1.
Indian subcontinent
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Geologically, the Indian subcontinent is related to the land mass that rifted from Gondwana and merged with the Eurasian plate nearly 55 million years ago. Geographically, it is the region in south-central Asia delineated by the Himalayas in the north, the Hindu Kush in the west. Politically, the Indian subcontinent usually includes Bangladesh, Bhutan, India, Maldives, Nepal, Pakistan, sometimes, the term South Asia is used interchangeably with Indian subcontinent. There is no consensus about which countries should be included in each and it is first attested in 1845 to refer to the North and South Americas, before they were regarded as separate continents. Its use to refer to the Indian subcontinent is seen from the twentieth century. It was especially convenient for referring to the region comprising both the British India and the states under British Paramountcy. The term Indian subcontinent also has a geological significance and it was, like the various continents, a part of the supercontinent of Gondwana. A series of tectonic splits caused formation of basins, each drifting in various directions. The geological region called the Greater India once included the Madagascar, Seychelles, Antartica, as a geological term, Indian subcontinent has meant that region formed from the collision of the Indian basin with Eurasia nearly 55 million years ago, towards the end of Paleocene. The Indian subcontinent has been a particularly common in the British Empire. The region, state Mittal and Thursby, has also labelled as India, Greater India. The BBC and some sources refer to the region as the Asian Subcontinent. Some academics refer to it as South Asian Subcontinent, the terms Indian subcontinent and South Asia are sometimes used interchangeably. There is no accepted definition on which countries are a part of South Asia or Indian subcontinent. In dictionary entries, the term subcontinent signifies a large, distinguishable subdivision of a continent, the region experienced high volcanic activity and plate subdivisions, creating Madagascar, Seychelles, Antartica, Austrolasia and the Indian subcontinent basin. The Indian subcontinent drifted northeastwards, colliding with the Eurasian plate nearly 55 million years ago and this geological region largely includes Bangladesh, Bhutan, India, Maldives, Nepal, Pakistan and Sri Lanka. The zone where the Eurasian and Indian subcontinent plates meet remains one of the active areas. The English term mainly continues to refer to the Indian subcontinent, physiographically, it is a peninsular region in south-central Asia delineated by the Himalayas in the north, the Hindu Kush in the west, and the Arakanese in the east
2.
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
3.
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
4.
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
5.
0 (number)
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0 is both a number and the numerical digit used to represent that number in numerals. The number 0 fulfills a role in mathematics as the additive identity of the integers, real numbers. As a digit,0 is used as a placeholder in place value systems, names for the number 0 in English include zero, nought or naught, nil, or—in contexts where at least one adjacent digit distinguishes it from the letter O—oh or o. Informal or slang terms for zero include zilch and zip, ought and aught, as well as cipher, have also been used historically. The word zero came into the English language via French zéro from Italian zero, in pre-Islamic time the word ṣifr had the meaning empty. Sifr evolved to mean zero when it was used to translate śūnya from India, the first known English use of zero was in 1598. The Italian mathematician Fibonacci, who grew up in North Africa and is credited with introducing the system to Europe. This became zefiro in Italian, and was contracted to zero in Venetian. The Italian word zefiro was already in existence and may have influenced the spelling when transcribing Arabic ṣifr, modern usage There are different words used for the number or concept of zero depending on the context. For the simple notion of lacking, the words nothing and none are often used, sometimes the words nought, naught and aught are used. Several sports have specific words for zero, such as nil in football, love in tennis and it is often called oh in the context of telephone numbers. Slang words for zero include zip, zilch, nada, duck egg and goose egg are also slang for zero. Ancient Egyptian numerals were base 10 and they used hieroglyphs for the digits and were not positional. By 1740 BC, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, was used to indicate the base level in drawings of tombs and pyramids. By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated sexagesimal positional numeral system, the lack of a positional value was indicated by a space between sexagesimal numerals. By 300 BC, a 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
6.
Negative numbers
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In mathematics, a negative number is a real number that is less than zero. If positive represents movement to the right, negative represents movement to the left, if positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal and they are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, if a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage, negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common idea of an opposite is reflected in arithmetic. For example, − −3 =3 because the opposite of an opposite is the original thing, negative numbers are usually written with a minus sign in front. For example, −3 represents a quantity with a magnitude of three, and is pronounced minus three or negative three. To help tell the difference between a subtraction operation and a number, occasionally the negative sign is placed slightly higher than the minus sign. Conversely, a number that is greater than zero is called positive, the positivity of a number may be emphasized by placing a plus sign before it, e. g. +3. In general, the negativity or positivity of a number is referred to as its sign, every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative numbers are referred to as integers. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, Liu Hui established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers, islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of numbers by the 17th century. Prior to the concept of numbers, mathematicians such as Diophantus considered negative solutions to problems false. Negative numbers can be thought of as resulting from the subtraction of a number from a smaller. For example, negative three is the result of subtracting three from zero,0 −3 = −3, in general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers
7.
Arithmetic
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Arithmetic is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations between them—addition, subtraction, multiplication and division. Arithmetic is an part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory and are still used to refer to a wider part of number theory. The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations as early as 2000 BC and these artifacts do not always reveal the specific process used for solving problems, but the characteristics of the particular numeral system strongly influence the complexity of the methods. The hieroglyphic system for Egyptian numerals, like the later Roman numerals, in both cases, this origin resulted in values that used a decimal base but did not include positional notation. Complex calculations with Roman numerals required the assistance of a board or the Roman abacus to obtain the results. Early number systems that included positional notation were not decimal, including the system for Babylonian numerals. Because of this concept, the ability to reuse the same digits for different values contributed to simpler. The continuous historical development of modern arithmetic starts with the Hellenistic civilization of ancient Greece, prior to the works of Euclid around 300 BC, Greek studies in mathematics overlapped with philosophical and mystical beliefs. For example, Nicomachus summarized the viewpoint of the earlier Pythagorean approach to numbers, Greek numerals were used by Archimedes, Diophantus and others in a positional notation not very different from ours. Because the ancient Greeks lacked a symbol for zero, they used three separate sets of symbols, one set for the units place, one for the tens place, and one for the hundreds. Then for the place they would reuse the symbols for the units place. Their addition algorithm was identical to ours, and their multiplication algorithm was very slightly different. Their long division algorithm was the same, and the square root algorithm that was taught in school was known to Archimedes. He preferred it to Heros method of successive approximation because, once computed, a digit doesnt change, and the square roots of perfect squares, such as 7485696, terminate immediately as 2736. For numbers with a part, such as 546.934. The ancient Chinese used a positional notation. Because they also lacked a symbol for zero, they had one set of symbols for the place
8.
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
9.
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
10.
Cosine
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In mathematics, the trigonometric functions are functions of an angle. They relate the angles of a triangle to the lengths of its sides, trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, more precise definitions are detailed below. Trigonometric functions have a range of uses including computing unknown lengths. In this use, trigonometric functions are used, for instance, in navigation, engineering, a common use in elementary physics is resolving a vector into Cartesian coordinates. In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another and that is, for any similar triangle the ratio of the hypotenuse and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides and it is these ratios that the trigonometric functions express. To define the functions for the angle A, start with any right triangle that contains the angle A. The three sides of the triangle are named as follows, The hypotenuse is the side opposite the right angle, the hypotenuse is always the longest side of a right-angled triangle. The opposite side is the side opposite to the angle we are interested in, in this side a. The adjacent side is the side having both the angles of interest, in this case side b, in ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180°. Therefore, in a triangle, the two non-right angles total 90°, so each of these angles must be in the range of as expressed in interval notation. The following definitions apply to angles in this 0° – 90° range and they can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions. For example, the figure shows sin for angles θ, π − θ, π + θ, and 2π − θ depicted on the unit circle and as a graph. The value of the sine repeats itself apart from sign in all four quadrants, and if the range of θ is extended to additional rotations, the trigonometric functions are summarized in the following table and described in more detail below. The angle θ is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case sin A = opposite hypotenuse = a h and this ratio does not depend on the size of the particular right triangle chosen, as long as it contains the angle A, since all such triangles are similar. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, in our case cos A = adjacent hypotenuse = b h
11.
Sanskrit
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Sanskrit is the primary liturgical language of Hinduism, a philosophical language of Hinduism, Buddhism, and Jainism, and a literary language and lingua franca of ancient and medieval South Asia. As a result of transmission of Hindu and Buddhist culture to Southeast Asia and parts of Central Asia, as one of the oldest Indo-European languages for which substantial written documentation exists, Sanskrit holds a prominent position in Indo-European studies. The body of Sanskrit literature encompasses a rich tradition of poetry and drama as well as scientific, technical, philosophical, the compositions of Sanskrit were orally transmitted for much of its early history by methods of memorization of exceptional complexity, rigor, and fidelity. Thereafter, variants and derivatives of the Brahmi script came to be used, Sanskrit is today one of the 22 languages listed in the Eighth Schedule of the Constitution of India, which mandates the Indian government to develop the language. It continues to be used as a ceremonial language in Hindu religious rituals and Buddhist practice in the form of hymns. The Sanskrit verbal adjective sáṃskṛta- may be translated as refined, elaborated, as a term for refined or elaborated speech, the adjective appears only in Epic and Classical Sanskrit in the Manusmṛti and the Mahabharata. The pre-Classical form of Sanskrit is known as Vedic Sanskrit, with the language of the Rigveda being the oldest and most archaic stage preserved, Classical Sanskrit is the standard register as laid out in the grammar of Pāṇini, around the fourth century BCE. Sanskrit, as defined by Pāṇini, evolved out of the earlier Vedic form, the present form of Vedic Sanskrit can be traced back to as early as the second millennium BCE. Scholars often distinguish Vedic Sanskrit and Classical or Pāṇinian Sanskrit as separate dialects, although they are quite similar, they differ in a number of essential points of phonology, vocabulary, grammar and syntax. Vedic Sanskrit is the language of the Vedas, a collection of hymns, incantations and theological and religio-philosophical discussions in the Brahmanas. Modern linguists consider the metrical hymns of the Rigveda Samhita to be the earliest, for nearly 2000 years, Sanskrit was the language of a cultural order that exerted influence across South Asia, Inner Asia, Southeast Asia, and to a certain extent East Asia. A significant form of post-Vedic Sanskrit is found in the Sanskrit of Indian epic poetry—the Ramayana, the deviations from Pāṇini in the epics are generally considered to be on account of interference from Prakrits, or innovations, and not because they are pre-Paninian. Traditional Sanskrit scholars call such deviations ārṣa, meaning of the ṛṣis, in some contexts, there are also more prakritisms than in Classical Sanskrit proper. There were four principal dialects of classical Sanskrit, paścimottarī, madhyadeśī, pūrvi, the predecessors of the first three dialects are attested in Vedic Brāhmaṇas, of which the first one was regarded as the purest. In the 2001 Census of India,14,035 Indians reported Sanskrit to be their first language, in India, Sanskrit is among the 14 original languages of the Eighth Schedule to the Constitution. The state of Uttarakhand in India has ruled Sanskrit as its official language. In October 2012 social activist Hemant Goswami filed a petition in the Punjab. More than 3,000 Sanskrit works have been composed since Indias independence in 1947, much of this work has been judged of high quality, in comparison to both classical Sanskrit literature and modern literature in other Indian languages
12.
Bakhshali Manuscript
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The Bakhshali Manuscript is a mathematical manuscript written on birch bark which was found near the village of Bakhshali in 1881. It is notable for being the oldest extant manuscript in Indian mathematics, the manuscript was discovered in 1881 by a peasant in the village of Bakhshali, which is near Peshawar, now in Pakistan. The first research on the manuscript was done by A. F. R. Hoernlé, after the death of Hoernle, it was examined by G. R. Kaye, who has edited the work and published it as a book in 1927. The extant manuscript is incomplete, consisting of seventy leaves of birch bark, the intended order of the 70 leaves is indeterminate. It is currently housed in the Bodleian Library at the University of Oxford and is said to be too fragile to be examined by scholars, the manuscript is a compendium of rules and illustrative example. Each example is stated as a problem, the solution is described, the sample problems are in verse and the commentary is in prose associated with calculations. The problems involve arithmetic, algebra and geometry, including mensuration, the manuscript is written in an earlier form of Śāradā script, which was mainly in use from the 8th to the 12th century, in the northwestern part of India, such as Kashmir and neighbouring regions. The language is the Gatha dialect, a colophon to one of the sections states that it was written by a brahmin identified as the son of Chajaka, a king of calculators, for the use of Vasiṣṭhas son Hasika. The brahmin might have been the author of the commentary as well as the scribe of the manuscript, the manuscript is a compilation of mathematical rules and examples, and prose commentaries on these verses. This is a similar to that of Bhāskara Is commentary on the gaṇita chapter of the Āryabhaṭīya. Its date is uncertain, and has generated considerable debate, most scholars agree that the physical manuscript is a copy of a more ancient text, so that the dating of that ancient text is possible only based on the content. Hoernle thought that the manuscript was from the 9th century, Kaye, on the other hand, thought the work was composed in the 12th century. Kayes assessment is discounted in the current scholarship, Indian scholars assign it an earlier date. Datta assigned it to the centuries of the Christian era. Channabasappa dates it to 200-400 CE, on the grounds that it uses mathematical terminology different from that of Aryabhata, hayashi has stated that it was no later than the 7th century. The dot symbol used as a zero the Bakhshali manuscript came to be called the shunya-bindu, references to the concept are found in Subandhus Vasavadatta, which has been dated between 385 and 465 CE by the scholar Maan Singh. Ratna Kumari Svadhyaya Sansthan M N Channabasappa, on the square root formula in the Bakhshali manuscript. 11, 112–124 David H. Bailey, Jonathan Borwein, a Quartically Convergent Square Root Algorithm, An Exercise in Forensic Paleo-Mathematics The Bakhshali manuscript 6 – The Bakhshali manuscript Hoernle, On the Bakhshali Manuscript,1887, archive. org
13.
