Advancement of science and mathematics.

AKS The Primality Test. .The AKS primality test is a deterministic primality-proving algorithm created and published by three Indian Institute of Technology Kanpur computer scientists, Manindra Agrawal, Neeraj Kayal, and Nitin Saxena on 6 August 2002 in a paper titled PRIMES is in P, Commenting on the impact of this discovery, Paul Leyland noted: "One reason for the excitement within the mathematical community is not only does this algorithm settle a long-standing problem, it also does so in a brilliantly simple manner. Everyone is now wondering what else has been similarly overlooked".

Baudhāyana, (fl. c. 800 BCE)[1] was the author of the Baudhayana sūtras, which cover dharma, daily ritual, mathematics, etc. He belongs to the Yajurveda school, and is older than the other sūtra author Āpastamba. He was the author of the earliest of the Shulba Sutras—appendices to the Vedas giving rules for the construction of altars—called the Baudhāyana Śulbasûtra. These are notable from the point of view of mathematics, for containing several important mathematical results, including giving a value of pi to some degree of precision, and stating a version of what is now known as the Pythagorean theorem. Sequences associated with primitive Pythagorean triples have been named Baudhayana sequences. These sequences have been used in cryptography as random sequences and for the generation of keys

Baudhāyana, (fl. c. 800 BCE)[1] was the author of the Baudhayana sūtras, which cover dharma, daily ritual, mathematics, etc. He belongs to the Yajurveda school, and is older than the other sūtra author Āpastamba. He was the author of the earliest of the Shulba Sutras—appendices to the Vedas giving rules for the construction of altars—called the Baudhāyana Śulbasûtra. These are notable from the point of view of mathematics, for containing several important mathematical results, including giving a value of pi to some degree of precision, and stating a version of what is now known as the Pythagorean theorem. Sequences associated with primitive Pythagorean triples have been named Baudhayana sequences. These sequences have been used in cryptography as random sequences and for the generation of keys

Finite Difference Interpolation: The Indian mathematician Brahmagupta presented what is possibly the first instance[97 of finite difference interpolation around 665 CE.

Algebraic abbreviations: The mathematician Brahmagupta had begun using abbreviations for unknowns by the 7th century. He employed abbreviations for multiple unknowns occurring in one complex problem. Brahmagupta also used abbreviations for square roots and cube roots.

Basu's theorem: The Basu's theorem, a result of Debabrata Basu (1955) states that any complete sufficient statistic is independent of any ancillary statistic.

Brahmagupta–Fibonacci identity, Brahmagupta formula, Brahmagupta matrix, and Brahmagupta theorem: Discovered by the Indian mathematician, Brahmagupta (598–668 CE).

Chakravala method: The Chakravala method, a cyclic algorithm to solve indeterminate quadratic equations is commonly attributed to Bhāskara II, (c. 1114 – 1185 CE) although some attribute it to Jayadeva (c. 950~1000 CE).Jayadeva pointed out that Brahmagupta’s approach to solving equations of this type would yield infinitely large number of solutions, to which he then described a general method of solving such equations. Jayadeva's method was later refined by Bhāskara II in his Bijaganita treatise to be known as the Chakravala method, chakra (derived from cakraṃ चक्रं) meaning 'wheel' in Sanskrit, relevant to the cyclic nature of the algorithm. With reference to the Chakravala method, E. O. Selenuis held that no European performances at the time of Bhāskara, nor much later, came up to its marvellous height of mathematical complexity.

Hindu number system: With decimal place-value and a symbol for zero, this system was the ancestor of the widely used Arabic numeral system. It was developed in the Indian subcontinent between the 1st and 6th centuries CE.

Fibonacci numbers: This sequence was first described by Virahanka (c. 700 AD), Gopāla (c. 1135), and Hemachandra (c as an outgrowth of the earlier writings on Sanskrit prosody by Pingala (c. 200 BC).

Zero, symbol: Indians were the first to use the zero as a symbol and in arithmetic operations, although Babylonians used zero to signify the 'absent'. In those earlier times a blank space was used to denote zero, later when it created confusion a dot was used to denote zero (could be found in Bakhshali manuscript). In 500 AD circa Aryabhata again gave a new symbol for zero (0).

