Great Mathematician Class 10 Mathematics Srinivsa Ramanujan And GH hardy

Great Mathematician Class 10 Art Integration Activity

1. Great Mathematician Name :- Avanish M Class:- 10 th ‘A’ Roll no:- 1008 To: Class teacher Pushpalatha Mam

2. Srinivasa Ramanujan S. Ramanujan was born on December 22,1887. He was born in Erode ,a city in Tamil Nadu. Ramanujan ’s father is K.Srinivasa Iyengar . Ramanujan’s mother is Komalatammal .

3. Srinivasa Ramanujan did not complete a formal university degree. His formal education was limited due to various factors, including financial difficulties and health issues. However, despite lacking a formal graduation, Ramanujan's mathematical genius and self-taught insights led him to make profound contributions to mathematics. Srinivasa Ramanujan (1887-1920) was an Indian mathematician who made significant contributions to mathematical analysis, number theory, infinite series, and continued fraction. Srinivasa Ramanujan became famous around 1913 when G.H. Hardy, a prominent mathematician at Cambridge University, recognized the extraordinary talent and potential in Ramanujan's mathematical work. Hardy invited Ramanujan to Cambridge, England, where he collaborated with him and other mathematicians. Through Hardy's efforts, Ramanujan's work gained recognition and began to be appreciated internationally, leading to his fame in the mathematical community during his lifetime and continuing posthumously.

4. Early Life and Education : Ramanujan was born on December 22, 1887, in Erode, Tamil Nadu, India. He showed an early aptitude for mathematics and was largely self-taught. Mathematical Discoveries : Ramanujan independently discovered many results in number theory, including the properties of highly composite numbers, partition functions, mock theta functions, and continued fractions. Collaboration with G.H. Hardy : In 1913, Ramanujan wrote to G.H. Hardy, a prominent British mathematician, who recognized his extraordinary talent. This led to Ramanujan moving to Cambridge, England, where he collaborated with Hardy and others. Ramanujan -Hardy Number : 1729 is known as the Hardy- Ramanujan number, as it is the smallest number that can be expressed as the sum of two cubes in two different ways (1729 = 1^3 + 12^3 = 9^3 + 10^3).

5. Contributions : Ramanujan made substantial contributions to mathematical analysis, infinite series, continued fractions, and number theory. His work has had a profound impact on various branches of mathematics. Recognition and Legacy : Despite his short life, Ramanujan left behind numerous notebooks filled with mathematical discoveries and conjectures. His work continues to inspire mathematicians and researchers worldwide, and he is considered one of the greatest mathematical prodigies of all time. Illness and Death : Ramanujan's health declined during his time in England, partly due to the climate and malnutrition. He returned to India in 1919 and died on April 26, 1920, at the age of 32.

6. Srinivasa Ramanujan was profoundly skilled and made significant contributions to several areas of mathematics, including: Number Theory : Ramanujan made groundbreaking discoveries in number theory, particularly in the areas of partitions of numbers, modular forms, and properties of numbers such as highly composite numbers and prime numbers. Infinite Series : He developed novel methods and formulas for computing infinite series, many of which were completely new to the mathematical community at the time. Special Functions : Ramanujan introduced new types of special functions, such as mock theta functions, which have since been studied extensively for their connections to modular forms and other areas of mathematics.

7. Modular Forms : His work on modular forms, which are important in areas like complex analysis and algebraic geometry, was ahead of its time and has had lasting impacts on these fields. Continued Fractions : Ramanujan studied continued fractions extensively and derived new results and identities related to them. Approximations and Equations : He developed remarkably accurate approximations and equations, some of which have been used in fields such as physics and engineering.

8. He born on February 7, in the year 1877. He was born in Cranlegih,Surrey,England . G. H. Hardy graduated from Trinity College, Cambridge. He completed his undergraduate studies there and subsequently became a fellow of the college in 1900. G. H. Hardy's father was Isaac Hardy. The mother of G. H. Hardy was Sophia Hardy G.H.HARDY

9. G. H. Hardy was first appointed as a professor in 1906 when he became the Cayley Lecturer in Mathematics at Cambridge. Later, in 1919, he took up the Savilian Chair of Geometry at the University of Oxford, a position he held until 1931. G. H. Hardy gained significant recognition in the mathematical community during the early 20th century. His fame grew particularly with his collaboration with John Edensor Littlewood , which began around 1911, and further increased with his mentorship of the Indian mathematician Srinivasa Ramanujan starting in 1914. The Hardy-Weinberg principle, formulated in 1908, also contributed to his early renown. However, his widespread fame solidified with the publication of "A Mathematician's Apology" in 1940, which became one of the most famous books on the nature of mathematical research.

10. G. H. Hardy's collaboration with John Edensor Littlewood was one of the most significant and productive partnerships in the history of mathematics. It began around 1911 and lasted for over 35 years. Together, they made substantial contributions to various areas of mathematics, particularly in analytic number theory. Additive Number Theory : They worked on the Waring's problem, which deals with representing natural numbers as sums of powers of integers. Prime Number Theorem : They provided important results and insights related to the distribution of prime numbers. Fourier Series : Hardy and Littlewood made significant contributions to the theory of Fourier series and integrals.

11. The collaboration between G. H. Hardy and Srinivasa Ramanujan is one of the most famous partnerships in the history of mathematics. It began in 1913 when Ramanujan , largely self-taught and working in isolation in India, sent a letter filled with his mathematical discoveries to Hardy in Cambridge. Hardy recognized the brilliance of Ramanujan's work and arranged for him to come to Cambridge in 1914. Partition Function : They developed the Hardy- Ramanujan asymptotic formula for the partition function p(n)p(n)p(n), which gives an approximation for the number of ways a positive integer can be expressed as the sum of positive integers. Ramanujan -Hardy Number : They worked together on properties of highly composite numbers and prime numbers, contributing to the field of analytic number theory.

12. Mock Theta Functions : Ramanujan introduced Hardy to his work on mock theta functions, which are a type of q-series that have since become an important area of research in number theory and mathematical physics. The Circle Method : They developed and refined the circle method, a technique used to estimate the number of solutions to certain equations in additive number theory. Continued Fractions and Infinite Series : They explored and expanded on Ramanujan's discoveries in the areas of continued fractions and infinite series, providing new insights and proofs for various results. Their collaboration was marked by mutual respect and admiration. Hardy considered Ramanujan to be a mathematical genius of the highest order, often comparing him to great mathematicians like Euler and Jacobi