Peshawar
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Peshawar is the capital of the Pakistani province of Khyber Pakhtunkhwa. It also serves as the centre and economic hub for the Federally Administered Tribal Areas. Situated in a valley near the eastern end of the historic Khyber Pass, close to the border with Afghanistan. Making it the oldest city in Pakistan and one of the oldest in South Asia, Peshawar is the largest city of Khyber Pakhtunkhwa province. According to the last census, it is also the ninth-largest city of Pakistan, the earliest settlement established in the area of Peshawar was called Puruṣapura, from which the current name Peshawar is derived. The Arab historian and geographer Al-Masudi noted that by the mid 10th century, after the Ghaznavid invasion, the citys name was again noted to be Parashāwar by Al-Biruni. The city became to be known as as Peshāwar by the era of Emperor Akbar, a name which is traditionally said to have been given by Akbar himself. The new name is said to have been based upon the Persian for frontier town, or more literally, forward city, though transcription errors and linguistic shifts may also account for the citys new name. Akbars bibliographer, Abul-Fazl ibn Mubarak, lists the name by both its former name Parashāwar, transcribed in Persian as پَرَشاوَر, and Peshāwar. Peshawar was founded as the ancient city of Puruṣapura, on the Gandhara Plains in the broad Valley of Peshawar, the city likely first existed as a small village in the 5th century BCE, within the cultural sphere of eastern ancient Persia. Puruṣapura was founded near the ancient Gandharan capital city of Pushkalavati, in the winter of 327-26 BCE, Alexander the Great subdued the Valley of Peshawar during his invasion of ancient India, as well as the nearby Swat and Buner valleys. Following Alexanders conquest, the Valley of Peshawar came under suzerainty of Seleucus I Nicator, a locally-made vase fragment that was found in Peshawar depicts a scene from Sophocles play Antigone. Following the Seleucid–Mauryan war, the region was ceded to the Mauryan Empire in 303 BCE, as Mauryan power declined, the Greco-Bactrian Kingdom based in modern Afghanistan declared its independence from the Seleucid Empire, and quickly seized Puruṣapura around 190 BCE. The city was ruled by several Iranic Parthian kingdoms. Puruṣapura was then captured by Gondophares, founder of the Indo-Parthian Kingdom, Gondophares established the nearby Takht-i-Bahi monastery in 46 CE. In the first century of the Common era, came under control of Kujula Kadphises, the city was made the empires winter capital. The Kushans summer capital at Kapisi was seen as the capital of the empire. Ancient Peshawars population was estimated to be 120,000, which would make it the seventh-most populous city in the world at the time, around 128 CE, Puruṣapura was made sole capital of the Kushan Empire under the rule of Kanishka
14.
Series (mathematics)
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In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a sequence has defined first and last terms. To emphasize that there are a number of terms, a series is often called an infinite series. In order to make the notion of an infinite sum mathematically rigorous, given an infinite sequence, the associated series is the expression obtained by adding all those terms together, a 1 + a 2 + a 3 + ⋯. These can be written compactly as ∑ i =1 ∞ a i, by using the summation symbol ∑. The sequence can be composed of any kind of object for which addition is defined. A series is evaluated by examining the finite sums of the first n terms of a sequence, called the nth partial sum of the sequence, and taking the limit as n approaches infinity. If this limit does not exist, the infinite sum cannot be assigned a value, and, in this case, the series is said to be divergent. On the other hand, if the partial sums tend to a limit when the number of terms increases indefinitely, then the series is said to be convergent, and the limit is called the sum of the series. An example is the series from Zenos dichotomy and its mathematical representation, ∑ n =1 ∞12 n =12 +14 +18 + ⋯. The study of series is a part of mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures, in addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For any sequence of numbers, real numbers, complex numbers, functions thereof. By definition the series ∑ n =0 ∞ a n converges to a limit L if and this definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k. When the index set is the natural numbers I = N, a series indexed on the natural numbers is an ordered formal sum and so we rewrite ∑ n ∈ N as ∑ n =0 ∞ in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers ∑ n =0 ∞ a n = a 0 + a 1 + a 2 + ⋯. When the semigroup G is also a space, then the series ∑ n =0 ∞ a n converges to an element L ∈ G if. This definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k, a series ∑an is said to converge or to be convergent when the sequence SN of partial sums has a finite limit
15.
Trigonometric function
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In mathematics, the trigonometric functions are functions of an angle. They relate the angles of a triangle to the lengths of its sides, trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, more precise definitions are detailed below. Trigonometric functions have a range of uses including computing unknown lengths. In this use, trigonometric functions are used, for instance, in navigation, engineering, a common use in elementary physics is resolving a vector into Cartesian coordinates. In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another and that is, for any similar triangle the ratio of the hypotenuse and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides and it is these ratios that the trigonometric functions express. To define the functions for the angle A, start with any right triangle that contains the angle A. The three sides of the triangle are named as follows, The hypotenuse is the side opposite the right angle, the hypotenuse is always the longest side of a right-angled triangle. The opposite side is the side opposite to the angle we are interested in, in this side a. The adjacent side is the side having both the angles of interest, in this case side b, in ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180°. Therefore, in a triangle, the two non-right angles total 90°, so each of these angles must be in the range of as expressed in interval notation. The following definitions apply to angles in this 0° – 90° range and they can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions. For example, the figure shows sin for angles θ, π − θ, π + θ, and 2π − θ depicted on the unit circle and as a graph. The value of the sine repeats itself apart from sign in all four quadrants, and if the range of θ is extended to additional rotations, the trigonometric functions are summarized in the following table and described in more detail below. The angle θ is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case sin A = opposite hypotenuse = a h and this ratio does not depend on the size of the particular right triangle chosen, as long as it contains the angle A, since all such triangles are similar. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, in our case cos A = adjacent hypotenuse = b h
16.
Arc tangent
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In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry. There are several notations used for the trigonometric functions. The most common convention is to name inverse trigonometric functions using a prefix, e. g. arcsin, arccos, arctan. This convention is used throughout the article, when measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ, where r is the radius of the circle. Similarly, in programming languages the inverse trigonometric functions are usually called asin, acos. The notations sin−1, cos−1, tan−1, etc, the confusion is somewhat ameliorated by the fact that each of the reciprocal trigonometric functions has its own name—for example, −1 = sec. Nevertheless, certain authors advise against using it for its ambiguity, since none of the six trigonometric functions are one-to-one, they are restricted in order to have inverse functions. There are multiple numbers y such that sin = x, for example, sin =0, when only one value is desired, the function may be restricted to its principal branch. With this restriction, for x in the domain the expression arcsin will evaluate only to a single value. These properties apply to all the trigonometric functions. The principal inverses are listed in the following table, if x is allowed to be a complex number, then the range of y applies only to its real part. Trigonometric functions of trigonometric functions are tabulated below. This is derived from the tangent addition formula tan = tan + tan 1 − tan tan , like the sine and cosine functions, the inverse trigonometric functions can be calculated using power series, as follows. For arcsine, the series can be derived by expanding its derivative,11 − z 2, as a binomial series, the series for arctangent can similarly be derived by expanding its derivative 11 + z 2 in a geometric series and applying the integral definition above. Arcsin = z + z 33 + z 55 + z 77 + ⋯ = ∑ n =0 ∞, for example, arccos x = π /2 − arcsin x, arccsc x = arcsin , and so on. Alternatively, this can be expressed, arctan z = ∑ n =0 ∞22 n 2. There are two cuts, from −i to the point at infinity, going down the imaginary axis and it works best for real numbers running from −1 to 1
17.
Calculus
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Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two branches, differential calculus, and integral calculus, these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the notions of convergence of infinite sequences. Generally, modern calculus is considered to have developed in the 17th century by Isaac Newton. Today, calculus has widespread uses in science, engineering and economics, Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, Calculus has historically been called the calculus of infinitesimals, or infinitesimal calculus. Calculus is also used for naming some methods of calculation or theories of computation, such as calculus, calculus of variations, lambda calculus. The ancient period introduced some of the ideas that led to integral calculus, the method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, indian mathematicians gave a non-rigorous method of a sort of differentiation of some trigonometric functions. 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, Cavalieris work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieris infinitesimals with the calculus of finite differences developed in Europe at around the same time, pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, in other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were considered disreputable. These ideas were arranged into a calculus of infinitesimals by Gottfried Wilhelm Leibniz. He is now regarded as an independent inventor of and contributor to calculus, unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. Leibniz and Newton are usually credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the used in calculus today
18.
Derivative
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The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a tool of calculus. For example, the derivative of the position of an object with respect to time is the objects velocity. The derivative of a function of a variable at a chosen input value. The tangent line is the best linear approximation of the function near that input value, for this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives may be generalized to functions of real variables. In this generalization, the derivative is reinterpreted as a transformation whose graph is the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables and it can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of variables, the Jacobian matrix reduces to the gradient vector. The process of finding a derivative is called differentiation, the reverse process is called antidifferentiation. The fundamental theorem of calculus states that antidifferentiation is the same as integration, differentiation and integration constitute the two fundamental operations in single-variable calculus. Differentiation is the action of computing a derivative, the derivative of a function y = f of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. It is called the derivative of f with respect to x, If x and y are real numbers, and if the graph of f is plotted against x, the derivative is the slope of this graph at each point. The simplest case, apart from the case of a constant function, is when y is a linear function of x. This formula is true because y + Δ y = f = m + b = m x + m Δ x + b = y + m Δ x. Thus, since y + Δ y = y + m Δ x and this gives an exact value for the slope of a line. If the function f is not linear, however, then the change in y divided by the change in x varies, differentiation is a method to find an exact value for this rate of change at any given value of x. The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the value of the ratio of the differences Δy / Δx as Δx becomes infinitely small
19.
Integral
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In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two operations of calculus, with its inverse, differentiation, being the other. The area above the x-axis adds to the total and that below the x-axis subtracts from the total, roughly speaking, the operation of integration is the reverse of differentiation. For this reason, the integral may also refer to the related notion of the antiderivative. In this case, it is called an integral and is written. The integrals discussed in this article are those termed definite integrals, a rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. A line integral is defined for functions of two or three variables, and the interval of integration is replaced by a curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space and this method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle. A similar method was developed in China around the 3rd century AD by Liu Hui. This method was used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi. The next significant advances in integral calculus did not begin to appear until the 17th century, further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the theorem of calculus. Wallis generalized Cavalieris method, computing integrals of x to a power, including negative powers. The major advance in integration came in the 17th century with the independent discovery of the theorem of calculus by Newton. The theorem demonstrates a connection between integration and differentiation and this connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the mathematical framework that both Newton and Leibniz developed
20.
Kerala
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Kerala historically known as Keralam, is an Indian state in South India on the Malabar Coast. It was formed on 1 November 1956 following the States Reorganisation Act by combining Malayalam-speaking regions, spread over 38,863 km2, it is bordered by Karnataka to the north and northeast, Tamil Nadu to the east and south, and the Lakshadweep Sea to the west. With 33,387,677 inhabitants as per the 2011 Census, Malayalam is the most widely spoken language and is also the official language of the state. The region has been a prominent spice exporter since 3000 BCE, the Chera Dynasty was the first prominent kingdom based in Kerala, though it frequently struggled against attacks by the neighbouring Cholas and Pandyas. In the 15th century, the spice trade attracted Portuguese traders to Kerala, after independence, Travancore and Cochin joined the Republic of India and Travancore-Cochin was given the status of a state in 1949. In 1956, Kerala state was formed by merging Malabar district, Travancore-Cochin, Hinduism is practised by more than half of the population, followed by Islam and Christianity. The culture is a synthesis of Aryan and Dravidian cultures, developed over millennia, under influences from other parts of India, the production of pepper and natural rubber contributes significantly to the total national output. In the agricultural sector, coconut, tea, coffee, cashew, the states coastline extends for 595 kilometres, and around 1.1 million people in the state are dependent on the fishery industry which contributes 3% to the states income. The state has the highest media exposure in India with newspapers publishing in nine languages, mainly English, Kerala is one of the prominent tourist destinations of India, with backwaters, beaches, Ayurvedic tourism and tropical greenery as its major attractions. The name Kerala has an uncertain etymology, One popular theory derives Kerala from Kera and alam is land, thus land of coconuts, this also happens to be a nickname for the state due to abundance of coconut trees and its use by the locals. The word Kerala is first recorded in a 3rd-century BCE rock inscription left by the Maurya emperor Ashoka, the inscription refers to the local ruler as Keralaputra, or son of Chera. This contradicts the theory that Kera is from coconut tree, at that time, one of three states in the region was called Cheralam in Classical Tamil, Chera and Kera are variants of the same word. The word Cheral refers to the oldest known dynasty of Kerala kings and is derived from the Proto-Tamil-Malayalam word for lake, the earliest Sanskrit text to mention Kerala is the Aitareya Aranyaka of the Rigveda. It is also mentioned in the Ramayana and the Mahabharata, the two Hindu epics, the Skanda Purana mentions the ecclesiastical office of the Thachudaya Kaimal who is referred to as Manikkam Keralar, synonymous with the deity of the Koodalmanikyam temple. Keralam may stem from the Classical Tamil cherive-alam or chera alam, the Greco-Roman trade map Periplus Maris Erythraei refers to Keralaputra as Celobotra. According to Hindu mythology, the lands of Kerala were recovered from the sea by the warrior sage Parasurama. Parasurama threw his axe across the sea, and the water receded as far as it reached, according to legend, this new area of land extended from Gokarna to Kanyakumari. The land which rose from sea was filled with salt and unsuitable for habitation, so Parasurama invoked the Snake King Vasuki, out of respect, Vasuki and all snakes were appointed as protectors and guardians of the land
21.
Harappa
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Harappa is an archaeological site in Punjab, Pakistan, about 24 km west of Sahiwal. The site takes its name from a village located near the former course of the Ravi River. The current village of Harappa is 6 km from the ancient site, although modern Harappa has a legacy railway station from the period of the British Raj, it is today just a small crossroads town of population 15,000. The site of the ancient city contains the ruins of a Bronze Age fortified city, which was part of the Cemetery H culture and the Indus Valley Civilization, centered in Sindh and the Punjab. Per archaeological convention of naming a previously unknown civilization by its first excavated site, the ancient city of Harappa was heavily damaged under British rule, when bricks from the ruins were used as track ballast in the construction of the Lahore-Multan Railway. In 2005, an amusement park scheme at the site was abandoned when builders unearthed many archaeological artifacts during the early stages of building work. A plea from the Pakistani archaeologist Ahmad Hasan Dani to the Ministry of Culture resulted in a restoration of the site, the Indus Valley Civilization has its earliest roots in cultures such as that of Mehrgarh, approximately 6000 BCE. The two greatest cities, Mohenjo-daro and Harappa, emerged circa 2600 BCE along the Indus River valley in Punjab, the bricks discovered were made of red sand, clay, stones and were baked at very high temperature. As early as 1826 Harappa located in west Punjab attracted the attention of a British officer in India, Indus Valley civilization was mainly an urban culture sustained by surplus agricultural production and commerce, the latter including trade with Sumer in southern Mesopotamia. Both Mohenjo-Daro and Harappa are generally characterized as having differentiated living quarters, flat-roofed brick houses, the weights and measures of the Indus Valley Civilization, on the other hand, were highly standardized, and conform to a set scale of gradations. Distinctive seals were used, among other applications, perhaps for identification of property, although copper and bronze were in use, iron was not yet employed. Wheel-made pottery—some of it adorned with animal and geometric motifs—has been found in profusion at all the major Indus sites, harappans had many trade routes along the Indus River that went as far as the Persian Gulf, Mesopotamia, and Egypt. Some of the most valuable things traded were carnelian and lapis lazuli, what is clear is that Harappan society was not entirely peaceful, with the human skeletal remains demonstrating some of the highest rates of injury found in South Asian prehistory. The excavators of the site have proposed the following chronology of Harappas occupation, Ravi Aspect of the Hakra phase, kot Dijian phase, c.2800 –2600 BC. Harappan Phase, c.2600 –1900 BC, transitional Phase, c.1900 –1800 BC. Late Harappan Phase, c.1800 –1300 BC, by far the most exquisite and obscure artifacts unearthed to date are the small, square steatite seals engraved with human or animal motifs. A large number of seals have been found at sites as Mohenjo-Daro. Many bear pictographic inscriptions generally thought to be a form of writing or script, despite the efforts of philologists from all parts of the world, and despite the use of modern cryptographic analysis, the signs remain undeciphered
22.