Law of signs in multiplication: The earliest use of notation for negative numbers, as subtrahend, is credited by scholars to the Chinese, dating back to the 2nd century BC. Like the Chinese, the Indians used negative numbers as subtrahend, but were the first to establish the "law of signs" with regards to the multiplication of positive and negative numbers, which did not appear in Chinese texts until 1299. Indian mathematicians were aware of negative numbers by the 7th century, and their role in mathematical problems of debt was understood. Mostly consistent and correct rules for working with negative numbers were formulated, and the diffusion of these rules led the Arab intermediaries to pass it on to Europe.

Madhava series: The infinite series for π and for the trigonometric sine, cosine, and arctangent is now attributed to Madhava of Sangamagrama (c. 1340 – 1425) and his Kerala school of astronomy and mathematics. He made use of the series expansion of \arctan x to obtain an infinite series expression for π.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 π. They used the improved series to derive a rational expression,104348/33215 for π correct up to eleven decimal places, i.e. 3.14159265359. Madhava of Sangamagrama and his successors at the Kerala school of astronomy and mathematics used geometric methods to derive large sum approximations for sine, cosin, and arttangent. They found a number of special cases of series later derived by Brook Taylor series. They also found the second-order Taylor approximations for these functions, and the third-order Taylor approximation for sine.

Pascal's triangle: Described in the 6th century CE by Varahamihira[, and in the 10th century by Halayudha,, commenting on an obscure reference by Pingala (the author of an earlier work on prosody) to the "Meru-prastaara", or the "Staircase of Mount Meru", in relation to binomial coefficients. (It was also independently discovered in the 10th or 11th century in Persia and China.)

Pell's equation, integral solution for: About a thousand years before Pell's time, Indian scholar Brahmagupta (598–668 CE) was able to find integral solutions to vargaprakṛiti (Pell's equation) \ x^2-Ny^2=1, where N is a nonsquare integer, in his Brâhma-sphuṭa-siddhânta treatise.

Ramanujan theta function, Ramanujan prime, Ramanujan summation, Ramanujan graph and Ramanujan's sum: Discovered by the Indian mathematician Srinivasa Ramanujan in the early 20th century.

Shrikhande graph: Graph invented by the Indian mathematician S.S. Shrikhande in 1959.

Sign convention: Symbols, signs and mathematical notation were employed in an early form in India by the 6th century when the mathematician-astronomer Aryabhata recommended the use of letters to represent unknown quantities. By the 7th century Brahmagupta had already begun using abbreviations for unknowns, even for multiple unknowns occurring in one complex problem. Brahmagupta also managed to use abbreviations for square roots and cube roots. By the 7th century fractions were written in a manner similar to the modern times, except for the bar separating the numerator and the denominator. A dot symbol for negative numbers was also employed. The Bakhshali Manuscript displays a cross, much like the modern '+' sign, except that it symbolized subtraction when written just after the number affected. The '=' sign for equality did not exist. Indian mathematics was transmitted to the Islamic world where this notation was seldom accepted initially and the scribes continued to write mathematics in full and without symbols.

Trigonometry was invented in India.* Trigonometric functions (adapted from Greek): * Trigonometric functions (adapted from Greek): The trigonometric functions sine and versine originated in Indian astronomy, adapted from the full-chord Greek versions (to the modern half-chord versions). They were described in detail by Aryabhata in the late 5th century, but were likely developed earlier in the Siddhantas, astronomical treatises of the 3rd or 4th century.Later, the 6th-century astronomer Varahamihira discovered a few basic trigonometric formulas and identities, such as sin^2(x) + cos^2(x) = 1. The first use of the idea of ‘sine’ in the way we use it today was in the work Aryabhatiyam by Aryabhata, in A.D. 500. Aryabhata used the word ardha-jya for the half-chord, which was shortened to jya or jiva in due course. When the Aryabhatiyam was translated into Arabic, the word jiva was retained as it is. The word jiva was translated into sinus, which means curve, when the Arabic version was translated into Latin. Soon the word sinus, also used as sine, became common in mathematical texts throughout Europe. An English Professor of astronomy Edmund Gunter (1581–1626), first used the abbreviated notation ‘sin’. The origin of the terms ‘cosine’ and ‘tangent’ was much later. The cosine function arose from the need to compute the sine of the complementary angle. Aryabhatta called it kotijya. The name cosinus originated with Edmund Gunter. In 1674, the English Mathematician Sir Jonas Moore first used the abbreviated notation ‘cos’.

