Aryabhata (476–550 CE) was the first in the line of greatmathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His most famous works are the Aryabhatiya (499 CE, when he was 23 years old) and the Arya-siddhanta. 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,[1] including Brahmagupta's references to him "in more than a hundred places by name".[2] Furthermore, in most instances "Aryabhatta" does not fit the metre either.[1] Birth Aryabhata mentions in the Aryabhatiya that it was composed 3,600 years into the Kali Yuga, when he was 23 years old. This corresponds to 499 CE, and implies that he was born in 476 CE.[1] Aryabhata provides no information about his place of birth. The only information comes from Bhāskara I, who describes Aryabhata as āśmakīya, "one belonging to the aśmaka country." While aśmaka was originally situated in the northwest of India, it is widely attested that, during the Buddha's time, a branch of the Aśmaka people settled in the region between the Narmada and Godavari rivers, in the South Gujarat–North Maharashtra region of central India. Aryabhata is believed to have been born there.[1][3] However, early Buddhist texts describe Ashmaka as being further south, in dakshinapath or the Deccan, while other texts describe the Ashmakas as having fought Alexander, which would put them further north.[3] Mathematics Place value system and zero The place-value system, first seen in the 3rd century Bakhshali Manuscript, was clearly in place in his work. While he did not use a symbol for zero, the French mathematician Georges Ifrah argues that knowledge of zero was implicit in Aryabhata's place-value system as a place holder for the powers of ten with null coefficients[10] However, Aryabhata did not use the brahmi numerals. Continuing the Sanskritic tradition from Vedic times, he used letters of the alphabet to denote numbers, expressing quantities, such as the table of sines in a mnemonic form.[11] ]Pi as irrational Aryabhata worked on the approximation for Pi (π), and may have come to the conclusion that π is irrational. In the second part of the Aryabhatiyam (gaṇitapāda10), he writes: caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ. "Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached."[12] This implies that the ratio of the circumference to the diameter is ((4+100)×8+62000)/20000 = 62832/20000 = 3.1416, which is accurate to five significant figures. It is speculated that Aryabhata used the word āsanna (approaching), to mean that not only is this an approximation but that the value is incommensurable (orirrational). If this is correct, it is quite a sophisticated insight, because the irrationality of pi was proved in Europe only in 1761 by Lambert).[13] After Aryabhatiya was translated into Arabic (ca. 820 CE) this approximation was mentioned in Al-Khwarizmi's book on algebra.[3] Mensuration and trigonometry In Ganitapada 6, Aryabhata gives the area of a triangle as tribhujasya phalashariram samadalakoti bhujardhasamvargah that translates to: "for a triangle, the result of a perpendicular with the half-side is the area Aryabhata discussed the concept of sine in his work by the name of ardha-jya. Literally, it means "half-chord". For simplicity, people started calling it jya. When Arabic writers translated his works from Sanskrit into Arabic, they referred it as jiba. However, in Arabic writings, vowels are omitted, and it was abbreviated as jb. Later writers substituted it with jiab, meaning "cove" or "bay." (In Arabic, jiba is a meaningless word.) Later in the 12th century, when Gherardo of Cremonatranslated these writings from Arabic into Latin, he replaced the...

...later mathematicians, especially Girolama Saccheri, tried to out do the work of Euclid but all eventually gave up when they realized that his theories were flawless due to his extensive proofs.
He even adventured into new branches of mathematics and science. First, he published his book, Optiks, which discussed perspective and how people view the world through their eyes. His influence in this realm, although overlooked by most, is extremely influential. He also studied catoptrics, or the mathematical functions of mirrors. He again applied deductive reasoning to understand the principles behind mirrors. He also became an important figure in the study of data, conics, and ratios through his work in arithmetic and geometry.
Euclid’s influence on modern mathematics and society are immeasurable. For students studying geometry worldwide, his influence is obvious. As the renowned Father of Geometry, Euclid created the foundation for the field in his Elements. He created a foundation which other mathematicians built off of for the next 2000 years. Without his work, the work of scientists and mathematicians, such as Ptolemy, Brahmagupta, Isaac Newton, Leonhard Euler, and Carl Friedrich Gauss, would not have been possible. Deductive reasoning strategies would also be much less common and popular. Therefore, geometry students would never have the opportunity to use proofs to come to conclusions about various geometric shapes...