Mohenjo-daro
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Mohenjo-daro is an archeological site in the province of Sindh, Pakistan. Mohenjo-daro was abandoned in the 19th century BCE as the Indus Valley Civilization declined, significant excavation has since been conducted at the site of the city, which was designated an UNESCO World Heritage Site in 1980. The site is threatened by erosion and improper restoration. Mohenjo-daro, the name for the site, has been variously interpreted as Mound of the Dead Men in Sindhi. The citys original name is unknown, based on his analysis of a Mohenjo-daro seal, Iravatham Mahadevan speculates that the citys ancient name could have been Kukkutarma. Cock-fighting may have had ritual and religious significance for the city, with domesticated chickens bred there for sacred purposes, Mohenjo-daro may also have been a point of diffusion for the eventual worldwide domestication of chickens. Mohenjo-daro is located west of the Indus River in Larkana District, Sindh, Pakistan and it is sited on a Pleistocene ridge in the middle of the flood plain of the Indus River Valley, around 28 kilometres from the town of Larkana. The Indus still flows east of the site, but the Ghaggar-Hakra riverbed on the side is now dry. Mohenjo-daro was built in the 26th century BCE and it was one of the largest cities of the ancient Indus Valley Civilization, also known as the Harappan Civilization, which developed around 3,000 BCE from the prehistoric Indus culture. Mohenjo-daro was the most advanced city of its time, with remarkably sophisticated civil engineering, when the Indus civilization went into sudden decline around 1900 BCE, Mohenjo-daro was abandoned. The ruins of the city remained undocumented for around 3,700 years until R. D and this led to large-scale excavations of Mohenjo-daro led by Kashinath Narayan Dikshit in 1924–25, and John Marshall in 1925–26. In the 1930s, major excavations were conducted at the site under the leadership of Marshall, D. K. Dikshitar, further excavations were carried out in 1945 by Ahmad Hasan Dani and Mortimer Wheeler. The last major series of excavations were conducted in 1964 and 1965 by Dr. George F. Dales, a dry core drilling conducted in 2015 by Pakistans National Fund for Mohenjo-daro revealed that the site is larger than the unearthed area. Mohenjo-daro has a layout based on a street grid of rectilinear buildings. Most were built of fired and mortared brick, some incorporated sun-dried mud-brick, the covered area of Mohenjo-daro is estimated at 300 hectares. The Oxford Handbook of Cities in World History offers an estimate of a peak population of around 40,000. The sheer size of the city, and its provision of buildings and facilities. The city is divided into two parts, the so-called Citadel and the Lower City
23.
Indus Valley Civilisation
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The Indus Valley Civilisation was a Bronze Age civilisation mainly in the northwestern regions of South Asia, extending from what today is northeast Afghanistan to Pakistan and northwest India. Along with ancient Egypt and Mesopotamia it was one of three early civilisations of the Old World, and of the three, the most widespread, at its peak, the Indus Civilisation may have had a population of over five million. Inhabitants of the ancient Indus river valley developed new techniques in handicraft, the Indus cities are noted for their urban planning, baked brick houses, elaborate drainage systems, water supply systems, and clusters of large non-residential buildings. The discovery of Harappa, and soon afterwards, Mohenjo-Daro, was the culmination of work beginning in 1861 with the founding of the Archaeological Survey of India in the British Raj, excavation of Harappan sites has been ongoing since 1920, with important breakthroughs occurring as recently as 1999. This Harappan civilisation is called the Mature Harappan culture to distinguish it from the cultures immediately preceding and following it. The early Harappan cultures were preceded by local Neolithic agricultural villages, as of 1999, over 1,056 cities and settlements had been found, of which 96 have been excavated, mainly in the general region of the Indus and Ghaggar-Hakra Rivers and their tributaries. Among the settlements were the urban centres of Harappa, Mohenjo-daro, Dholavira, Ganeriwala in Cholistan. The Harappan language is not directly attested and its affiliation is uncertain since the Indus script is still undeciphered, a relationship with the Dravidian or Elamo-Dravidian language family is favoured by a section of scholars. Recently, Indus sites have been discovered in Pakistans northwestern Frontier Province as well, other IVC colonies can be found in Afghanistan while smaller isolated colonies can be found as far away as Turkmenistan and in Maharashtra. The largest number of colonies are in the Punjab, Sindh, Rajasthan, Haryana, Indus Valley sites have been found most often on rivers, but also on the ancient seacoast, for example, Balakot, and on islands, for example, Dholavira. There is evidence of dry river beds overlapping with the Hakra channel in Pakistan, many Indus Valley sites have been discovered along the Ghaggar-Hakra beds. Among them are, Rupar, Rakhigarhi, Sothi, Kalibangan, Harappan Civilisation remains the correct one, according to the common archaeological usage of naming a civilisation after its first findspot. John wrote, I was much exercised in my mind how we were to get ballast for the line of the railway and they were told of an ancient ruined city near the lines, called Brahminabad. Visiting the city, he found it full of hard well-burnt bricks, and, convinced there was a grand quarry for the ballast I wanted. These bricks now provided ballast along 93 miles of the track running from Karachi to Lahore. In 1872–75, Alexander Cunningham published the first Harappan seal and it was half a century later, in 1912, that more Harappan seals were discovered by J. J. H. MacKay, and Marshall. By 1931, much of Mohenjo-Daro had been excavated, but excavations continued, such as that led by Sir Mortimer Wheeler, director of the Archaeological Survey of India in 1944. Among other archaeologists who worked on IVC sites before the independence in 1947 were Ahmad Hasan Dani, Brij Basi Lal, Nani Gopal Majumdar, and Sir Marc Aurel Stein
24.
Geometrical
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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
25.
Barrel
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Balloon Array for RBSP Relativistic Electron Losses is a NASA mission operated out of Dartmouth College that works with the Van Allen Probes mission. The BARREL project has launched 20 balloons from Antarctica during each of two campaigns in January 2013 and January 2014. Unlike the football-field-sized balloons typically launched at the Poles, these are each just 90 feet in diameter, BARREL will help study the Van Allen Radiation Belts and why they wax and wane over time. Each BARREL balloon carries instruments to measure particles ejected from the Belts which make it all the way to Earths atmosphere and this will help distinguish between various theories of what causes electron loss in the Belts. The Principal Investigator is Robyn Millan at Dartmouth College, co-Investigator institutions are University of Washington, U. C. BARREL is part of NASAs Living With a Star program, support for the Antarctica balloon campaigns is provided by the National Science Foundation, British Antarctic Survey, and South African National Antarctic Program
26.
Cone (geometry)
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A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex. A cone is formed by a set of segments, half-lines, or lines connecting a common point. If the enclosed points are included in the base, the cone is a solid object, otherwise it is a two-dimensional object in three-dimensional space. In the case of an object, the boundary formed by these lines or partial lines is called the lateral surface, if the lateral surface is unbounded. In the case of line segments, the cone does not extend beyond the base, while in the case of half-lines, in the case of lines, the cone extends infinitely far in both directions from the apex, in which case it is sometimes called a double cone. Either half of a cone on one side of the apex is called a nappe. The axis of a cone is the line, passing through the apex. If the base is right circular the intersection of a plane with this surface is a conic section, in general, however, the base may be any shape and the apex may lie anywhere. Contrasted with right cones are oblique cones, in which the axis passes through the centre of the base non-perpendicularly, a cone with a polygonal base is called a pyramid. Depending on the context, cone may also mean specifically a convex cone or a projective cone, cones can also be generalized to higher dimensions. The perimeter of the base of a cone is called the directrix, the base radius of a circular cone is the radius of its base, often this is simply called the radius of the cone. The aperture of a circular cone is the maximum angle between two generatrix lines, if the generatrix makes an angle θ to the axis, the aperture is 2θ. A cone with a region including its apex cut off by a plane is called a cone, if the truncation plane is parallel to the cones base. An elliptical cone is a cone with an elliptical base, a generalized cone is the surface created by the set of lines passing through a vertex and every point on a boundary. The slant height of a circular cone is the distance from any point on the circle to the apex of the cone. It is given by r 2 + h 2, where r is the radius of the cirf the cone and this application is primarily useful in determining the slant height of a cone when given other information regarding the radius or height. The volume V of any conic solid is one third of the product of the area of the base A B and the height h V =13 A B h. In modern mathematics, this formula can easily be computed using calculus – it is, up to scaling, the integral ∫ x 2 d x =13 x 3
27.
Cylinder (geometry)
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In its simplest form, a cylinder is the surface formed by the points at a fixed distance from a given straight line called the axis of the cylinder. It is one of the most basic curvilinear geometric shapes, commonly the word cylinder is understood to refer to a finite section of a right circular cylinder having a finite height with circular ends perpendicular to the axis as shown in the figure. If the ends are open, it is called an open cylinder, if the ends are closed by flat surfaces it is called a solid cylinder. The formulae for the area and the volume of such a cylinder have been known since deep antiquity. The area of the side is known as the lateral area. An open cylinder does not include either top or bottom elements, the surface area of a closed cylinder is made up the sum of all three components, top, bottom and side. Its surface area is A = 2πr2 + 2πrh = 2πr = πd=L+2B, for a given volume, the closed cylinder with the smallest surface area has h = 2r. Equivalently, for a surface area, the closed cylinder with the largest volume has h = 2r. Cylindric sections are the intersections of cylinders with planes, for a right circular cylinder, there are four possibilities. A plane tangent to the cylinder meets the cylinder in a straight line segment. Moved while parallel to itself, the plane either does not intersect the cylinder or intersects it in two line segments. All other planes intersect the cylinder in an ellipse or, when they are perpendicular to the axis of the cylinder, a cylinder whose cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder respectively. Elliptic cylinders are also known as cylindroids, but that name is ambiguous, as it can also refer to the Plücker conoid. The volume of a cylinder with height h is V = ∫0 h A d x = ∫0 h π a b d x = π a b ∫0 h d x = π a b h. Even more general than the cylinder is the generalized cylinder. The cylinder is a degenerate quadric because at least one of the coordinates does not appear in the equation, an oblique cylinder has the top and bottom surfaces displaced from one another. There are other unusual types of cylinders. Let the height be h, internal radius r, and external radius R, the volume is given by V = π h
28.
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
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Lothal
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Lothal is one of the most prominent cities of the ancient Indus valley civilisation, located in the Bhāl region of the modern state of Gujarāt and dating from 3700 BCE. Discovered in 1954, Lothal was excavated from 13 February 1955 to 19 May 1960 by the Archaeological Survey of India, the official Indian government agency for the preservation of ancient monuments. It was a vital and thriving trade centre in ancient times, with its trade of beads, gems and valuable ornaments reaching the far corners of West Asia, the techniques and tools they pioneered for bead-making and in metallurgy have stood the test of time for over 4000 years. Lothal is situated near the village of Saragwala in the Dholka Taluka of Ahmedabad district and it is six kilometres south-east of the Lothal-Bhurkhi railway station on the Ahmedabad-Bhavnagar railway line. It is also connected by roads to the cities of Ahmedabad, Bhavnagar, Rajkot. The nearest cities are Dholka and Bagodara, the findings consist of a mound, a township, a marketplace, and the dock. Adjacent to the excavated areas stands the Archaeological Museum, where some of the most prominent collections of Indus-era antiquities in India are displayed, when British India was partitioned in 1947, most Indus sites, including Mohenjo-daro and Harappa, became part of Pakistan. The Archaeological Survey of India undertook a new program of exploration, many sites were discovered across northwestern India. Lothal stands 670 kilometers from Mohenjo-daro, which is in Sindh, the meaning of Lothal in Gujarati to be the mound of the dead is not unusual, as the name of the city of Mohenjo-daro in Sindhi means the same. People in villages neighbouring to Lothal had known of the presence of an ancient town, as recently as 1850, boats could sail up to the mound. In 1942, timber was shipped from Broach to Saragwala via the mound, a silted creek connecting modern Bholad with Lothal and Saragwala represents the ancient flow channel of a river or creek. Speculation suggests that owing to the small dimensions of the main city, Lothal was not a large settlement at all. However, the ASI and other contemporary archaeologists assert that the city was a part of a river system on the trade route of the ancient peoples from Sindh to Saurashtra in Gujarat. Lothal provides with the largest collection of antiquities in the archaeology of modern India and it is essentially a single culture site—the Harappan culture in all its variances is evidenced. An indigenous micaceous Red Ware culture also existed, which is believed to be autochthonous, two sub-periods of Harappan culture are distinguished, the same period is identical to the exuberant culture of Harappa and Mohenjo-daro. After the core of the Indus civilisation had decayed in Mohenjo-daro and Harappa, Lothal seems not only to have survived and its constant threats - tropical storms and floods - caused immense destruction, which destabilised the culture and ultimately caused its end. Topographical analysis also shows signs that at about the time of its demise, thus the cause for the abandonment of the city may have been changes in the climate as well as natural disasters, as suggested by environmental magnetic records. Lothal is based upon a mound that was a salt marsh inundated by tide, small channel widths when compared to the lower reaches suggest the presence of a strong tidal influence upon the city—tidal waters ingressed up to and beyond the city
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Dholavira
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This village is 165 km from Radhanpur. Also known locally as Kotada timba, the site contains ruins of an ancient Indus Valley Civilization/Harappan city and it is one of the five largest Harappan sites and most prominent archaeological sites in India belonging to the Indus Valley Civilization. It is also considered as having been the grandest of cities of its time and it is located on Khadir bet island in the Kutch Desert Wildlife Sanctuary in the Great Rann of Kutch. The 47 ha quadrangular city lay between two streams, the Mansar in the north and Manhar in the south. The site was occupied from c.2650 BCE, declining slowly after about 2100 BCE and it was briefly abandoned then reoccupied until c.1450 BCE. The site was discovered in 1967-1968 by J. P. Joshi ex, D. G. of A. S. I. and is the fifth largest of eight major Harappan sites. It has been under excavation since 1990 by the Archaeological Survey of India, the other major Harappan sites discovered so far are, Harappa, Mohenjo-daro, Ganeriwala, Rakhigarhi, Kalibangan, Rupnagar and Lothal. R. S. S. Bisht, and there were 13 field excavations between 1990 and 2005, archaeologists believe that Dholavira was an important centre of trade between settlements in south Gujarat, Sindh and Punjab and Western Asia. Estimated to be older than the port-city of Lothal, the city of Dholavira has a shape and organization. The area measures 771.1 m in length, and 616.85 m in width, unlike Harappa and Mohenjo-daro, the city was constructed to a pre-existing geometrical plan consisting of three divisions – the citadel, the middle town, and the lower town. The acropolis and the town had been furnished with their own defence-work, gateways, built-up areas, street system, wells. The acropolis is the most thoroughly fortified and complex area in the city, the towering castle stands is defended by double ramparts. Next to this stands a place called the bailey where important officials lived, the city within the general fortifications accounts for 48 ha. There are extensive structure-bearing areas which are outside yet integral to the fortified settlement, beyond the walls, another settlement has been found. Dholavira is flanked by two water channels, the Mansar in the north, and the Manhar in the south. S. Bist. One of the features of Dholavira is the sophisticated water conservation system of channels and reservoirs. The city had massive reservoirs, three of which are exposed and they were used for storing fresh water brought by rains or to store water diverted from two nearby rivulets. This clearly came in response to the climate and conditions of Kutch
31.
