When it comes to mathematical genius there is one name you need to know and that is of Srinivasa Ramanujan.
He was a remarkable mathematician and could see connections between numbers that nobody else could see.
Ramanujan was a self-taught Indian man who, despite having no formal education in mathematics,
reached out to a Cambridge professor in the form of this letter and described some of his ideas.
That professor, G. H. Hardy was able to recognize Ramanujan's genius
and invited him from India to come over to England and work with him.
They had a really productive
collaboration that spanned many years
And since this letter is what started such a remarkable story, I thought today I would show you through
exactly what Ramanujan said to Hardy
that enabled hardy to recognize his genius and be so excited about it.
The letter starts off with a cover page followed by about 11 pages of
mathematics.
Now, a couple of these pages have now been lost along with the original cover letter
but what I have here is a transcription published by Hardy.
Let's just have a read of this first paragraph here it begins: Dear Sir,
I beg to introduce myself to you as a clerk in the accounts department of the Port Trust Office of Madras
on a salary of only 20 pounds per annum.
I am now about 23 years of age. I have had no university education
but I have undergone the ordinary school course.
After leaving school, I have been employing the spare time at my disposal
to work at
mathematics.
I have not trodden through the conventional regular course, which is followed in a university course
but I am striking out a new path for myself. I have made a special
Investigation of divergent series in general and the results I get are termed by the local mathematicians as "startling".
Now G. H. Hardy whose first initials stand for Godfrey Harold is one of England's leading pure mathematicians
And Ramanujan is trying to introduce himself to Hardy in such a way that Hardy
understands where he's coming from that he's kind of on the back foot with his situation
and tries to interest Hardy in taking him seriously.
Hardy, like many mathematicians back then as well as now,
would probably receive many such letters from people claiming to have discovered great things or to be
their own kind of genius and it's probably quite hard to tell who is the real deal.
Let's now have a look at the second section of the letter and it was actually rather unpromising.
Ramanujan goes on to describe a result that he must have thought was one of his best but was actually well known at the time.
He is describing something called the Gamma function, which is really an extension of the
factorial function. For the factorial function which we would write as n factorial,
if n was say five,
then this would be equal to
five times four times three times two times one.
The gamma function could also be written as n minus one factorial.
So if we were to do the Gamma of five it would be equal to
four times three times two times one.
If this is our complex plane with real numbers and imaginary numbers
the gamma function is well defined on this whole right hand area
for all complex numbers with positive real part.
What Ramanujan is describing with this equation here is a way to extend this function
to also be defined in this left-hand side area.
In this case, that would be for all negative values of z except for the negative integers.
If we draw a plot of z versus the Gamma of z we see it is indeed
well defined on all of these positive numbers
and for all of the fractional negative numbers, but these dashed lines are the negative integers where the function is not defined.
Now, this trick of extending the domain of a function actually is called analytic continuation
and was well known as a technique at the time and doing analytic continuation
we would get exactly what Ramanujan has written here.
So, although it is not a new result, Ramanujan has independently come up with something that is useful to
mathematicians without having access to the papers in which it would have been published.
Let's take a look at this final paragraph here
in which Ramanujan says that he's come across an article written by Hardy in which there is a statement that no
definite expression has been as yet found for the number of primes less than any given number.
I have found an expression which very nearly approximates to the real result the error being negligible.
I would request you to go through the enclosed papers.
Being poor if you are convinced that there is anything of value. I would like to have my theorems published.
I have not given the actual investigations,
nor the expressions that I get but I have indicated the lines on which I proceed.
Being inexperienced, I would very highly value any advice you give me.
Requesting to be excused for the trouble I give you.
I remain, Dear Sir, Yours truly, S. Ramanujan
So what he is talking about here in the third paragraph is about the distribution of prime numbers.
If we want to find how many prime numbers there are less than or equal to a certain given number x,
then we would denote that with the prime counting function denoted pi of x.
What we have is a bit of a step function where every time we find or count a new prime number
we add one to the tally and many great mathematicians have tried to come up with a function that would accurately
approximate this.
Ramanujan goes into a bit more detail about his technique in the first couple of pages of his working.
These are copies that I have here, the originals being stored at the Cambridge University library.
Ramanujan claimed that he could easily calculate the number of prime numbers up to 100 millions
with generally no error and in some cases with an error of one or two.
Now, it turns out that his prime counting technique was not quite as precise as he claimed it to be
but still was some really great mathematics.
So perhaps those first two results mentioned in the cover letter were not so promising
But Ramanujan then goes on to give over a hundred more results
that shows just how prolific he is in coming up with ideas and just how much work
he's already done. On this third page here
we have the formulas for many sums and integrals that ramanujan considers to be interesting.
On this last page
we have talk of divergent series and we have written down the now infamous result
that one plus two plus three plus four
and so on is equal to minus one over twelve.
And you might have heard of this seemingly absurd result. I'll link to some videos about it in the description as well.
Ramanujan himself acknowledges that to add up all of the positive integers and get a result that isn't positive infinity
seems quite absurd, but his confidence in his own methods
meant that he didn't shy away from presenting this to Hardy. After receiving the letter from Ramanujan,
Hardy consulted with his friend and collaborator Littlewood. A quote from Littlewood
was that these results must be true because no one would have the
imagination to invent them.
It was Littlewood who also went on to later say that every positive integer was one of Ramanujan's personal friends
In Hardy's first response to Ramanujan
he says that he was exceedingly interested by his letter and by the theorems which he stated.
However, he says that before he can judge them, he must see proofs of some of the assertations.
Hardy then groups the results into three categories.
These are, number one, results which are already known, number two,
results which are new but not that important
and number three, results that are new and potentially important.
But Hardy insists that all of this relies on being able to see some proofs of this work.
He put quite a few of the results into that first category of things that were already known
and ramanujan later said that that actually made him really happy because to see that some of his results were indeed already known
verified them as being correct and true because even he himself might have been a bit unsure of a few of them.
In category 2 these are the ones that are new but not very
important. And the only thing that makes it into category three being new and important
is this expression about finding the number of primes less than a given x.
He asks Ramanujan to send him some proofs as quickly as possible
and acknowledges that he has done a good deal of work worth publication.
He also includes at the end of his reply
some notes suggested by Littlewood.
After a bit more back and forth, Ramanujan is invited over to England to join Hardy and Littlewood
and work on publishing some of these results.
Ramanujan unfortunately only lived a short life and so
the things that were able to be published while he was alive were only a small portion of the work that he'd done.
But many years on I believe that mathematicians can still find lots of gems written down in
Ramanujan's notebooks.
Some of his results might seem at first like random mathematical facts
but many of them indeed give insight into
unexpected and surprising relationships
between patterns of numbers.
I hope you have found it interesting to take a look at this letter, which revealed a genius.
Thank you for watching and a special thank you to the viewers who support me on Patreon.
It is your support that makes these videos possible to make at all.
Also, a special shout out to today's patron cat of the day, Stella.
