The Spirit of Ramanujan
Go back to the Math Jam ArchiveAoPS instructor David Patrick will discuss the Spirit of Ramanujan Talent Initiative. We will discuss Ramanujan's life and some of the mathematics that he produced, and we will discuss the SoR program and how to apply. AoPS is a partner of SoR in 2020-21. We will be joined by Professor Ken Ono of the University of Virginia, who is the Director of the SoR program and was an associate producer and mathematical consultant for The Man Who Knew Infinity, a feature film about Ramanujan starring Dev Patel.
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Facilitator: Dave Patrick
DPatrick
2021-12-02 19:26:01
Welcome to the 2021-22 Spirit of Ramanujan Math Jam! We'll get started in about 5 minutes.
Welcome to the 2021-22 Spirit of Ramanujan Math Jam! We'll get started in about 5 minutes.
smileapple
2021-12-02 19:26:45
cool, is it about the movie or the organization/scholarship?
cool, is it about the movie or the organization/scholarship?
DPatrick
2021-12-02 19:27:07
We'll mention the movie, but it's mostly about Ramanujan himself, some of the math he did, and we'll talk about the scholarship program at the end.
We'll mention the movie, but it's mostly about Ramanujan himself, some of the math he did, and we'll talk about the scholarship program at the end.
medical_ordnance
2021-12-02 19:27:17
Let me guess DPatrick David Patrick?
Let me guess DPatrick David Patrick?
DPatrick
2021-12-02 19:27:23
Yep, that's me!
Yep, that's me!
peterguo
2021-12-02 19:28:14
why dont you host the math jam on his birthday
why dont you host the math jam on his birthday
peterguo
2021-12-02 19:28:14
its in like less than a month
its in like less than a month
DPatrick
2021-12-02 19:28:27
True...I think there will be other events around his birthday though!
True...I think there will be other events around his birthday though!
DPatrick
2021-12-02 19:28:40
This was a good night on our schedule and on our special guest's schedule.
This was a good night on our schedule and on our special guest's schedule.
Kaito471
2021-12-02 19:29:00
Special guest?
Special guest?
suvamkonar
2021-12-02 19:29:00
who's the guest?
who's the guest?
DPatrick
2021-12-02 19:29:08
You'll find out in a minute or two!
You'll find out in a minute or two!
smileapple
2021-12-02 19:29:45
ken ono?
ken ono?
ThinkThink
2021-12-02 19:29:45
Let me guess, the guest is KenOno?
Let me guess, the guest is KenOno?
DPatrick
2021-12-02 19:29:59
Good deducing. I'll introduce him properly in a minute or so.
Good deducing. I'll introduce him properly in a minute or so.
DPatrick
2021-12-02 19:30:14
Time to get started!
Time to get started!
DPatrick
2021-12-02 19:30:20
Welcome to the 2021-22 Spirit of Ramanujan Math Jam!
Welcome to the 2021-22 Spirit of Ramanujan Math Jam!
DPatrick
2021-12-02 19:30:29
DPatrick
2021-12-02 19:30:40
I'm Dave Patrick, and I'll be leading our discussion tonight. Many of you know me from around AoPS: I've taught dozens of AoPS classes over the past 17 years, and I've written or co-written a few of our textbooks.
I'm Dave Patrick, and I'll be leading our discussion tonight. Many of you know me from around AoPS: I've taught dozens of AoPS classes over the past 17 years, and I've written or co-written a few of our textbooks.
DPatrick
2021-12-02 19:30:56
I'm pleased to have a very special guest here with us tonight. Ken Ono (KenOno) is the Thomas Jefferson Professor and Chair of the Department of Mathematics at the University of Virginia.
I'm pleased to have a very special guest here with us tonight. Ken Ono (KenOno) is the Thomas Jefferson Professor and Chair of the Department of Mathematics at the University of Virginia.
KenOno
2021-12-02 19:31:05
Hi everyone!
Hi everyone!
DPatrick
2021-12-02 19:31:14
Ken is also the Director of the "Spirit of Ramanujan" project, with which AoPS is a partner, and which we'll be talking about tonight.
Ken is also the Director of the "Spirit of Ramanujan" project, with which AoPS is a partner, and which we'll be talking about tonight.
KenOno
2021-12-02 19:31:34
So happy to be here in AoPs world.
So happy to be here in AoPs world.
DPatrick
2021-12-02 19:31:41
Before we get started I would like to take a moment to explain our virtual classroom procedures to those who have not previously participated in a Math Jam or one of our online classes.
Before we get started I would like to take a moment to explain our virtual classroom procedures to those who have not previously participated in a Math Jam or one of our online classes.
DPatrick
2021-12-02 19:31:59
The classroom is moderated, meaning that students can type into the classroom, but these comments will not go directly into the room. These comments go to the moderators, who may choose to share your comments with the room, or to respond to them privately if we can.
The classroom is moderated, meaning that students can type into the classroom, but these comments will not go directly into the room. These comments go to the moderators, who may choose to share your comments with the room, or to respond to them privately if we can.
DPatrick
2021-12-02 19:32:14
This helps keep the discussion organized and on track. This also means that only well-written comments are likely to be dropped into the classroom, so please take time writing responses that are complete and easy to read.
This helps keep the discussion organized and on track. This also means that only well-written comments are likely to be dropped into the classroom, so please take time writing responses that are complete and easy to read.
DPatrick
2021-12-02 19:32:26
Today we also have a teaching assistant: Tudor Sarpe (Snakes).
Today we also have a teaching assistant: Tudor Sarpe (Snakes).
DPatrick
2021-12-02 19:32:43
Tudor first joined AoPS in 2015, though he has been browsing the AoPS contest collections and forums since 2012. Tudor is from Moldova, a small country in Eastern Europe that too few people know about. During High School, he has been highly involved in mathematics and science olympiads. He has won top places at national maths, biology, physics and chemistry olympiads, has a couple bronze medals from the Balkan Mathematical Olympiad, and has been on the IMO team.
Tudor first joined AoPS in 2015, though he has been browsing the AoPS contest collections and forums since 2012. Tudor is from Moldova, a small country in Eastern Europe that too few people know about. During High School, he has been highly involved in mathematics and science olympiads. He has won top places at national maths, biology, physics and chemistry olympiads, has a couple bronze medals from the Balkan Mathematical Olympiad, and has been on the IMO team.
Snakes
2021-12-02 19:32:52
Hi, everyone
Hi, everyone
DPatrick
2021-12-02 19:33:08
Snakes is here to try to answer any questions that you have, and Ken and I will also be answering questions at the end. We may have a lot of students here tonight, so please be patient! We may not be able to get to every question.
Snakes is here to try to answer any questions that you have, and Ken and I will also be answering questions at the end. We may have a lot of students here tonight, so please be patient! We may not be able to get to every question.
DPatrick
2021-12-02 19:33:24
Tonight, we're going to talk about the life and mathematics of Srinivasa Ramanujan.
Tonight, we're going to talk about the life and mathematics of Srinivasa Ramanujan.
DPatrick
2021-12-02 19:33:29
DPatrick
2021-12-02 19:33:54
And we'll be talking about a project called "Spirit of Ramanujan" that Ken Ono founded, the goal of which is to help find and nurture future mathematical and scientific talent.
