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Kuṭṭaka
Kuṭṭaka is an algorithm for finding integer solutions of linear Diophantine equations. A linear Diophantine equation is an equation of the form ''ax'' + ''by'' = ''c'' where ''x'' and ''y'' are unknown quantities and ''a'', ''b'', and ''c'' are known quantities with integer values. The algorithm was originally invented by the Indian astronomer-mathematician Āryabhaṭa (476–550 CE) and is described very briefly in his Āryabhaṭīya. Āryabhaṭa did not give the algorithm the name ''Kuṭṭaka'', and his description of the method was mostly obscure and incomprehensible. It was Bhāskara I (c. 600 – c. 680) who gave a detailed description of the algorithm with several examples from astronomy in his ''Āryabhatiyabhāṣya'', who gave the algorithm the name ''Kuṭṭaka''. In Sanskrit, the word Kuṭṭaka means ''pulverization'' (reducing to powder), and it indicates the nature of the algorithm. The algorithm in essence is a process where the coefficients in a given line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bhāskara II
Bhāskara II (c. 1114–1185), also known as Bhāskarāchārya ("Bhāskara, the teacher"), and as Bhāskara II to avoid confusion with Bhāskara I, was an Indian mathematician and astronomer. From verses, in his main work, Siddhānta Shiromani (सिद्धांतशिरोमणी), it can be inferred that he was born in 1114 in Vijjadavida (Vijjalavida) and living in the Sahyadri mountain ranges of Western Ghats, believed to be the town of Patan in Chalisgaon, located in present-day Khandesh region of Maharashtra by scholars. He is the only ancient mathematician who has been immortalized on a monument. 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. Colebrooke who was the first European to translate (1817) Bhaskaracharya II's mathematical classics refers to the family as Maharashtrian Brahmins residing on the ban ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kuṭṭākāra Śirōmaṇi
The ''Kuṭṭākāra Śirōmaṇi'' is a medieval Indian treatise in Sanskrit devoted exclusively to the study of the Kuṭṭākāra, or Kuṭṭaka, an algorithm for solving linear Diophantine equations. It is authored by one Dēvarāja about whom little is known. From statements given by the author at the end of the book, one can infer that the name of Dēvarāja's father was Varadarājācārya, then famously known as Siddhāntavallabha. Since the book contains a few verses from the Lilavati, it should have been composed during a period after the Lilavati was composed, that is after 1150 CE. Treatises such as the Kuṭṭākāra Śirōmaṇi devoted exclusively to specialized topics are very rare in Indian mathematical literature. The algorithm was first formulated by Aryabhata I and given in verses in the Ganitapada of his Aryabhatiya. Aryabhata's description of the algorithm was brief and hence obscure and incomprehensible. However, from the interpretations of the verses b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantine Equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents. Diophantine problems have fewer equations than unknowns and involve finding integers that solve simultaneously all equations. As such systems of equations define algebraic curves, algebraic surfaces, or, more generally, algebraic sets, their study is a part of algebraic geometry that is called ''Diophantine geometry''. The word ''Diophantine'' refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra. The mathematical study of Diophantine problems that Di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Āryabhaṭa
Aryabhata (ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the '' Aryabhatiya'' (which mentions that in 3600 '' Kali Yuga'', 499 CE, he was 23 years old) and the ''Arya-siddhanta.'' Aryabhata created a system of phonemic number notation in which numbers were represented by consonant-vowel monosyllables. Later commentators such as Brahmagupta divide his work into ''Ganita ("Mathematics"), Kalakriya ("Calculations on Time") and Golapada ("Spherical Astronomy")''. His pure mathematics discusses topics such as determination of square and cube roots, geometrical figures with their properties and mensuration, arithmetric progression problems on the shadow of the gnomon, quadratic equations, linear and indeterminate equations. Aryabhata calculated the value of pi (''π)'' to the fourth decimal digit and was likely aware th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aryabhatiya
''Aryabhatiya'' (IAST: ') or ''Aryabhatiyam'' ('), a Sanskrit astronomical treatise, is the ''magnum opus'' and only known surviving work of the 5th century Indian mathematician Aryabhata. Philosopher of astronomy Roger Billard estimates that the book was composed around 510 CE based on historical references it mentions. Structure and style Aryabhatiya is written in Sanskrit and divided into four sections; it covers a total of 121 verses describing different moralitus via a mnemonic writing style typical for such works in India (see definitions below): 1. Gitikapada (13 verses): large units of time—kalpa, manvantara, and yuga—which present a cosmology different from earlier texts such as Lagadha's Vedanga Jyotisha (ca. 1st century BCE). There is also a table of ine (jya), given in a single verse. The duration of the planetary revolutions during a mahayuga is given as 4.32 million years. 2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra); arithmetic and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extended Euclidean Algorithm
