Madhav of Sangamgram
Madhav of Sangamgram (c.1350 - c 1425) was a famous Kerala mathematician-astronomer who was from Iranalakkuta, a town located near the Cochin district of the Indian state of Kerala. He is considered the founder of the Kerala Mathematical Society (Kerala School of Astronomy and Mathematics). He was the first person who had developed many infinite series, which is called "a decisive step ahead of the infinite method of ancient mathematics in transiting the trans-transformation to infinity". His discovery opened the path which is today known as Mathematical Analysis. Madhavan made a leading contribution in the study of infinite series, calculus, trigonometry, geometry and algebra. He was one of the greatest mathematicians-astronomers in the Middle Ages.
Some scholars have also given the idea that Madhav's work was broadcast through the Kerala School, Jesuit missionaries and traders, who were quite active around the ancient port of Kochi at that time, to Europe. As a result, its effect would have been on the subsequent European development order in analysis and calculus.
Name
Madhava was born in the form of InnerAppleli or Ininivavalli Madhavan Namboodiri. He had written that the name of his house was related to a vihar, where a plant named "Bakulam" was planted. According to Achyuta Pisharti, (who wrote a comment on Veneravaram written by Madhavan) Bakulam was known locally as "Irani". Dr. K.V. Sharma, who is the official of the work related to Madhavan, is of the opinion that his house is named Irinnarappilli or Irinnavalli.
Irininglukkitya was once known as Irinationalkutal. Sangamgramam (Literature Sangamgam = Ekta, Village = Village) The Dravidian word Irinikutak is a temporary translation in Sanskrit, this word means 'Iru (two) Ananti (market) kutal (unity)' or unity of two markets. Historiography
Although before Madhav, there are evidence of some math related work in Kerala (e.g., Sadratnamala c. 1300, a set of fragmented results), it is clear from the mention that during medieval Kerala, Madhav developed for the development of rich mathematical tradition. Provided a creative impulse.
However, most of the main functions of Madhav (except for some) have lost. He has been mentioned in the works of his subsequent mathematicians, especially in the Neuralanth Somaiya's systematic system (K. 1500) as the spread of infinite categories with many sinθ and arctanθ. In the sixth century article Mahajanan type, Madhav has been formulated as a source of many types of derivatives of π. Jindindev's tuition language (c.115) which is written in the Malayalam language, in this category presented with evidence in the form of Taylor category spread for polygons like 1 / (1 + x), where x = tan θ etc. Have gone.
In this way, which work has actually been done by Madhava, this is a matter of controversy. Tip: Deepika (also known as System of Contemporary Art), is probably composed by Jyeshtha Deva's disciple, Sanjay Variyar, it presents many forms of category expansion for sin θ, cos θ and arctan θ, as well as radius and arc There are also some products of length, most of which are also seen in Yuktibhashan. The category, which Rajgopal and Rangachari argued about, is not widely quoted in the original Sanskrit language, and since some of these categories have given credit to Madhava for Madhav, possibly some other format is also Madhava Can be of the work itself.
Other people have speculated that the early composition Karanapaddhati (c. 1375-1475), or Mahanagyanyan type may possibly be written by Madhava, but its probability is very low.
Karaparthi, along with other ancient mathematical compositions of Kerala, Saturnathnamala and system of literature and strategic language, in 1834 an article of Charles Matthew Whish has been considered, the discovery of this flute (the name given by Newton for the calculation). Newton was the first article to attract attention to his priorities. In the middle of the twentieth century, Russian scholar Zhushkevich reviewed the legendary work of Madhava and in 1972, a Vernacular Inspection of Kerala School was also provided by Sarma. Genealogy Explanation of the rules
There are many known astronomers who had come before the era of Madhavan, in which there were Kuthalur Kejhar (2ns Century. Ref: Purananuru 229), Vararuki (4th century), Sankaranarayana (866 AD). Other unknown people may also have existed in their anteroom. However, we have a clear account of the post-Madhavan period. God Namboodiri was a direct disciple of him. According to the palm-manuscript handwriting of a Malayalam commentary on the Surya Siddhanta, Nilakandha Somayaji was the disciple of Parameswar's son Damodar (c.1400-1500). Jyeshtha was a disciple of Nilkanth. Achyut Pisharti, who belonged to Trichkantiyur, also referred to him as a disciple of Jyeshtha Dev and was the disciple of Grammarian Melpatur Narayana Bhattathiri Achyut. Contribution
If we look at mathematics as a range of algebraic processes from the process of infinity, then the first step towards this change will ideally start with the spread of infinite series. It is the transformation towards infinite series, for which Madhva is credited. In Europe, such a first class was developed by James Gregory in 1667. Madhav's work in relation to the category is notable, but the thing which is really extraordinary, is the evaluation of the error post (or revision post) he has done. This suggests that he understood the border nature of the infinite range very well. Therefore, Madhav must have invented the ideas of the underlying ideas of infinite series spread, the power range, the trigonometric range, and the rational realization of infinite series.