Vedas
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The Vedas are a large body of knowledge texts originating in the ancient Indian subcontinent. Composed in Vedic Sanskrit, the texts constitute the oldest layer of Sanskrit literature, Hindus consider the Vedas to be apauruṣeya, which means not of a man, superhuman and impersonal, authorless. Vedas are also called śruti literature, distinguishing them from religious texts. The Veda, for orthodox Indian theologians, are considered revelations seen by ancient sages after intense meditation, in the Hindu Epic the Mahabharata, the creation of Vedas is credited to Brahma. The Vedic hymns themselves assert that they were created by Rishis, after inspired creativity. There are four Vedas, the Rigveda, the Yajurveda, the Samaveda, each Veda has been subclassified into four major text types – the Samhitas, the Aranyakas, the Brahmanas, and the Upanishads. Some scholars add a fifth category – the Upasanas, the various Indian philosophies and denominations have taken differing positions on the Vedas. Schools of Indian philosophy which cite the Vedas as their authority are classified as orthodox. Other śramaṇa traditions, such as Lokayata, Carvaka, Ajivika, Buddhism and Jainism, despite their differences, just like the texts of the śramaṇa traditions, the layers of texts in the Vedas discuss similar ideas and concepts. The Sanskrit word véda knowledge, wisdom is derived from the root vid- to know and this is reconstructed as being derived from the Proto-Indo-European root *u̯eid-, meaning see or know. The noun is from Proto-Indo-European *u̯eidos, cognate to Greek εἶδος aspect, not to be confused is the homonymous 1st and 3rd person singular perfect tense véda, cognate to Greek οἶδα oida I know. Root cognates are Greek ἰδέα, English wit, etc, the Sanskrit term veda as a common noun means knowledge. The term in some contexts, such as hymn 10.93.11 of the Rigveda, means obtaining or finding wealth, property, a related word Vedena appears in hymn 8.19.5 of the Rigveda. It was translated by Ralph T. H. Griffith as ritual lore, as studying the Veda by the 14th century Indian scholar Sayana, as bundle of grass by Max Müller, Vedas are called Maṛai or Vaymoli in parts of South India. Marai literally means hidden, a secret, mystery, in some south Indian communities such as Iyengars, the word Veda includes the Tamil writings of the Alvar saints, such as Divya Prabandham, for example Tiruvaymoli. The Vedas are among the oldest sacred texts, the Samhitas date to roughly 1700–1100 BC, and the circum-Vedic texts, as well as the redaction of the Samhitas, date to c. 1000-500 BC, resulting in a Vedic period, spanning the mid 2nd to mid 1st millennium BC, or the Late Bronze Age, Michael Witzel gives a time span of c.1500 to c. Witzel makes special reference to the Near Eastern Mitanni material of the 14th century BC the only record of Indo-Aryan contemporary to the Rigvedic period
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Vedic Period
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The Vedic period was the period in Indian history during which the Vedas, the oldest scriptures of Hinduism, were composed. During the early part of the Vedic period, the Indo-Aryans settled into northern India, scholars consider Vedic civilisation to have been a composite of the Indo-Aryan and Harappan cultures. The end of the Vedic period witnessed the rise of large, around the beginning of the Common Era, the Vedic tradition formed one of the main constituents of the so-called Hindu synthesis. The commonly proposed period of earlier Vedic age is dated back to 2nd millennium BCE, after the collapse of the Indus Valley Civilisation, which ended c.1900 BCE, groups of Indo-Aryan peoples migrated into north-western India and started to inhabit the northern Indus Valley. The knowledge about the Aryans comes mostly from the Rigveda-samhita, which was composed between c and they brought with them their distinctive religious traditions and practices. The Vedic beliefs and practices of the era were closely related to the hypothesised Proto-Indo-European religion. According to Anthony, the Old Indic religion probably emerged among Indo-European immigrants in the zone between the Zeravshan River and Iran. It was a mixture of old Central Asian and new Indo-European elements. At least 383 non-Indo-European words were borrowed from this culture, including the god Indra, Indra was the subject of 250 hymns, a quarter of the Rig Veda. He was associated more than any other deity with Soma, a stimulant drug probably borrowed from the BMAC religion and his rise to prominence was a peculiar trait of the Old Indic speakers. These migrations may have been accompanied with violent clashes with the people who inhabited this region. The Rig Veda contains accounts of conflicts between the Aryas and the Dasas and Dasyus, the Rig Veda describes Dasas and Dasyus as people who do not perform sacrifices or obey the commandments of gods. Their speech is described as mridhra which could variously mean soft, uncouth, hostile, other adjectives which describe their physical appearance are subject to many interpretations. Internecine military conflicts between the tribes of Vedic Aryans are also described in the Rig Veda. Most notable of such conflicts was the Battle of Ten Kings, which took place on the banks of the river Parushni. The battle was fought between the tribe Bharatas, led by their chief Sudas, against a confederation of ten tribes— Puru, Yadu, Turvasha, Anu, Druhyu, Alina, Bhalanas, Paktha, Siva, Vishanin. Bharatas lived around the regions of the river Saraswati, while Purus, their western neighbours. The other tribes dwelt north-west of the Bharatas in the region of Punjab, division of the waters of Ravi could have been a reason for the war
33.
History of large numbers
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Different cultures used different traditional numeral systems for naming large numbers. The extent of large numbers used varied in each culture, the Indians had a passion for high numbers, which is intimately related to their religious thought. For example, in belonging to the Vedic literature, we find individual Sanskrit names for each of the powers of 10 up to a trillion. One of these Vedic texts, the Yajur Veda, even discusses the concept of infinity, stating that if you subtract purna from purna. The last number at which he arrived after going through nine successive counting systems was 10421, that is, there is also an analogous system of Sanskrit terms for fractional numbers, capable of dealing with both very large and very small numbers. Larger number in Buddhism works up to Bukeshuo bukeshuo zhuan 107 ×2122 or 1037218383881977644441306597687849648128, the Ancient Greeks used a system based on the myriad, that is ten thousand, and their largest named number was a myriad myriad, or one hundred million. In The Sand Reckoner, Archimedes devised a system of naming large numbers reaching up to 108 ×1016 and this largest number appears because it equals a myriad myriad to the myriad myriadth power, all taken to the myriad myriadth power. Archimedes only used his system up to 1064, much later, but still in antiquity, the Hellenistic mathematician Diophantus used a similar notation to represent large numbers. The Indians, who invented the positional system, along with negative numbers. By the 7th century, Indian mathematicians were familiar enough with the notion of infinity as to define it as the quantity whose denominator is zero, far larger finite numbers than any of these occur in modern mathematics. See for instance Grahams number which is too large to express using exponentiation or even tetration, for more about modern usage for large numbers see Large numbers. The ultimate in large numbers was, until recently, the concept of infinity, a number defined by being greater than any finite number, of these transfinite numbers, perhaps the most extraordinary, and arguably, if they exist, largest, are the large cardinals. The concept of numbers, however, was first considered by Indian Jaina mathematicians as far back as 400 BC
34.
Yajurveda
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The Yajurveda is the Veda of prose mantras. An ancient Vedic Sanskrit text, it is a compilation of ritual offering formulas that were said by a priest while an individual performed ritual actions such as those before the yajna fire, Yajurveda is one of the four Vedas, and one of the scriptures of Hinduism. The exact century of Yajurvedas composition is unknown, and estimated by scholars to be around 1200 to 1000 BCE, the Yajurveda is broadly grouped into two – the black Yajurveda and the white Yajurveda. The term black implies the un-arranged, unclear, motley collection of verses in Yajurveda, in contrast to the white which implies the well arranged, the black Yajurveda has survived in four recensions, while two recensions of white Yajurveda have survived into the modern times. The earliest and most ancient layer of Yajurveda samhita includes about 1,875 verses, the middle layer includes the Satapatha Brahmana, one of the largest Brahmana texts in the Vedic collection. The youngest layer of Yajurveda text includes the largest collection of primary Upanishads and these include the Brihadaranyaka Upanishad, the Isha Upanishad, the Taittiriya Upanishad, the Katha Upanishad, the Shvetashvatara Upanishad and the Maitri Upanishad. Yajurveda is a compound Sanskrit word, composed of yajus and veda, monier-Williams translates yajus as religious reverence, veneration, worship, sacrifice, a sacrificial prayer, formula, particularly mantras muttered in a peculiar manner at a sacrifice. Johnson states yajus means prose formulae or mantras, contained in the Yajur Veda, michael Witzel interprets Yajurveda to mean a knowledge text of prose mantras used in Vedic rituals. Ralph Griffith interprets the name to mean knowledge of sacrifice or sacrificial texts, carl Olson states that Yajurveda is a text of mantras that are repeated and used in rituals. The Yajurveda text includes Shukla Yajurveda of which about 16 recensions are known, only two recensions of the Shukla Yajurveda have survived, Madhyandina and Kanva, and others are known by name only because they are mentioned in other texts. These two recensions are nearly the same, except for few differences, in contrast to Shukla Yajurveda, the four surviving recensions of Krishna Yajurveda are very different versions. The samhita in the Shukla Yajurveda is called the Vajasaneyi Samhita, the name Vajasaneyi is derived from Vajasaneya, patronymic of sage Yajnavalkya, and the founder of the Vajasaneyi branch. There are two surviving recensions of the Vajasaneyi Samhita, Vajasaneyi Madhyandina and Vajasaneyi Kanva, there are four surviving recensions of the Krishna Yajurveda – Taittirīya saṃhitā, Maitrayani saṃhitā, Kaṭha saṃhitā and Kapiṣṭhala saṃhitā. A total of eighty six recensions are mentioned to exist in Vayu Purana, the Katha school is referred to as a sub-school of Carakas in some ancient texts of India, because they did their scholarship as they wandered from place to place. The best known and best preserved of these recensions is the Taittirīya saṃhitā, some attribute it to Tittiri, a pupil of Yaksa and mentioned by Panini. The text is associated with the Taittiriya school of the Yajurveda, the Kāṭhaka saṃhitā or the Caraka-Kaṭha saṃhitā, according to tradition was compiled by Katha, a disciple of Vaisampayana. Like the Maitrayani Samhita, it offers more detailed discussion of some rituals than the younger Taittiriya samhita that frequently summarizes such accounts. The Kapiṣṭhala saṃhitā or the Kapiṣṭhala-Kaṭha saṃhitā, named after the sage Kapisthala is extant only in some large fragments and this text is practically a variant of the Kāṭhaka saṃhitā
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Mantra
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A Mantra is a sacred utterance, a numinous sound, a syllable, word or phonemes, or group of words in Sanskrit believed by practitioners to have psychological and spiritual powers. A mantra may or may not have syntactic structure or literal meaning, the earliest mantras were composed in Vedic Sanskrit by Hindus in India, and are at least 3000 years old. Mantras now exist in various schools of Hinduism, Buddhism, Jainism and Sikhism, in Japanese Shingon tradition, the word Shingon means mantra. Similar hymns, chants, compositions and concepts are found in Zoroastrianism, Taoism, Christianity, the use, structure, function, importance, and types of mantras vary according to the school and philosophy of Hinduism and of Buddhism. Mantras serve a role in tantra. In this school, mantras are considered to be a sacred formula, in other schools of Hinduism, Buddhism, Jainism or Sikhism, initiation is not a requirement. Mantras come in forms, including ṛc and sāman. They are typically melodic, mathematically structured meters, believed to be resonant with numinous qualities, at its simplest, the word ॐ serves as a mantra. In more sophisticated forms, mantras are melodic phrases with spiritual interpretations such as a longing for truth, reality, light, immortality, peace, love, knowledge. Some mantras have no meaning, yet are musically uplifting. The Sanskrit word mantra- consists of the root man- to think, scholars consider mantras to be older than 1000 BC. By the middle Vedic period—1000 BC to 500 BC—claims Frits Staal, mantras in Hinduism had developed into a blend of art, the Chinese translation is zhenyan 眞言, 真言, literally true words, the Japanese onyomi reading of the Chinese being shingon. Mantras are neither unique to Hinduism, nor to other Indian religions such as Buddhism, similar creative constructs developed in Asian, mantras, suggests Frits Staal, may be older than language. There is no accepted definition of mantra. Renou has defined mantra as thought, mantras are structured formulae of thoughts, claims Silburn. Farquhar concludes that mantras are a religious thought, prayer, sacred utterance, zimmer defines mantra as a verbal instrument to produce something in one’s mind. There is no universally applicable uniform definition of mantra because mantras are used in different religions, in some schools of Hinduism for example, suggests Gonda, mantra is sakti to the devotee in the form of formulated and expressed thought. Staal clarifies that mantras are not rituals, they are what is recited or chanted during a ritual, There is a long history of scholarly disagreement on the meaning of mantras and whether they are really instruments of mind, as implied by the etymological origin of the word mantra
36.
Ashvamedha
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The Ashvamedha is a horse sacrifice ritual followed by the Śrauta tradition of Vedic religion. It was used by ancient Indian kings to prove their imperial sovereignty, in the territory traversed by the horse, any rival could dispute the kings authority by challenging the warriors accompanying it. After one year, if no enemy had managed to kill or capture the horse and it would be then sacrificed, and the king would be declared as an undisputed sovereign. A stallion is selected and released for a year of wandering in the company of a hundred or more warriors, the king remains at home and turns over rulership to the adhvaryu priest who makes daily ghee libations into a footprint of the absent horse. The hotr priest recites evening narrations of the exploits of past kings, the release of the horse is considered an invitation to dispute the authority of the king. If the horse wanders into a controlled by someone who doesnt recognize the authority of the king. After almost a year the horse is guided home, if no defending army has managed to kill or capture it, soma pressings and various animal sacrifices are performed during the building of a great altar. The four major priests symbolically receive the four quarters of space, on the second of three pressing days, the horse, a hornless goat and gayal are dedicated to Prajapati. Other animals are dedicated to a variety of deities, three of the queens wash the horse and adorn it with jewelry and ghee. The horse, hornless goat, and gayal are asphyxiated, the chief queen lies down and the adhvaryu guides the horses penis against the queens vagina, signifying the birth of a new king. The king ascends the throne while the Purusha Sukta is recited, the adhvaryu takes the dismembered parts of the three chief animals and assembles them on the ground with the head of the goat facing west, the other two animals facing east. All the parts are offered into the ahavaniya. The adhvaryu makes three additional offerings into the throat, on the right front hoof of the horse. A final offering is made using a leper who stands in water as an altar, sanskrit epics and Puranas mention numerous legendary performances of the horse sacrifice. He again performed a thousand Ashvamedha on different locations and a hundred Rajasuya, a quotation of the Cārvāka from Madhavacharyas Sarva-Darsana-Sangraha states, The three authors of the Vedas were buffoons, knaves, and demons. All the well-known formulae of the pandits, jarphari, turphari, griffith omits verses VSM23. 20–31, protesting that they are not reproducible even in the semi-obscurity of a learned European language. Keiths 1914 translation also omits verses, while others such has Manohar L. Varadpande, praised the ritual as social occasions of great magnitude. In respect to the adhyatma paksha, the Prajapati-Agni, or the Purusha, the Creator, is the Ashva, He is the same as the Varuna, the Most Supreme
37.
