute angles, respectively, theorems on the lengths of chords are applications of the law of sines. And Archimedes theorem on broken chords is equivalent to formulas for sines of sums, the first trigonometric table was apparently compiled by Hipparchus of Nicaea, who is now consequently known as the father of trigonometry. Hipparchus was the first to tabulate the corresponding values of arc and it seems that the systematic use of the 360° circle is largely due to Hipparchus and his table of chords
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Nasir al-Din al-Tusi
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Khawaja Muhammad ibn Muhammad ibn al-Hasan al-Tūsī, better known as Nasīr al-Dīn Tūsī, was a Persian polymath, architect, philosopher, physician, scientist, theologian and Marja Taqleed. He was of the Twelver Shī‘ah Islamic belief, the Muslim scholar Ibn Khaldun considered Tusi to be the greatest of the later Persian scholars. Nasir al-Din Tusi was born in the city of Tus in medieval Khorasan in the year 1201, in Hamadan and Tus he studied the Quran, Hadith, Shia jurisprudence, logic, philosophy, mathematics, medicine and astronomy. He was apparently born into a Shī‘ah family and lost his father at a young age, at a young age he moved to Nishapur to study philosophy under Farid al-Din Damad and mathematics under Muhammad Hasib. He met also Farid al-Din Attar, the legendary Sufi master who was killed by Mongol invaders. In Mosul he studied mathematics and astronomy with Kamal al-Din Yunus and he was captured after the invasion of the Alamut castle by the Mongol forces. Tusi has about 150 works, of which 25 are in Persian and the remaining are in Arabic, here are some of his major works, Kitāb al-Shakl al-qattāʴ Book on the complete quadrilateral. A five volume summary of trigonometry, al-Tadhkirah fiilm al-hayah – A memoir on the science of astronomy. Many commentaries were written about this work called Sharh al-Tadhkirah - Commentaries were written by Abd al-Ali ibn Muhammad ibn al-Husayn al-Birjandi, akhlaq-i Nasiri – A work on ethics. Al-Risalah al-Asturlabiyah – A Treatise on astrolabe, Zij-i ilkhani – A major astronomical treatise, completed in 1272. Sharh al-isharat Awsaf al-Ashraf a short work in Persian Tajrīd al-iʿtiqād – A commentary on Shia doctrines. During his stay in Nishapur, Tusi established a reputation as an exceptional scholar, tusi’s prose writing, which number over 150 works, represent one of the largest collections by a single Islamic author. Writing in both Arabic and Persian, Nasir al-Din Tusi dealt with religious topics and non-religious or secular subjects. His works include the definitive Arabic versions of the works of Euclid, Archimedes, Ptolemy, Autolycus, Tusi convinced Hulegu Khan to construct an observatory for establishing accurate astronomical tables for better astrological predictions. Beginning in 1259, the Rasad Khaneh observatory was constructed in Azarbaijan, south of the river Aras, and to the west of Maragheh, the capital of the Ilkhanate Empire. Based on the observations in this for the time being most advanced observatory and this book contains astronomical tables for calculating the positions of the planets and the names of the stars. His model for the system is believed to be the most advanced of his time. Between Ptolemy and Copernicus, he is considered by many to be one of the most eminent astronomers of his time, for his planetary models, he invented a geometrical technique called a Tusi-couple, which generates linear motion from the sum of two circular motions
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Abu al-Wafa' Buzjani
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Abū al-Wafāʾ, Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī or Abū al-Wafā Būzhjānī was a Persian mathematician and astronomer who worked in Baghdad. He made important innovations in spherical trigonometry, and his work on arithmetics for businessmen contains the first instance of using numbers in a medieval Islamic text. He is also credited with compiling the tables of sines and tangents at 15 intervals and he also introduced the secant and cosecant functions, as well studied the interrelations between the six trigonometric lines associated with an arc. His Almagest was widely read by medieval Arabic astronomers in the centuries after his death and he is known to have written several other books that have not survived. He was born in Buzhgan, in Khorasan, at age 19, in 959 AD, he moved to Baghdad and remained there for the next forty years, and died there in 998. In Baghdad, he received patronage by members of the Buyid court, abu Al-Wafa was the first to build a wall quadrant to observe the sky. It has been suggested that he was influenced by the works of Al-Battani as the latter describes a quadrant instrument in his Kitāb az-Zīj, in 997, he participated in an experiment to determine the difference in local time between his location and that of al-Biruni. The result was close to present-day calculations, showing a difference