The Indian astronomer Brahmagupta who lived in the 7th century is credited with introducing what fundamental mathematical concept?
"work exploring the longitudes of the planets, diurnal rotation, lunar and solar eclipses, risings and settings, the moon's crescent and conjunctions of the planets, are discussed in his treatise Khandakhadyaka. Brahmagupta Brahmagupta () (born , died ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ""Brāhmasphuṭasiddhānta"" (BSS, ""correctly established doctrine of Brahma"", dated 628), a theoretical treatise, and the ""Khaṇḍakhādyaka"" (""edible bite"", dated 665), a more practical text. Brahmagupta was the first to give rules to compute with ""zero"". The texts composed by Brahmagupta were composed in elliptic verse"
"crystals"". In the 7th century, Brahmagupta identified the Brahmagupta theorem, Brahmagupta's identity and Brahmagupta's formula, and for the first time, in ""Brahma-sphuta-siddhanta"", he lucidly explained the use of zero as both a placeholder and decimal digit, and explained the Hindu–Arabic numeral system. It was from a translation of this Indian text on mathematics (c. 770) that Islamic mathematicians were introduced to this numeral system, which they adapted as Arabic numerals. Islamic scholars carried knowledge of this number system to Europe by the 12th century, and it has now displaced all older number systems throughout the world. Various symbol sets are"
"Brahmagupta's interpolation formula Brahmagupta's interpolation formula is a second-order polynomial interpolation formula developed by the Indian mathematician and astronomer Brahmagupta (598–668 CE) in the early 7th century CE. The Sanskrit couplet describing the formula can be found in the supplementary part of ""Khandakadyaka"" a work of Brahmagupta completed in 665 CE. The same couplet appears in Brahmagupta's earlier ""Dhyana-graha-adhikara"", which was probably written ""near the beginning of the second quarter of the 7th century CE, if not earlier."" Brahmagupta was the one of the first to describe and use an interpolation formula using second-order differences. Brahmagupa's interpolation formula is equivalent"
"wrote two texts on mathematics and astronomy: The Brahma Sphuta Siddhanta in 628, and the Khandakhadyaka in 665. He made seminal contributions to mathematics, including the first mathematical treatment of zero, rules for manipulating positive and negative numbers, as well as algorithms for algebraic operations on decimal numbers. His work on astronomy and mathematics was transmitted to the court of the Abbasid Caliph Al-Mansur (r. 754-775 CE), who had the Indian astronomical texts translated into Arabic. Through these texts, the decimal number system spread through the Arab world and later Europe. The Sanskrit poet Magha, the author of ""Sisupalavadha,"" lived"
"Brāhmasphuṭasiddhānta The Brāhmasphuṭasiddhānta (""Correctly Established Doctrine of Brahma"", abbreviated BSS) is the main work of Brahmagupta, written c. 628. Τhis text of mathematical astronomy contains significant mathematical content, including a good understanding of the role of zero, rules for manipulating both negative and positive numbers, a method for computing square roots, methods of solving linear and quadratic equations, and rules for summing series, Brahmagupta's identity, and Brahmagupta’s theorem. The book was written completely in verse and does not contain any kind of mathematical notation. Nevertheless, it contained the first clear description of the quadratic formula (the solution of the quadratic"
"In 628 AD, Brahmagupta suggested that gravity was a force of attraction. He also lucidly explained the use of zero as both a placeholder and a decimal digit, along with the Hindu-Arabic numeral system now used universally throughout the world. Arabic translations of the two astronomers' texts were soon available in the Islamic world, introducing what would become Arabic numerals to the Islamic world by the 9th century. During the 14th–16th centuries, the Kerala school of astronomy and mathematics made significant advances in astronomy and especially mathematics, including fields such as trigonometry and analysis. In particular, Madhava of Sangamagrama is"
"fundamental principles of mathematics in 332 shlokas. The treatise contained: Aryabhata also wrote the ""Arya Siddhanta"", which is now lost. Aryabhata's contributions include: Trigonometry: Arithmetic: Algebra: Mathematical astronomy: Varahamihira (505–587) produced the ""Pancha Siddhanta"" (""The Five Astronomical Canons""). He made important contributions to trigonometry, including sine and cosine tables to 4 decimal places of accuracy and the following formulas relating sine and cosine functions: In the 7th century, two separate fields, arithmetic (which included measurement) and algebra, began to emerge in Indian mathematics. The two fields would later be called ' (literally ""mathematics of algorithms"") and ' (lit. ""mathematics of"
"mathematician Brahmagupta in the 7th century CE. Several centuries later, his ideas were extended to a complete solution of Pell's equation known as the chakravala method, attributed to either of the Indian mathematicians Jayadeva or Bhāskara II. The problem of representing integers by sums of two squares was considered in the 3rd century by Diophantus. In the 17th century, inspired while reading Diophantus's Arithmetica, Fermat made several observations about representations by specific quadratic forms including that which is now known as Fermat's theorem on sums of two squares. Euler provided the first proofs of Fermat's observations and added some new"