And we'll be talking about a project called "Spirit of Ramanujan" that Ken Ono founded, the goal of which is to help find and nurture future mathematical and scientific talent.
DPatrick
2021-12-02 19:34:07
But first, let's talk about Ramanujan himself.
But first, let's talk about Ramanujan himself.
DPatrick
2021-12-02 19:34:13
Anybody know about him?
Anybody know about him?
CalvinAndDobbs
2021-12-02 19:34:41
My dad said he could do complex math problems in his head and he didn't go to a universinty or anything
My dad said he could do complex math problems in his head and he didn't go to a universinty or anything
Ladka13
2021-12-02 19:34:41
the most famous Indian mathematician.
the most famous Indian mathematician.
aquagold
2021-12-02 19:34:41
yes he found a fraction that is better than 22/7 for pi
yes he found a fraction that is better than 22/7 for pi
srihaas
2021-12-02 19:34:41
yes, he is a famous indian mathmaticean
yes, he is a famous indian mathmaticean
ZJ42
2021-12-02 19:34:41
I know the story about the taxi
I know the story about the taxi
Kaito471
2021-12-02 19:34:48
He only lived for 33 years
According to the post stamp
He only lived for 33 years
According to the post stamp
P3Tan
2021-12-02 19:34:48
He was born in 1887 and died in 1920
He was born in 1887 and died in 1920
KenOno
2021-12-02 19:35:09
Lots of knowledgeable folks out tonight.
Lots of knowledgeable folks out tonight.
DPatrick
2021-12-02 19:35:11
As you may know, there was recently (2015) a motion picture about Ramanujan's life titled The Man Who Knew Infinity, starring Dev Patel and Jeremy Irons. The film is based on a biography of the same title written in 1991 by Robert Kanigel.
As you may know, there was recently (2015) a motion picture about Ramanujan's life titled The Man Who Knew Infinity, starring Dev Patel and Jeremy Irons. The film is based on a biography of the same title written in 1991 by Robert Kanigel.
DPatrick
2021-12-02 19:35:17
Our guest Ken Ono was one of the mathematical consultants on the film.
Our guest Ken Ono was one of the mathematical consultants on the film.
DPatrick
2021-12-02 19:35:21
Did anybody here see the movie?
Did anybody here see the movie?
krishiam
2021-12-02 19:35:44
I did it was good
I did it was good
Jc426
2021-12-02 19:35:44
I saw the trailer
I saw the trailer
ZJ42
2021-12-02 19:35:44
I didn't but now I want to
I didn't but now I want to
dBIT
2021-12-02 19:35:49
Yes I have seen the movie
Yes I have seen the movie
Printrbot
2021-12-02 19:35:49
No, but I want to now!
No, but I want to now!
DPatrick
2021-12-02 19:35:56
Unfortunately as of right now it does not appear to be available for free on any streaming service in the U.S.
It is available for rent for $2.99 on a bunch of different sites.
Unfortunately as of right now it does not appear to be available for free on any streaming service in the U.S.
It is available for rent for $2.99 on a bunch of different sites.
DPatrick
2021-12-02 19:36:14
Ramanujan was born in Erode, India in 1887. He grew up relatively poor and worked as a clerk. But from an early age he clearly had a love and aptitude for mathematics.
Ramanujan was born in Erode, India in 1887. He grew up relatively poor and worked as a clerk. But from an early age he clearly had a love and aptitude for mathematics.
DPatrick
2021-12-02 19:36:30
By age 11 he was the mathematical equal of university students who were boarding in his family home. He was able to continue his advanced study of math by reading as many books as he could get his hands on. He also, around age 15, discovered on his own how to solve a quartic equation.
By age 11 he was the mathematical equal of university students who were boarding in his family home. He was able to continue his advanced study of math by reading as many books as he could get his hands on. He also, around age 15, discovered on his own how to solve a quartic equation.
DPatrick
2021-12-02 19:36:46
(We'll talk more about this in a few minutes when we start talking about Ramanujan's mathematical achievements. And don't worry if you don't know what "quartic equation" means -- I'll explain that too!)
(We'll talk more about this in a few minutes when we start talking about Ramanujan's mathematical achievements. And don't worry if you don't know what "quartic equation" means -- I'll explain that too!)
Ladka13
2021-12-02 19:37:03
but how did he get so good with no training. he can't just wake up and know everything
but how did he get so good with no training. he can't just wake up and know everything
DPatrick
2021-12-02 19:37:23
Well, he was fortunate in that he had access to lots of books. And he was a really hard worker!
Well, he was fortunate in that he had access to lots of books. And he was a really hard worker!
DPatrick
2021-12-02 19:37:41
A particular book that furthered his study was A Synopsis of Elementary Results in Pure and Applied Mathematics by G.S. Carr, that Ramanujan obtained in 1903 at age 16. This book was written in 1886. It contained over 5000 theorems and was an attempt to summarize all the "basic" mathematics known at the time. (Last time I checked, you can find it in Google Books!) Ramanujan read and studied it in great detail and it formed the foundation of his mathematical thought.
A particular book that furthered his study was A Synopsis of Elementary Results in Pure and Applied Mathematics by G.S. Carr, that Ramanujan obtained in 1903 at age 16. This book was written in 1886. It contained over 5000 theorems and was an attempt to summarize all the "basic" mathematics known at the time. (Last time I checked, you can find it in Google Books!) Ramanujan read and studied it in great detail and it formed the foundation of his mathematical thought.
DPatrick
2021-12-02 19:38:09
Ramanujan spent so much time on mathematics that he generally did poorly in other subjects in school, and as such failed to graduate from university. He unfortunately also suffered from various illness (much of which were probably made worse due to poverty, and to the lack of "modern" medicine in India at the time), which also interfered with his formal studies.
Ramanujan spent so much time on mathematics that he generally did poorly in other subjects in school, and as such failed to graduate from university. He unfortunately also suffered from various illness (much of which were probably made worse due to poverty, and to the lack of "modern" medicine in India at the time), which also interfered with his formal studies.
srihaas
2021-12-02 19:38:21
did he write any books?
did he write any books?
DPatrick
2021-12-02 19:38:35
Indeed he wrote a lot!
Indeed he wrote a lot!
DPatrick
2021-12-02 19:38:51
He started publishing papers containing his original work in India, small articles at first, but eventually his first "full" paper titled "Some Properties of Bernoulli's Numbers" in 1911 in the Journal of the Indian Mathematical Society. This paper, in particular, started establishing Ramanujan's reputation as a mathematical genius in India.
He started publishing papers containing his original work in India, small articles at first, but eventually his first "full" paper titled "Some Properties of Bernoulli's Numbers" in 1911 in the Journal of the Indian Mathematical Society. This paper, in particular, started establishing Ramanujan's reputation as a mathematical genius in India.
DPatrick
2021-12-02 19:39:11
But, even though his mathematical talent was becoming acknowledged within India, he had trouble finding employment as a mathematician because of his lack of a formal university degree. Instead, he had to work as a clerk for low pay and rely on the additional financial support of others in the Indian mathematics community.
But, even though his mathematical talent was becoming acknowledged within India, he had trouble finding employment as a mathematician because of his lack of a formal university degree. Instead, he had to work as a clerk for low pay and rely on the additional financial support of others in the Indian mathematics community.
DPatrick
2021-12-02 19:39:25
Anybody know what changed for him in 1913?