Algebraic abbreviations: The mathematician Brahmagupta had begun using abbreviations for unknowns by the 7th century. He employed abbreviations for multiple unknowns occurring in one complex problem. Brahmagupta also used abbreviations for square roots and cube roots.

Basu's theorem: The Basu's theorem, a result of Debabrata Basu (1955) states that any complete sufficient statistic is independent of any ancillary statistic.

Brahmagupta–Fibonacci identity, Brahmagupta formula, Brahmagupta matrix, and Brahmagupta theorem: Discovered by the Indian mathematician, Brahmagupta (598–668 CE).

Chakravala method: The Chakravala method, a cyclic algorithm to solve indeterminate quadratic equations is commonly attributed to Bhāskara II, (c. 1114 – 1185 CE) although some attribute it to Jayadeva (c. 950~1000 CE).Jayadeva pointed out that Brahmagupta’s approach to solving equations of this type would yield infinitely large number of solutions, to which he then described a general method of solving such equations. Jayadeva's method was later refined by Bhāskara II in his Bijaganita treatise to be known as the Chakravala method, chakra (derived from cakraṃ चक्रं) meaning 'wheel' in Sanskrit, relevant to the cyclic nature of the algorithm. With reference to the Chakravala method, E. O. Selenuis held that no European performances at the time of Bhāskara, nor much later, came up to its marvellous height of mathematical complexity.

Hindu number system: With decimal place-value and a symbol for zero, this system was the ancestor of the widely used Arabic numeral system. It was developed in the Indian subcontinent between the 1st and 6th centuries CE.

Fibonacci numbers: This sequence was first described by Virahanka (c. 700 AD), Gopāla (c. 1135), and Hemachandra (c as an outgrowth of the earlier writings on Sanskrit prosody by Pingala (c. 200 BC).

Zero, symbol: Indians were the first to use the zero as a symbol and in arithmetic operations, although Babylonians used zero to signify the 'absent'. In those earlier times a blank space was used to denote zero, later when it created confusion a dot was used to denote zero (could be found in Bakhshali manuscript). In 500 AD circa Aryabhata again gave a new symbol for zero (0).

Law of signs in multiplication: The earliest use of notation for negative numbers, as subtrahend, is credited by scholars to the Chinese, dating back to the 2nd century BC. Like the Chinese, the Indians used negative numbers as subtrahend, but were the first to establish the "law of signs" with regards to the multiplication of positive and negative numbers, which did not appear in Chinese texts until 1299. Indian mathematicians were aware of negative numbers by the 7th century, and their role in mathematical problems of debt was understood. Mostly consistent and correct rules for working with negative numbers were formulated, and the diffusion of these rules led the Arab intermediaries to pass it on to Europe.

Madhava series: The infinite series for π and for the trigonometric sine, cosine, and arctangent is now attributed to Madhava of Sangamagrama (c. 1340 – 1425) and his Kerala school of astronomy and mathematics. He made use of the series expansion of \arctan x to obtain an infinite series expression for π.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 π. They used the improved series to derive a rational expression,104348/33215 for π correct up to eleven decimal places, i.e. 3.14159265359. Madhava of Sangamagrama and his successors at the Kerala school of astronomy and mathematics used geometric methods to derive large sum approximations for sine, cosin, and arttangent. They found a number of special cases of series later derived by Brook Taylor series. They also found the second-order Taylor approximations for these functions, and the third-order Taylor approximation for sine.

Pascal's triangle: Described in the 6th century CE by Varahamihira[, and in the 10th century by Halayudha,, commenting on an obscure reference by Pingala (the author of an earlier work on prosody) to the "Meru-prastaara", or the "Staircase of Mount Meru", in relation to binomial coefficients. (It was also independently discovered in the 10th or 11th century in Persia and China.)

Pell's equation, integral solution for: About a thousand years before Pell's time, Indian scholar Brahmagupta (598–668 CE) was able to find integral solutions to vargaprakṛiti (Pell's equation) \ x^2-Ny^2=1, where N is a nonsquare integer, in his Brâhma-sphuṭa-siddhânta treatise.

