One of the most remarkable things about the
mathematician Ramanujan was that he was mostly
self taught. Today I would like to show you
the math book that Ramanujan read when he
was 16, and it is said to have been one of
the key factors in awakening his genius. That
book is A Synopsis of Elementary Results in
Pure and Applied Mathematics by G. S. Carr.
Now, this book might not have been the best
choice but it’s what Ramanujan could get
his hands on. He was quite poor and a friend
lent him a library copy of this book. The
book was a collection of around 5000 mathematical
theorems and Ramanujan studied it in great
detail. The author was a private tutor for
students preparing for exams such as the Cambridge
University Mathematical Tripos and the book
was written to be kind of an aid for these
students in preparing for the exam, and it
wasn’t really intended to be a stand alone
teaching resource. The author mentions this
in the preface of the book, saying that it
is arranged with the view of assisting the
student in the task of revision. To this end
I have, in many cases, merely indicated the
salient points of a demonstration, or merely
referred to the theorems by which the proposition
is proved. He says that it should be a similar
pleasure to a traveller who is discovering
the roads for themselves with the assistance
of a map. The concise nature of the book is
quite important because it may have influenced
Ramanujan's own approach to writing down mathematics.
He was famous for not really giving any proofs
of his discoveries and in the first letter
that he wrote to professor Hardy, where he
laid out pages and pages of his mathematical
discoveries, he didn’t give any proofs of
them or indicate where they really came from,
and in Hardy’s response to Ramanujan he
said, you know, these look promising but before
I can believe them I need to see your proofs.
And much of the collaboration in the years
that followed between Hardy and Ramanujan
was about Hardy teaching Ramanujan how to
write these formal proofs such that his work
could get published. So maybe that was the
influence of the writing style in this book
but it could have been some other factors
as well. Paper would have been quite expensive
for Ramanujan, so maybe that’s why he chose
to only write down his final results, and
he did his working elsewhere,. He might have
also thought that it was unnecessary to write
down any proofs because maybe to him if someone
asked him where an equation came from, he
could just see it in his mind. So let's have
a look at some of the pages of this book,
and we’ll have a look at how the author
describes things. The book is huge and it
starts off with some basic algebra. Factors
are the very first thing that is mentioned.
But then it accelerates through the theory
of equations, trigonometry, geometry, conics.
It lays a foundation for differential and
integral calculus and differential equations.
It also covers some pretty advanced ideas
that even I don’t understand and there are
some words here that I don't think I've seen
before. Ramanujan published his first journal
paper when he was 24. It was published in
the Journal of the Indian Mathematical Society
and it was about properties of the Bernoulli
numbers. Now, Bernoulli numbers are defined
in the text but there isn’t any mention
of their properties, so perhaps this inspired
Ramanujan to want to fill in the blanks. On
page 359 we can see where Ramanujan likely
first learned about the gamma function, and
the gamma function went on to feature a lot
in Ramanujan's work, so much so that he mentioned
it as one of the results he was most proud
of in his first letter to Hardy. There’s
also a mention of things like infinite series
and their convergence or divergence and that’s
another thing that I think was really important
to Ramanujan’s thinking and perhaps this
is where he first got inspired. So do I think
that you should go and try to read this book?
Well, no, I don’t. I don’t think there
is anything particularly special about this
book, and I think in the case of Ramanujan,
what was special was the reader. I think that
the best book to learn math from is the one
that you can get your hands on. Whether it’s
from the local library or from a second hand
book store or lent from a friend like in the
case of Ramanujan, then that’s probably
going to do you just fine, as long as it’s
pitched at the level that you’re looking
for. I think that this synopsis of mathematics
that we’ve spoken about is probably a bit
difficult for most people, certainly not the
easiest book to learn math from, but in the
description I’ve linked to a free PDF of
it if you really do want to use it as your
resource, or just have a little look at it
a bit more out of curiosity. If you do want
some book recommendations, I’ve made videos
on books for learning mathematics and books
for learning physics so I’ll link them down
in the description along with a few other
resources including the subreddit r/learnmath.
Now, you can read about maths all you want,
but it would be doing practice problems and
exercises for yourself that will really cement
the ideas. Another resource is Brilliant.org
who have been a supporter of my channel for
a long time and are the sponsor of this video.
If you are looking for an interactive resource
where you can learn at your own pace, then
Brilliant is a great option. Brilliant have
over 60 courses in math, science and computer
science and you can sign up for free at Brilliant.org/Tibees
and see if it’s a good fit for you. If it
is, then by using my link you can get 20 percent
off a premium membership. So thanks Brilliant
and thanks to my patrons, including today’s
patron cat of the day, Holly. Thanks for watching
and I hope to see you next time.