Vedic Sanskrit
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Vedic Sanskrit is an Indo-European language, more specifically one branch of the Indo-Iranian group. It is the ancient language of the Vedas of Hinduism, texts compiled over the period of the mid-2nd to mid-1st millennium BCE and it was orally preserved, predating the advent of Brahmi script by several centuries. Vedic Sanskrit is a language, whose consensus translation has been challenging. Extensive ancient literature in Vedic Sanskrit language has survived into the era. Quite early in the era, Sanskrit separated from the Avestan language. The exact century of separation is unknown but this separation of Sanskrit, Avestan language developed in ancient Persia, was the language of Zoroastrianism, but was a dead language in the Sasanian period. The separation of language into Avestan and Vedic Sanskrit is estimated, on linguistic grounds. The date of composition of the oldest hymns of the Rigveda is vague at best, both Asko Parpola and J. P. Mallory place the locus of the division of Indo-Aryan from Iranian in the Bronze Age culture of the Bactria–Margiana Archaeological Complex. Parpola elaborates the model and has Proto-Rigvedic Indo-Aryans intrude the BMAC around 1700 BCE and he assumes early Indo-Aryan presence in the Late Harappan horizon from about 1900 BCE, and Proto-Rigvedic intrusion to the Punjab as corresponding to the Gandhara grave culture from about 1700 BCE. According to this model, Rigvedic within the larger Indo-Aryan group is the ancestor of the Dardic languages. The Rigveda must have been complete by around the 12th century BCE. The pre-1200 BCE layers mark a change in Vedic Sanskrit. Mantra language This period includes both the mantra and prose language of the Atharvaveda, the Rigveda Khilani, the Samaveda Samhita, and these texts are largely derived from the Rigveda, but have undergone certain changes, both by linguistic change and by reinterpretation. For example, the more ancient injunctive verb system is no longer in use, Samhita prose An important linguistic change is the disappearance of the injunctive, subjunctive, optative, imperative. New innovation in Vedic Sanskrit appear such as the development of periphrastic aorist forms and this must have occurred before the time of Pāṇini because Panini makes a list of those from northwestern region of India who knew these older rules of Vedic Sanskrit. Brahmana prose In this layer of Vedic literature, the archaic Vedic Sanskrit verb system has been abandoned, the Yajñagāthās texts provide a probable link between Vedic Sanskrit, Classical Sanskrit and languages of the Epics. Complex meters such as Anuṣṭubh and rules of Sanskrit prosody had been or were being innovated by this time, sutra language This is the last stratum of Vedic literature, comprising the bulk of the Śrautasūtras and Gṛhyasūtras and some Upanishads such as the Katha Upanishad and Maitrayaniya Upanishad. Grammar of the Vedic language Vedic meter Vedic period A Vedic Word Concordance Delbrück, Berthold, Windisch, die Altindische Wortfolge Aus Dem Catapathabrâhmaòa, Dargestellt Von B
38.
Pythagorean Theorem
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In mathematics, the Pythagorean theorem, also known as Pythagorass theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the two sides. There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework, Mesopotamian, Indian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases. The theorem has been given numerous proofs – possibly the most for any mathematical theorem and they are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The Pythagorean theorem was known long before Pythagoras, but he may well have been the first to prove it, in any event, the proof attributed to him is very simple, and is called a proof by rearrangement. The two large squares shown in the figure each contain four triangles, and the only difference between the two large squares is that the triangles are arranged differently. Therefore, the space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean theorem and that Pythagoras originated this very simple proof is sometimes inferred from the writings of the later Greek philosopher and mathematician Proclus. Several other proofs of this theorem are described below, but this is known as the Pythagorean one, If the length of both a and b are known, then c can be calculated as c = a 2 + b 2. If the length of the c and of one side are known. The Pythagorean equation relates the sides of a triangle in a simple way. Another corollary of the theorem is that in any triangle, the hypotenuse is greater than any one of the other sides. A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation. This theorem may have more known proofs than any other, the book The Pythagorean Proposition contains 370 proofs, Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. Draw the altitude from point C, and call H its intersection with the side AB, point H divides the length of the hypotenuse c into parts d and e. By a similar reasoning, the triangle CBH is also similar to ABC, the proof of similarity of the triangles requires the triangle postulate, the sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the leads to the equality of ratios of corresponding sides. The first result equates the cosines of the angles θ, whereas the second result equates their sines, the role of this proof in history is the subject of much speculation
39.
First Babylonian Dynasty
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The chronology of the first dynasty of Babylonia is debated as there is a Babylonian King List A and a Babylonian King List B. In this chronology, the years of List A are used due to their wide usage. The reigns in List B are longer, in general, thus any evidence must come from surrounding regions and written records. Not much is known about the kings from Sumuabum through Sin-muballit other than the fact they were Amorites rather than indigenous Akkadians, what is known, however, is that they accumulated little land. When Hammurabi ascended the throne of Babylon, the empire consisted of a few towns in the surrounding area, Dilbat, Sippar, Kish. Once Hammurabi was king, his military victories gained land for the empire, however, Babylon remained but one of several important areas in Mesopotamia, along with Assyria, then ruled by Shamshi-Adad I, and Larsa, then ruled by Rim-Sin I. In Hammurabis thirtieth year as king, he began to establish Babylon as the center of what would be a great empire. In that year, he conquered Larsa from Rim-Sin I, thus gaining control over the urban centers of Nippur, Ur, Uruk. In essence, Hammurabi gained control over all of south Mesopotamia, the other formidable political power in the region in the 2nd millennium was Eshnunna, which Hammurabi succeeded in capturing in c. Babylon exploited Eshnunnas well-established commercial trade routes and the stability that came with them. It was not long before Hammurabis army took Assyria and parts of the Zagros Mountains, Hammurabis other name was Hammurapi-ilu, meaning Hammurapi the god or perhaps Hammurapi is god. He could have been Amraphel king of Shinar or Sinear in the Jewish records and the Bible, Abraham lived from 1871 to 1784, according to modern interpretations of the Old Testaments figures that have been usually reckoned in modern half years before the Exodus, from equinox to equinox. The Venus tablets of Ammisaduqa are famous, and several books had been published about them, several dates have been offered but the old dates of many sourcebooks seems to be outdated and incorrect. A few sources, some printed almost a century ago, claim that the text mentions an occultation of the Venus by the moon. However, this may be a misinterpretation, calculations support 1659 for the fall of Babylon, based on the statistical probability of dating based on the planets observations. The presently accepted middle chronology is too low from the point of view. A text about the fall of Babylon by the Hittites of Mursilis I at the end of Samsuditanas reign which tells about an eclipse is crucial for a correct Babylonian chronology. The pair of lunar and solar eclipses occurred in the month Shimanu, the lunar eclipse took place on February 9,1659 BC
40.
Pythagorean triples
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A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written, and an example is. If is a Pythagorean triple, then so is for any integer k. A primitive Pythagorean triple is one in which a, b and c are coprime, a right triangle whose sides form a Pythagorean triple is called a Pythagorean triangle. However, right triangles with non-integer sides do not form Pythagorean triples, for instance, the triangle with sides a = b =1 and c = √2 is right, but is not a Pythagorean triple because √2 is not an integer. Moreover,1 and √2 do not have a common multiple because √2 is irrational. There are 16 primitive Pythagorean triples with c ≤100, Note, for example, each of these low-c points forms one of the more easily recognizable radiating lines in the scatter plot. The formula states that the integers a = m 2 − n 2, b =2 m n, c = m 2 + n 2 form a Pythagorean triple. The triple generated by Euclids formula is primitive if and only if m and n are coprime, every primitive triple arises from a unique pair of coprime numbers m, n, one of which is even. It follows that there are infinitely many primitive Pythagorean triples and this relationship of a, b and c to m and n from Euclids formula is referenced throughout the rest of this article. Despite generating all primitive triples, Euclids formula does not produce all triples—for example and this can be remedied by inserting an additional parameter k to the formula. That these formulas generate Pythagorean triples can be verified by expanding a2 + b2 using elementary algebra, many formulas for generating triples with particular properties have been developed since the time of Euclid. A proof of the necessity that a, b, c be expressed by Euclids formula for any primitive Pythagorean triple is as follows, all such triples can be written as where a2 + b2 = c2 and a, b, c are coprime. Thus a, b, c are pairwise coprime, as a and b are coprime, one is odd, and one may suppose that it is a, by exchanging, if needed, a and b. This implies that b is even and c is odd, from a 2 + b 2 = c 2 we obtain c 2 − a 2 = b 2 and hence = b 2. Since b is rational, we set it equal to m n in lowest terms, thus b = n m, as being the reciprocal of b. As m n is fully reduced, m and n are coprime, and they cannot be both even. If they were odd, the numerator of m 2 − n 22 m n would be a multiple of 4
41.
Diophantine equations
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In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied. A linear Diophantine equation is an equation between two sums of monomials of degree zero or one, an exponential Diophantine equation is one in which exponents on terms can be unknowns. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations, in more technical language, they define an algebraic curve, algebraic surface, or more general object, and ask about the lattice points on it. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis, the solutions are described by the following theorem, This Diophantine equation has a solution if and only if c is a multiple of the greatest common divisor of a and b. Moreover, if is a solution, then the solutions have the form, where k is an arbitrary integer. Proof, If d is this greatest common divisor, Bézouts identity asserts the existence of integers e and f such that ae + bf = d, If c is a multiple of d, then c = dh for some integer h, and is a solution. On the other hand, for pair of integers x and y. Thus, if the equation has a solution, then c must be a multiple of d. If a = ud and b = vd, then for every solution, we have a + b = ax + by + k = ax + by + k = ax + by, showing that is another solution. Finally, given two solutions such that ax1 + by1 = ax2 + by2 = c, one deduces that u + v =0. As u and v are coprime, Euclids lemma shows that exists a integer k such that x2 − x1 = kv. Therefore, x2 = x1 + kv and y2 = y1 − ku, the system to be solved may thus be rewritten as B = UC. Calling yi the entries of V−1X and di those of D = UC and it follows that the system has a solution if and only if bi, i divides di for i ≤ k and di =0 for i > k. If this condition is fulfilled, the solutions of the system are V. Hermite normal form may also be used for solving systems of linear Diophantine equations, however, Hermite normal form does not directly provide the solutions, to get the solutions from the Hermite normal form, one has to successively solve several linear equations. Nevertheless, Richard Zippel wrote that the Smith normal form is more than is actually needed to solve linear diophantine equations. Instead of reducing the equation to diagonal form, we only need to make it triangular, the Hermite normal form is substantially easier to compute than the Smith normal form. Integer linear programming amounts to finding some integer solutions of systems that include also inequations
42.
Squaring the circle
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Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the area as a given circle by using only a finite number of steps with compass. It may be taken to ask whether specified axioms of Euclidean geometry concerning the existence of lines and circles entail the existence of such a square. It had been known for decades before then that the construction would be impossible if π were transcendental. Approximate squaring to any given non-perfect accuracy, in contrast, is possible in a number of steps. The expression squaring the circle is used as a metaphor for trying to do the impossible. The term quadrature of the circle is used to mean the same thing as squaring the circle. Methods to approximate the area of a circle with a square were known already to Babylonian mathematicians. Indian mathematicians also found a method, though less accurate. Archimedes showed that the value of pi lay between 3 + 1/7 and 3 + 10/71, see Numerical approximations of π for more on the history. The first known Greek to be associated with the problem was Anaxagoras, Hippocrates of Chios squared certain lunes, in the hope that it would lead to a solution — see Lune of Hippocrates. Even then there were skeptics—Eudemus argued that magnitudes cannot be divided up without limit, the problem was even mentioned in Aristophaness play The Birds. It is believed that Oenopides was the first Greek who required a plane solution, james Gregory attempted a proof of its impossibility in Vera Circuli et Hyperbolae Quadratura in 1667. Although his proof was faulty, it was the first paper to attempt to solve the problem using algebraic properties of pi and it was not until 1882 that Ferdinand von Lindemann rigorously proved its impossibility. The Victorian-age mathematician, logician and author, Charles Lutwidge Dodgson also expressed interest in debunking illogical circle-squaring theories, in one of his diary entries for 1855, Dodgson listed books he hoped to write including one called Plain Facts for Circle-Squarers. The value my friend selected for Pi was 3.2, more than a score of letters were interchanged before I became sadly convinced that I had no chance. A ridiculing of circle-squaring appears in Augustus de Morgans A Budget of Paradoxes published posthumously by his widow in 1872, originally published as a series of articles in the Athenæum, he was revising them for publication at the time of his death. Circle squaring was very popular in the century, but hardly anyone indulges in it today
43.
Square root of two
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The square root of 2, or the th power of 2, written in mathematics as √2 or 2 1⁄2, is the positive algebraic number that, when multiplied by itself, gives the number 2. Technically, it is called the square root of 2. Geometrically the square root of 2 is the length of a diagonal across a square sides of one unit of length. It was probably the first number known to be irrational, the rational approximation of the square root of two,665, 857/470,832, derived from the fourth step in the Babylonian algorithm starting with a0 =1, is too large by approx. 1. 6×10−12, its square is 2. 0000000000045… The rational approximation 99/70 is frequently used, despite having a denominator of only 70, it differs from the correct value by less than 1/10,000. The numerical value for the root of two, truncated to 65 decimal places, is,1. 41421356237309504880168872420969807856967187537694807317667973799….41421296 ¯. That is,1 +13 +13 ×4 −13 ×4 ×34 =577408 =1.4142156862745098039 ¯. This approximation is the seventh in a sequence of increasingly accurate approximations based on the sequence of Pell numbers, despite having a smaller denominator, it is only slightly less accurate than the Babylonian approximation. Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned. For a while, the Pythagoreans treated as a secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it. The square root of two is occasionally called Pythagoras number or Pythagoras constant, for example by Conway & Guy, there are a number of algorithms for approximating √2, which in expressions as a ratio of integers or as a decimal can only be approximated. The most common algorithm for this, one used as a basis in many computers and calculators, is the Babylonian method of computing square roots, which is one of many methods of computing square roots. It goes as follows, First, pick a guess, a0 >0, then, using that guess, iterate through the following recursive computation, a n +1 = a n +2 a n 2 = a n 2 +1 a n. The more iterations through the algorithm, the approximation of the square root of 2 is achieved. Each iteration approximately doubles the number of correct digits, starting with a0 =1 the next approximations are 3/2 =1.5 17/12 =1.416. The value of √2 was calculated to 137,438,953,444 decimal places by Yasumasa Kanadas team in 1997, in February 2006 the record for the calculation of √2 was eclipsed with the use of a home computer. Shigeru Kondo calculated 1 trillion decimal places in 2010, for a development of this record, see the table below. Among mathematical constants with computationally challenging decimal expansions, only π has been calculated more precisely, such computations aim to check empirically whether such numbers are normal
44.
