Ramanujan has been described as a somewhat shy and quiet person, a dignified man with pleasant demeanor. He lived a simple life in Cambridge. Ramanujan's first Indian biographers describe him as a staunchly orthodox Hindu. He credits his skill to his family goddess of Namakkal, Namagiri Thayar (Goddess Mahalakshmi). He looked to her for inspiration in his work and said that he dreamed of drops of blood which symbolized his wife Narasimha. Later, scrolls of complex mathematical material appeared before his eyes. He would often say, "An equation makes no sense to me unless it expresses the idea of God."
Hardy quoted Ramanujan as saying that all religions seemed to him to be equally true. Hardy further argued that Ramanujan's religious belief was romanticized by Westerners and exaggerated—in the context of his belief, practice—by Indian biographers. Also he commented on Ramanujan's strict vegetarianism.
Similarly, in an interview with Frontline, Berndt said, "Many people falsely attribute mystical powers to Ramanujan's mathematical thinking. This is not true. He has meticulously recorded every result in his three notebooks," Further anticipating that Ramanujan prepared intermediate results on the slate. that he could not afford to record the paper more permanently.
In mathematics, there is a difference between formulating or working through insight and proof. Ramanujan proposed a plethora of sources that could be examined in depth later. G h. Hardy stated that Ramanujan's discoveries are unusually rich and that there is much more to him than the discoveries at the beginning. As a byproduct of his work, new directions of research were opened. The most interesting examples of these formulas include the infinite series for , one of which is given below:
1
I
,
2
2
9801
I
K
,
0
I
,
4
K
,
,
,
1103
,
26390
K
,
,
K
,
,
4
396
4
K
,
{\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {( 4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}.}
This result is based on the negative fundamental discriminant d = −4 × 58 = −232 square number h(d) = 2. Also, 26390 = 5 × 7 × 13 × 58 and 16 × 9801 = 3962, which is related to the fact that
I
I
58
,
396
4
,
104.000000177
,
,
{\textstyle e^{\pi {\sqrt {58}}}=396^{4}-104.000000177\dots .}
This can be compared with Heigner numbers, which have an orbital number of 1 and get the same formula.
Ramanujan's series converges exceptionally fast and forms the basis of some of the fastest algorithms currently used for computing k. Reducing the sum to the first term also gives an approximation
9801√2
,
4412
for , which is correct to six decimal places; Shortening it to the first two terms gives the correct value to 14 decimal places. See also the more general Ramanujan-Sato series.
One of Ramanujan's notable abilities was quick solving of problems, which is illustrated by the following anecdote about an incident in which P. C. Mahalanobis posed a problem:
Imagine you are on a street with houses marked from 1 to n. (x) There is a house in the middle such that the sum of the numbers of the houses to its left is equal to the sum of the numbers of the houses to its right. If n is between 50 and 500, then what are n and x?' This is a bivariate problem with many solutions. Ramanujan thought about it and replied with a twist: He gave a continued fraction. The unusual part was that it was the solution to a whole class of problems. Mahalanobis was amazed and asked how he did it. 'This is simple. The moment I heard the problem, I knew the answer was a continuous fraction. Which part continued, I asked myself. Then the answer came to my mind', replied Ramanujan."
His intuition led him to derive some previously unknown identities, such as