of approximately 1 hour between the two longitudes. Abu al-Wafa is also known to have worked with Abū Sahl al-Qūhī, while what is extant from his works lacks theoretical innovation, his observational data were used by many later astronomers, including al-Biruni. Among his works on astronomy, only the first seven treatises of his Almagest are now extant, the work covers numerous topics in the fields of plane and spherical trigonometry, planetary theory, and solutions to determine the direction of Qibla. He established several trigonometric identities such as sin in their modern form, some sources suggest that he introduced the tangent function, although other sources give the credit for this innovation to al-Marwazi. A book of zij called Zīj al‐wāḍiḥ, no longer extant, a Book on Those Geometric Constructions Which Are Necessary for a Craftsman. This text contains over one hundred geometric constructions, including for a regular heptagon, the legacy of this text in Latin Europe is still debated. A Book on What Is Necessary from the Science of Arithmetic for Scribes and this is the first book where negative numbers have been used in the medieval Islamic texts. He also wrote translations and commentaries on the works of Diophantus, al-Khwārizmī. The crater Abul Wáfa on the Moon is named after him, oConnor, John J. Robertson, Edmund F. Mohammad Abul-Wafa Al-Buzjani, MacTutor History of Mathematics archive, University of St Andrews. Būzjānī, Abū al‐Wafāʾ Muḥammad ibn Muḥammad ibn Yaḥyā al‐Būzjānī. A Study of Method, Historia Mathematica,39, 34–83, doi,10. 1016/j. hm.2011.09.001 Youschkevitch, A. P. Abūl-Wafāʾ Al-Būzjānī, Muḥammad Ibn Muḥammad Ibn Yaḥyā Ibn Ismāʿīl Ibn Al-ʿAbbās
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Ismail al-Jazari
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Badīʿ az-Zaman Abū l-ʿIzz ibn Ismāʿīl ibn ar-Razāz al-Jazarī was an Islamic polymath, a scholar, inventor, mechanical engineer, artisan, artist and mathematician. Nothing is known about early life. He is famed for authoring the Book of Knowledge of Ingenious Mechanical Devices, according to Mayr, the books style resembles that of a modern do-it-yourself book. Some of his devices were inspired by earlier devices, such as one of his water clocks. He also cites the influence of the Banū Mūsā brothers for his fountains, al-Saghani for the design of a candle clock, and Hibatullah ibn al-Husayn for musical automata. A camshaft, a shaft to which cams are attached, was introduced in 1206 by al-Jazari, the cam and camshaft also appeared in European mechanisms from the 14th century. The eccentrically mounted handle of the rotary quern-stone in fifth century BCE Spain that spread across the Roman Empire constitutes a crank, the earliest evidence of a crank and connecting rod mechanism dates to the 3rd century AD Hierapolis sawmill in the Roman Empire. The crank also appears in the century in several of the hydraulic devices described by the Banū Mūsā brothers in their Book of Ingenious Devices. In 1206, al-Jazari invented an early crankshaft, which he incorporated with a crank-connecting rod mechanism in his twin-cylinder pump and he used the crankshaft with a connecting rod in two of his water-raising machines, the crank-driven saqiya chain pump and the double-action reciprocating piston suction pump. His water pump also employed the first known crank-slider mechanism, al-Jazari invented a method for controlling the speed of rotation of a wheel using an escapement mechanism. According to Donald Hill, al-Jazari described several early mechanical controls, including a metal door, a combination lock. A segmental gear is a piece for receiving or communicating reciprocating motion from or to a cogwheel, consisting of a sector of a gear, or ring, having cogs on the periphery. It was in these machines that he introduced his most important ideas. The first known use of a crankshaft in a pump was in one of al-Jazaris saqiya machines. The concept of minimizing intermittent working is also first implied in one of al-Jazaris saqiya chain pumps, al-Jazari also constructed a water-raising saqiya chain pump which was run by hydropower rather than manual labour, though the Chinese were also using hydropower for chain pumps prior to him. Saqiya machines like the ones he described have been supplying water in Damascus since the 13th century up until modern times, and were in everyday use throughout the medieval Islamic world. This pump is driven by a wheel, which drives, through a system of gears. The pistons work in horizontally opposed cylinders, each provided with valve-operated suction, the delivery pipes are joined above the centre of the machine to form a single outlet into the irrigation system
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Ibn al-Haytham