"Aryabhata Aryabhata (IAST: ) 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"" (499 CE, when he was 23 years old) and the ""Arya-siddhanta"". 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 meter either. Aryabhata"
"work ""Khanda-khādyaka"", a practical manual of Indian astronomy in the ""karana"" category meant to be used by students. Brahmagupta lived beyond 665 CE. He is believed to have died in Ujjain. Brahmagupta directed a great deal of criticism towards the work of rival astronomers, and his ""Brahmasphutasiddhanta"" displays one of the earliest schisms among Indian mathematicians. The division was primarily about the application of mathematics to the physical world, rather than about the mathematics itself. In Brahmagupta's case, the disagreements stemmed largely from the choice of astronomical parameters and theories. Critiques of rival theories appear throughout the first ten astronomical"
"known to the Indian scientists. Presumably, Bhaskara did not invent it, but he was the first having no compunctions to use the Brahmi numerals in a scientific contribution in Sanskrit. Bhaskara wrote three astronomical contributions. In 629 he annotated the ""Aryabhatiya"", written in verses, about mathematical astronomy. The comments referred exactly to the 33 verses dealing with mathematics. There he considered variable equations and trigonometric formulae. His work ""Mahabhaskariya"" divides into eight chapters about mathematical astronomy. In chapter 7, he gives a remarkable approximation formula for sin x, that is which he assigns to Aryabhata. It reveals a relative error"
"equation). Brāhmasphuṭasiddhānta is one of the first books to provide concrete ideas on positive numbers, negative numbers, and zero. He wrote the following rules: The last two of these rules are notable as the earliest attempt to define division by zero, even though they are not compatible with modern number theory (division by zero is undefined for a field). Brāhmasphuṭasiddhānta The Brāhmasphuṭasiddhānta (""Correctly Established Doctrine of Brahma"", abbreviated BSS) is the main work of Brahmagupta, written c. 628. Τhis text of mathematical astronomy contains significant mathematical content, including a good understanding of the role of zero, rules for manipulating both"
"from India to Europe. (See Zero: Etymology.) In addition to ""Surya Prajnapti"", important Jain works on mathematics included the ""Sthananga Sutra"" (c. 300 BCE – 200 CE); the ""Anuyogadwara Sutra"" (c. 200 BCE – 100 CE); and the ""Satkhandagama"" (c. 2nd century CE). Important Jain mathematicians included Bhadrabahu (d. 298 BCE), the author of two astronomical works, the ""Bhadrabahavi-Samhita"" and a commentary on the ""Surya Prajinapti""; Yativrisham Acharya (c. 176 BCE), who authored a mathematical text called ""Tiloyapannati""; and Umasvati (c. 150 BCE), who, although better known for his influential writings on Jain philosophy and metaphysics, composed a mathematical work"
"to be so until the Colonial period. The main authors of classical Indian mathematics (400 CE to 1200 CE) were scholars like Aryabhata, Brahmagupta, and Bhaskara II. Indian mathematicians made early contributions to the study of the decimal number system, zero, negative numbers, arithmetic, and algebra. In addition, trigonometry, having evolved in the Hellenistic world and having been introduced into ancient India through the translation of Greek works, was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China, and Europe and led"
"influence on the development of Islamic astronomy. Its contents are preserved to some extent in the works of Varahamihira (flourished c. 550), Bhaskara I (flourished c. 629), Brahmagupta (598 – c. 665), and others. It is one of the earliest astronomical works to assign the start of each day to midnight. ""Aryabhatiya"" was particularly popular in South India, where numerous mathematicians over the ensuing millennium wrote commentaries. The work was written in verse couplets and deals with mathematics and astronomy. Following an introduction that contains astronomical tables and Aryabhata’s system of phonemic number notation in which numbers are represented by"
"computation of areas and volumes. Brahmagupta wrote his astronomical work """" in 628. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: ""basic operations"" (including cube roots, fractions, ratio and proportion, and barter) and ""practical mathematics"" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In the latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral: Brahmagupta's theorem: If a cyclic quadrilateral has diagonals that are perpendicular to each other, then the perpendicular line drawn from the point of intersection of the diagonals to any side of"
"Bhāskara I Bhāskara () (commonly called Bhaskara I to avoid confusion with the 12th-century mathematician Bhāskara II) was a 7th-century mathematician, who was the first to write numbers in the Hindu decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's work. This commentary, ""Āryabhaṭīyabhāṣya"", written in 629 CE, is among the oldest known prose works in Sanskrit on mathematics and astronomy. He also wrote two astronomical works in the line of Aryabhata's school, the ""Mahābhāskarīya"" and the ""Laghubhāskarīya"". On 7 June 1979 the"
