Vidhi Lalchand
Doctoral Student @ Cambridge Bayesian Machine Learning
Knowing the man who knew Infinity
June 8, 2020
Tea Talk
S. Ramanujan
1887-1920
~ 2 years ago
....while waiting for my 1st year viva to commence at CMS.
~ 2 years ago
....while waiting for my 1st year viva to commence at CMS.
Ramanujan
(1887-1920)
~ 2 years ago
....while waiting for my 1st year viva to commence at CMS.
Ramanujan
(1887-1920)
~ 2 years ago
....while waiting for my 1st year viva to commence at CMS.
Ramanujan
(1887-1920)
....a mysterious letter
arrived at Trinity College, Cambridge addressed to mathematician G.H. Hardy.
....a mysterious letter
Initial Responses
-
".....is this is a prank"
-
"a cruel joke"
Initial Responses
-
".....is this is a prank"
-
"a cruel joke"
....after taking a second look at some of the formulae in the letter
1 + 2 + 3 + .... = -\dfrac{1}{12}
\sqrt{1 + 2\sqrt{1 + 3\sqrt{1 + 4}....}} = 3
Some of the formula he recognised to be true and some others seemed ludicrous but he concluded that they must be true....as no one would have the imagination to invent them.
(divergent series)
(nested radicals)
Origin Story
- Born in a small village in the Madras Presidency (British India).
- Wasn't a remarkable student in school and showed little to no interest in subjects like languages or history.
- He did stumble onto the book "Elementary Results in Pure and Applied Mathematics" by G.S. Carr - notably his first tryst with mathematics.
- He didn't respond to systematic instruction so he dropped out of college with no degree.
After a fair few letters were exchanged, Hardy arranged for Ramanujan to come to Cambridge.
He set sail for England in March 1914.
"A two-time college drop-out, with no formal education beyond high-school. Here he was pitting his brains against the accumulated wisdom of Europe".
- from "The Man who knew Infinity" by Robert Kanigel
Hardy & Ramanujan
"he arrived at results through a process of mingled argument, intuition and induction of which he was unable to give a coherent account "
Despite this, they had an astounding collaboration publishing over 21 papers in 4 years (1914-1919).
He managed to elect Ramanujan to a Trinity fellowship in 1917 making him the first Indian to hold that distinction.
Partition Numbers
\begin{aligned}
p(1) &= 1 \hspace{2mm} (1)\\
p(2) &= 2 \hspace{2mm} (2, 1 + 1) \\
p(3) &= 3 \hspace{2mm} (3, 2 + 1, 1 + 1 + 1) \\
p(4) &= 5 \hspace{2mm} (4, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1)\\
p(5) &= 7 \\
\vdots & \\
p(10) &= 42\\
p(100) &= 190569292 \\
\end{aligned}
p(n) = \dfrac{1}{4n\sqrt{3}}e^{\pi\sqrt{2n/3}}
n
p(n)
Eponyms
Taxicab Numbers
I remember once going to see him [Ramanujan] when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two [positive] cubes in two different ways.
As told by Hardy:
Taxicab Numbers
(also called the Hardy-Ramanujan numbers)
Smallest integer expressible as a sum of two positive integer cubes in n distinct ways
Ta(1) = 2 = 1^{3} + 1^{3}
\begin{aligned}
Ta(2) = 1729 &= 9^{3} + 10^{3} \\
&=1^{3} + 12^{3} \\
\end{aligned}
Taxicab Numbers
(also called the Hardy-Ramanujan numbers)
Smallest integer expressible as a sum of two positive integer cubes in n distinct ways
Ta(1) = 2 = 1^{3} + 1^{3}
\begin{aligned}
Ta(2) = 1729 &= 9^{3} + 10^{3} \\
&=1^{3} + 12^{3} \\
\end{aligned}
\begin{aligned}
{Ta}(3)=87539319&=167^{3}+436^{3}
\\&=228^{3}+423^{3}
\\&=255^{3}+414^{3}
\end{aligned}
The "Lost" Notebooks
(Work he did between 1919 and 1920 in Madras)
In popular culture
In popular culture
In popular culture
The broadway production "A First Class Man" was based on his life.
"A disappearing number" - a British play based on the relationship between Hardy and Ramanujan.
In Popular Culture
On G.H.Hardy
On G.H.Hardy
"one cannot do well in more than one thing, so if what one wants is to excel, he would be right. But if what one wants is fulfillment, I think it’s a different story"
My own thoughts.
\infty
Knowing the man who knew Infinity
By Vidhi Lalchand
Knowing the man who knew Infinity
CBL Tea Talk