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Abū ʿAlī al-Ḥasan ibn al-Ḥasan ibn al-Haytham, also known by the Latinization Alhazen or Alhacen, was an Arab Muslim scientist, mathematician, astronomer, and philosopher. Ibn al-Haytham made significant contributions to the principles of optics, astronomy, mathematics and he was the first to explain that vision occurs when light bounces on an object and then is directed to ones eyes. He spent most of his close to the court of the Fatimid Caliphate in Cairo and earned his living authoring various treatises. In medieval Europe, Ibn al-Haytham was honored as Ptolemaeus Secundus or simply called The Physicist and he is also sometimes called al-Baṣrī after his birthplace Basra in Iraq, or al-Miṣrī. Ibn al-Haytham was born c.965 in Basra, which was part of the Buyid emirate. Alhazen arrived in Cairo under the reign of Fatimid Caliph al-Hakim, Alhazen continued to live in Cairo, in the neighborhood of the famous University of al-Azhar, until his death in 1040. Legend has it that after deciding the scheme was impractical and fearing the caliphs anger, during this time, he wrote his influential Book of Optics and continued to write further treatises on astronomy, geometry, number theory, optics and natural philosophy. Among his students were Sorkhab, a Persian from Semnan who was his student for three years, and Abu al-Wafa Mubashir ibn Fatek, an Egyptian prince who learned mathematics from Alhazen. Alhazen made significant contributions to optics, number theory, geometry, astronomy, Alhazens work on optics is credited with contributing a new emphasis on experiment. In al-Andalus, it was used by the prince of the Banu Hud dynasty of Zaragossa and author of an important mathematical text. A Latin translation of the Kitab al-Manazir was made probably in the twelfth or early thirteenth century. His research in catoptrics centred on spherical and parabolic mirrors and spherical aberration and he made the observation that the ratio between the angle of incidence and refraction does not remain constant, and investigated the magnifying power of a lens. His work on catoptrics also contains the known as Alhazens problem. Alhazen wrote as many as 200 books, although only 55 have survived, some of his treatises on optics survived only through Latin translation. During the Middle Ages his books on cosmology were translated into Latin, Hebrew, the crater Alhazen on the Moon is named in his honour, as was the asteroid 59239 Alhazen. In honour of Alhazen, the Aga Khan University named its Ophthalmology endowed chair as The Ibn-e-Haitham Associate Professor, Alhazen, by the name Ibn al-Haytham, is featured on the obverse of the Iraqi 10, 000-dinar banknote issued in 2003, and on 10-dinar notes from 1982. The 2015 International Year of Light celebrated the 1000th anniversary of the works on optics by Ibn Al-Haytham, Alhazens most famous work is his seven-volume treatise on optics Kitab al-Manazir, written from 1011 to 1021. Optics was translated into Latin by a scholar at the end of the 12th century or the beginning of the 13th century
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Arabic numerals
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In this numeral system, a sequence of digits such as 975 is read as a single number, using the position of the digit in the sequence to interpret its value. The symbol for zero is the key to the effectiveness of the system, the system was adopted by Arab mathematicians in Baghdad and passed on to the Arabs farther west. There is some evidence to suggest that the numerals in their current form developed from Arabic letters in the Maghreb, the current form of the numerals developed in North Africa, distinct in form from the Indian and eastern Arabic numerals. The use of Arabic numerals spread around the world through European trade, books, the term Arabic numerals is ambiguous. It most commonly refers to the widely used in Europe. Arabic numerals is also the name for the entire family of related numerals of Arabic. It may also be intended to mean the numerals used by Arabs and it would be more appropriate to refer to the Arabic numeral system, where the value of a digit in a number depends on its position. The decimal Hindu–Arabic numeral system was developed in India by AD700, the development was gradual, spanning several centuries, but the decisive step was probably provided by Brahmaguptas formulation of zero as a number in AD628. The system was revolutionary by including zero in positional notation, thereby limiting the number of digits to ten. It is considered an important milestone in the development of mathematics, one may distinguish between this positional system, which is identical throughout the family, and the precise glyphs used to write the numerals, which varied regionally. The glyphs most commonly used in conjunction with the Latin script since early modern times are 0123456789. The first universally accepted inscription containing the use of the 0 glyph in India is first recorded in the 9th century, in an inscription at Gwalior in Central India dated to 870. Numerous Indian documents on copper plates exist, with the symbol for zero in them, dated back as far as the 6th century AD. Inscriptions in Indonesia and Cambodia dating