"is uncertain. LV Gurjar dates it no later than the 4th century, Hoernle dates it between the third and fourth centuries, Ayyangar and Pingree dates it to the 8th or 9th centuries, and George Gheverghese Joseph dates it to about AD 400 and no later than the early 7th century, During the 7th century AD, negative numbers were used in India to represent debts. The Indian mathematician Brahmagupta, in ""Brahma-Sphuta-Siddhanta"" (written c. AD 630), discussed the use of negative numbers to produce the general form quadratic formula that remains in use today. He also found negative solutions of quadratic equations"
"used spherical trigonometry in his calculations to improve the calendar system and Chinese astronomy. The mathematical science of the Chinese would incorporate the work and teaching of Arab missionaries with knowledge of spherical trigonometry who had come to China in the course of the thirteenth century. Although the origin of our present system of numerical notation is ancient, there is no doubt that it was in use among the Hindus over two thousand years ago. The algebraic notation of the Indian mathematician, Brahmagupta, was syncopated. Addition was indicated by placing the numbers side by side, subtraction by placing a dot"
"Hindu mythology. Madhava of Sangamagrama (c. 1340 – 1425) and his Kerala school of astronomy and mathematics developed and founded mathematical analysis. The infinite series for π was stated by him, and he made use of the series expansion of formula_7 to obtain an infinite series expression, now known as the ""Madhava-Gregory series"", for formula_8. Their rational approximation of the ""error"" for the finite sum of their series are of particular interest. They manipulated the error term to derive a faster converging series for formula_8. They used the improved series to derive a rational expression, formula_10 for formula_8 correct up"
"east. The mathematical astronomy text ""Brahmasiddhanta"" of Brahmagupta (598-668) was received in the court of Al-Mansur (753–774). It was translated by Alfazari into Arabic as ""Az-Zīj ‛alā Sinī al-‛Arab"", popularly called ""Sindhind"". This translation was the means by which the Hindu numerals were transmitted from the Sanskrit to the Arabic tradition. According to Al-Biruni, Al-Fazari also translated the ""Khandakhadyaka (Arakand) of"" Brahmagupta. Through the resulting Arabic translations of ""Sindhind"" and ""Arakand"", the use of Indian numerals became established in the Islamic world. The etymology of the word ""sine"" comes from the Latin mistranslation of the word ""jiba"", which is an"
"yields the following approximate value for : This is Stirling's interpolation formula truncated at the second-order differences. It is not known how Brahmagupta arrived at his interpolation formula. Brahmagupta has given a separate formula for the case where the values of the independent variable are not equally spaced. Brahmagupta's interpolation formula Brahmagupta's interpolation formula is a second-order polynomial interpolation formula developed by the Indian mathematician and astronomer Brahmagupta (598–668 CE) in the early 7th century CE. The Sanskrit couplet describing the formula can be found in the supplementary part of ""Khandakadyaka"" a work of Brahmagupta completed in 665 CE. The"
"Indian Space Research Organisation launched Bhaskara I honouring the mathematician. Little is known about Bhāskara's life. He was probably a Marathi astronomer. He was born at Bori, in Parbhani district of Maharashtra state in India in 7th century. His astronomical education was given by his father. Bhaskara is considered the most important scholar of Aryabhata's astronomical school. He and Brahmagupta are two of the most renowned Indian mathematicians who made considerable contributions to the study of fractions. Bhaskara's probably most important mathematical contribution concerns the representation of numbers in a positional system. The first positional representations had been known to"
"from three different centuries: from 224-383 AD, 680-779 AD, and 885-993 AD. It is not known how fragments from different centuries came to be packaged together. This period is often known as the golden age of Indian Mathematics. This period saw mathematicians such as Aryabhata, Varahamihira, Brahmagupta, Bhaskara I, Mahavira, Bhaskara II, Madhava of Sangamagrama and Nilakantha Somayaji give broader and clearer shape to many branches of mathematics. Their contributions would spread to Asia, the Middle East, and eventually to Europe. Unlike Vedic mathematics, their works included both astronomical and mathematical contributions. In fact, mathematics of that period was included"
"During the 6th century, astronomy was influenced by the Greek and Byzantine astronomical traditions. Aryabhata (476–550), in his magnum opus ""Aryabhatiya"" (499), propounded a computational system based on a planetary model in which the Earth was taken to be spinning on its axis and the periods of the planets were given with respect to the Sun. He accurately calculated many astronomical constants, such as the periods of the planets, times of the solar and lunar eclipses, and the instantaneous motion of the Moon. Early followers of Aryabhata's model included Varahamihira, Brahmagupta, and Bhaskara II. Astronomy was advanced during the Shunga"