...Died: Jan 12, 1665 (at age 60 or 61), in Castres, France
Nationality: French
Famous For: Fermat’s Last Theorem
Pierre de Fermat (1601-1665)
Fermat’s contributions to mathematics
Fermat mathematician made significant contributions to number theory, probability theory, analytic geometry and the early development of infinitesimal calculus. He ventured into the areas of mathematics which included pre-evolved calculus and trigonometry.
Fermat’s primary contribution to mathematics was in the field of number theory. C.G. Bachet’s translation of Diophantus of Alexandria inspired his interest in the Theory of Numbers. He introduced Fermat’s “Last Theorem,” which states that there is no solution in integers of the equation xn + yn = zn (xyz#0, n>2).Fermat contributed to the development of calculus through his work on the properties of curves. Sir Isaac Newton said that his invention of calculus was based on Fermat’s methods of tangents. Fermat’s work on calculus was an aid in developing the differential calculus.
John Napier (1550-1617)
Born: 1550 in Merchiston Tower, Edinburgh
Died: April 4, 1617 (at age 66 or 67) in Edinburgh
Nationality: Scottish
Famous For: Discovering logarithms
Napier’s contributions to mathematics
John Napier was a Scottish mathematician who found lasting fame as the inventor of logarithms. He also invented at least one war weapon. His position as a member of the Scottish nobility allowed him to more...

...Mathematicians of the 17th Century
Jacob Bernoulli (also known as James or Jacques) (27 December 1654/6 January 1655 – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family.
He became familiar with calculus through a correspondence with Gottfried Leibniz, then collaborated with his brother Johann on various applications, notably publishing papers ontranscendental curves (1696) and isoperimetry(1700, 1701). In 1690, Jacob Bernoulli became the first person to develop the technique for solving separable differential equations.
Upon returning to Basel in 1682, he founded a school for mathematics and the sciences. He was appointed professor of mathematics at theUniversity of Basel in 1687, remaining in this position for the rest of his life.
Jacob Bernoulli is best known for the work Ars Conjectandi (The Art of Conjecture), published eight years after his death in 1713 by his nephew Nicholas. In this work, he described the known results in probability theory and in enumeration, often providing alternative proofs of known results. This work also includes the application of probability theory to games of chance and his introduction of the theorem known as the law of large numbers. The terms Bernoulli trial and Bernoulli numbers result from this work. The lunar crater Bernoulli is also named after him jointly with his brother Johann.
John Craig (1663 – October 11, 1731) was...

...Bengal Legislative Council and inflicted defeats on three ministries. The Calcutta Municipal Act of 1923 was a major landmark in the history of local self-government in India. The Swarajists were elected to the Calcutta Corporation in a majority in 1924. Deshbandhu was elected mayor and Subash Chandra Bose was appointedChief Executive Officer. The leaders of Swaraj Party began to advocate fordominion status to India. Many of the elected deputies soon forgot about obstruction and began cooperating with the government (tariff autonomy bill passed, 1923). In 1924 Gandhi was released from prison due to poor health and was elected President of the Indian National Congress. 1925 saw the first woman becoming the president of Indian National Congress when Sarojini Naidu was elected President for the Kanpur session.
Revolutionary Movement in India during 1920s and 1930s
The revolutionaries in northern India organized under the leadership of the old veterans, Ramprasad Bismil, Jogesh Chatterjee, Chandrashekhar Azad and Sachindranath Sanyal whose ‘Bandi Jiwani’ served as a textbook to the revolutionary movement. They met in Kanpur in October 1924 and founded the Hindustan Republican Association (HRA) to organize armed revolution to overthrow colonial rule and establish in its place a Federal Republic of the United States of India.
Gopinath Saha in January 1924 tried to assassinate Charles Tegart, the hated...