Ramanujan theta function, Ramanujan prime, Ramanujan summation, Ramanujan graph and Ramanujan's sum: Discovered by the Indian mathematician Srinivasa Ramanujan in the early 20th century.

Shrikhande graph: Graph invented by the Indian mathematician S.S. Shrikhande in 1959.

Sign convention: Symbols, signs and mathematical notation were employed in an early form in India by the 6th century when the mathematician-astronomer Aryabhata recommended the use of letters to represent unknown quantities. By the 7th century Brahmagupta had already begun using abbreviations for unknowns, even for multiple unknowns occurring in one complex problem. Brahmagupta also managed to use abbreviations for square roots and cube roots. By the 7th century fractions were written in a manner similar to the modern times, except for the bar separating the numerator and the denominator. A dot symbol for negative numbers was also employed. The Bakhshali Manuscript displays a cross, much like the modern '+' sign, except that it symbolized subtraction when written just after the number affected. The '=' sign for equality did not exist. Indian mathematics was transmitted to the Islamic world where this notation was seldom accepted initially and the scribes continued to write mathematics in full and without symbols.

Trigonometry was invented in India.* Trigonometric functions (adapted from Greek): * Trigonometric functions (adapted from Greek): The trigonometric functions sine and versine originated in Indian astronomy, adapted from the full-chord Greek versions (to the modern half-chord versions). They were described in detail by Aryabhata in the late 5th century, but were likely developed earlier in the Siddhantas, astronomical treatises of the 3rd or 4th century.Later, the 6th-century astronomer Varahamihira discovered a few basic trigonometric formulas and identities, such as sin^2(x) + cos^2(x) = 1. The first use of the idea of ‘sine’ in the way we use it today was in the work Aryabhatiyam by Aryabhata, in A.D. 500. Aryabhata used the word ardha-jya for the half-chord, which was shortened to jya or jiva in due course. When the Aryabhatiyam was translated into Arabic, the word jiva was retained as it is. The word jiva was translated into sinus, which means curve, when the Arabic version was translated into Latin. Soon the word sinus, also used as sine, became common in mathematical texts throughout Europe. An English Professor of astronomy Edmund Gunter (1581–1626), first used the abbreviated notation ‘sin’. The origin of the terms ‘cosine’ and ‘tangent’ was much later. The cosine function arose from the need to compute the sine of the complementary angle. Aryabhatta called it kotijya. The name cosinus originated with Edmund Gunter. In 1674, the English Mathematician Sir Jonas Moore first used the abbreviated notation ‘cos’.

Medicine

Cataract in the Human Eye—magnified view seen on examination with a slit lamp. Indian surgeon Susruta performed cataract surgery by the 6th century BCE.

Cataract in the Human Eye—magnified view seen on examination with a slit lamp. Indian surgeon Susruta performed cataract surgery by the 6th century BCE.

Amastigotes in a chorionic villus. Upendranath Brahmachari (19 December 1873 – February 6, 1946) discovered Urea Stibamine, a treatment which helped nearly eradicate Visceral leishmaniasis.

Ayurvedic and Siddha medicine: Ayurveda and Siddha are ancient and traditional systems of medicine. Ayurveda dates back to Iron Age India (1st millennium BC) and still practiced today as a form of complementary and alternative medicine. It means "knowledge for longevity". Siddha medicine is mostly prevalent in South India. Herbs and minerals are basic raw materials of the Siddha system which dates back to the period of siddha saints around the 5th century BC.

Cataract surgery: Cataract surgery was known to the Indian physician Sushruta (6th century BCE). In India, cataract surgery was performed with a special tool called the Jabamukhi Salaka, a curved needle used to loosen the lens and push the cataract out of the field of vision] The eye would later be soaked with warm butter and then bandaged. Though this method was successful, Susruta cautioned that cataract surgery should only be performed when absolutely necessary. Greek philosophers and scientists traveled to India where these surgeries were performed by physicians. The removal of cataract by surgery was also introduced into China from India.

Cure for Leprosy: Kearns & Nash (2008) state that the first mention of leprosy is described in the Indian medical treatise Sushruta Samhita (6th century BCE). However, The Oxford Illustrated Companion to Medicine holds that the mention of leprosy, as well as ritualistic cures for it, were described in the Atharva-veda (1500–1200 BCE), written before the Sushruta Samhita.

