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Pell's Equation
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive Square number, nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinates, the equation is represented by a hyperbola; solutions occur wherever the curve passes through a point whose ''x'' and ''y'' coordinates are both integers, such as the Triviality (mathematics), trivial solution with ''x'' = 1 and ''y'' = 0. Joseph Louis Lagrange proved that, as long as ''n'' is not a square number, perfect square, Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accurately Diophantine approximation, approximate the square root of ''n'' by rational numbers of the form ''x''/''y''. This equation was first studied extensively Indian mathematics, in India starting with Brahmagupta, who found an integer solution to 92x^2 + 1 = y^2 in his '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chakravala Method
The ''chakravala'' method () is a cyclic algorithm to solve indeterminate quadratic equations, including Pell's equation. It is commonly attributed to Bhāskara II, (c. 1114 – 1185 CE)Hoiberg & Ramchandani – Students' Britannica India: Bhaskaracharya II, page 200Kumar, page 23 although some attribute it to Jayadeva (c. 950 ~ 1000 CE).Plofker, page 474 Jayadeva pointed out that Brahmagupta's approach to solving equations of this type could be generalized, and he then described this general method, which was later refined by Bhāskara II in his '' Bijaganita'' treatise. He called it the Chakravala method: ''chakra'' meaning "wheel" in Sanskrit, a reference to the cyclic nature of the algorithm.Goonatilake, page 127 – 128 C.-O. Selenius held that no European performances at the time of Bhāskara, nor much later, exceeded its marvellous height of mathematical complexity. This method is also known as the cyclic method and contains traces of mathematical induction. Histor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proclus
Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor (, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers of late antiquity. He set forth one of the most elaborate and fully developed systems of Neoplatonism and, through later interpreters and translators, exerted an influence on Byzantine philosophy, early Islamic philosophy, scholastic philosophy, and German idealism, especially G. W. F. Hegel, who called Proclus's ''Platonic Theology'' "the true turning point or transition from ancient to modern times, from ancient philosophy to Christianity." Biography The primary source for the life of Proclus is the eulogy ''Proclus'', ''or On Happiness'' that was written for him upon his death by his successor, Marinus, Marinus' biography set out to prove that Proclus reached the peak of virtue and attained eudaimonia. There are also a few details about the time in which he lived in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagoreanism
Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the Ancient Greece, ancient Greek colony of Crotone, Kroton, in modern Calabria (Italy) circa 530 BC. Early Pythagorean communities spread throughout Magna Graecia. Already during Pythagoras' life it is likely that the distinction between the ''akousmatikoi'' ("those who listen"), who is conventionally regarded as more concerned with religious, and ritual elements, and associated with the oral tradition, and the ''mathematikoi'' ("those who learn") existed. The ancient biographers of Pythagoras, Iamblichus () and his master Porphyry (philosopher), Porphyry ( ) seem to make the distinction of the two as that of 'beginner' and 'advanced'. As the Pythagorean cenobites practiced an esoteric path, like the Greco-Roman mysteries, mystery schools of antiquity, the adherents, ''akou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like Number (sports), jersey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root Of 2
The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as \sqrt or 2^. It is an algebraic number, and therefore not a transcendental number. Technically, it should be called the ''principal'' square root of 2, to distinguish it from the negative number with the same property. Geometrically, the square root of 2 is the length of a diagonal across a Unit square, square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational number, irrational. The fraction (≈ 1.4142857) is sometimes used as a good Diophantine approximation, rational approximation with a reasonably small denominator. Sequence in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 60 decimal places: : History The Babylonian clay tablet YBC 7289 (–1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greece
Greece, officially the Hellenic Republic, is a country in Southeast Europe. Located on the southern tip of the Balkan peninsula, it shares land borders with Albania to the northwest, North Macedonia and Bulgaria to the north, and Turkey to the east. The Aegean Sea lies to the east of the Geography of Greece, mainland, the Ionian Sea to the west, and the Sea of Crete and the Mediterranean Sea to the south. Greece has the longest coastline on the Mediterranean Basin, spanning List of islands of Greece, thousands of islands and nine Geographic regions of Greece, traditional geographic regions. It has a population of over 10 million. Athens is the nation's capital and List of cities and towns in Greece, largest city, followed by Thessaloniki and Patras. Greece is considered the cradle of Western culture, Western civilisation and the birthplace of Athenian democracy, democracy, Western philosophy, Western literature, historiography, political science, major History of science in cl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