Anybody know what changed for him in 1913?
DPatrick
2021-12-02 19:39:34
(If you saw the movie you probably know)
(If you saw the movie you probably know)
HobbsClass
2021-12-02 19:39:56
He wrote a letter to Hardy
He wrote a letter to Hardy
Ladka13
2021-12-02 19:39:56
he meta famous mathematician I believe
he meta famous mathematician I believe
Kaito471
2021-12-02 19:39:56
Did he gain an opportunity of some sort?
Did he gain an opportunity of some sort?
ZJ42
2021-12-02 19:39:56
He met Hardy
He met Hardy
srihaas
2021-12-02 19:40:00
he met G.H Hardy? (did not see the movie)
he met G.H Hardy? (did not see the movie)
Arcticturn
2021-12-02 19:40:00
He worked with GH hardy
He worked with GH hardy
DPatrick
2021-12-02 19:40:09
Right! A major turning point in Ramanujan's life was his writing a letter to the prominent British mathematician G. H. Hardy in January 1913. His letter began:
Right! A major turning point in Ramanujan's life was his writing a letter to the prominent British mathematician G. H. Hardy in January 1913. His letter began:
DPatrick
2021-12-02 19:40:14
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'.
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'.
DPatrick
2021-12-02 19:40:28
Enclosed with the letter was pages of Ramanujan's work -- he was hoping that Hardy would be able to mentor him and help him get his work improved and published to the wider world. He had written to other mathematicians in Britain, who had basically ignored his letters.
Enclosed with the letter was pages of Ramanujan's work -- he was hoping that Hardy would be able to mentor him and help him get his work improved and published to the wider world. He had written to other mathematicians in Britain, who had basically ignored his letters.
DPatrick
2021-12-02 19:40:42
But Hardy took the letter seriously. Although a lot of the work was unproved and was poorly written, it was clear to Hardy that there was a lot of really good math in Ramanujan's letter! Hardy started a professional relationship with Ramanujan and was able to provide financial support.
But Hardy took the letter seriously. Although a lot of the work was unproved and was poorly written, it was clear to Hardy that there was a lot of really good math in Ramanujan's letter! Hardy started a professional relationship with Ramanujan and was able to provide financial support.
KenOno
2021-12-02 19:40:45
David, you type really fast!!!
David, you type really fast!!!
DPatrick
2021-12-02 19:40:59
Magic.
Magic.
ZJ42
2021-12-02 19:41:10
cntrl C/V?
cntrl C/V?
DPatrick
2021-12-02 19:41:19
It is possible I may have written some of this beforehand...
It is possible I may have written some of this beforehand...
DPatrick
2021-12-02 19:41:28
Anyway...In 1914, Hardy arranged for Ramanujan to be able to travel and visit Trinity College at Cambridge University in England.
Anyway...In 1914, Hardy arranged for Ramanujan to be able to travel and visit Trinity College at Cambridge University in England.
DPatrick
2021-12-02 19:41:44
Any idea how long it took Ramanujan to travel from India to London? Remember, this was slightly over 100 years ago!
Any idea how long it took Ramanujan to travel from India to London? Remember, this was slightly over 100 years ago!
awesomeaadharsa
2021-12-02 19:42:09
a month or 2
a month or 2
peterguo
2021-12-02 19:42:09
like a few weeks possibly
like a few weeks possibly
plantLover
2021-12-02 19:42:09
a month?
a month?
CerealTurtle
2021-12-02 19:42:09
3 weeks?
3 weeks?
QueenRaven337
2021-12-02 19:42:09
a month? or maybe 2 weeks?
a month? or maybe 2 weeks?
Arcticturn
2021-12-02 19:42:09
3 weeks?
3 weeks?
DPatrick
2021-12-02 19:42:19
These are all pretty close. It took 27 days.
These are all pretty close. It took 27 days.
DPatrick
2021-12-02 19:42:27
Ramanujan left India on March 17, 1914 by boat, and arrived in London on April 14.
Ramanujan left India on March 17, 1914 by boat, and arrived in London on April 14.
DPatrick
2021-12-02 19:42:42
This began a 5-year collaboration with Hardy that produced many important mathematical results. Also, Ramanujan was finally able to finish his formal education, and graduated from Cambridge with the equivalent of a Ph.D. in 1916.
This began a 5-year collaboration with Hardy that produced many important mathematical results. Also, Ramanujan was finally able to finish his formal education, and graduated from Cambridge with the equivalent of a Ph.D. in 1916.
Arcticturn
2021-12-02 19:43:01
in 2 years?!
in 2 years?!
DPatrick
2021-12-02 19:43:19
He had most of the knowledge already -- what the university did mostly was just make the degree official.
He had most of the knowledge already -- what the university did mostly was just make the degree official.
DPatrick
2021-12-02 19:43:38
With financial stability and a group of peers that respected his work, Ramanujan was able to professionally thrive in England. Many honors followed including perhaps the most prestigious: in 1918 he was elected a Fellow of the Royal Society, probably the highest honor that can be given for science in Britain. Ramanujan was only the second Indian so elected, and one of the youngest at age 31.
With financial stability and a group of peers that respected his work, Ramanujan was able to professionally thrive in England. Many honors followed including perhaps the most prestigious: in 1918 he was elected a Fellow of the Royal Society, probably the highest honor that can be given for science in Britain. Ramanujan was only the second Indian so elected, and one of the youngest at age 31.
Morgan456
2021-12-02 19:43:46
Did he ever go back to India?
Did he ever go back to India?
DPatrick
2021-12-02 19:44:11
Yes: In early 1919, World War I was over so (now age 31) he decided to return to his native India, with his reputation firmly established and his financial situation stable. The hope was that he would continue his career as a full member of India's mathematical community.
Yes: In early 1919, World War I was over so (now age 31) he decided to return to his native India, with his reputation firmly established and his financial situation stable. The hope was that he would continue his career as a full member of India's mathematical community.
KenOno
2021-12-02 19:44:14
And we are still trying to figure out stuff in 3 notebooks he left behind. I've held them in my hands.
And we are still trying to figure out stuff in 3 notebooks he left behind. I've held them in my hands.
DPatrick
2021-12-02 19:44:32
Sadly, upon returning to India, he almost immediately fell into poor health, and after about a year, he died on April 26, 1920, at an age of only 32.
Sadly, upon returning to India, he almost immediately fell into poor health, and after about a year, he died on April 26, 1920, at an age of only 32.
ChrisalonaLiverspur
2021-12-02 19:44:51
srihaas
2021-12-02 19:44:51
he had so much to live for
he had so much to live for
dBIT
2021-12-02 19:45:03
I got the opportunity ti visit the Ramanujan Museum in Chennai. Heartwarming to see photos and his letters displayed
I got the opportunity ti visit the Ramanujan Museum in Chennai. Heartwarming to see photos and his letters displayed
DPatrick
2021-12-02 19:45:09
If you're interested in reading a more thorough biography of Ramanujan, there is one on the "History of Mathematics" site maintained by the University of St. Andrews, at
http://www-history.mcs.st-and.ac.uk/Biographies/Ramanujan.html
The quotes that I've used above are from this page.
If you're interested in reading a more thorough biography of Ramanujan, there is one on the "History of Mathematics" site maintained by the University of St. Andrews, at
http://www-history.mcs.st-and.ac.uk/Biographies/Ramanujan.html
The quotes that I've used above are from this page.
