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Chakravala
The ''chakravala'' method () is a cyclic algorithm to solve Indeterminate equation, 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 (mathematician), 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 contai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Bhāskara II
Bhāskara II ('; 1114–1185), also known as Bhāskarāchārya (), was an Indian people, Indian polymath, Indian mathematicians, mathematician, astronomer and engineer. From verses in his main work, Siddhānta Śiromaṇi, it can be inferred that he was born in 1114 in Vijjadavida (Vijjalavida) and living in the Satpura mountain ranges of Western Ghats, believed to be the town of Patana in Chalisgaon, located in present-day Khandesh region of Maharashtra by scholars. In a temple in Maharashtra, an inscription supposedly created by his grandson Changadeva, lists Bhaskaracharya's ancestral lineage for several generations before him as well as two generations after him. Henry Thomas Colebrooke, Henry Colebrooke who was the first European to translate (1817) Bhaskaracharya II's mathematical classics refers to the family as Maharashtrian Brahmins residing on the banks of the Godavari River, Godavari. Born in a Hindu Deshastha Brahmin family of scholars, mathematicians and astrono ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bhaskara's Lemma
''Bhaskara's'' Lemma is an identity used as a lemma during the 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: Bhask .... It states that: :\, Nx^2 + k = y^2\implies \,N\left(\frac\right)^2 + \frac = \left(\frac\right)^2 for integers m,\, x,\, y,\, N, and non-zero integer k. Proof The proof follows from simple algebraic manipulations as follows: multiply both sides of the equation by m^2-N, add N^2x^2+2Nmxy+Ny^2, factor, and divide by k^2. :\, Nx^2 + k = y^2\implies Nm^2x^2-N^2x^2+k(m^2-N) = m^2y^2-Ny^2 :\implies Nm^2x^2+2Nmxy+Ny^2+k(m^2-N) = m^2y^2+2Nmxy+N^2x^2 :\implies N(mx+y)^2+k(m^2-N) = (my+Nx)^2 :\implies \,N\left(\frac\right)^2 + \frac = \left(\frac\right)^2. So long as neither k nor m^2-N are zero, the implication goes in both directions. (The lemma holds f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brahmagupta's Identity
In algebra, Brahmagupta's identity says that, for given n, the product of two numbers of the form a^2+nb^2 is itself a number of that form. In other words, the set of such numbers is closed under multiplication. Specifically: :\begin \left(a^2 + nb^2\right)\left(c^2 + nd^2\right) & = \left(ac-nbd\right)^2 + n\left(ad+bc\right)^2 & & & (1) \\ & = \left(ac+nbd\right)^2 + n\left(ad-bc\right)^2, & & & (2) \end Both (1) and (2) can be verified by expanding each side of the equation. Also, (2) can be obtained from (1), or (1) from (2), by changing ''b'' to −''b''. This identity holds in both the ring of integers and the ring of rational numbers, and more generally in any commutative ring. History The identity is a generalization of the so-called Fibonacci identity (where ''n''=1) which is actually found in Diophantus' ''Arithmetica'' (III, 19). That identity was rediscovered by Brahmagupta (598–668), an Indian mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bijaganita
''Bijaganita'' ( iːd͡ʒəgəɳit̪ᵊ, -ɪt̪ᵊ IAST: ') was treatise on algebra by the Indian mathematician Bhāskara II. It is the second volume of his main work '' Siddhānta Shiromani (''"Crown of treatises") alongside '' Lilāvati'', ''Grahaganita'' and ''Golādhyāya''. Meaning The title of the work, , which literally translates to "mathematics () using seeds ()", is one of the two main branches of mediaeval Indian mathematics, the other being , or "mathematics using algorithms". derives its name from the fact that "it employs algebraic equations () which are compared to seeds () of plants since they have the potentiality to generate solutions to mathematical problems." Contents The book is divided into six parts, mainly indeterminate equations, quadratic equations, simple equations, surds. The contents are: * Introduction * On Simple Equations * On Quadratic Equations * On Equations involving indeterminate Questions of the 1st Degree * On Equations involving inde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use Conditional (computer programming), conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning). In contrast, a Heuristic (computer science), heuristic is an approach to solving problems without well-defined correct or optimal results.David A. Grossman, Ophir Frieder, ''Information Retrieval: Algorithms and Heuristics'', 2nd edition, 2004, For example, although social media recommender systems are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation. As an e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root
In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4^2 = (-4)^2 = 16. Every nonnegative real number has a unique nonnegative square root, called the ''principal square root'' or simply ''the square root'' (with a definite article, see below), which is denoted by \sqrt, where the symbol "\sqrt" is called the '' radical sign'' or ''radix''. For example, to express the fact that the principal square root of 9 is 3, we write \sqrt = 3. The term (or number) whose square root is being considered is known as the ''radicand''. The radicand is the number or expression underneath the radical sign, in this case, 9. For non-negative , the principal square root can also be written in exponent notation, as x^. Every positive number has two square roots: \sqrt (which is positive) and -\sqrt (which i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantine Equations
''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name: *Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated ... * Diophantine equation * Diophantine quintuple * Diophantine set {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Science+Business Media
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-European Roots Of Mathematics
The Non-European Unity Movement (NEUM) was a Trotskyist organisation formed in South Africa in 1943. It had links to the Workers Party of South Africa (WPSA), the first countrywide Trotskyist organisation, and was initially conceived as a broad protest front. It proposed a 10 Point Programme of radical reforms. It stressed non-racialism, meaning that it rejected race-based organising (and the concept of race itself), unlike the main nationalist groups of the time, was highly critical of the South African Communist Party and the African National Congress, and made a principle of non-collaboration with the apartheid regime and its allies The movement developed a substantial influence in the Cape Province, including Pondoland, and had some role in the 1950-1961 Pondoland peasant revolt, but split in 1957. The faction around Isaac Bangani Tabata formed a new African Peoples' Democratic Union of Southern Africa (APDUSA) in 1961, and the Unity Movement of South Africa (UMSA) in ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The American Mathematical Monthly
''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. The editor-in-chief is Vadim Ponomarenko (San Diego State University). The journal gives the Lester R. Ford Award annually to "authors of articles of expository excellence" published in the journal. Editors-in-chief The following persons are or have been editor-in-chief: See also *''Mathematics Magazine'' *''Notices of the American Mathematical Society ''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Florian Cajori
Florian Cajori (February 28, 1859 – August 14 or 15, 1930) was a Swiss-American historian of mathematics. Biography Florian Cajori was born in Zillis, Switzerland, as the son of Georg Cajori and Catherine Camenisch. He attended schools first in Zillis and later in Chur. In 1875, Florian Cajori emigrated to the United States at the age of sixteen, and attended the State Normal school in Whitewater, Wisconsin. After graduating in 1878, he taught in a country school, and later began studying mathematics at University of Wisconsin–Madison. In 1883, Cajori received both his bachelor's and master's degrees from the University of Wisconsin–Madison, briefly attended Johns Hopkins University for 8 months in between degrees. He taught for a few years at Tulane University, before being appointed as professor of applied mathematics there in 1887. He was then driven north by tuberculosis. He founded the Colorado College Scientific Society and taught at Colorado College where he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