BCE
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Common Era or Current Era is a year-numbering system for the Julian and Gregorian calendars that refers to the years since the start of this era, i. e. since AD1. The preceding era is referred to as before the Common or Current Era, the Current Era notation system can be used as a secular alternative to the Dionysian era system, which distinguishes eras as AD and BC. The two notation systems are equivalent, thus 2017 CE corresponds to AD2017 and 400 BCE corresponds to 400 BC. The year-numbering system for the Gregorian calendar is the most widespread civil calendar used in the world today. For decades, it has been the standard, recognized by international institutions such as the United Nations. The expression has been traced back to Latin usage to 1615, as vulgaris aerae, the term Common Era can be found in English as early as 1708, and became more widely used in the mid-19th century by Jewish academics. He attempted to number years from a reference date, an event he referred to as the Incarnation of Jesus. Dionysius labeled the column of the table in which he introduced the new era as Anni Domini Nostri Jesu Christi, numbering years in this manner became more widespread in Europe with its usage by Bede in England in 731. Bede also introduced the practice of dating years before what he supposed was the year of birth of Jesus, in 1422, Portugal became the last Western European country to switch to the system begun by Dionysius. The first use of the Latin term vulgaris aerae discovered so far was in a 1615 book by Johannes Kepler, Kepler uses it again in a 1616 table of ephemerides, and again in 1617. A1635 English edition of that book has the title page in English – so far, a 1701 book edited by John LeClerc includes Before Christ according to the Vulgar Æra,6. A1716 book in English by Dean Humphrey Prideaux says, before the beginning of the vulgar æra, a 1796 book uses the term vulgar era of the nativity. The first so-far-discovered usage of Christian Era is as the Latin phrase aerae christianae on the page of a 1584 theology book. In 1649, the Latin phrase æræ Christianæ appeared in the title of an English almanac, a 1652 ephemeris is the first instance so-far-found for English usage of Christian Era. The English phrase common Era appears at least as early as 1708, a 1759 history book uses common æra in a generic sense, to refer to the common era of the Jews. The first-so-far found usage of the phrase before the era is in a 1770 work that also uses common era and vulgar era as synonyms. The 1797 edition of the Encyclopædia Britannica uses the terms vulgar era, the Catholic Encyclopedia in at least one article reports all three terms being commonly understood by the early 20th century. Thus, the era of the Jews, the common era of the Mahometans, common era of the world
45.
Plimpton 322
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Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics. It has number 322 in the G. A and this tablet, believed to have been written about 1800 BC, has a table of four columns and 15 rows of numbers in the cuneiform script of the period. This table lists what are now called Pythagorean triples, i. e. integers a, b, from a modern perspective, a method for constructing such triples is a significant early achievement, known long before the Greek and Indian mathematicians discovered solutions to this problem. Although the tablet was interpreted in the past as a table, more recently published work sees this as anachronistic. For readable popular treatments of this tablet see Robson or, more briefly, Robson is a more detailed and technical discussion of the interpretation of the tablets numbers, with an extensive bibliography. Plimpton 322 is partly broken, approximately 13 cm wide,9 cm tall, according to Banks, the tablet came from Senkereh, a site in southern Iraq corresponding to the ancient city of Larsa. More specifically, based on formatting similarities with other tablets from Larsa that have explicit dates written on them, Robson points out that Plimpton 322 was written in the same format as other administrative, rather than mathematical, documents of the period. The main content of Plimpton 322 is a table of numbers, with four columns and fifteen rows, the fourth column is just a row number, in order from 1 to 15. The second and third columns are visible in the surviving tablet. Conversion of these numbers from sexagesimal to decimal raises additional ambiguities, the sixty sexigesimal entries are exact, no truncations or rounding off. In each row, the number in the column can be interpreted as the shortest side s of a right triangle. The number in the first column is either the fraction s 2 l 2 or d 2 l 2 =1 + s 2 l 2, scholars still differ, however, on how these numbers were generated. Below is the translation of the tablet. Otto E. Neugebauer argued for an interpretation, pointing out that this table provides a list of Pythagorean triples. For instance, line 11 of the table can be interpreted as describing a triangle with short side 3/4 and hypotenuse 5/4, forming the side, hypotenuse ratio of the familiar right triangle. If p and q are two numbers, one odd and one even, then form a Pythagorean triple. For instance, line 11 can be generated by this formula with p =2 and q =1, as Neugebauer argues, each line of the tablet can be generated by a pair that are both regular numbers, integer divisors of a power of 60. This property of p and q being regular leads to a denominator that is regular, neugebauers explanation is the one followed e. g. by Conway & Guy
46.
Boolean logic
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In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively. It is thus a formalism for describing logical relations in the way that ordinary algebra describes numeric relations. Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic, according to Huntington, the term Boolean algebra was first suggested by Sheffer in 1913. Boolean algebra has been fundamental in the development of digital electronics and it is also used in set theory and statistics. Booles algebra predated the modern developments in algebra and mathematical logic. In an abstract setting, Boolean algebra was perfected in the late 19th century by Jevons, Schröder, Huntington, in fact, M. H. Stone proved in 1936 that every Boolean algebra is isomorphic to a field of sets. Shannon already had at his disposal the abstract mathematical apparatus, thus he cast his switching algebra as the two-element Boolean algebra, in circuit engineering settings today, there is little need to consider other Boolean algebras, thus switching algebra and Boolean algebra are often used interchangeably. Efficient implementation of Boolean functions is a problem in the design of combinational logic circuits. Logic sentences that can be expressed in classical propositional calculus have an equivalent expression in Boolean algebra, thus, Boolean logic is sometimes used to denote propositional calculus performed in this way. Boolean algebra is not sufficient to capture logic formulas using quantifiers, the closely related model of computation known as a Boolean circuit relates time complexity to circuit complexity. Whereas in elementary algebra expressions denote mainly numbers, in Boolean algebra they denote the truth values false and these values are represented with the bits, namely 0 and 1. Addition and multiplication then play the Boolean roles of XOR and AND respectively, Boolean algebra also deals with functions which have their values in the set. A sequence of bits is a commonly used such function, another common example is the subsets of a set E, to a subset F of E is associated the indicator function that takes the value 1 on F and 0 outside F. The most general example is the elements of a Boolean algebra, as with elementary algebra, the purely equational part of the theory may be developed without considering explicit values for the variables. The basic operations of Boolean calculus are as follows, AND, denoted x∧y, satisfies x∧y =1 if x = y =1 and x∧y =0 otherwise. OR, denoted x∨y, satisfies x∨y =0 if x = y =0, NOT, denoted ¬x, satisfies ¬x =0 if x =1 and ¬x =1 if x =0. Alternatively the values of x∧y, x∨y, and ¬x can be expressed by tabulating their values with truth tables as follows, the first operation, x → y, or Cxy, is called material implication. If x is then the value of x → y is taken to be that of y
47.
Programming languages
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A programming language is a formal computer language designed to communicate instructions to a machine, particularly a computer. Programming languages can be used to programs to control the behavior of a machine or to express algorithms. From the early 1800s, programs were used to direct the behavior of such as Jacquard looms. Thousands of different programming languages have created, mainly in the computer field. Many programming languages require computation to be specified in an imperative form while other languages use forms of program specification such as the declarative form. The description of a language is usually split into the two components of syntax and semantics. Some languages are defined by a document while other languages have a dominant implementation that is treated as a reference. Some languages have both, with the language defined by a standard and extensions taken from the dominant implementation being common. A programming language is a notation for writing programs, which are specifications of a computation or algorithm, some, but not all, authors restrict the term programming language to those languages that can express all possible algorithms. For example, PostScript programs are created by another program to control a computer printer or display. More generally, a language may describe computation on some, possibly abstract. It is generally accepted that a specification for a programming language includes a description, possibly idealized. In most practical contexts, a programming language involves a computer, consequently, abstractions Programming languages usually contain abstractions for defining and manipulating data structures or controlling the flow of execution. Expressive power The theory of computation classifies languages by the computations they are capable of expressing, all Turing complete languages can implement the same set of algorithms. ANSI/ISO SQL-92 and Charity are examples of languages that are not Turing complete, markup languages like XML, HTML, or troff, which define structured data, are not usually considered programming languages. Programming languages may, however, share the syntax with markup languages if a computational semantics is defined, XSLT, for example, is a Turing complete XML dialect. Moreover, LaTeX, which is used for structuring documents. The term computer language is used interchangeably with programming language
48.
Music theory
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Music theory is the study of the practices and possibilities of music. The term is used in three ways in music, though all three are interrelated. The first is what is otherwise called rudiments, currently taught as the elements of notation, of key signatures, of time signatures, of rhythmic notation, Theory in this sense is treated as the necessary preliminary to the study of harmony, counterpoint, and form. The second is the study of writings about music from ancient times onwards, Music theory is frequently concerned with describing how musicians and composers make music, including tuning systems and composition methods among other topics. However, this medieval discipline became the basis for tuning systems in later centuries, Music theory as a practical discipline encompasses the methods and concepts composers and other musicians use in creating music. The development, preservation, and transmission of music theory in this sense may be found in oral and written music-making traditions, musical instruments, and other artifacts. In ancient and living cultures around the world, the deep and long roots of music theory are clearly visible in instruments, oral traditions, and current music making. Many cultures, at least as far back as ancient Mesopotamia and ancient China, have also considered music theory in more formal ways such as written treatises, in modern academia, music theory is a subfield of musicology, the wider study of musical cultures and history. Etymologically, music theory is an act of contemplation of music, from the Greek θεωρία, a looking at, viewing, contemplation, speculation, theory, also a sight, a person who researches, teaches, or writes articles about music theory is a music theorist. University study, typically to the M. A. or Ph. D level, is required to teach as a music theorist in a US or Canadian university. Methods of analysis include mathematics, graphic analysis, and especially analysis enabled by Western music notation, comparative, descriptive, statistical, and other methods are also used. See for instance Paleolithic flutes, Gǔdí, and Anasazi flute, several surviving Sumerian and Akkadian clay tablets include musical information of a theoretical nature, mainly lists of intervals and tunings. The scholar Sam Mirelman reports that the earliest of these dates from before 1500 BCE. Further, All the Mesopotamian texts are united by the use of a terminology for music that, much of Chinese music history and theory remains unclear. The earliest texts about Chinese music theory are inscribed on the stone and they include more than 2800 words describing theories and practices of music pitches of the time. The bells produce two intertwined pentatonic scales three tones apart with additional pitches completing the chromatic scale, Chinese theory starts from numbers, the main musical numbers being twelve, five and eight. Twelve refers to the number of pitches on which the scales can be constructed, the Lüshi chunqiu from about 239 BCE recalls the legend of Ling Lun. On order of the Yellow Emperor, Ling Lun collected twelve bamboo lengths with thick, blowing on one of these like a pipe, he found its sound agreeable and named it huangzhong, the Yellow Bell
49.
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
50.
Binomial coefficient
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In mathematics, any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient. Commonly, a coefficient is indexed by a pair of integers n ≥ k ≥0 and is written. It is the coefficient of the xk term in the expansion of the binomial power n. The value of the coefficient is given by the expression n. k, arranging binomial coefficients into rows for successive values of n, and in which k ranges from 0 to n, gives a triangular array called Pascals triangle. The properties of binomial coefficients have led to extending the definition to beyond the case of integers n ≥ k ≥0. Andreas von Ettingshausen introduced the notation in 1826, although the numbers were known centuries earlier, the earliest known detailed discussion of binomial coefficients is in a tenth-century commentary, by Halayudha, on an ancient Sanskrit text, Pingalas Chandaḥśāstra. In about 1150, the Indian mathematician Bhaskaracharya gave an exposition of binomial coefficients in his book Līlāvatī, alternative notations include C, nCk, nCk, Ckn, Cnk, and Cn, k in all of which the C stands for combinations or choices. Many calculators use variants of the C notation because they can represent it on a single-line display, in this form the binomial coefficients are easily compared to k-permutations of n, written as P, etc. For natural numbers n and k, the binomial coefficient can be defined as the coefficient of the monomial Xk in the expansion of n, the same coefficient also occurs in the binomial formula, which explains the name binomial coefficient. This shows in particular that is a number for any natural numbers n and k. Most of these interpretations are easily seen to be equivalent to counting k-combinations, several methods exist to compute the value of without actually expanding a binomial power or counting k-combinations. It also follows from tracing the contributions to Xk in n−1, as there is zero Xn+1 or X−1 in n, one might extend the definition beyond the above boundaries to include =0 when either k > n or k <0. This recursive formula then allows the construction of Pascals triangle, surrounded by white spaces where the zeros, or the trivial coefficients, a more efficient method to compute individual binomial coefficients is given by the formula = n k _ k. = n ⋯ k ⋯1 = ∏ i =1 k n +1 − i i and this formula is easiest to understand for the combinatorial interpretation of binomial coefficients. The numerator gives the number of ways to select a sequence of k distinct objects, retaining the order of selection, the denominator counts the number of distinct sequences that define the same k-combination when order is disregarded. Due to the symmetry of the binomial coefficient with regard to k and n−k, calculation may be optimised by setting the limit of the product above to the smaller of k. This formula follows from the formula above by multiplying numerator and denominator by. As a consequence it involves many factors common to numerator and denominator and it is less practical for explicit computation unless common factors are first cancelled
51.
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
52.
Fibonacci number
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The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics, the sequence described in Liber Abaci began with F1 =1. Fibonacci numbers are related to Lucas numbers L n in that they form a complementary pair of Lucas sequences U n = F n and V n = L n. They are intimately connected with the ratio, for example. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is a journal dedicated to their study. The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody, in the Sanskrit tradition of prosody, there was interest in enumerating all patterns of long syllables that are 2 units of duration, and short syllables that are 1 unit of duration. Counting the different patterns of L and S of a given duration results in the Fibonacci numbers, susantha Goonatilake writes that the development of the Fibonacci sequence is attributed in part to Pingala, later being associated with Virahanka, Gopāla, and Hemachandra. He dates Pingala before 450 BC, however, the clearest exposition of the sequence arises in the work of Virahanka, whose own work is lost, but is available in a quotation by Gopala, Variations of two earlier meters. For example, for four, variations of meters of two three being mixed, five happens, in this way, the process should be followed in all mātrā-vṛttas. The sequence is also discussed by Gopala and by the Jain scholar Hemachandra, outside India, the Fibonacci sequence first appears in the book Liber Abaci by Fibonacci. The puzzle that Fibonacci posed was, how many pairs will there be in one year, at the end of the first month, they mate, but there is still only 1 pair. At the end of the month the female produces a new pair. At the end of the month, the original female produces a second pair. At the end of the month, the original female has produced yet another new pair. At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs plus the number of pairs alive last month and this is the nth Fibonacci number. The name Fibonacci sequence was first used by the 19th-century number theorist Édouard Lucas, the most common such problem is that of counting the number of compositions of 1s and 2s that sum to a given total n, there are Fn+1 ways to do this. For example, if n =5, then Fn+1 = F6 =8 counts the eight compositions, 1+1+1+1+1 = 1+1+1+2 = 1+1+2+1 = 1+2+1+1 = 2+1+1+1 = 2+2+1 = 2+1+2 = 1+2+2, all of which sum to 5. The Fibonacci numbers can be found in different ways among the set of strings, or equivalently
53.