KenOno
2021-12-02 19:45:13
Nice dBit.
Nice dBit.
DPatrick
2021-12-02 19:45:29
As an aside, this website is a great resource: it has biographies of over 3,000 mathematicians spanning over 35 centuries, from Ahmes (1680-1620 BC) to Mirzakhani (1977-2017), and includes many living mathematicians as well.
As an aside, this website is a great resource: it has biographies of over 3,000 mathematicians spanning over 35 centuries, from Ahmes (1680-1620 BC) to Mirzakhani (1977-2017), and includes many living mathematicians as well.
DPatrick
2021-12-02 19:45:50
Now let's talk about some the mathematics that Ramanujan studied.
Now let's talk about some the mathematics that Ramanujan studied.
DPatrick
2021-12-02 19:46:07
Ramanujan did a lot of mathematics in his relatively short life, and a great deal of it was quite advanced. Way too advanced for us to talk about here! But there are some objects of his study that we can discuss tonight.
Ramanujan did a lot of mathematics in his relatively short life, and a great deal of it was quite advanced. Way too advanced for us to talk about here! But there are some objects of his study that we can discuss tonight.
DPatrick
2021-12-02 19:46:16
First, I mentioned earlier that one of Ramanujan's earliest advanced studies was, in his early teen years, his study of solutions to quartic and quintic equations.
First, I mentioned earlier that one of Ramanujan's earliest advanced studies was, in his early teen years, his study of solutions to quartic and quintic equations.
DPatrick
2021-12-02 19:46:26
To start, many of you already know how to solve quadratic equations, which is an equation involving a polynomial of degree 2. In particular, how can we solve $ax^2 + bx + c = 0$?
To start, many of you already know how to solve quadratic equations, which is an equation involving a polynomial of degree 2. In particular, how can we solve $ax^2 + bx + c = 0$?
karthic7073
2021-12-02 19:47:01
quadratic formula
quadratic formula
PWT
2021-12-02 19:47:01
quadratic formula
quadratic formula
RedFireTruck
2021-12-02 19:47:01
quadratic formula
quadratic formula
tacowizard
2021-12-02 19:47:01
Comeplete the square
Comeplete the square
QueenRaven337
2021-12-02 19:47:01
Factoring, complete the square, or quadratic formula
Factoring, complete the square, or quadratic formula
ChrisalonaLiverspur
2021-12-02 19:47:01
quadratic formula? complete the square? factor?
quadratic formula? complete the square? factor?
DPatrick
2021-12-02 19:47:13
Certainly there are many different methods!
Certainly there are many different methods!
DPatrick
2021-12-02 19:47:25
But one that always works is to use the well-known quadratic formula: $$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$
But one that always works is to use the well-known quadratic formula: $$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$
DPatrick
2021-12-02 19:47:54
It is pretty straightforward to prove that this always works, using basic algebra. (We prove it in AoPS's Introduction to Algebra textbook and classes.)
It is pretty straightforward to prove that this always works, using basic algebra. (We prove it in AoPS's Introduction to Algebra textbook and classes.)
HonestCat
2021-12-02 19:48:01
I prefer completing the square
I prefer completing the square
Ladka13
2021-12-02 19:48:01
completing the square is so underrated
completing the square is so underrated
DPatrick
2021-12-02 19:48:05
I agree!
I agree!
DPatrick
2021-12-02 19:48:13
The next step up is solving the cubic equation $ax^3 + bx^2 + cx + d = 0$.
The next step up is solving the cubic equation $ax^3 + bx^2 + cx + d = 0$.
QueenRaven337
2021-12-02 19:48:23
But, those types of formulas has to be pretty long if applied to cubic, quartic, quintic etc.
But, those types of formulas has to be pretty long if applied to cubic, quartic, quintic etc.
DPatrick
2021-12-02 19:48:32
Indeed -- there is a formula for solving this as well, but it's a lot more complicated.
Indeed -- there is a formula for solving this as well, but it's a lot more complicated.
KenOno
2021-12-02 19:48:37
Nice QueenRaven337.
Nice QueenRaven337.
DPatrick
2021-12-02 19:48:42
If we set
$$\begin{array}{rcl}
p &=& -\frac{b}{3a} \\
q &=& p^3 + \frac{bc-3ad}{6a^2} \\
r &=& \frac{c}{3a}
\end{array}$$
If we set
$$\begin{array}{rcl}
p &=& -\frac{b}{3a} \\
q &=& p^3 + \frac{bc-3ad}{6a^2} \\
r &=& \frac{c}{3a}
\end{array}$$
DPatrick
2021-12-02 19:48:53
then we get
$$x = \sqrt[3]{q + \sqrt{q^2 + (r-p^2)^3}} + \sqrt[3]{q - \sqrt{q^2 + (r-p^2)^3}} + p.$$
then we get
$$x = \sqrt[3]{q + \sqrt{q^2 + (r-p^2)^3}} + \sqrt[3]{q - \sqrt{q^2 + (r-p^2)^3}} + p.$$
vedp4008
2021-12-02 19:49:08
wow
wow
ZJ42
2021-12-02 19:49:08
woah
woah
DPatrick
2021-12-02 19:49:16
Yeah. Most people don't bother to ever learn this formula nowadays: it's way too complicated to accurately memorize. (I certainly don't have it memorized -- I looked it up.)
Yeah. Most people don't bother to ever learn this formula nowadays: it's way too complicated to accurately memorize. (I certainly don't have it memorized -- I looked it up.)
pikapika007
2021-12-02 19:49:34
how would you go about proving it?
how would you go about proving it?
DPatrick
2021-12-02 19:49:46
It's pretty complicated -- way too complicated to quickly discuss tonight I'm afraid.
It's pretty complicated -- way too complicated to quickly discuss tonight I'm afraid.
DPatrick
2021-12-02 19:50:02
But it's been around a while. This formula, and other similar formulas for solving cubics, were discovered in the 1500s. Ramanujan would have learned this in his early teens.
But it's been around a while. This formula, and other similar formulas for solving cubics, were discovered in the 1500s. Ramanujan would have learned this in his early teens.
DPatrick
2021-12-02 19:50:24
What Ramanujan then undertook, and eventually successfully solved himself, was solving the quartic equation $ax^4 + bx^3 + cx^2 + dx + e = 0$.
What Ramanujan then undertook, and eventually successfully solved himself, was solving the quartic equation $ax^4 + bx^3 + cx^2 + dx + e = 0$.
srihaas
2021-12-02 19:50:44
oh no...
oh no...
Coldapple
2021-12-02 19:50:44
wow
wow
DPatrick
2021-12-02 19:50:54
I'm not even going to write the formula down -- it would take way way too much space.
I'm not even going to write the formula down -- it would take way way too much space.
DPatrick
2021-12-02 19:51:06
You can find it here if you're curious: https://commons.wikimedia.org/wiki/File:Quartic_Formula.svg
You can find it here if you're curious: https://commons.wikimedia.org/wiki/File:Quartic_Formula.svg
DPatrick
2021-12-02 19:51:15
(Basically just look for it on Wikipedia)
(Basically just look for it on Wikipedia)
DPatrick
2021-12-02 19:51:38
It's possible that Ramanujan thought that he had discovered something new. He almost certainly didn't know that the quartic had also been solved in the 1500s.