to AD683 have also been found and their work was principally responsible for the diffusion of the Indian system of numeration in the Middle East and the West. In the 10th century, Middle-Eastern mathematicians extended the decimal system to include fractions. The decimal point notation was introduced by Sind ibn Ali, who wrote the earliest treatise on Arabic numerals. Ghubar numerals themselves are probably of Roman origin, some popular myths have argued that the original forms of these symbols indicated their numeric value through the number of angles they contained, but no evidence exists of any such origin. In 825 Al-Khwārizmī wrote a treatise in Arabic, On the Calculation with Hindu Numerals, Algoritmi, the translators rendition of the authors name, gave rise to the word algorithm
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History of calculus
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Calculus, known in its early history as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series. Isaac Newton and Gottfried Leibniz independently discovered calculus in the mid-17th century, however, each inventor claimed the other stole his work in a bitter dispute that continued until the end of their lives. The ancient period introduced some of the ideas that led to integral calculus, india had a long history of trigonometry as witnessed by the 8th century BC treatise Sulba Sutras, or rules of the chord, where the sine, cosine, and tangent were conceived. Indian mathematicians gave a method of differentiation of some trigonometric functions. Babylonians may have discovered the trapezoidal rule while doing astronomical observations of Jupiter, the method of exhaustion was later reinvented in China by Liu Hui in the 4th century AD in order to find the area of a circle. In the 5th century AD, Zu Chongzhi established a method that would later be called Cavalieris principle to find the volume of a sphere, Greek mathematicians are also credited with a significant use of infinitesimals. At approximately the time, Zeno of Elea discredited infinitesimals further by his articulation of the paradoxes which they create. It should not be thought that infinitesimals were put on a rigorous footing during this time, only when it was supplemented by a proper geometric proof would Greek mathematicians accept a proposition as true. It was not until the 17th century that the method was formalized by Cavalieri as the method of Indivisibles, Archimedes was the first to find the tangent to a curve, other than a circle, in a method akin to differential calculus. The pioneers of the such as Isaac Barrow and Johann Bernoulli were diligent students of Archimedes. In the Middle East, Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, the mathematical study of continuity was revived in the 14th century by the Oxford Calculators and French collaborators such as Nicole Oresme. They proved the Merton mean speed theorem, that an accelerated body travels the same distance as a body with uniform speed whose speed is half the final velocity of the accelerated body. In the 17th century, European mathematicians Isaac Barrow, René Descartes, Pierre de Fermat, Blaise Pascal, John Wallis, Isaac Newton would later write that his own early ideas about calculus came directly from Fermats way of drawing tangents. Torricelli extended this work to other such as the cycloid. In a 1659 treatise, Fermat is credited with a trick for evaluating the integral of any power function directly. Fermat also obtained a technique for finding the centers of gravity of various plane and solid figures, the first full proof of the fundamental theorem of calculus was given by Isaac Barrow. Newton and Leibniz, building on work, independently developed the surrounding theory of infinitesimal calculus in the late 17th century
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History of geometry
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Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of mathematics, the other being the study of numbers. Classic geometry was focused in compass and straightedge constructions, geometry was revolutionized by Euclid, who introduced mathematical rigor and the axiomatic method still in use today. His book, The Elements is widely considered the most influential textbook of all time, the earliest recorded beginnings of geometry can be traced to early peoples, who discovered obtuse triangles in the ancient Indus Valley, and ancient Babylonia from around 3000 BC. Among these were some surprisingly sophisticated principles, and a mathematician might be hard put to derive some of them without the use of calculus. For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras and the Indian Sulba Sutras around 800 B. C. Problem 30 of the Ahmes papyrus uses these methods to calculate the area of a circle and this assumes that π is 4×², with an error of slightly over 0.63 percent. Problem 48 involved using a square with side 9 units and this square was cut into a 3x3 grid. The diagonal of the squares