Plastic surgery: Plastic surgery was being carried out in India by 2000 BCE. The system of punishment by deforming a miscreant's body may have led to an increase in demand for this practice.The surgeon Sushruta contributed mainly to the field of plastic and cataract surgery. The medical works of both Sushruta and Charak were translated into Arabic language during the Abbasid Caliphate (750 CE). These translated Arabic works made their way into Europe via intermediaries. In Italy the Branca family of Sicily and Gaspare Tagliacozzi of Bologna became familiar with the techniques of Sushruta.

Lithiasis treatment: The earliest operation for treating lithiasis, or the formations of stones in the body, is also given in the Sushruta Samhita (6th century BCE). The operation involved exposure and going up through the floor of the bladder.

Visceral leishmaniasis, treatment of: The Indian (Bengali) medical practitioner Upendranath Brahmachari (19 December 1873 – 6 February 1946) was nominated for the Nobel Prize in Physiology or Medicine in 1929 for his discovery of 'ureastibamine (antimonial compound for treatment of kala azar) and a new disease, post-kalaazar dermal leishmanoid.' Brahmachari's cure for Visceral leishmaniasis was the urea salt of para-amino-phenyl stibnic acid which he called Urea Stibamine. Following the discovery of Urea Stibamine, Visceral leishmaniasis was largely eradicated from the world, except for some underdeveloped regions.

Ayurvedic and Siddha medicine: Ayurveda and Siddha are ancient and traditional systems of medicine. Ayurveda dates back to Iron Age India (1st millennium BC) and still practiced today as a form of complementary and alternative medicine. It means "knowledge for longevity". Siddha medicine is mostly prevalent in South India. Herbs and minerals are basic raw materials of the Siddha system which dates back to the period of siddha saints around the 5th century BC.

Cataract surgery: Cataract surgery was known to the Indian physician Sushruta (6th century BCE). In India, cataract surgery was performed with a special tool called the Jabamukhi Salaka, a curved needle used to loosen the lens and push the cataract out of the field of vision] The eye would later be soaked with warm butter and then bandaged. Though this method was successful, Susruta cautioned that cataract surgery should only be performed when absolutely necessary. Greek philosophers and scientists traveled to India where these surgeries were performed by physicians. The removal of cataract by surgery was also introduced into China from India.

Cure for Leprosy: Kearns & Nash (2008) state that the first mention of leprosy is described in the Indian medical treatise Sushruta Samhita (6th century BCE). However, The Oxford Illustrated Companion to Medicine holds that the mention of leprosy, as well as ritualistic cures for it, were described in the Atharva-veda (1500–1200 BCE), written before the Sushruta Samhita.

Plastic surgery: Plastic surgery was being carried out in India by 2000 BCE. The system of punishment by deforming a miscreant's body may have led to an increase in demand for this practice.The surgeon Sushruta contributed mainly to the field of plastic and cataract surgery. The medical works of both Sushruta and Charak were translated into Arabic language during the Abbasid Caliphate (750 CE). These translated Arabic works made their way into Europe via intermediaries. In Italy the Branca family of Sicily and Gaspare Tagliacozzi of Bologna became familiar with the techniques of Sushruta.

Lithiasis treatment: The earliest operation for treating lithiasis, or the formations of stones in the body, is also given in the Sushruta Samhita (6th century BCE). The operation involved exposure and going up through the floor of the bladder.

Visceral leishmaniasis, treatment of: The Indian (Bengali) medical practitioner Upendranath Brahmachari (19 December 1873 – 6 February 1946) was nominated for the Nobel Prize in Physiology or Medicine in 1929 for his discovery of 'ureastibamine (antimonial compound for treatment of kala azar) and a new disease, post-kalaazar dermal leishmanoid.' Brahmachari's cure for Visceral leishmaniasis was the urea salt of para-amino-phenyl stibnic acid which he called Urea Stibamine. Following the discovery of Urea Stibamine, Visceral leishmaniasis was largely eradicated from the world, except for some underdeveloped regions.

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Some Images: idatapix.com

Some Images: idatapix.com

Wikipedia The Free Encyclopedia.