శాస్త్ర విజ్ఞానము

5 comments

1. gaddeswarup Says:
   Posted on March 26, 2010 at 2:57 AM

2. I think that it was Newton who used binomial series first (at least in the West) and the convergence was discussed by Abel later. Gauss developed a test later. Newton also used the binomial seires to obtain series for sin(x). He knew the the integral formula for inverse sin (x), expanded the integrand by binomial series and integrated term by term. He then inverted the series to obtain the series expansion for sin (x)! Many of the series expansions were obtained by Madhava nearly three hundred years before. Madhava also had his version of the error term in Taylor series.

    
3. V Srinivasa Chakravarthy Says:
   Posted on March 26, 2010 at 8:02 AM

4. Dear Gaddeswarup garu
   
   Thank you for the insightful comments. My sourcebook for this article is "Men of mathematics" by ET Bell.
   
   It may be true that Newton used binomial series to get series expansions. But this book says earlier work (before that of Gauss) was more of applied maths kind and lacked rigor that an analyst demands.
   To quote from the book:
   
   "The rigor which Gauss imposed on analysis gradually overshadowed the whole of mathematics, both in his own habits and in those of his contemporaries - Abel, Cauchy - and his successors - Weierstrass, Dedekind, and mathematics after Gauss became a totally different thing from the mathematics of Newton, Euler and Lagrange."
   
   It is true that ancient Indians knew about binomial theorem. I think I wrote about it in an article on Pascal's triangle/Meru Prastaram. But dont know much about our work on infinite series. Will try to look for some info. If you have some references please let me know.

    
5. gaddeswarup Says:
   Posted on March 26, 2010 at 9:00 AM

6. Chakravarti garu,
   I think what you say about Gauss is true. I think Abel supplied proofs and extended it to comlex exponents,but died too young. Gauss was a bigger influence. My information about Newton is from (read long ago) Edwards's book on the development of Calculus. I do not know whether Indians had binomial series but many of Indian contributions are now in a possibly definitive book by Kim Plofker
   http://www.thesouthasian.org/archives/2006/situation_in_dantewara_distric.html
   I have not read the book but read her earlier article in "The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook".
   I too was inspired by Bell's book to get in to mathematics but many say for history of mathematics, it is not too reliable. But what you said seems to correct in broad oultline. Thanks for very interesting and readable posts.

    
7. Salahuddin Says:
   Posted on September 15, 2011 at 12:04 AM

8. A proof of fermat's little theorem (my own proof) using binomial expansion is here: http://salahuddin-knowledgehobbyist.blogspot.com/2010/08/i-could-also-prove-fermats-theorem.html. This is my blog, contains many recreational mathematics stuff. I talked about perfect numbers as well, and some math problems for school kids, please visit the maths section there.

    
9. V Srinivasa Chakravarthy Says:
   Posted on September 15, 2011 at 1:09 PM

10. Salahuddin@
    The math part of your blog is awesome. Do you plan to translate it into Telugu?