It's possible that Ramanujan thought that he had discovered something new. He almost certainly didn't know that the quartic had also been solved in the 1500s.
DPatrick
2021-12-02 19:51:50
But it is an extremely complicated formula, and it's amazing that Ramanujan was able to discover it, essentially working on his own, at age 15.
But it is an extremely complicated formula, and it's amazing that Ramanujan was able to discover it, essentially working on his own, at age 15.
Printrbot
2021-12-02 19:51:56
You wern't kidding about taking too much space
You wern't kidding about taking too much space
DPatrick
2021-12-02 19:52:04
I know, right?
I know, right?
peterguo
2021-12-02 19:52:12
would the quintic be even longer
would the quintic be even longer
DPatrick
2021-12-02 19:52:25
Yeah, Ramanujan attempted to solve the quintic equation:
$$ ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0 $$
Yeah, Ramanujan attempted to solve the quintic equation:
$$ ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0 $$
mop
2021-12-02 19:52:40
Quintic doesn't exist
Quintic doesn't exist
srihaas
2021-12-02 19:52:40
there is no quintic formula
there is no quintic formula
Arcticturn
2021-12-02 19:52:40
I don't think you can solve the quintic equation...
I don't think you can solve the quintic equation...
mop
2021-12-02 19:52:40
The quintic formula doesn't exist.
The quintic formula doesn't exist.
DPatrick
2021-12-02 19:52:47
He did not have success.
He did not have success.
DPatrick
2021-12-02 19:52:55
That's because in the early 1800s, the mathematicians Ruffini and Abel actually proved that a solution was impossible! That is, there is no possible formula (no matter how complicated) that can be used to plug in the coefficients of a quintic and output a solution. (Ruffini first came up with the result in 1799, but his proof was incomplete. Abel completed the proof in 1824.)
That's because in the early 1800s, the mathematicians Ruffini and Abel actually proved that a solution was impossible! That is, there is no possible formula (no matter how complicated) that can be used to plug in the coefficients of a quintic and output a solution. (Ruffini first came up with the result in 1799, but his proof was incomplete. Abel completed the proof in 1824.)
DPatrick
2021-12-02 19:53:19
A little later in the 1800s, the mathematician Galois developed a new branch of mathematics (that today is called Galois theory) that provided a more elegant proof that there is no formula to solve the quintic. It's part of a branch of mathematics called field theory, which you typically first learn about in an undergraduate abstract algebra course. (Or in AoPS's Group Theory course!)
A little later in the 1800s, the mathematician Galois developed a new branch of mathematics (that today is called Galois theory) that provided a more elegant proof that there is no formula to solve the quintic. It's part of a branch of mathematics called field theory, which you typically first learn about in an undergraduate abstract algebra course. (Or in AoPS's Group Theory course!)
DPatrick
2021-12-02 19:53:43
But Ramanujan probably didn't know any of this at the time. And more importantly, he didn't let his lack of success in this particular problem deter him from his continued pursuit of mathematics.
But Ramanujan probably didn't know any of this at the time. And more importantly, he didn't let his lack of success in this particular problem deter him from his continued pursuit of mathematics.
DPatrick
2021-12-02 19:54:00
That's a good lesson for us all -- a lot of problems are hard. We may not be able to solve them. But that shouldn't deter us from trying! We often learn a lot from our attempts to solve problems, even if we don't come up with the solution we want.
That's a good lesson for us all -- a lot of problems are hard. We may not be able to solve them. But that shouldn't deter us from trying! We often learn a lot from our attempts to solve problems, even if we don't come up with the solution we want.
DPatrick
2021-12-02 19:54:37
So on to something else...someone mentioned a taxi a little bit eariler...
So on to something else...someone mentioned a taxi a little bit eariler...
KenOno
2021-12-02 19:54:40
I actually took 20 years to solve a problem.
I actually took 20 years to solve a problem.
DPatrick
2021-12-02 19:55:01
Indeed, that's not all that unusual for professional mathematicians!
Indeed, that's not all that unusual for professional mathematicians!
ZJ42
2021-12-02 19:55:13
I mentioned the taxi
I mentioned the taxi
DPatrick
2021-12-02 19:55:20
One of the most famous anecdotes regarding Ramanujan occurred when Hardy was visiting Ramanujan in the hospital during one of his many illnesses.
One of the most famous anecdotes regarding Ramanujan occurred when Hardy was visiting Ramanujan in the hospital during one of his many illnesses.
Kaito471
2021-12-02 19:55:31
Wasn't there an interesting number on the top of a taxi or something that made Ramanujan think?
Wasn't there an interesting number on the top of a taxi or something that made Ramanujan think?
dBIT
2021-12-02 19:55:45
yes thr taxicab numbers
yes thr taxicab numbers
DPatrick
2021-12-02 19:55:47
That's right -- Hardy recalled telling Ramanujan that he traveled to the hospital in taxicab number $1729$, and was disappointed that the cab number was so dull. Ramanujan immediately replied that on the contrary, $1729$ was a very interesting number. Do you know why?
That's right -- Hardy recalled telling Ramanujan that he traveled to the hospital in taxicab number $1729$, and was disappointed that the cab number was so dull. Ramanujan immediately replied that on the contrary, $1729$ was a very interesting number. Do you know why?
HonestCat
2021-12-02 19:56:24
Sum of 2 cubes in 2 different ways
Sum of 2 cubes in 2 different ways
SparklyFlowers
2021-12-02 19:56:24
it's the sum of two cubes or smething
it's the sum of two cubes or smething
karthic7073
2021-12-02 19:56:24
it was the smallest number that can be written as the sum of two cubes in two different ways
it was the smallest number that can be written as the sum of two cubes in two different ways
DPatrick
2021-12-02 19:56:36
$1729$ is the smallest number that can be written as a sum of two positive cubes in two different ways! Can you find them?
$1729$ is the smallest number that can be written as a sum of two positive cubes in two different ways! Can you find them?
KenOno
2021-12-02 19:56:47
Nice guys! It is also my license plate number in Virginia.
Nice guys! It is also my license plate number in Virginia.
srihaas
2021-12-02 19:57:05
1729 = 12^3 +1^3 = 10^3 + 9^3
1729 = 12^3 +1^3 = 10^3 + 9^3
HonestCat
2021-12-02 19:57:05
10^3+9^3 and 12^3+1^3
10^3+9^3 and 12^3+1^3
Lionking212
2021-12-02 19:57:05
1729=$12^3+1^3=10^3+9^3$
1729=$12^3+1^3=10^3+9^3$
karthic7073
2021-12-02 19:57:05
10^3+9^3 and 12^3+1^3
10^3+9^3 and 12^3+1^3
twwig
2021-12-02 19:57:05
1000 and 729 is one
1000 and 729 is one
mop
2021-12-02 19:57:05
$1^3+12^3=10^3+9^3=1729$
$1^3+12^3=10^3+9^3=1729$
QueenRaven337
2021-12-02 19:57:05
1000 + 729 and 1728 + 1
1000 + 729 and 1728 + 1
DPatrick
2021-12-02 19:57:16
Nice!
Nice!