were used to make an irregular octagon with an area of 63 units. This gave a value for π of 3.111. The two problems together indicate a range of values for π between 3.11 and 3.16. Problem 14 in the Moscow Mathematical Papyrus gives the only ancient example finding the volume of a frustum of a pyramid, describing the correct formula, the Babylonians may have known the general rules for measuring areas and volumes. They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, which would be correct if π is estimated as 3, the Pythagorean theorem was also known to the Babylonians. Also, there was a recent discovery in which a tablet used π as 3, the Babylonians are also known for the Babylonian mile, which was a measure of distance equal to about seven miles today. This measurement for distances eventually was converted to a used for measuring the travel of the Sun, therefore. There have been recent discoveries showing that ancient Babylonians may have discovered astronomical geometry nearly 1400 years before Europeans did, the Indian Vedic period had a tradition of geometry, mostly expressed in the construction of elaborate altars. Early Indian texts on this include the Satapatha Brahmana and the Śulba Sūtras. According to, the Śulba Sūtras contain the earliest extant verbal expression of the Pythagorean Theorem in the world, the diagonal rope of an oblong produces both which the flank and the horizontal <ropes> produce separately
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Science in the medieval Islamic world
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Science in the medieval Islamic world was the science developed and practised during the Islamic Golden Age under the Abbasid Caliphate. Islamic scientific achievements encompassed a range of subject areas, especially astronomy, mathematics. Other subjects of scientific inquiry included alchemy and chemistry, botany, geography and cartography, ophthalmology, pharmacology, physics, in the 8th century, scholars had translated Indian, Assyrian, Sassanian and Greek knowledge, including the works of Aristotle, into Arabic. These translations became a wellspring for advances by scientists from Muslim-ruled areas during the Middle Ages, major religious and cultural works of the empire were translated into Arabic. The culture inherited Greek, Indic, Assyrian and Persian influences, an era of high culture and innovation ensued. Islamic scientific achievements encompass a range of subject areas, especially mathematics, astronomy. Other subjects of scientific inquiry included physics, alchemy and chemistry, ophthalmology, alchemists supposed that gold was the noblest metal, and that other metals formed a series down to the basest, such as lead. They believed, too, that an element, the elixir. Jabir ibn Hayyan wrote on alchemy, based on his own experiments and he described laboratory techniques and experimental methods that would continue to be used when alchemy had transformed into chemistry. Ibn Hayyan identified many substances including sulfuric and nitric acids and he described processes including sublimation, reduction and distillation. He utilized equipment such as the alembic and the retort stand, there is considerable uncertainty as to the actual provenance of many works that are ascribed to him. Astronomy was one of the disciplines within Islamic science. Al-Battani accurately determined the length of the solar year and he contributed to numeric tables, such as the Tables of Toledo, used by astronomers to predict the movements of the sun, moon and planets across the sky. Some of his tables were later used by Copernicus. Al-Zarqali developed a more accurate astrolabe, used for centuries afterwards and he constructed a water clock in Toledo. He discovered that the Suns apogee moves slowly relative to the fixed stars, nasir al-Din al-Tusi wrote an important revision to Ptolemys celestial model. When he became Helagus astrologer, he was given an observatory and gained access to Chinese techniques and he developed trigonometry as a separate field, and compiled the most accurate astronomical tables available up to that time. The study of the world extended to a detailed examination of plants
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Timeline of Islamic science and technology
