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\title{All about Ramanujan and his Contributions in Math~}
\author[1]{Maritza}%
\affil[1]{Affiliation not available}%
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Srinivasa Ramanujan was born in Erode, India on December 22, 1887 to
parents~K. Srinivasa Iyengar, and mother Komalatammal. Growing up, his
family did not have the best financial stability due to his father
working as an accounting clerk and his mother as a singer at a temple.
However, this did not stop Ramanujan from gaining so much knowledge. By
the age of 10, Ramanujan was already ahead of his classmates and was
attending high school. His passion for math began when he put his hands
on a book called,~ ~\emph{A Synopsis of Elementary Results in Pure and
Applied Mathematics}~. With no prior knowledge of advanced math,
Ramanujan taught himself many mathematical concepts from this book and
soon became a genius of the topic itself.~ After high school, Ramanujan
had~received a scholarship to the Government Arts College in Kumbakonam.
Because most of his attention revolved around math, he failed to
complete his other subjects in school and lost the scholarship
altogether. This led Ramanujan to obtain education elsewhere.~
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\includegraphics[width=0.28\columnwidth]{figures/ram3/ram3}
\caption{{Ramanujan~
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\begin{figure}[h!]
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\caption{{Hardy
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At the age of 19, Ramanujan found himself very poor with no job.
However, this did not stop Ramanujan from trying to pursue a career in
math. While trying to look for any job at this point, Ramanujan's mother
had arranged a marriage for him to a 10 year old girl, at this time he
was about 21 years old. In spite of luck, after marriage Ramanujan found
himself talking to an official named, Ramaswamy Aiyer who was also a
mathematician. Ramanujan had showed his findings for the first time to
someone else who appreciated math, Aiyer was immediately impressed and
wanted to help Ramanujan publish his findings. In 1911 Aiyer had finally
helped him publish his first piece of work on Bernoulli numbers in the
~\emph{Journal of the Indian Mathematical Society}.~ ~ ~A team of people
were on Ramanujan's side trying to help him succeed. By this time,
Ramanujan had written a letter and had sent it off to many British
mathematicians.~ The only one to take an interest in his letter was
Godfrey Harold Hardy from the University of Cambridge. Ramanujan was now
25, and did not know that he had impressed Hardy to the extent that he
did.
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Finally in 1914, Ramanujan found himself traveling to England where
Hardy was awaiting to work with him. it only took two years for
Ramanujan to be recognized for his contributions and was given an
award.~ Although his success finally had come through, Ramanujan's
health was~ not in good shape. In the year of 1917, Ramanujan was
diagnosed with Tuberculosis. The disease itself forced Ramanujan to
receive a lot of bed rest. However, while in the hospital, Hardy and
Ramanujan kept a great friendship. A ear later Ramanujan recovered
enough to go back to India. In 1919, Ramanujan would pass way at the age
of 32.~
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\textbf{Contributions in Math~}
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One of the greatest contributions that Ramanujan had dealt with was the
Goldbach conjecture. The Goldbach conjecture states, the statement is
``Every even integer greater that two is the sum of two primes.'' Even
though the conjecture itself was never solved, Hardy and Ramanujan had
showed that every larger integer could be written of at most 4 numbers.
In addition to this discovery, Ramanujan was also trying to proof
Fermat's last theorem. Fermat's Last Theorem states, ``that no three
positive integers~\(a,b\) and~\(c\) satisfy the
equation~\(a^n+b^n=c^n\) for any integer value of~\(n\)
greater than~\(2\).'' The neat thing that Ramanujan was
able to do with this was find many near counter examples. In other
words, Ramanujan was able to come up with counter examples that missed a
cube by just one. One of the counterexamples
include~\(135^3+138^3=172^3-1\).~ We can see that these sort of examples will
always have a~\(+1\) or~\(-1\) at the end which
comes from \(\left(-1\right)^n\) in our generating function.
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\includegraphics[width=0.70\columnwidth]{figures/r1/r1}
\caption{{Fermat's Last Theorem~
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\caption{{Ramanujan's writing for near misses~
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Ramanujan also made a series for calculating pi. The series itself as we
can see has a denominator of each term is a power of 2. By this
obeservation, we can see how the partial sum of the series can be
expressed as binary. By using Sterling's approximation, Ramanujan came
up with~\(\binom{2n}{n}\equiv\frac{2^{2n}}{\sqrt{\pi n}}\). By taking a closer look, we can see how the
nth term in the series makes a rational number. The rational number has
a numerator that can be expressed as~\(2^{6n}\)~along with the
denominator as~\(2^{-6n-4}\). Overall we can see tha the series
will converge quickly and we will be left with a series that looks
like~\(\frac{1}{\pi}=\frac{5}{16}+\frac{376}{65536}+\frac{19224}{268435456}+...\)~And in the end, Ramanujan was able to come up
with an approximation for pi.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/r5/r5}
\caption{{Full series for calculating pi
{\label{103805}}%
}}
\end{center}
\end{figure}
One of the coolest things that ramanujan created was his magic square.~
Some of the coll properties that the square holds is that the sum of an
column is 139, the sum of any row is 139, the sum of the diagnal
elements is 139, and the sum of any 2x2 box is 139, excluding the ,
middle two columns. What is even more interesting about this magic
sqaure is that the very top row is ramanujan's birthday/international
mathematics day. Below I have provided a link explaining how to create
your own magic sqaure!~
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\begin{figure}[h!]
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\includegraphics[width=0.70\columnwidth]{figures/r7/r7}
\caption{{Properties for Ramanujan's Magic Square~
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\begin{figure}[h!]
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\includegraphics[width=0.70\columnwidth]{figures/r6/r6}
\caption{{Creating your own magic square
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\end{center}
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\textbf{Fun Facts About Ramanujan:}
\textbf{1.) In high school he devoured books on mathematics and taught
himself advanced theorems about math. ~}
\textbf{2.) He read~\emph{A Synopsis of Elementary Results in Pure and
Applied Mathematics~}by G.S.Carr and This was the book that had started
it all.}
\textbf{3.)~ He spent a total of five years in Cambridge University and
was eventually~ received a PhD in mathematics. ~~}
\textbf{4.) In 1918 he was elected Fellow of the Royal Society for his
investigation in Elliptic function and the Theory of Numbers. ~}
\textbf{5.) One of Ramanujan's famous quotes is, " An equation means
nothing to me unless it expresses a thought of God.~ "}
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References :~
~
\url{https://math.stackexchange.com/questions/49628/hardy-ramanujan-asymptotic-formula-for-the-partition-number}
~~
\url{http://ramanujan.sirinudi.org/}
\url{https://cosmosmagazine.com/mathematics/ramanujan-humble-maths-genius}
\url{https://www.thebetterindia.com/52974/srinivasa-ramanujan-mathematician-biopic/}
~
\url{http://www-history.mcs.st-and.ac.uk/Biographies/Ramanujan.html}
\url{https://www.famousscientists.org/srinivasa-ramanujan/}
~
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