DPatrick
2021-12-02 19:57:20
$1729 = 1^3 + 12^3 = 1 + 1728$
$1729 = 9^3 + 10^3 = 729 + 1000$
$1729 = 1^3 + 12^3 = 1 + 1728$
$1729 = 9^3 + 10^3 = 729 + 1000$
DPatrick
2021-12-02 19:57:39
For this reason, $1729$ is called the 2nd taxicab number. It's also sometimes called the Hardy-Ramanujan number.
For this reason, $1729$ is called the 2nd taxicab number. It's also sometimes called the Hardy-Ramanujan number.
sdash314
2021-12-02 19:57:51
What's the first?
What's the first?
DPatrick
2021-12-02 19:57:56
The first is $2 = 1^3 + 1^3$. It's the smallest number that can be written as a sum of two positive cubes in one way.
The first is $2 = 1^3 + 1^3$. It's the smallest number that can be written as a sum of two positive cubes in one way.
DPatrick
2021-12-02 19:58:10
Then, the 3rd taxicab number is the smallest number that can be written as a sum of two positive cubes in three different ways. Any guess as to how big it is? (How many digits do you think it has?)
Then, the 3rd taxicab number is the smallest number that can be written as a sum of two positive cubes in three different ways. Any guess as to how big it is? (How many digits do you think it has?)
boing123
2021-12-02 19:58:44
9?
9?
Coldapple
2021-12-02 19:58:44
10
10
HonestCat
2021-12-02 19:58:44
8
8
krithikrokcs
2021-12-02 19:58:44
9
9
srihaas
2021-12-02 19:58:44
7 digits
7 digits
HonestCat
2021-12-02 19:58:44
x*10^8 or something
x*10^8 or something
arjken
2021-12-02 19:58:44
Maybe 7 digits?
Maybe 7 digits?
DPatrick
2021-12-02 19:58:50
You're in the right ballpark.
You're in the right ballpark.
DPatrick
2021-12-02 19:58:56
It's the 8-digit number $87{,}539{,}319$, which is $167^3 + 436^3 = 228^3 + 423^3 = 255^3 + 414^3$.
It's the 8-digit number $87{,}539{,}319$, which is $167^3 + 436^3 = 228^3 + 423^3 = 255^3 + 414^3$.
DPatrick
2021-12-02 19:59:03
It's likely that this was first discovered as recently as 1957.
It's likely that this was first discovered as recently as 1957.
P3Tan
2021-12-02 19:59:09
so the 4th would be the smallest number that can be written as a sum of two positive cubes in 4 ways?
so the 4th would be the smallest number that can be written as a sum of two positive cubes in 4 ways?
DPatrick
2021-12-02 19:59:14
Exactly!
Exactly!
DPatrick
2021-12-02 19:59:21
And more generally, the $n$th taxicab number is the smallest positive integer that can be written as a sum of two positive cubes in $n$ different ways.
And more generally, the $n$th taxicab number is the smallest positive integer that can be written as a sum of two positive cubes in $n$ different ways.
algebrakitty318
2021-12-02 19:59:31
What about the next one? Was that found?
What about the next one? Was that found?
karthic7073
2021-12-02 19:59:37
whats the highest taxi cab number found?
whats the highest taxi cab number found?
DPatrick
2021-12-02 19:59:50
As of right now, the first 6 taxicab numbers are known:
$2$
$1729$
$87539319$
$6963472309248$
$48988659276962496$
$24153319581254312065344$
As of right now, the first 6 taxicab numbers are known:
$2$
$1729$
$87539319$
$6963472309248$
$48988659276962496$
$24153319581254312065344$
Lionking212
2021-12-02 19:59:59
how do u calculate the nth taxikab number?
how do u calculate the nth taxikab number?
P3Tan
2021-12-02 19:59:59
what is the 10th?
what is the 10th?
DPatrick
2021-12-02 20:00:04
Nobody knows! (Yet!)
Nobody knows! (Yet!)
ZJ42
2021-12-02 20:00:13
computers could probably find more, right?
computers could probably find more, right?
algebrakitty318
2021-12-02 20:00:13
Those are some big numbers
Those are some big numbers
DPatrick
2021-12-02 20:00:24
Yeah...these are big numbers even for computers.
Yeah...these are big numbers even for computers.
DPatrick
2021-12-02 20:00:40
The 6th taxicab number was only discovered by computer search in 2008.
The 6th taxicab number was only discovered by computer search in 2008.
peterguo
2021-12-02 20:00:57
how many digits do you think the seventh taxicab number would have
how many digits do you think the seventh taxicab number would have
KenOno
2021-12-02 20:00:59
Great questions....This is about elliptic curves...and is closely related to the $1 million dollar prize problem.
Great questions....This is about elliptic curves...and is closely related to the $1 million dollar prize problem.
KenOno
2021-12-02 20:01:11
Birch and Swinnerton-Dyer Conjecture.
Birch and Swinnerton-Dyer Conjecture.
DPatrick
2021-12-02 20:01:17
It is known that $24885189317885898975235988544$ can be written as a sum of two perfect cubes in $7$ different ways. So the 7th taxicab number is at most this number.
It is known that $24885189317885898975235988544$ can be written as a sum of two perfect cubes in $7$ different ways. So the 7th taxicab number is at most this number.
DPatrick
2021-12-02 20:01:23
But we don't know if this is actually the 7th taxicab number, because there might be a smaller number that works that we haven't found yet.
But we don't know if this is actually the 7th taxicab number, because there might be a smaller number that works that we haven't found yet.
DPatrick
2021-12-02 20:01:51
And as Ken mentions, this is not just fun and games. This is tied to some really important unanswered questions in mathematics!
And as Ken mentions, this is not just fun and games. This is tied to some really important unanswered questions in mathematics!
DPatrick
2021-12-02 20:02:14
Just recently in 2015, our guest Ken Ono and my AoPS colleague Sarah Trebat-Leder wrote a paper titled The 1729 K3 Surface which tied Ramanujan's observation about the number 1729 to some important curves and surfaces in the modern field of algebraic geometry.
Just recently in 2015, our guest Ken Ono and my AoPS colleague Sarah Trebat-Leder wrote a paper titled The 1729 K3 Surface which tied Ramanujan's observation about the number 1729 to some important curves and surfaces in the modern field of algebraic geometry.
KenOno
2021-12-02 20:02:32
And String theory...and quantum gravity.
And String theory...and quantum gravity.
DPatrick
2021-12-02 20:02:52
Mathematics is connected in so many interesting and mysterious ways.
Mathematics is connected in so many interesting and mysterious ways.
DPatrick
2021-12-02 20:03:07
One more fun note about taxicab numbers: How many people here are fans of the TV show Futurama?
One more fun note about taxicab numbers: How many people here are fans of the TV show Futurama?
Lionking212
2021-12-02 20:03:34
me!!
me!!
DPatrick
2021-12-02 20:03:52
It's a somewhat older show by the same people that created The Simpsons.
It's a somewhat older show by the same people that created The Simpsons.
DPatrick
2021-12-02 20:04:01
Apparently the creators of Futurama are big fans of taxicab numbers.
Apparently the creators of Futurama are big fans of taxicab numbers.
DPatrick
2021-12-02 20:04:12
In particular, the number $1729$ is hidden in several episodes. It's the registration number of the Planet Express ship on the show, for one thing.
In particular, the number $1729$ is hidden in several episodes. It's the registration number of the Planet Express ship on the show, for one thing.