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This timeline of science and engineering in the Islamic world covers the time period from the eighth century AD to the introduction of European science to the Islamic world in the nineteenth century. All year dates are according to the Gregorian calendar except where noted. 721 –815 Chemistry, glass, Jabir ibn Hayyan, first chemist known to produce sulfuric acid, as well as many other chemicals and instruments. Wrote on adding color to glass by adding small quantities of metallic oxides to the glass and this was a new advance in glass industry unknown in antiquity. His works include The Elaboration of the Grand Elixir, The Chest of Wisdom in which he writes on nitric acid, Kitab al-istitmam,780 –850 – Mathematics, al-Khwarizmi Developed the calculus of resolution and juxtaposition, more briefly referred to as al-jabr, or algebra. C.810 Bayt al-Hikma set up in Baghdad, there Greek and Indian mathematical and astronomy works are translated into Arabic. 801 –873 Chemistry, Al-Kindi writes on the distillation of wine as that of water and gives 107 recipes for perfumes. According to one account written seven centuries after his death, Ibn Firnas was injured during an elevated winged trial flight,826 –901 Mathematics, Thabit ibn Qurra Studied at Baghdads House of Wisdom under the Banu Musa brothers. Discovered a theorem which enables pairs of numbers to be found. Later, al-Baghdadi a developed variant of the theorem, the location where the meat took the longest to rot was the one he chose for building the hospital. Advocated that patients not be told their real condition so that fear or despair do not affect the healing process, Wrote on alkali, caustic soda, soap and glycerine. Gave descriptions of equipment processes and methods in his book Kitab al-Asrar in 925, by this century, three systems of counting are used in the Arab world. Its arithmetic at first required the use of a dust board because the methods required moving the numbers around in the calculation, modified arithmetic methods for the Indian numeral system to make it possible for pen and paper use. Hitherto, doing calculations with the Indian numerals necessitated the use of a dust board as noted earlier, Wrote several treatises using the finger-counting system of arithmetic, and was also an expert on the Indian numerals system. About the Indian system he wrote, did not find application in business circles, using the Indian numeral system, abul Wafa was able to extract roots. 957 Chemistry, Abul Hasan Ali Al-Masudi, wrote on the reaction of water with zaj water giving sulfuric acid. 980 Mathematics, al-Baghdadi Studied a slight variant of Thabit ibn Qurras theorem on amicable numbers, al-Baghdadi also wrote about and compared the three systems of counting and arithmetic used in the region during this period. 1048 –1131 Mathematics, Omar Khayyam, gave a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections
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Adolph P. Yushkevich
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Adolph-Andrei Pavlovich Yushkevich was a Soviet historian of mathematics, leading expert in medieval mathematics of the East and the work of Leonhard Euler. He is a winner of George Sarton Medal by the History of Science Society for a lifetime of scholarly achievement, Yushkevich was born in Odessa, then Russian Empire, in todays Ukraine, to a Jewish family. His father was a Sorbonne-educated philosopher and a mathematician, active in politics and he was a Menshevik who was in ssylka in Siberia, and later in France. His uncle, Semen Solomonovich Yushkevich was a well-known Jewish writer, Yushkevich grew up in St. Petersburg and later in Paris where he lived until Russian Revolution of 1917, when Yushkevich family returned to Odessa. For a time, Sofya Yanovskaya was one of Adolfs teachers in a gymnasium, in 1923, Yushkevich started his studies at the Department of Mathematics of Moscow State University. His doctoral advisor was Dmitri Egorov, but he was awarded a Ph. D. degree without a defense, from 1930 to 1952 he worked at Bauman Technical University where he rose to professorship in 1940 and head of the department of mathematics in 1941. In the years 1941–1943 he was evacuated to Izhevsk, together with the whole Bauman Technical University, starting 1952, he became a full-time researcher at Vavilov Institute of Natural History, where he worked until retirement. Yushkevich published over 300 works in history of mathematics and he was a member of several foreign academies, including German Academy of Sciences Leopoldina, and president of the International Academy of the History of Science. Yushkevich died in Moscow in 1993 and he bequeathed his personal library to the Vavilov Institute. Isabella Bashmakova, A. N. Bogolyubov, S. S. Demidov, B. V. Gnedenko, E. Knobloch, Galina Matvievskaya, D. E. Rowe, B. A. Rozenfeld, IN MEMORIAM, Adolph Andrei Pavlovich Yushkevich. CS1 maint, Uses authors parameter I. G, rybnikov, S. A. Yanovskaya, Adolf Pavlovich Yushkevich is 60, Russian Mathematical Surveys 22, 187–194. Adolf-Andrei Pavlovich Yushkevich, Russian Mathematical Surveys,49,4, K. Shelma, An interview with Adolf-Andrei Pavlovich Iushkevich, in Voprosy Istorii Estestvoznaniya i Techniki, Vol.94, No. S. S. Demidov, T. A. Tokareva, Adolf P. Yushkevich and formation of the society of mathematical historians, 6-th Tambov All-Russian Conference in History of Mathematics, Pershin Publ. OConnor, John J. Robertson, Edmund F, Adolph Andrei Pavlovich Yushkevich, MacTutor History of Mathematics archive, University of St Andrews. Adolph Andrei Yushkevich at the Mathematics Genealogy Project A. P. Yushkevich, a Brief History of the Kenneth O. May Prize in the History of Mathematics
63.