However, as mentioned above, it is difficult to determine the results given by Madhava and the consequences given by his successors purely. Below is a summary of the results for which Madhav has been credited by various scholars. Infinite range Main article: Madhav Range
Under his many contributions, he discovered infinite series for the trigonometric functions of sine, cosine, tangent and arctangent, and also extracted several methods for calculating the circumference of a circle. The dialect is a range of known Madhava from the book, which includes the proof of the inverse range of tangent and the merits of production, was discovered by Madhava only. In the book, Jyeshtha Deva interprets this category as follows.
Will get from r & # x03B8; = r without & # x2061; & # x03B8; cos & # x2061; & # x03B8; & # x2212; ( 1 / 3 ) r ( without & # x2061; & # x03B8; ) 3 ( cos & # x2061; & # x03B8; ) 3 + ( 1 / 5 ) r ( without & # x2061; & # x03B8; ) 5 ( cos & # x2061; & # x03B8; ) 5 & # x2212; ( 1 / 7 ) r ( without & # x2061; & # x03B8; ) 7 ( cos & # x2061; & # x03B8; ) 7 + . . . {\displaystyle r\theta ={\frac {r\sin \theta }{\cos \theta }}-(1/3)\,r\,{\frac {\left(\sin \theta \right)^{3}}{\left(\cos \theta \right)^{3}}}+(1/5)\,r\,{\frac {\left(\sin \theta \right)^{5}}{\left(\cos \theta \right)^{5}}}-(1/7)\,r\,{\frac {\left(\sin \theta \right)^{7}}{\left(\cos \theta \right)^{7}}}+...}
Which gives a result later: & # x03B8; = tan & # x2061; & # x03B8; & # x2212; ( 1 / 3 ) tan 3 & # x2061; & # x03B8; + ( 1 / 5 ) tan 5 & # x2061; & # x03B8; & # x2212; & # x2026; {\displaystyle \theta =\tan \theta -(1/3)\tan ^{3}\theta +(1/5)\tan ^{5}\theta -\ldots }
This class was traditionally known as Gregory (after the name of James Gregory, who discovered it three centuries later). Even if we consider this category to be a discovery of Jyeshtha Deva, it will be a matter of centuries before the time of Gregorian, and of course, other endless categories of this sort are discovered by Madhava. Today, it is known as the Madhava-Gregory-Leibniz category. Trigonometry
Madhav also gave the most suitable timetable for sine, which was defined as the values of the semi-living organisms which were drawn at equal intervals on the quadrant of the given circle. It is believed that they would have obtained this very accurate story based on the following categories of promotions: without q = q - q / 3! + q / 5! - ... cos q = 1 - q/2! + q/4! - ... Pie (π) value
In the following verses, Madhava has described the relation of the circle and its diameter (i.e. the value of pi) in this verse, which has been expressed through the verse in this stanza - Disinterest New York City: New York City
It means - the circumference of the circle with 9 x 10 diameter will be 2872433388233.
In relation to the value of π, the mention of Madhva's work has been found in the Mahyagnya type ("Methods for the Great Science") where some scholars such as Sarma, believe that the book itself may have been written by Madhav itself, On the other hand it is more likely to be written by the 16th century successors. This book gives credit to Madhava for many broadcasts and gives the following infinite series expansion for π, now known as Madhava-Leibniz category: Vyasaghatinita as Vyasa Vyasasagara Behavior Trishadavidishvashmashamamamatamamrana self in different order kuryat Nirvatyatra Haritstha Jamitaya to lose the yatra. Problems with an increase in the number of issues. & # x03C0; 4 = 1 & # x2212; 1 3 + 1 5 & # x2212; 1 7 + & # X22EF; + ( & # x2212; 1 ) n 2 n + 1 + & # X22EF; {\displaystyle {\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +{\frac {(-1)^{n}}{2n+1}}+\cdots }
which he had received from the power division of arc-tangent function. However, the most impacting thing is that he also gave an amended post, Rn, which was for n error after the calculation for n terms.