...Indian Mathematicians
RAMANUJAN
He was born on 22na of December 1887 in a small village of Tanjore district, Madras. He failed in English in Intermediate, so his formal studies were stopped but his self-study of mathematics continued.
He sent a set of 120 theorems to Professor Hardy of Cambridge. As a result he invited Ramanujan to England.
Ramanujan showed that any big number can be written as sum of not more than four prime numbers.
He showed that how to divide the number into two or more squares or cubes.
When Mr .Litlewood came to see Ramanujan in taxi number 1729, Ramanujan said that 1729 is the smallest number which can be written in the form of sum of cubes of two numbers in two ways, i.e. 1729 = 93 + 103 = 13 + 123 since then the number 1729 is called Ramanujan’s number.
In the third century B.C, Archimedes noted that the ratio of circumference of a circle to its diameter is constant. The ratio is now called ‘pi ( Π )’ (the 16th letter in the Greek alphabet series)
The largest numbers the Greeks and the Romans used were 106 whereas Hindus used numbers as big as 1053 with specific names as early as 5000 B.C. during the Vedic period.
Srinivasa Ramanujan Aiyangar was an Indian Mathematician who was born in Erode, India in 1887 on December 22. He was born into a family that was not very well to do. He went to school at the nearby place, Kumbakonam. Ramanujan is very well known for his efforts on continued fractions and series...

...ARYABHATA..
Aryabhata (IAST: Āryabhaṭa, Sanskrit: आर्यभट) (476–550 CE) was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His most famous works are the Āryabhaṭīya (499 CE, when he was 23 years old) and the Arya-siddhanta. Place value system and zero
The place-value system, first seen in the 3rd century Bakhshali Manuscript, was clearly in place in his work. While he did not use a symbol for zero, the French mathematician Georges Ifrah explains that knowledge of zero was implicit in Aryabhata's place-value system as a place holder for the powers of ten with null coefficients[7]
However, Aryabhata did not use the Brahmi numerals. Continuing the Sanskritic tradition from Vedic times, he used letters of the alphabet to denote numbers, expressing quantities, such as the table of sines in a mnemonic form.[8]
Approximation of π
Aryabhata worked on the approximation for pi (), and may have come to the conclusion that is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes:
caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.
"Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached." [9]
This implies that the ratio of the circumference to the diameter is ((4 + 100) × 8 + 62000)/20000 = 62832/20000 = 3.1416, which...

...price rates.
5. Another example of Pepsi, when the Pepsi factory was under its setting up process in the Punjab, the farmers of Punjab protested because the Pepsi authority committed to purchase the potatoes of the farmer but later on the reject and told the farmer that there potatoes are not up to their standards which is used in making up wafers chips and they have imported potatoes from the foreign countries.
6. Only 30 % is mandatory to purchase from Small and medium enterprises (SME's). What about rest 70%?
7. The large supermarket will import rest 70% which will increase current account deficit of the India. It will demotivate the small and medium scale industries. Hence it will increase unemployment.
In a developing country like India allowing FDI may help in booming the economy. But, this will only be up to a certain period of time only. Initially, India will get employment but that may also affect the middle class people who are mostly depended on small business. The employment provided by FDI will be of low class like, sweepers, security guards, etc. FDI saying that the farmers will be benefited but, their land used for cultivating the crops by spraying large amount of pesticides will be infected.
Ex. FDI using one farmers land for one year. It will cultivate large amount of crops by using pesticides indirectly affecting the land's quality. After one year that land will be poisoned. Then FDI will shift to...

...-
- - -
Structure
13.1 . Illtroduction
13.2 Theory of Fseedotn and Self-Realisatiotl
13.3 Emphasis on Human Reason
13.4 Critique of Nationalism
13.5 Differences with Gandlii
'13.6 A ~ ~ a l y sof Bolshevism is 13.7 Summary
13.8 Exercises
Rabindranath 'Tagorc (1861-1941) was an outstanding litcrnry figure of India who exerted consiclerable inf uence on human thinking in the contcml>oraryworld. T l ~ i s influence extcnded to the political arena as well by his lilcid elucidation of inlpartant conccpts like nationalis~m, freedom, human ratiollality and l ~ i s many dil'fcsences with Mahatma Gnntlhi's (1 869-1948) philosopl~y strategies. ancl Wliile Gatidhi was a political and social activist and Tagore was a poet, there was renlarkable consistency in tile enunciation of their ~ilgjorpolitical tlietnes, which they developed and refined reflecting on major cvents OF their time. I~urthermot-c,in Tagore there was a quest of (z poet for hitrnat~perfection and conlplcteness and 1101merely a pragmatic analysis of a particular probleln or a sitnatian, His expression was an elocll~cntappcal of his faith i11 the human spirit and the opti~nislnby which the entire humankind could tliinlicl-~ wanted to be based both he on reason and a concern fbr the masses, He criticised Gandlli whenever he felt that the Mahatma was deviating from these planks. He not only criticised but also provided an alternative perception to that of Gandhi. He acknowledges his...

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