DPatrick
2021-12-02 20:04:19
DPatrick
2021-12-02 20:04:24
Image from mathworld.wolfram.com
Image from mathworld.wolfram.com
DPatrick
2021-12-02 20:04:35
And in one scene someone hails a taxi. Can you make out the number on the cab's roof?
And in one scene someone hails a taxi. Can you make out the number on the cab's roof?
DPatrick
2021-12-02 20:04:39
DPatrick
2021-12-02 20:04:45
Image from Gawker Media
Image from Gawker Media
srihaas
2021-12-02 20:05:07
the 3rd taxicab number
the 3rd taxicab number
karthic7073
2021-12-02 20:05:07
87539319
87539319
P3Tan
2021-12-02 20:05:07
87539319?
87539319?
ZJ42
2021-12-02 20:05:07
87539319
87539319
twwig
2021-12-02 20:05:07
its the third taxicab nimber
its the third taxicab nimber
Ladka13
2021-12-02 20:05:07
81539319
81539319
aquagold
2021-12-02 20:05:07
875399319
875399319
DPatrick
2021-12-02 20:05:12
It's hard to make out, but it's the 3rd taxicab number $87539319$.
It's hard to make out, but it's the 3rd taxicab number $87539319$.
DPatrick
2021-12-02 20:05:21
A little joke for those in the know.
A little joke for those in the know.
DPatrick
2021-12-02 20:05:28
So, it may not surprise you to learn that Futurama's head writer did math contests in high school and has a degree in physics from Harvard!
So, it may not surprise you to learn that Futurama's head writer did math contests in high school and has a degree in physics from Harvard!
mathcountisgreat
2021-12-02 20:05:43
Wow!
Wow!
SparklyFlowers
2021-12-02 20:05:43
oo
oo
DPatrick
2021-12-02 20:06:09
Finally for tonight's math, let's talk about partitions.
Finally for tonight's math, let's talk about partitions.
DPatrick
2021-12-02 20:06:23
A partition is a way a writing a positive integer as a sum of positive integers, where the order of the numbers in the sum doesn't matter.
A partition is a way a writing a positive integer as a sum of positive integers, where the order of the numbers in the sum doesn't matter.
DPatrick
2021-12-02 20:06:32
For example, all of the partitions of $3$ are:
$3$
$2+1$
$1+1+1$
(We don't list $1+2$ separately because the order of the terms doesn't matter. We also consider the "sum" $3$ as a partition of $3$.)
For example, all of the partitions of $3$ are:
$3$
$2+1$
$1+1+1$
(We don't list $1+2$ separately because the order of the terms doesn't matter. We also consider the "sum" $3$ as a partition of $3$.)
DPatrick
2021-12-02 20:06:42
One classic mathematical problem is to count the number of partitions of some number.
One classic mathematical problem is to count the number of partitions of some number.
DPatrick
2021-12-02 20:06:49
For example, how many partitions of $6$ are there?
For example, how many partitions of $6$ are there?
DPatrick
2021-12-02 20:07:14
To count them, we can try to list them.
To count them, we can try to list them.
plantLover
2021-12-02 20:07:48
6, 5+1, 4+2, 3+3
6, 5+1, 4+2, 3+3
wamofan
2021-12-02 20:07:55
15, 24, 222, 114, 123, 1122, 1311, 121111, 111111
15, 24, 222, 114, 123, 1122, 1311, 121111, 111111
DPatrick
2021-12-02 20:08:10
Indeed, if we were careful and systematic about it, we could end up with the entire list:
Indeed, if we were careful and systematic about it, we could end up with the entire list:
DPatrick
2021-12-02 20:08:15
$6$
$5+1$
$4+2$
$4+1+1$
$3+3$
$3+2+1$
$3+1+1+1$
$2+2+2$
$2+2+1+1$
$2+1+1+1+1$
$1+1+1+1+1+1$
$6$
$5+1$
$4+2$
$4+1+1$
$3+3$
$3+2+1$
$3+1+1+1$
$2+2+2$
$2+2+1+1$
$2+1+1+1+1$
$1+1+1+1+1+1$
karthic7073
2021-12-02 20:08:27
11
11
HonestCat
2021-12-02 20:08:27
11?
11?
Mkd20
2021-12-02 20:08:36
11
11
DPatrick
2021-12-02 20:08:42
Looks like there are 11 partitions of $6$.
Looks like there are 11 partitions of $6$.
peterguo
2021-12-02 20:08:50
is there a formula for this for a question like how many partitions of n are there
is there a formula for this for a question like how many partitions of n are there
DPatrick
2021-12-02 20:08:56
An excellent question!
An excellent question!
DPatrick
2021-12-02 20:09:05
Hardy and Ramanujan asked the same question!
Hardy and Ramanujan asked the same question!
DPatrick
2021-12-02 20:09:22
By brute force listing them, we could make a table of the number of partitions of the first 10 positive integers:
By brute force listing them, we could make a table of the number of partitions of the first 10 positive integers:
DPatrick
2021-12-02 20:09:27
$$\begin{array}{r|l}
n & \text{number of partitions of $n$} \\ \hline
1 & 1 \\
2 & 2 \\
3 & 3 \\
4 & 5 \\
5 & 7 \\
6 & 11 \\
7 & 15 \\
8 & 22 \\
9 & 30 \\
10 & 42
\end{array}$$
$$\begin{array}{r|l}
n & \text{number of partitions of $n$} \\ \hline
1 & 1 \\
2 & 2 \\
3 & 3 \\
4 & 5 \\
5 & 7 \\
6 & 11 \\
7 & 15 \\
8 & 22 \\
9 & 30 \\
10 & 42
\end{array}$$
DPatrick
2021-12-02 20:09:40
The number of partitions of $n$ gets big pretty quick. For example, do you have a rough guess for the number of partitions of $100$?
The number of partitions of $n$ gets big pretty quick. For example, do you have a rough guess for the number of partitions of $100$?
Kaito471
2021-12-02 20:09:54
a lot.
a lot.
DPatrick
2021-12-02 20:09:59
Indeed.
Indeed.
jason543
2021-12-02 20:10:17
over 190 million
over 190 million
DPatrick
2021-12-02 20:10:25
Yeah, it gets big quickly.
Yeah, it gets big quickly.
KenOno
2021-12-02 20:10:28
Nice guess.
Nice guess.
DPatrick
2021-12-02 20:10:37
It turns out that there are $190{,}569{,}292$ different partitions of $100$.
It turns out that there are $190{,}569{,}292$ different partitions of $100$.
smileapple
2021-12-02 20:10:54
is it exponential?
is it exponential?
DPatrick
2021-12-02 20:11:02
It does seem to be growing super-fast.
It does seem to be growing super-fast.
DPatrick
2021-12-02 20:11:07
Ramanujan and Hardy wondered if they could come up with a formula for the number of partitions of $n$.
Ramanujan and Hardy wondered if they could come up with a formula for the number of partitions of $n$.
DPatrick
2021-12-02 20:11:21
They partially succeeded. They didn't come up with an exact formula, but they did come up with a formula that provides a good approximation.
They partially succeeded. They didn't come up with an exact formula, but they did come up with a formula that provides a good approximation.