Madhav had given three formats of Rn, which used to modify Nikatagman, it has names Rn = 1 / (4n), Rn = n / (4n + 1); Rn = (n + 1) / (4n + 5n).
Where the third amendment results in a very accurate calculation of π.
How to reach this modification post. The most reliable fact is that they come in the form of the first three convergence of continuous variation, which can be derived from the standard of π itself itself, it is 62832/20000 (see Basic Fifth C. for calculation, Aryabhatta) .
By giving an infinite series by the conversion of the original infinite series of π, they gave a more rapidly converging category & # x03C0; = 12 ( 1 & # x2212; 1 3 & # x22C5; 3 + 1 5 & # x22C5; 3 2 & # x2212; 1 7 & # x22C5; 3 3 + & # X22EF; ) {\displaystyle \pi ={\sqrt {12}}\left(1-{1 \over 3\cdot 3}+{1 \over 5\cdot 3^{2}}-{1 \over 7\cdot 3^{3}}+\cdots \right)}
By using the first 21 terms for the calculation of the nearest value of π, they received a value that was correct to 11 decimal places (3.14159265359). The value of 3.1415926535898 is sometimes attributed to 13 decimal places, for which Madhav is sometimes credited. But this is probably given by some of his disciples. This was among the most proximal values of π from the fifth century (see, History of Numerical Appropriation of π).
The book Sadrnamnama, which is generally considered to be a book prior to the times of Madhava, also surprisingly gives a very precise value of π, π = 3.14159265358979324 (correct to 17 decimal places). On this basis, R. Gupta has argued that this book may also be written by Madhava. Algebra
Madhav also researched the other categories of arc line length and the approximate value of rational numbers of π related to π, they searched methods of polynomialization, searched the convergence test for infinite series and analyzed the infinite continuous different numbers. He also solved the mysterious equations by repetition, and also received the closest value of the esoteric numbers by continuous different numbers. Calculus (calculus)
Madhav laid the foundation for the development of calculus, which his successors developed further in the Kerala School of Astronomy and Mathematics. (It is important to keep in mind that some of the principles of calculus were known to ancient mathematicians.) Madhava also carried forward some results from the ancient works, which included the works of Bhaskar II.
In calculus, he has used the initial format of differential, integration and he or his disciples developed the integration of simple functions. Works of Madhava
K.V. Sharma has described Madhava as the author of the following books: : En: Keralite Mathematical Society Main article: Keralite Mathematical Society
After the Madhavan, the Keralite Mathematical community flourished for at least two centuries. We got the idea of integration from Jyeshtha Deva, which was called as compilatum (Hindi Earth Collection), as in this statement: Ekadey kothar pad sankalitam samam padravargarthinte pakuti,
which translates integration into a variable (verse) which will be equal to half of the square of the variable; That is, the integration of x dx equals x / 2. This is clearly the beginning of integration. Another result related to this is that the area inside a curve is equal to its integral. Most of these results from the existence of similar results in Europe many centuries ago. In many ways, Jyeshtha's trilingual language can be considered as the world's first book on calculus. effect Trade route between the southwestern part of India and the Roman empire
Madhava has been called "the greatest mathematician of the Middle Ages-astronomers", or "founder of mathematical analysis; some of his research in this field reveals that he had extraordinary intuition talent within." According to O'Connor and Robertson, Madhav's proper assessment will be in the following words: he took a decisive step towards modern classical analysis. Potential spread in Europe
During the first contact with European directional directors on the Malabar beach, Kerala School was quite famous in the 15th-16th century. At that time, the Kochi port located near Sangamgram was a major center of maritime trade and many Jesuit missionaries and businessmen were active in this area. Looking at the fame of the Kerala school and during this period, some scholars, in which Yu, as a result of the interest shown by some of the Jesuit group by local scholars, Of Manchester, J. Joseph also said that during this period, articles from Kerala School have reached Europe, which was a century before the time of Newton. Although no European translation of these articles is in existence, it is possible that these ideas have influenced subsequent European developments in analysis and calculus. (For more details, see Kerala School). This is due to not understanding the author's views properly. In the 16th century, the people of Jesuits, who were familiar with the prestige of Madhavan and his disciples, it was almost impossible for them to study Sanskrit and Malayalam, to bring it to European mathematicians, instead of doing this searching themselves Claims. Also see them
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