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
64.
Online Etymology Dictionary
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The Online Etymology Dictionary is a free online dictionary that describes the origins of English-language words. Douglas Harper compiled the etymology dictionary to record the history and evolution of more than 30,000 words, including slang, the core body of its etymology information stems from Ernest Weekleys An Etymological Dictionary of Modern English. Other sources include the Middle English Dictionary and the Barnhart Dictionary of Etymology, in producing his large dictionary, Douglas Harper says that he is essentially and for the most part a compiler, an evaluator of etymology reports which others have made. Harper works as a Copy editor/Page designer for LNP Media Group, as of June 2015, there were nearly 50,000 entries in the dictionary. It is cited in articles as a source for explaining the history
65.
Jan Gullberg
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Jan Gullberg was a Swedish surgeon and anaesthesiologist, but became known as a writer on popular science and medical topics. He is best known outside Sweden as the author of Mathematics, From the Birth of Numbers, Gullberg grew up and was trained as a surgeon in Sweden. He qualified in medicine at the University of Lund in 1964 and he practised as a surgeon in Saudi Arabia, Norway and Virginia Mason Hospital, Seattle in the United States, as well as in Sweden. Gullberg saw himself as a rather than a writer. His first book, on science, won the Swedish Medical Societys Jubilee Prize in 1980 and he died of a stroke in Nordfjordeid, Norway at the hospital where he was working. He was twice married, first to Anne-Marie Hallin, with whom he had three children, and Ann, with whom he adopted two sons, Gullbergs second book, Mathematics, From the Birth of Numbers, took ten years to write, consuming all of Gullbergs spare time. It proved a success, its first edition of 17,000 copies was virtually sold out within six months. I take it with me everywhere I go, allen says the book has special charm, making innovative use of the margin and providing excellent quotes and quips throughout. His favourite chapter is Cornerstones of Mathematics which he believes should appeal both to beginners and old hands and he admires the efficient Babylonian method of finding square roots, using division and averaging. He learns from Gullberg how to multiply and divide using an abacus, allen is delighted by the chapter on combinatorics, with its approach to graph theory and magic squares, complete with 1740 map of the seven bridges of Königsberg. And he loved the chapter on probability and he records that he finds its introductory accounts useful for engineers who use maths only occasionally, and suggests how the book could be used for undergraduate students. He concludes by describing the book as gigantic, in every sense and was 10 years in the making, and calls it a giant leap forward for mathematics and all those who love it. The book was reviewed in Scientific American, but more reservedly in New Scientist. Kevin Kelly comments that the book is an able to provide answers on obscure mathematical concepts, in his view The book has wit and humor. Gullberg commented At the start no real mathematician would accept my book, and perhaps it was a bit crazy of me to write a book on mathematics, as it would be for a mathematician to write a book on surgery. Vätska Gas Energi – Kemi och Fysik med tillämpningar i vätskebalans-, blodgas- och näringslära Kiruna
66.