India
India, officially the Republic of India, is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area; the List of countries by population (United Nations), most populous country since 2023; and, since its independence in 1947, the world's most populous democracy. Bounded by the Indian Ocean on the south, the Arabian Sea on the southwest, and the Bay of Bengal on the southeast, it shares land borders with Pakistan to the west; China, Nepal, and Bhutan to the north; and Bangladesh and Myanmar to the east. In the Indian Ocean, India is near Sri Lanka and the Maldives; its Andaman and Nicobar Islands share a maritime border with Thailand, Myanmar, and Indonesia. Modern humans arrived on the Indian subcontinent from Africa no later than 55,000 years ago., "Y-Chromosome and Mt-DNA data support the colonization of South Asia by modern humans originating in Africa. ... Coalescence dates for most non-European populations averag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Pell (mathematician)
John Pell (1 March 1611 – 12 December 1685) was an English mathematician and political agent abroad. He was made Royal Chair of Mathematics at Orange College by the Prince of Orange, and was under the patronage of Sir Charles Cavendish. He was also a compeer and correspondent of René Descartes and Thomas Hobbes. Early life He was born at Southwick in West Sussex, England. His father, also named John Pell, was from Southwick, and his mother was Mary Holland, from Halden in Kent. The second of two sons, Pell's older brother was Thomas Pell. By the time he was six, they were orphans, their father dying in 1616 and their mother the following year. John Pell the elder had a fine library, which proved valuable to the young Pell as he grew up. He was educated at Steyning Grammar School and entered Trinity College, Cambridge, at the age of 13. During his university career he became an accomplished linguist; even before taking a B.A. degree in 1629, he corresponded with Henry B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonhard Euler
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential discoveries in many other branches of mathematics, such as analytic number theory, complex analysis, and infinitesimal calculus. He also introduced much of modern mathematical terminology and Mathematical notation, notation, including the notion of a mathematical function. He is known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory. Euler has been called a "universal genius" who "was fully equipped with almost unlimited powers of imagination, intellectual gifts and extraordinary memory". He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Kingdom of Prussia, Prussia. Euler is credited for popularizing the Greek letter \pi (lowercase Pi (letter), pi) to denote Pi, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Brouncker, 2nd Viscount Brouncker
William Brouncker, 2nd Viscount Brouncker FRS ( – 5 April 1684) was an Anglo-Irish peer and mathematician who served as the president of the Royal Society from 1662 to 1677. Best known for introducing Brouncker's formula, he also worked as a civil servant, serving as a commissioner in the Royal Navy. Brouncker was a friend and colleague of Samuel Pepys, and features prominently in the Pepys' diary. Biography Brouncker was born in Castlelyons, County Cork, the elder son of William Brouncker (1585–1649), 1st Viscount Brouncker and Winifred, daughter of Sir William Leigh of Newnham. His family came originally from Melksham in Wiltshire. His grandfather Sir Henry Brouncker (died 1607) had been Lord President of Munster 1603–1607, and settled his family in Ireland. His father was created a viscount in the Peerage of Ireland in 1645 for his services to the Crown. Although the first viscount had fought for the Crown in the Anglo-Scots war of 1639, malicious gossip sai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greek Mathematics
Ancient Greek mathematics refers to the history of mathematical ideas and texts in Ancient Greece during Classical antiquity, classical and late antiquity, mostly from the 5th century BC to the 6th century AD. Greek mathematicians lived in cities spread around the shores of the ancient Mediterranean, from Anatolia to Italy and North Africa, but were united by Greek culture and the Ancient Greek, Greek language. The development of mathematics as a theoretical discipline and the use of deductive reasoning in Mathematical proof, proofs is an important difference between Greek mathematics and those of preceding civilizations. The early history of Greek mathematics is obscure, and traditional narratives of Theorem, mathematical theorems found before the fifth century BC are regarded as later inventions. It is now generally accepted that treatises of deductive mathematics written in Greek began circulating around the mid-fifth century BC, but the earliest complete work on the subje ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