Mount Meru (mythology)
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Mount Meru is a sacred mountain with five peaks in Hindu, Jain and Buddhist cosmology and is considered to be the center of all the physical, metaphysical and spiritual universes. Meru, also called මහා මේරු පර්වතය Sumeru or Sineru, to which can be added the approbatory prefix su-, many famous Hindu and similar Jain as well as Buddhist temples have been built as symbolic representations of this mountain. The highest point on the pyatthat, a Burmese-style multi-tiered roof, some researchers identify Mount Meru or Sumeru with the Pamirs, northwest of Kashmir. The Suryasiddhanta mentions that Mt. Meru lies in the middle of the Earth in the land of the Jambunad, narpatijayacharyā, a ninth-century text, based on mostly unpublished texts of Yāmal Tantr, mentions Sumeruḥ Prithvī-madhye shrūyate drishyate na tu. Vārāhamihira, in his Pancha-siddhāntikā, claims Mt. Meru to be at the North Pole, suryasiddhānta, however, mentions a Mt. Meru in the middle of Earth, besides a Sumeru and a Kumeru at both the Poles. There exist several versions of Cosmology in existing Hindu texts, the Matsya Purana and the Bhāgvata Purāna along with some other Hindu texts consistently give the height of 84,000 yojanas to Mount Meru which translates into 672,000 miles or 1,082,000 kilometers. Mount Meru was said to be the residence of King Padamja Brahma in antiquity. The Puranas and Hindu epics, often state that Surya, i. e. the Sun God, along all its planets and stars taken together as one unit. Mount Meru is also the abode of Lord Brahma and the Demi-Gods, according to the Epic, Mahabharata, Pandavas and Draupadi climbed this mountain to attain heaven. Draupadi and other four Pandavas were cast down for their sins, only Yudhishthira along with his faithful dog climbed the mountain, making him the only one to reach the Divine door. According to Vasubandhus Abhidharmakośabhāṣyam, Sumeru is 80,000 yojanas tall, the exact measure of one yojana is uncertain, but some accounts put it at about 24,000 feet, or approximately 4.5 miles, but other accounts put it between 7–9 miles. It also descends beneath the surface of the waters to a depth of 80,000 yojanas. Sumeru is often used as a simile for both size and stability in Buddhist texts, Sumeru is said to be shaped like an hourglass, with a top and base of 80,000 yojanas square, but narrowing in the middle to 20,000 yojanas square. Sumeru is the center of a mandala-like complex of seas. There are seven seas and seven surrounding mountain-walls, until one comes to the vast outer sea which forms most of the surface of the world, the known world, which is on the continent of Jambudvipa, is directly south of Sumeru. The next 40,000 yojanas below this heaven consist of a sheer precipice, from this point Sumeru expands again, going down in four terraced ledges, each broader than the one above. The first terrace constitutes the heaven of the Four Great Kings and is divided into four parts, facing north, south, each section is governed by one of the Four Great Kings, who face outward toward the quarter of the world that they supervise. 40,000 yojanas is also the height at which the Sun, half a day later, when the Sun has moved to the south, it is noon in Jambudvīpa, dusk in Pūrvavideha, dawn in Aparagodānīya, and midnight in Uttarakuru
54.
Combinatorics
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Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general methods were developed. One of the oldest and most accessible parts of combinatorics is graph theory, Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms. A mathematician who studies combinatorics is called a combinatorialist or a combinatorist, basic combinatorial concepts and enumerative results appeared throughout the ancient world. Greek historian Plutarch discusses an argument between Chrysippus and Hipparchus of a rather delicate enumerative problem, which was shown to be related to Schröder–Hipparchus numbers. In the Ostomachion, Archimedes considers a tiling puzzle, in the Middle Ages, combinatorics continued to be studied, largely outside of the European civilization. The Indian mathematician Mahāvīra provided formulae for the number of permutations and combinations, later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations. During the Renaissance, together with the rest of mathematics and the sciences, works of Pascal, Newton, Jacob Bernoulli and Euler became foundational in the emerging field. In modern times, the works of J. J. Sylvester and Percy MacMahon helped lay the foundation for enumerative, graph theory also enjoyed an explosion of interest at the same time, especially in connection with the four color problem. In the second half of the 20th century, combinatorics enjoyed a rapid growth, in part, the growth was spurred by new connections and applications to other fields, ranging from algebra to probability, from functional analysis to number theory, etc. These connections shed the boundaries between combinatorics and parts of mathematics and theoretical science, but at the same time led to a partial fragmentation of the field. Enumerative combinatorics is the most classical area of combinatorics and concentrates on counting the number of combinatorial objects. Although counting the number of elements in a set is a rather broad mathematical problem, fibonacci numbers is the basic example of a problem in enumerative combinatorics. The twelvefold way provides a framework for counting permutations, combinations and partitions. Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis, in contrast with enumerative combinatorics, which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae. Partition theory studies various enumeration and asymptotic problems related to integer partitions, originally a part of number theory and analysis, it is now considered a part of combinatorics or an independent field. It incorporates the bijective approach and various tools in analysis and analytic number theory, graphs are basic objects in combinatorics
55.
Pythagorean theorem
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In mathematics, the Pythagorean theorem, also known as Pythagorass theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the two sides. There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework, Mesopotamian, Indian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases. The theorem has been given numerous proofs – possibly the most for any mathematical theorem and they are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The Pythagorean theorem was known long before Pythagoras, but he may well have been the first to prove it, in any event, the proof attributed to him is very simple, and is called a proof by rearrangement. The two large squares shown in the figure each contain four triangles, and the only difference between the two large squares is that the triangles are arranged differently. Therefore, the space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean theorem and that Pythagoras originated this very simple proof is sometimes inferred from the writings of the later Greek philosopher and mathematician Proclus. Several other proofs of this theorem are described below, but this is known as the Pythagorean one, If the length of both a and b are known, then c can be calculated as c = a 2 + b 2. If the length of the c and of one side are known. The Pythagorean equation relates the sides of a triangle in a simple way. Another corollary of the theorem is that in any triangle, the hypotenuse is greater than any one of the other sides. A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation. This theorem may have more known proofs than any other, the book The Pythagorean Proposition contains 370 proofs, Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. Draw the altitude from point C, and call H its intersection with the side AB, point H divides the length of the hypotenuse c into parts d and e. By a similar reasoning, the triangle CBH is also similar to ABC, the proof of similarity of the triangles requires the triangle postulate, the sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the leads to the equality of ratios of corresponding sides. The first result equates the cosines of the angles θ, whereas the second result equates their sines, the role of this proof in history is the subject of much speculation
56.
Square root of 2
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The square root of 2, or the th power of 2, written in mathematics as √2 or 2 1⁄2, is the positive algebraic number that, when multiplied by itself, gives the number 2. Technically, it is called the square root of 2. Geometrically the square root of 2 is the length of a diagonal across a square sides of one unit of length. It was probably the first number known to be irrational, the rational approximation of the square root of two,665, 857/470,832, derived from the fourth step in the Babylonian algorithm starting with a0 =1, is too large by approx. 1. 6×10−12, its square is 2. 0000000000045… The rational approximation 99/70 is frequently used, despite having a denominator of only 70, it differs from the correct value by less than 1/10,000. The numerical value for the root of two, truncated to 65 decimal places, is,1. 41421356237309504880168872420969807856967187537694807317667973799….41421296 ¯. That is,1 +13 +13 ×4 −13 ×4 ×34 =577408 =1.4142156862745098039 ¯. This approximation is the seventh in a sequence of increasingly accurate approximations based on the sequence of Pell numbers, despite having a smaller denominator, it is only slightly less accurate than the Babylonian approximation. Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned. For a while, the Pythagoreans treated as a secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it. The square root of two is occasionally called Pythagoras number or Pythagoras constant, for example by Conway & Guy, there are a number of algorithms for approximating √2, which in expressions as a ratio of integers or as a decimal can only be approximated. The most common algorithm for this, one used as a basis in many computers and calculators, is the Babylonian method of computing square roots, which is one of many methods of computing square roots. It goes as follows, First, pick a guess, a0 >0, then, using that guess, iterate through the following recursive computation, a n +1 = a n +2 a n 2 = a n 2 +1 a n. The more iterations through the algorithm, the approximation of the square root of 2 is achieved. Each iteration approximately doubles the number of correct digits, starting with a0 =1 the next approximations are 3/2 =1.5 17/12 =1.416. The value of √2 was calculated to 137,438,953,444 decimal places by Yasumasa Kanadas team in 1997, in February 2006 the record for the calculation of √2 was eclipsed with the use of a home computer. Shigeru Kondo calculated 1 trillion decimal places in 2010, for a development of this record, see the table below. Among mathematical constants with computationally challenging decimal expansions, only π has been calculated more precisely, such computations aim to check empirically whether such numbers are normal
57.
Jainism
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Jainism, traditionally known as Jain Dharma, is an ancient Indian religion belonging to the śramaṇa tradition. The central tenet is non-violence and respect all living beings. The three main principles of Jainism are ahimsa, anekantavada and aparigraha, followers of Jainism take five main vows, ahimsa, satya, asteya, brahmacharya and aparigraha. Jain monks and nuns observe these vows absolutely whereas householders observe them within their practical limitations, self-discipline and asceticism are thus major focuses of Jainism. The word Jain derives from the Sanskrit word jina, a human being who has conquered all inner passions like attachment, desire, anger, pride, greed, etc. is called Jina. Followers of the path practiced and preached by the jinas are known as Jains, Parasparopagraho Jivanam is the motto of Jainism. Jains trace their history through a succession of teachers and revivers of the Jain path known as Tirthankaras. In the current era, this started with Rishabhdeva and concluded with Mahavira, Jains believe that Jainism is eternal and while it may be forgotten, it will be revived from time to time. With 6-7 million followers, the majority of Jains reside in India, outside of India, some of the largest Jain communities are present in Canada, Europe, Kenya, the UK, Suriname, Fiji, and the United States. Contemporary Jainism is divided into two sects, Digambara and Śvētāmbara. Namokar Mantra is the most common and basic prayer in Jainism, major Jain festivals include Paryushana and Daslakshana, Mahavir Jayanti, and Diwali. The principle of ahimsa is the most fundamental and well-known aspect of Jainism, the everyday implementation of the principle of non-violence is more comprehensive than in other religions and is the hallmark for Jain identity. Jains believe in avoiding harm to others thoughts, speech. According to the Jain text, Purushartha Siddhyupaya, killing any living being out of passions is hiṃsā, Jains extend the practice of nonviolence and kindness not only towards other humans but towards all living beings. For this reason, vegetarianism is a hallmark of Jain identity, if there is violence against animals during the production of dairy products, veganism is encouraged. Jainism has an elaborate framework on types of life and includes life-forms that may be invisible. Therefore, after humans and animals, insects are the living being offered protection in Jain practice. For example, insects in the home are often escorted out instead of killed, Jainism teaches that intentional harm and the absence of compassion make an action more violent
58.
Jain
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Jainism, traditionally known as Jain Dharma, is an ancient Indian religion belonging to the śramaṇa tradition. The central tenet is non-violence and respect all living beings. The three main principles of Jainism are ahimsa, anekantavada and aparigraha, followers of Jainism take five main vows, ahimsa, satya, asteya, brahmacharya and aparigraha. Jain monks and nuns observe these vows absolutely whereas householders observe them within their practical limitations, self-discipline and asceticism are thus major focuses of Jainism. The word Jain derives from the Sanskrit word jina, a human being who has conquered all inner passions like attachment, desire, anger, pride, greed, etc. is called Jina. Followers of the path practiced and preached by the jinas are known as Jains, Parasparopagraho Jivanam is the motto of Jainism. Jains trace their history through a succession of teachers and revivers of the Jain path known as Tirthankaras. In the current era, this started with Rishabhdeva and concluded with Mahavira, Jains believe that Jainism is eternal and while it may be forgotten, it will be revived from time to time. With 6-7 million followers, the majority of Jains reside in India, outside of India, some of the largest Jain communities are present in Canada, Europe, Kenya, the UK, Suriname, Fiji, and the United States. Contemporary Jainism is divided into two sects, Digambara and Śvētāmbara. Namokar Mantra is the most common and basic prayer in Jainism, major Jain festivals include Paryushana and Daslakshana, Mahavir Jayanti, and Diwali. The principle of ahimsa is the most fundamental and well-known aspect of Jainism, the everyday implementation of the principle of non-violence is more comprehensive than in other religions and is the hallmark for Jain identity. Jains believe in avoiding harm to others thoughts, speech. According to the Jain text, Purushartha Siddhyupaya, killing any living being out of passions is hiṃsā, Jains extend the practice of nonviolence and kindness not only towards other humans but towards all living beings. For this reason, vegetarianism is a hallmark of Jain identity, if there is violence against animals during the production of dairy products, veganism is encouraged. Jainism has an elaborate framework on types of life and includes life-forms that may be invisible. Therefore, after humans and animals, insects are the living being offered protection in Jain practice. For example, insects in the home are often escorted out instead of killed, Jainism teaches that intentional harm and the absence of compassion make an action more violent
59.
Infinity
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Infinity is an abstract concept describing something without any bound or larger than any number. In mathematics, infinity is treated as a number but it is not the same sort of number as natural or real numbers. Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th, in the theory he developed, there are infinite sets of different sizes. For example, the set of integers is countably infinite, while the set of real numbers is uncountable. Ancient cultures had various ideas about the nature of infinity, the ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept. The earliest recorded idea of infinity comes from Anaximander, a pre-Socratic Greek philosopher who lived in Miletus and he used the word apeiron which means infinite or limitless. However, the earliest attestable accounts of mathematical infinity come from Zeno of Elea, aristotle called him the inventor of the dialectic. He is best known for his paradoxes, described by Bertrand Russell as immeasurably subtle, however, recent readings of the Archimedes Palimpsest have found that Archimedes had an understanding about actual infinite quantities. The Jain mathematical text Surya Prajnapti classifies all numbers into three sets, enumerable, innumerable, and infinite, on both physical and ontological grounds, a distinction was made between asaṃkhyāta and ananta, between rigidly bounded and loosely bounded infinities. European mathematicians started using numbers in a systematic fashion in the 17th century. John Wallis first used the notation ∞ for such a number, euler used the notation i for an infinite number, and exploited it by applying the binomial formula to the i th power, and infinite products of i factors. In 1699 Isaac Newton wrote about equations with an number of terms in his work De analysi per aequationes numero terminorum infinitas. The infinity symbol ∞ is a symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ infinity and in LaTeX as \infty and it was introduced in 1655 by John Wallis, and, since its introduction, has also been used outside mathematics in modern mysticism and literary symbology. Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers, in real analysis, the symbol ∞, called infinity, is used to denote an unbounded limit. X → ∞ means that x grows without bound, and x → − ∞ means the value of x is decreasing without bound. ∑ i =0 ∞ f = ∞ means that the sum of the series diverges in the specific sense that the partial sums grow without bound. Infinity can be used not only to define a limit but as a value in the real number system
60.
