







"A mathematician, like a painter or a poet, is a maker of patterns. If their patterns are more permanent than those made by others, it is because they are made with ideas."

-G.H.Hardy









There have been many who have studied and fallen in love with the language of mathematics, such that the numbers form a poetry in their mind. One such extraordinary mathematician was Srinivasa Ramanujan.









A picture of young Ramanujan .







He was a very intuitive mathematician, and even though his life was short , his contributions to maths has been rather astonishing , and really really interesting. One of his amazing insights into the world of mathematics is this...





Imagine if I give you a series say, 1+2+3+4+.......∞ now, this series has infinite number of terms, so if I take a calculator and add all the terms , I could never reach the last term, since there are infinite number of terms. Now take a rough guess , what do you think is the answer to that series? Some really big number right? Well that's what you and I would think, but it turns out , if you play with the numbers long enough , you will see some subtle insight into the miracles of mathematics, one such miracle is what we call Thee Ramanujan Series.





But before understanding this, we first need to grasp 2 of his amazing results, if you are not really into mathematics but still curious about this mathematician's brilliance, just hang in there a little, it will be a little math heavy journey, but we will do it together, we will go slow, it will be fun!





"To preserve my brains I want food and this is now my first consideration. Any sympathetic letter from you will be helpful to me here to get a scholarship…" — Srinivasa Ramanujan. Letter to G.H. Hardy (27 Feb 1913)









THE FIRST RAMANUJAN SERIES.





So here we add 1-1+1-1+1-1+1......∞ , and the result is surprising .









So now we , know the result is 1-1+1-1+....... ∞=1/2, surprising right? This may seem really counterintuitive , and personally for me that is the most interesting part.





THE SECOND RAMANUJAN SERIES.





Now, we look at the second series, 1-2+3-4+5-6+....... ∞













The calculation reveals the result of 1-2+3-4+5-6...... =1/4





THE THIRD RAMANUJAN SERIES.





The series is 1+2+3+4+5.....∞





One of the most interesting thing about this, is that it's really easy, try it on your own now! The basic idea is the same, and the solution involves using the first two series and shuffling the numbers to finally create the pattern, and thus pops out the answer, a single , unique, amazing and unprecedented answer .Try it on your own , play with the numbers , it's fun, it's easy , invest some time in learning this language and it will surely make you appreciate the various intricacies in nature!



















This to me at first was profoundly funny. There are many other ways to prove these results , some of which are given below...





Establishing the result of the third series by only using the second result.

















PROOF BY EULER RIEMANN ZETA FUNCTION.









(This can also be proved by binomial expansion).







Now using this expansion, we find the solution to the second series...











Now defining a function helps us prove the third Ramanujan Series, using the result we obtained for the second one. This function is what we call the Euler Riemann zeta function







And thus the Euler Riemann zeta function can be used to produce the results of the Ramanujan Series. And thus the Euler Riemann zeta function can be used to produce the results of the Ramanujan Series.









The point of this article is to show the beauty of mathematics, and how fun it can be to play with numbers and series and integral and differential calculus and so on. It can be rigorous, difficult but the elegance of the mathematical ideas , just like poetry , is timeless. Thus a number of men and women have been , and will be contributing to it and reveal the hidden secretes of nature. Just like learning any new language, it takes a lot of effort in the beginning, but at some point we start speaking it fluently. If the Universe communicates through mathematics , isn't it worth it to invest some time into it, for the book of nature awaits a curious reader !





"Mathematics , rightly viewed, possesses not only truth, but supreme beauty. A beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." -Bertrand Russell.

Article by : Shanawaz Shaikh F.Y.BSc , Jai Hind College, Mumbai.

It literally means the Universe itself. Mathematics for more than the last 400 years(Since the beginning of Physics) has been the most effective tool to describe the Universe, we do not know why or how but it does seem like it is somehow an intrinsic feature of Nature itself. It allows us to understand the laws of Gravity, the relationship of electricity and magnetism , the nature of sound waves, from the tiniest quark inside a proton to the size of the massive event horizons of 2 colliding Black Holes, and gives you a simple and elegant equation that you could write on the back of an envelope,, its truly humbling and astounding. Although , this raises a question, is mathematics really an intrinsic property of the Universe? Why is it that something that we invent from our brains would for some reason be a feature of nature. And if it really isn't true, then could there be limitations to the appropriate of mathematics in describing the physical world, and thus a limitation in our attempts to contemplate it....