In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers ''a'' and ''b'', also the coefficients of Bézout's identity, which are integers ''x'' and ''y'' such that : ax + by = \gcd(a, b). This is a certifying algorithm, because the gcd is the only number that can simultaneously satisfy this equation and divide the inputs. It allows one to compute also, with almost no extra cost, the quotients of ''a'' and ''b'' by their greatest common divisor. also refers to a very similar algorithm for computing the polynomial greatest common divisor and the coefficients of Bézout's identity of two univariate polynomials. The extended Euclidean algorithm is particularly useful when ''a'' and ''b'' are coprime. With that provision, ''x'' is the modular multiplicative inverse of ''a'' modulo ''b'', and ''y'' is the modular multiplicative inverse of ''b'' modul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of 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 perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code execution through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a Heuristic (computer science), heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result. As an effective method, an algorithm ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bibhutibhushan Datta
Bibhutibhushan Datta (also Bibhuti Bhusan Datta; Bengali : বিভূতিভূষণ দত্ত, Bibhūtibhūṣaṇ Datta) (28 June 1888 – 6 October 1958) was a historian of Indian mathematics. Datta came from a poor Bengali family. He was a student of Ganesh Prasad, studied at University of Calcutta and secured the master's degree in mathematics in 1914 and doctorate degree in 1920 in applied mathematics. He taught at Calcutta University where he was lecturer at University Science College, and during 1924–1929 he was Rhashbehari Ghosh Professor of Applied Mathematics. During the 1920s and 1930s he created a reputation as an authority on the history of Indian mathematics. He was also deeply interested in Indian philosophy and religion. In 1929 he retired from his professorship and left the university in 1933, and became a '' sannyasin'' (an ascetic, a person who has renounced worldly pleasures) in 1938 under the name Swami Vidyaranya. ''History of Hindu Mathematics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Division
In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than the absolute value of the divisor. A fundamental property is that the quotient and the remainder exist and are unique, under some conditions. Because of this uniqueness, ''Euclidean division'' is often considered without referring to any method of computation, and without explicitly computing the quotient and the remainder. The methods of computation are called integer division algorithms, the best known of which being long division. Euclidean division, and algorithms to compute it, are fundamental for many questions concerning integers, such as the Euclidean algorithm for finding the greatest common divisor of two integers, and modular arithmetic, for which only remainders are considered. The operation consisting of computing only th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aryabhata II
Āryabhaṭa (c. 920 – c. 1000) was an Indian mathematician and astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, moons, comets and galaxies – in either ..., and the author of the '' Maha-Siddhanta''. The numeral II is given to him to distinguish him from the earlier and more influential Āryabhaṭa I. Scholars are unsure of when exactly he was born, though some give dates of his main publications being between 950–1100. Maha Siddhanta Aryabhata's most eminent work was Maha Siddhanta. The treatise consists of eighteen chapters and was written in the form of verse in Sanskrit. The initial twelve chapters deals with topics related to mathematical astronomy and covers the topics that Indian mathematicians of that period had already worked on. The various topics that have been included in these tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mahavira
Mahavira (Sanskrit: महावीर) also known as Vardhaman, was the 24th ''tirthankara'' (supreme preacher) of Jainism. He was the spiritual successor of the 23rd ''tirthankara'' Parshvanatha. Mahavira was born in the early part of the 6th century BCE into a royal Kshatriya Jain family in ancient India. His mother's name was Trishala and his father's name was Siddhartha. They were lay devotees of Parshvanatha. Mahavira abandoned all worldly possessions at the age of about 30 and left home in pursuit of spiritual awakening, becoming an ascetic. Mahavira practiced intense meditation and severe austerities for twelve and a half years, after which he attained '' Kevala Jnana'' (omniscience). He preached for 30 years and attained Moksha (liberation) in the 6th century BCE, although the year varies by sect. Historically, Mahavira, who revived and preached Jainism in ancient India, was an older contemporary of Gautama Buddha. Jains celebrate ''Mahavir Janma Kalyanak'' every ye ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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