Sanskrit language
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Sanskrit is the primary liturgical language of Hinduism, a philosophical language of Hinduism, Buddhism, and Jainism, and a literary language and lingua franca of ancient and medieval South Asia. As a result of transmission of Hindu and Buddhist culture to Southeast Asia and parts of Central Asia, as one of the oldest Indo-European languages for which substantial written documentation exists, Sanskrit holds a prominent position in Indo-European studies. The body of Sanskrit literature encompasses a rich tradition of poetry and drama as well as scientific, technical, philosophical, the compositions of Sanskrit were orally transmitted for much of its early history by methods of memorization of exceptional complexity, rigor, and fidelity. Thereafter, variants and derivatives of the Brahmi script came to be used, Sanskrit is today one of the 22 languages listed in the Eighth Schedule of the Constitution of India, which mandates the Indian government to develop the language. It continues to be used as a ceremonial language in Hindu religious rituals and Buddhist practice in the form of hymns. The Sanskrit verbal adjective sáṃskṛta- may be translated as refined, elaborated, as a term for refined or elaborated speech, the adjective appears only in Epic and Classical Sanskrit in the Manusmṛti and the Mahabharata. The pre-Classical form of Sanskrit is known as Vedic Sanskrit, with the language of the Rigveda being the oldest and most archaic stage preserved, Classical Sanskrit is the standard register as laid out in the grammar of Pāṇini, around the fourth century BCE. Sanskrit, as defined by Pāṇini, evolved out of the earlier Vedic form, the present form of Vedic Sanskrit can be traced back to as early as the second millennium BCE. Scholars often distinguish Vedic Sanskrit and Classical or Pāṇinian Sanskrit as separate dialects, although they are quite similar, they differ in a number of essential points of phonology, vocabulary, grammar and syntax. Vedic Sanskrit is the language of the Vedas, a collection of hymns, incantations and theological and religio-philosophical discussions in the Brahmanas. Modern linguists consider the metrical hymns of the Rigveda Samhita to be the earliest, for nearly 2000 years, Sanskrit was the language of a cultural order that exerted influence across South Asia, Inner Asia, Southeast Asia, and to a certain extent East Asia. A significant form of post-Vedic Sanskrit is found in the Sanskrit of Indian epic poetry—the Ramayana, the deviations from Pāṇini in the epics are generally considered to be on account of interference from Prakrits, or innovations, and not because they are pre-Paninian. Traditional Sanskrit scholars call such deviations ārṣa, meaning of the ṛṣis, in some contexts, there are also more prakritisms than in Classical Sanskrit proper. There were four principal dialects of classical Sanskrit, paścimottarī, madhyadeśī, pūrvi, the predecessors of the first three dialects are attested in Vedic Brāhmaṇas, of which the first one was regarded as the purest. In the 2001 Census of India,14,035 Indians reported Sanskrit to be their first language, in India, Sanskrit is among the 14 original languages of the Eighth Schedule to the Constitution. The state of Uttarakhand in India has ruled Sanskrit as its official language. In October 2012 social activist Hemant Goswami filed a petition in the Punjab. More than 3,000 Sanskrit works have been composed since Indias independence in 1947, much of this work has been judged of high quality, in comparison to both classical Sanskrit literature and modern literature in other Indian languages
61.
Vaishali (ancient city)
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Vaishali was a city in Bihar, India, and is now an archeological site. It is a part of the Tirhut Division and it was the capital city of the Licchavi, considered one of the first example of a republic, in the Vajjian Confederacy mahajanapada, around the 6th century BCE. It was here in 599 BCE the 24th Jain Tirthankara, Bhagwan Mahavira was born and brought up in Kundalagrama in Vaiśālī republic and it contains one of the best-preserved of the Pillars of Ashoka, topped by a single Asiatic lion. At the time of the Buddha, Vaiśālī, which he visited on occasions, was a very large city, rich and prosperous, crowded with people. There were 7,707 pleasure grounds and an number of lotus ponds. Its courtesan, Amrapali, was famous for her beauty, the city had three walls, each one gāvuta away from the other, and at three places in the walls were gates with watch towers. Outside the town, leading uninterruptedly up to the Himalaya, was the Mahavana, nearby were other forests, such as Gosingalasāla. Vaishali derives its name from King Vishal of the Mahabharata age, the city was also called Visālā. Buddhaghosa, the a 5th-century Indian Theravadin Buddhist commentator and scholar says, even before the advent of Buddhism and Jainism, Vaiśālī was the capital of the vibrant republican Licchavi state. In that period, Vaiśālī was an ancient metropolis and the city of the republic of the Vaiśālī state. However, very little is known about the history of Vaiśālī. The last among the 34 was Sumati, who is considered a contemporary of Dasaratha, father of the Hindu god, in the republic of Vaiśālī, Lord Mahavira was born. Gautama Buddha delivered his last sermon at Vaiśālī and announced his Parinirvana there, Vaiśālī is also renowned as the land of Amrapali, the great Indian courtesan, who appears in many folktales, as well as in Buddhist literature. Ambapali became a disciple of Buddha. Manudev was a king of the illustrious Lichchavi clan of the confederacy. A kilometer away is Abhishek Pushkarini, the coronation tank, the sacred waters of the tank anointed the elected representatives of Vaiśālī. Next to it stands the Japanese temple and the Vishwa Shanti Stupa built by the Nipponzan Myohoji sect of Japan, a small part of the Buddhas relics found in Vaiśālī have been enshrined in the foundation and in the chhatra of the Stupa. Near the coronation tank is Stupa 1 or the Relic Stupa, here the Lichchavis reverentially encased one of the eight portions of the Masters relics, which they received after the Mahaparinirvana. After his last discourse the Awakened One set out for Kushinagar, Buddha gave them his alms bowl but they still refused to return
62.
Bhadrabahu
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Bhadrabahu was, according to the Digambara sect of Jainism, the last Shruta Kevalin in Jainism. He was the last acharya of the undivided Jain sangha and he was the last spiritual teacher of Chandragupta Maurya. According to the Digambara sect of Jainism, there were five Shruta Kevalins in Jainism - Govarddhana Mahamuni, Vishnu, Nandimitra, Aparajita, Bhadrabahu was born in Pundravardhana to a Brahmin family during which time the secondary capital of the Mauryas was Ujjain. When he was seven, Govarddhana Mahamuni predicted that he will be the last Shruta Kevali and he was then initiated as a Jain Muni and by practicing gyan, dhyan, tap and sanyam got the Acharya pad. According to Śvētāmbara tradition, he lived from 433 BC to 357 BC, Digambara tradition dates him to have died in 365 BC. On the night of full moon in the month of Kartik, Chandragupta Maurya saw sixteen dreams, Bhadrabahu decided the famine would make it harder for monks to survive and migrated with a group of twelve thousand disciples to South India, bringing with him Chandragupta, turned Digambara monk. According to the inscriptions at Shravanabelgola, Bhadrabahu died after taking the vow of Sallekhana, according to Svetambaras, Bhadrabahu was the author of Kalpa Sūtra, four Chedda sutras, commentaries on ten scriptures, Bhadrabahu Samhita and Vasudevcharita. Bhadrabahu was the last acharya of the undivided Jain sangha, after him, the Sangha split into two separate teacher-student lineages of monks. Digambara monks belong to the lineage of Acharya Vishakha and Svetambara monks follow the tradition of Acharya Sthulibhadra, two inscriptions of about 900 AD on the Kaveri near Seringapatam describe the summit of a hill called Chandragiri as marked by the footprints of Bhadrabahu and Chandragupta munipati. A Shravanabelagola inscription of 1129 mentions Bhadrabahu Shrutakevali, and Chandragupta who acquired such merit that he was worshipped by the forest deities, another inscription of 1163 similarly couples and describes them. A third inscription of the year 1432 speaks of Yatindra Bhadrabahu, and his disciple Chandragupta, bhadrabahu-charitra was written by Ratnanandi of about 1450 AD. Wiley, Kristi L, The a to Z of Jainism, p
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Umasvati
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Umaswami, also known as Umaswati, was an early 1st-millennium Indian scholar, possibly 2nd-century CE, and the chief disciple of Acharya Kundakunda of Jainism. Umaswati authored the Jain text Tattvartha Sutra, Umaswatis work was the first Sanskrit language text on Jain philosophy, and is the earliest extant comprehensive Jain philosophy text accepted as authoritative by all four Jain traditions. His text has the importance in Jainism as Vedanta Sutras and Yogasutras have in Hinduism. Umaswati is claimed by both the Digambara and Śvētāmbara sects of Jainism as their own, on the basis of his genealogy, he was also called Nagaravachka. Umaswati was influential not only in Jainism, but also other Indian traditions over the centuries, the 13th- to 14th-century Madhvacharya, founder of Dvaita Vedanta school of Hindu philosophy, for example referred to Umaswati in his works as Umasvati-Vachakacharya. Umaswati, also spelled Umasvati, was an Acharya and therefore one of the Pañca-Parameṣṭhi in Jaina tradition, the main philosophy in Umaswatis Tatvartha Sutra aphorisms is that all life, both human and non-human, is sacred. Umaswati was born in Nyagrodhika village and his father was Svati and his mother was Uma. Umaswati was thus called as Svatitanaya after his fathers name and as Vatsisuta after his mothers lineage and his name is a combination of the names of his parents. Umaswati is also known as Vacaka-sramana and Nagaravacaka, according to Vidyabhusanas book published in 1920, Umaswati lived in the 1st-century CE and died in 85 CE. More recent scholarship, such as by Padmanabh Jaini on the hand, places him later. Umaswati authored his scriptural work the Tattvartha Sutra when he was in Pataliputra or Kusumapura and he was the first Jain thinker to have written a philosophical work in the sutra style. Umaswati in his Tattvartha Sutra, an aphoristic sutra text in Sanskrit language and he includes the doctrines on the subjects of non-violence or ahimsa, Anekantavada, and non-possession. The text, states Jaini, summarizes religious, ethical and philosophical themes of Jainism in the second century India, the Sūtras or verses have found ready acceptance with all the sects of Jainas, and on which bhasya have been written. Umaswati states that these beliefs are essential to achieving moksha or emancipation and his sutra have been variously translated. The first verse of Tattvartha Sutra has been translated as follows, in chapter 2, Umaswati presents sutras on soul. In chapter 3 through 6, Umaswati presents sutras for his first three categories of truth, in chapter 7, Umaswati presents the Jaina vows and explains their value in stopping karmic particle inflow to the soul. The vows, translates Nathmal Tatia, are ahimsa, anirta, asteya, brahmacharya, Umaswati, in chapter 8 of Tattvartha Sutra presents his sutras on how karma affects rebirths. He asserts that accumulated karma in life determine the length of life and realm of rebirth for each soul in each of four states – infernal beings, plants and animals, human beings and as gods
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Metaphysics
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Metaphysics is a branch of philosophy exploring the fundamental nature of reality. In this vein, metaphysics seeks to answer two questions, Ultimately, what is there. Topics of metaphysical investigation include existence, objects and their properties, space and time, cause and effect, a central branch of metaphysics is ontology, the investigation into the basic categories of being and how they relate to one another. Another branch is metaphysical cosmology, which seeks to understand the origin, there are two broad conceptions about what world is studied by metaphysics. The strong, classical view assumes that the objects studied by metaphysics exist independently of any observer, the weaker, more modern view assumes that the objects studied by metaphysics exist inside the mind of an observer, so the subject becomes a form of introspection and conceptual analysis. Some philosophers, notably Kant, discuss both of these worlds and what can be inferred about each one, some philosophers and scientists, such as the logical positivists, reject the entire subject of metaphysics as meaningless, while others disagree and think that it is legitimate. Ontology deals with the determination whether categories of being are fundamental and it is the inquiry into being in so much as it is being, or into beings insofar as they exist—and not insofar as particular facts may be obtained about them or particular properties belong to them. Most ontologies assume or assert the existence of categories including objects, properties, space, immediate questions arising from this include the nature of objects. Only properties can be observed directly, so what does it mean for an object to exist, how can we be sure that such objects exist at all. The word is has two uses in English, separated out in ontology. It can denote existence as in there is an elephant in the room, some philosophers also include sub-classing as a third form of is-ness or being, as in the elephant is a mammal. Some philosophers, notably of the Platonic school, contend that all refer to existent entities. Other philosophers contend that nouns do not always name entities, between these poles of realism and nominalism, stand a variety of other positions. An ontology may give an account of which refer to entities, which do not, why. Other controversial categories of objects and properties which may be argued to exist or not include aesthetic and moral properties, stances about the status of such things may form the foundation for other branches of philosophy such as aesthetics, ethics and political philosophy. Identity is a fundamental metaphysical issue, metaphysicians investigating identity are tasked with the question of what, exactly, it means for something to be identical to itself. Other issues of identity arise in the context of time, what does it mean for something to be itself across two moments in time, how do we account for this. Another question of identity arises when we ask what our criteria ought to be for determining identity, and how does the reality of identity interface with linguistic expressions
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Vedic period
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The Vedic period was the period in Indian history during which the Vedas, the oldest scriptures of Hinduism, were composed. During the early part of the Vedic period, the Indo-Aryans settled into northern India, scholars consider Vedic civilisation to have been a composite of the Indo-Aryan and Harappan cultures. The end of the Vedic period witnessed the rise of large, around the beginning of the Common Era, the Vedic tradition formed one of the main constituents of the so-called Hindu synthesis. The commonly proposed period of earlier Vedic age is dated back to 2nd millennium BCE, after the collapse of the Indus Valley Civilisation, which ended c.1900 BCE, groups of Indo-Aryan peoples migrated into north-western India and started to inhabit the northern Indus Valley. The knowledge about the Aryans comes mostly from the Rigveda-samhita, which was composed between c and they brought with them their distinctive religious traditions and practices. The Vedic beliefs and practices of the era were closely related to the hypothesised Proto-Indo-European religion. According to Anthony, the Old Indic religion probably emerged among Indo-European immigrants in the zone between the Zeravshan River and Iran. It was a mixture of old Central Asian and new Indo-European elements. At least 383 non-Indo-European words were borrowed from this culture, including the god Indra, Indra was the subject of 250 hymns, a quarter of the Rig Veda. He was associated more than any other deity with Soma, a stimulant drug probably borrowed from the BMAC religion and his rise to prominence was a peculiar trait of the Old Indic speakers. These migrations may have been accompanied with violent clashes with the people who inhabited this region. The Rig Veda contains accounts of conflicts between the Aryas and the Dasas and Dasyus, the Rig Veda describes Dasas and Dasyus as people who do not perform sacrifices or obey the commandments of gods. Their speech is described as mridhra which could variously mean soft, uncouth, hostile, other adjectives which describe their physical appearance are subject to many interpretations. Internecine military conflicts between the tribes of Vedic Aryans are also described in the Rig Veda. Most notable of such conflicts was the Battle of Ten Kings, which took place on the banks of the river Parushni. The battle was fought between the tribe Bharatas, led by their chief Sudas, against a confederation of ten tribes— Puru, Yadu, Turvasha, Anu, Druhyu, Alina, Bhalanas, Paktha, Siva, Vishanin. Bharatas lived around the regions of the river Saraswati, while Purus, their western neighbours. The other tribes dwelt north-west of the Bharatas in the region of Punjab, division of the waters of Ravi could have been a reason for the war