"of ""formula_6"" and ""formula_7"" makes this expression especially pleasing as a mnemonic. Equivalent forms of Bhaskara I's formula have been given by almost all subsequent astronomers and mathematicians of India. For example, Brahmagupta's (598 – 668 CE) ""Brhma-Sphuta-Siddhanta"" (verses 23 – 24, Chapter XIV) gives the formula in the following form: Also, Bhaskara II (1114 – 1185 CE) has given this formula in his Lilavati (Kshetra-vyavahara, Soka No.48) in the following form: The formula is applicable for values of ""x""° in the range from 0 to 180. The formula is remarkably accurate in this range. The graphs of sin ("
"Euclidean algorithm, which was probably discovered independently in India. Āryabhaṭa seems to have had in mind applications to astronomical calculations. Brahmagupta (628 CE) started the systematic study of indefinite quadratic equations—in particular, the misnamed Pell equation, in which Archimedes may have first been interested, and which did not start to be solved in the West until the time of Fermat and Euler. Later Sanskrit authors would follow, using Brahmagupta's technical terminology. A general procedure (the chakravala, or ""cyclic method"") for solving Pell's equation was finally found by Jayadeva (cited in the eleventh century; his work is otherwise lost); the earliest"
"the Gurjaradesa, the second largest kingdom of Western India, comprising southern Rajasthan and northern Gujarat in modern-day India. It was also a centre of learning for mathematics and astronomy. Brahmagupta became an astronomer of the ""Brahmapaksha"" school, one of the four major schools of Indian astronomy during this period. He studied the five traditional ""siddhanthas"" on Indian astronomy as well as the work of other astronomers including Aryabhata I, Latadeva, Pradyumna, Varahamihira, Simha, Srisena, Vijayanandin and Vishnuchandra. In the year 628, at an age of 30, he composed the ""Brāhmasphuṭasiddhānta"" (the improved treatise of Brahma) which is believed to be"
"was the spread of the decimal number system used in the texts. The mathematician Al-Khwarizmi (800–850 CE) wrote a text called ""al-Jam wal-tafriq bi hisal-al-Hind"" (Addition and Subtraction in Indian Arithmetic), which was translated into Latin in the 13th century as ""Algorithmi de numero indorum"". Through these texts, the decimal number system and Brahmagupta's algorithms for arithmetic have spread throughout the world. Al-Khwarizmi also wrote his own version of ""Sindhind"", drawing on Al-Fazari's version and incorporating Ptolemaic elements. Indian astronomic material circulated widely for centuries, even passing into medieval Latin texts. Brahmagupta gave the solution of the general linear equation"
"seeds,"" with ""seeds""—like the seeds of plants—representing unknowns with the potential to generate, in this case, the solutions of equations). Brahmagupta, in his astronomical work """" (628 CE), included two chapters (12 and 18) devoted to these fields. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: ""basic operations"" (including cube roots, fractions, ratio and proportion, and barter) and ""practical mathematics"" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In the latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral: Brahmagupta's theorem: If a cyclic quadrilateral"
"the period following the 7th-century works of Brahmagupta and Bhāskara I. It generally treats the same astronomical subject matter and demonstrates the same computational techniques as earlier authors, although there are some significant innovations, such that Lalla’s treatise offers a compromise between the rival astronomical schools of his predecessors, Āryabhaṭa I and Brahmagupta. Although Lalla is known to be a follower of Āryabhaṭa, he combines techniques from both. It is within ""Śiṣyadhīv ṛddhidatantra"" that the earliest known description of a ""perpetuum mobile"" is described. Lalla's other works such as his text, "" Jyotiṣaratnakośa"" or ""Treasury of Jewels"", is his treatise"
"Haridatta Haridatta (c. 683 CE) was an astronomer-mathematician of Kerala, India, who is believed to be the promulgator of the Parahita system of astronomical computations. This system of computations is widely popular in Kerala and Tamil Nadu. According to legends, Haridatta promulgated the Parahita system on the occasion of the ""Mamankam"" held in the year 683 CE. Mamankam was a 12-yearly festival held in Thirunnavaya on the banks of the Bharathapuzha river. The distinctive contribution of Harfidatta, apart from his resolving the Aryabhatiya calculations and using the Katapayadi system of numerals is the corrections he introduced to the values of"
"include: Arithmetic: Algebra: Geometry: Calculus: Trigonometry: The Kerala school of astronomy and mathematics was founded by Madhava of Sangamagrama in Kerala, South India and included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. It flourished between the 14th and 16th centuries and the original discoveries of the school seems to have ended with Narayana Bhattathiri (1559–1632). In attempting to solve astronomical problems, the Kerala school astronomers ""independently"" created a number of important mathematics concepts. The most important results, series expansion for trigonometric functions, were given in Sanskrit verse in a book by Neelakanta"