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Trairāśika
Trairāśika is the Sanskrit term used by Indian astronomers and mathematicians of the pre-modern era to denote what is known as the " rule of three" in elementary mathematics and algebra. In the contemporary mathematical literature, the term "rule of three" refers to the principle of cross-multiplication which states that if \tfrac=\tfrac then ad=bc or a=\tfrac. The antiquity of the term ''trairāśika'' is attested by its presence in the Bakhshali manuscript, a document believed to have been composed in the early centuries of the Common Era. The ''trairāśika'' rule Basically ''trairāśika'' is a rule which helps to solve the following problem: :"If p produces h what would i produce?" Here p is referred to as ''pramāṇa'' ("argument"), h as ''phala'' ("fruit") and i as ''ichcā'' ("requisition"). The ''pramāṇa'' and ''icchā'' must be of the same denomination, that is, of the same kind or type like weights, money, time, or numbers of the same objects. ''Phala'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cross-multiplication
In mathematics, specifically in elementary arithmetic and elementary algebra, given an equation between two Fraction (mathematics), fractions or rational fraction, rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable. The method is also occasionally known as the "cross your heart" method because lines resembling a heart outline can be drawn to remember which things to multiply together. Given an equation like : \frac a b = \frac c d, where and are not zero, one can cross-multiply to get : ad = bc \quad \text \quad a = \fracd. In Euclidean geometry the same calculation can be achieved by considering the ratios as those of similar triangles. Procedure In practice, the method of cross-multiplying means that we multiply the numerator of each (or one) side by the denominator of the other side, effectively crossing the terms over: : \frac a b \nwarrow \frac c d, \quad \frac a b \nearrow \frac c d. The mathematical justific ... [...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] |
Sanskrit
Sanskrit (; stem form ; nominal singular , ,) is a classical language belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. It arose in northwest South Asia after its predecessor languages had Trans-cultural diffusion, diffused there from the northwest in the late Bronze Age#South Asia, Bronze Age. Sanskrit is the sacred language of Hinduism, the language of classical Hindu philosophy, and of historical texts of Buddhism and Jainism. It was a lingua franca, link language in ancient and medieval South Asia, and upon transmission of Hindu and Buddhist culture to Southeast Asia, East Asia and Central Asia in the early medieval era, it became a language of religion and high culture, and of the political elites in some of these regions. As a result, Sanskrit had a lasting effect on the languages of South Asia, Southeast Asia and East Asia, especially in their formal and learned vocabularies. Sanskrit generally connotes several Indo-Aryan languages# ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aryabhata
Aryabhata ( ISO: ) or Aryabhata I (476–550 CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the '' Āryabhaṭīya'' (which mentions that in 3600 '' Kali Yuga'', 499 CE, he was 23 years old) and the ''Arya- siddhanta''. For his explicit mention of the relativity of motion, he also qualifies as a major early physicist. Biography Name While there is a tendency to misspell his name as "Aryabhatta" by analogy with other names having the " bhatta" suffix, his name is properly spelled Aryabhata: every astronomical text spells his name thus, including Brahmagupta's references to him "in more than a hundred places by name". Furthermore, in most instances "Aryabhatta" would not fit the metre either. Time and place of birth Aryabhata mentions in the ''Aryabhatiya'' that he was 23 years old 3,600 years into the '' Kali Yuga'', but this is not to mean that the text was composed at that ti ... [...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] |
Yuktibhāṣā
''Yuktibhāṣā'' (), also known as Gaṇita-yukti-bhāṣā and ( English: ''Compendium of Astronomical Rationale''), is a major treatise on mathematics and astronomy, written by the Indian astronomer Jyesthadeva of the Kerala school of mathematics around 1530. The treatise, written in Malayalam, is a consolidation of the discoveries by Madhava of Sangamagrama, Nilakantha Somayaji, Parameshvara, Jyeshtadeva, Achyuta Pisharati, and other astronomer-mathematicians of the Kerala school. It also exists in a Sanskrit version, with unclear author and date, composed as a rough translation of the Malayalam original. The work contains proofs and derivations of the theorems that it presents. Modern historians used to assert, based on the works of Indian mathematics that first became available, that early Indian scholars in astronomy and computation lacked in proofs, but demonstrates otherwise. Some of its important topics include the infinite series expansions of functions; pow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kerala School Of Astronomy And Mathematics
The Kerala school of astronomy and mathematics or the Kerala school was a school of Indian mathematics, mathematics and Indian astronomy, astronomy founded by Madhava of Sangamagrama in Kingdom of Tanur, Tirur, Malappuram district, Malappuram, Kerala, India, which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. The school flourished between the 14th and 16th centuries and its original discoveries seem to have ended with Melpathur Narayana Bhattathiri, Narayana Bhattathiri (1559–1632). In attempting to solve astronomical problems, the Kerala school independently discovered a number of important mathematical concepts. Their most important results—series expansion for trigonometric functions—were described in Sanskrit verse in a book by Neelakanta called ''Tantrasangraha'' (around 1500), and again in a commentary on this work, called ''Tantrasangraha-vakhya'', of unknown authors ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Līlāvatī
''Līlāvatī'' is a treatise by Indian mathematician Bhāskara II on mathematics, written in 1150 AD. It is the first volume of his main work, the ''Siddhānta Shiromani'', alongside the ''Bijaganita'', the ''Grahaganita'' and the ''Golādhyāya''. Name Bhaskara II's book on arithmetic is the subject of interesting legends that assert that it was written for his daughter, Lilavati. As the story goes, the author had studied Lilavati's horoscope and predicted that she would remain both childless and unmarried. To avoid this fate, he ascertained an auspicious moment for his daughter's wedding. To alert his daughter at the correct time, he placed a cup with a small hole at the bottom of a vessel filled with water, arranged so that the cup would sink at the beginning of the propitious hour. He put the device in a room with a warning to Lilavati to not go near it. In her curiosity, though, she went to look at the device. A pearl from her bridal dress accidentally dropped into it, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractions (mathematics)
A fraction (from , "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A ''common'', ''vulgar'', or ''simple'' fraction (examples: and ) consists of an integer numerator, displayed above a line (or before a slash like ), and a non-zero integer denominator, displayed below (or after) that line. If these integers are positive, then the numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. For example, in the fraction , the numerator 3 indicates that the fraction represents 3 equal parts, and the denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates of a cake. Fractions can be used to represent ratios and division. Thus the fraction can be used to represent the ratio 3:4 (the ratio of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic
Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. Arithmetic systems can be distinguished based on the type of numbers they operate on. Integer arithmetic is about calculations with positive and negative integers. Rational number arithmetic involves operations on fractions of integers. Real number arithmetic is about calculations with real numbers, which include both rational and irrational numbers. Another distinction is based on the numeral system employed to perform calculations. Decimal arithmetic is the most common. It uses the basic numerals from 0 to 9 and their combinations to express numbers. Binary arithmetic, by contrast, is used by most computers and represents numbers as combinations of the basic numerals 0 and 1. Computer arithmetic deals with the specificities of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indian Mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, Bhaskara II, Varāhamihira, and Madhava of Sangamagrama, Madhava. The Decimal, decimal number system in use today: "The measure of the genius of Indian civilisation, to which we owe our modern (number) system, is all the greater in that it was the only one in all history to have achieved this triumph. Some cultures succeeded, earlier than the Indian, in discovering one or at best two of the characteristics of this intellectual feat. But none of them managed to bring together into a complete and coherent system the necessary and sufficient conditions for a number-system with the same potential as our own." was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of 0 (number), ze ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