University of St Andrews
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The University of St Andrews is a British public research university in St Andrews, Fife, Scotland. It is the oldest of the four ancient universities of Scotland, St Andrews was founded between 1410 and 1413, when the Avignon Antipope Benedict XIII issued a papal bull to a small founding group of Augustinian clergy. St Andrews is made up from a variety of institutions, including three constituent colleges and 18 academic schools organised into four faculties, the university occupies historic and modern buildings located throughout the town. The academic year is divided into two terms, Martinmas and Candlemas, in term time, over one-third of the towns population is either a staff member or student of the university. It is ranked as the third best university in the United Kingdom in national league tables, the Times Higher Education World Universities Ranking names St Andrews among the worlds Top 50 universities for Social Sciences, Arts and Humanities. St Andrews has the highest student satisfaction amongst all multi-faculty universities in the United Kingdom, St Andrews has many notable alumni and affiliated faculty, including eminent mathematicians, scientists, theologians, philosophers, and politicians. Six Nobel Laureates are among St Andrews alumni and former staff, a charter of privilege was bestowed upon the society of masters and scholars by the Bishop of St Andrews, Henry Wardlaw, on 28 February 1411. Wardlaw then successfully petitioned the Avignon Pope Benedict XIII to grant the university status by issuing a series of papal bulls. King James I of Scotland confirmed the charter of the university in 1432, subsequent kings supported the university with King James V confirming privileges of the university in 1532. A college of theology and arts called St Johns College was founded in 1418 by Robert of Montrose, St Salvators College was established in 1450, by Bishop James Kennedy. St Leonards College was founded in 1511 by Archbishop Alexander Stewart, St Johns College was refounded by Cardinal James Beaton under the name St Marys College in 1538 for the study of divinity and law. Some university buildings that date from this period are still in use today, such as St Salvators Chapel, St Leonards College Chapel, at this time, the majority of the teaching was of a religious nature and was conducted by clerics associated with the cathedral. During the 17th and 18th centuries, the university had mixed fortunes and was beset by civil. He described it as pining in decay and struggling for life, in the second half of the 19th century, pressure was building upon universities to open up higher education to women. In 1876, the University Senate decided to allow women to receive an education at St Andrews at a roughly equal to the Master of Arts degree that men were able to take at the time. The scheme came to be known as the L. L. A and it required women to pass five subjects at an ordinary level and one at honours level and entitled them to hold a degree from the university. In 1889 the Universities Act made it possible to admit women to St Andrews. Agnes Forbes Blackadder became the first woman to graduate from St Andrews on the level as men in October 1894
67.
JSTOR
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JSTOR is a digital library founded in 1995. Originally containing digitized back issues of journals, it now also includes books and primary sources. It provides full-text searches of almost 2,000 journals, more than 8,000 institutions in more than 160 countries have access to JSTOR, most access is by subscription, but some older public domain content is freely available to anyone. William G. Bowen, president of Princeton University from 1972 to 1988, JSTOR originally was conceived as a solution to one of the problems faced by libraries, especially research and university libraries, due to the increasing number of academic journals in existence. Most libraries found it prohibitively expensive in terms of cost and space to maintain a collection of journals. By digitizing many journal titles, JSTOR allowed libraries to outsource the storage of journals with the confidence that they would remain available long-term, online access and full-text search ability improved access dramatically. Bowen initially considered using CD-ROMs for distribution, JSTOR was initiated in 1995 at seven different library sites, and originally encompassed ten economics and history journals. JSTOR access improved based on feedback from its sites. Special software was put in place to make pictures and graphs clear, with the success of this limited project, Bowen and Kevin Guthrie, then-president of JSTOR, wanted to expand the number of participating journals. They met with representatives of the Royal Society of London and an agreement was made to digitize the Philosophical Transactions of the Royal Society dating from its beginning in 1665, the work of adding these volumes to JSTOR was completed by December 2000. The Andrew W. Mellon Foundation funded JSTOR initially, until January 2009 JSTOR operated as an independent, self-sustaining nonprofit organization with offices in New York City and in Ann Arbor, Michigan. JSTOR content is provided by more than 900 publishers, the database contains more than 1,900 journal titles, in more than 50 disciplines. Each object is identified by an integer value, starting at 1. In addition to the site, the JSTOR labs group operates an open service that allows access to the contents of the archives for the purposes of corpus analysis at its Data for Research service. This site offers a facility with graphical indication of the article coverage. Users may create focused sets of articles and then request a dataset containing word and n-gram frequencies and they are notified when the dataset is ready and may download it in either XML or CSV formats. The service does not offer full-text, although academics may request that from JSTOR, JSTOR Plant Science is available in addition to the main site. The materials on JSTOR Plant Science are contributed through the Global Plants Initiative and are only to JSTOR