DPatrick
2021-12-02 20:11:36
Their formula, today known as the Hardy-Ramanujan formula, is
$$p(n) \sim \frac{1}{4n\sqrt3}\exp\left(\pi\sqrt{\frac{2n}{3}}\right).$$
Their formula, today known as the Hardy-Ramanujan formula, is
$$p(n) \sim \frac{1}{4n\sqrt3}\exp\left(\pi\sqrt{\frac{2n}{3}}\right).$$
DPatrick
2021-12-02 20:11:47
In the formula $\exp(x)$ is an alternative way of writing $e^x$, where $e = 2.718281828\ldots$ is the Euler number.
In the formula $\exp(x)$ is an alternative way of writing $e^x$, where $e = 2.718281828\ldots$ is the Euler number.
DPatrick
2021-12-02 20:12:15
Using this formula, $p(100) \sim 199{,}280{,}893$. That's not too far off the exact answer that I showed you a moment ago.
Using this formula, $p(100) \sim 199{,}280{,}893$. That's not too far off the exact answer that I showed you a moment ago.
SmartGroot
2021-12-02 20:12:24
what is the p at the beginning
what is the p at the beginning
DPatrick
2021-12-02 20:12:39
Sorry, I should have said that. $p(n)$ is our notation for the number of partitions of $n$.
Sorry, I should have said that. $p(n)$ is our notation for the number of partitions of $n$.
Ladka13
2021-12-02 20:12:47
few hundred thousand off. not much
few hundred thousand off. not much
DPatrick
2021-12-02 20:13:09
The key is that the percentage error that the formula is off by gets smaller and smaller as $n$ grows large.
The key is that the percentage error that the formula is off by gets smaller and smaller as $n$ grows large.
DPatrick
2021-12-02 20:13:19
For instance, we see that for $n=100$ the formula is a little less than $9$ million too high. That seems like a lot, but it's less than $5\%$ higher than the actual count.
For instance, we see that for $n=100$ the formula is a little less than $9$ million too high. That seems like a lot, but it's less than $5\%$ higher than the actual count.
DPatrick
2021-12-02 20:13:29
If we go to $n=1000$, the actual count is about $2.406 \cdot 10^{31}$ and the formula gives $2.440 \cdot 10^{31}$. The formula is only off by about $1.5\%$.
If we go to $n=1000$, the actual count is about $2.406 \cdot 10^{31}$ and the formula gives $2.440 \cdot 10^{31}$. The formula is only off by about $1.5\%$.
DPatrick
2021-12-02 20:13:41
At $n=10000$ the actual count is about $3.617 \cdot 10^{106}$ (that's more than a googol!) and the formula gives about $3.633 \cdot 10^{106}$. The formula is only off by about $0.5\%$.
At $n=10000$ the actual count is about $3.617 \cdot 10^{106}$ (that's more than a googol!) and the formula gives about $3.633 \cdot 10^{106}$. The formula is only off by about $0.5\%$.
JaiT
2021-12-02 20:13:46
and what is the ~ symbol
and what is the ~ symbol
DPatrick
2021-12-02 20:13:58
That indicates that's it's not an exact formula -- it gives an approximation.
That indicates that's it's not an exact formula -- it gives an approximation.
HonestCat
2021-12-02 20:14:02
Wait, how does pi show up here?
Wait, how does pi show up here?
DPatrick
2021-12-02 20:14:08
That's a great question.
That's a great question.
DPatrick
2021-12-02 20:14:23
It turns out that $\pi$ shows up in number theory a lot.
It turns out that $\pi$ shows up in number theory a lot.
DPatrick
2021-12-02 20:14:38
As I said before...a lot of math is connected in very surprising ways!
As I said before...a lot of math is connected in very surprising ways!
DPatrick
2021-12-02 20:14:59
But you'll have to learn some more advanced math before we can explain exactly why this happens with $\pi$.
But you'll have to learn some more advanced math before we can explain exactly why this happens with $\pi$.
DPatrick
2021-12-02 20:15:09
Ramanujan studied partitions of integers quite a bit and produced a lot of results. There are a lot of open questions about partitions that are still studied to this day. Indeed, this is one of the areas of study of our guest Professor Ken Ono, and a lot of his work builds on work first started by Ramanujan nearly 100 years ago.
Ramanujan studied partitions of integers quite a bit and produced a lot of results. There are a lot of open questions about partitions that are still studied to this day. Indeed, this is one of the areas of study of our guest Professor Ken Ono, and a lot of his work builds on work first started by Ramanujan nearly 100 years ago.
KenOno
2021-12-02 20:15:30
Yup...I still like to add and count.
Yup...I still like to add and count.
DPatrick
2021-12-02 20:15:34
So that seems like a good note on which to transition into the Spirit of Ramanujan initiative. Here's an excerpt from an article called "Why Ramanujan Matters" written by Ken Ono and Robert Schneider:
So that seems like a good note on which to transition into the Spirit of Ramanujan initiative. Here's an excerpt from an article called "Why Ramanujan Matters" written by Ken Ono and Robert Schneider:
DPatrick
2021-12-02 20:15:43
Ramanujan matters because he represents endless curiosity and untapped potential, which we all have to believe in to proceed in the sciences. Science usually advances on the work of thousands, over generations, fine-tuning and extending the scope of understanding. But from time to time, creative fireballs like Ramanujan burst onto the scene propelling human thought forward. Yet what if Ramanujan had not reached out to, or been taken seriously by Hardy? The loss of scientific understanding is something our modern world could not absorb. He matters because science matters: curiosity and creativity drive scientific inquiry.
Ramanujan matters because he represents endless curiosity and untapped potential, which we all have to believe in to proceed in the sciences. Science usually advances on the work of thousands, over generations, fine-tuning and extending the scope of understanding. But from time to time, creative fireballs like Ramanujan burst onto the scene propelling human thought forward. Yet what if Ramanujan had not reached out to, or been taken seriously by Hardy? The loss of scientific understanding is something our modern world could not absorb. He matters because science matters: curiosity and creativity drive scientific inquiry.
DPatrick
2021-12-02 20:16:13
That's what the Spirit of Ramanujan (which I'm going to abbreviate as SoR from now on) is all about: finding young people who have the potential, with the right resources and mentorship, to contribute to humankind's knowledge like Ramanujan did.
That's what the Spirit of Ramanujan (which I'm going to abbreviate as SoR from now on) is all about: finding young people who have the potential, with the right resources and mentorship, to contribute to humankind's knowledge like Ramanujan did.
DPatrick
2021-12-02 20:16:29
We at AoPS are pleased that Ken asked us to be a part of this search. SoR's goals are also part of AoPS's goals: to create a broader community of aspiring mathematicians and problems solvers, to make connections between people who might not otherwise have the opportunity to contact, and to provide resources via our website to all corners of the country and of the world.
We at AoPS are pleased that Ken asked us to be a part of this search. SoR's goals are also part of AoPS's goals: to create a broader community of aspiring mathematicians and problems solvers, to make connections between people who might not otherwise have the opportunity to contact, and to provide resources via our website to all corners of the country and of the world.
KenOno
2021-12-02 20:16:48
We are pleased to partner with AoPS.
We are pleased to partner with AoPS.
DPatrick
2021-12-02 20:16:51
Specifically, SoR provides two types of awards:
Specifically, SoR provides two types of awards:
DPatrick
2021-12-02 20:16:55
The Templeton-Ramanujan Development Awards will provide books and monetary grants to offset the costs of online resources.
The Templeton-Ramanujan Development Awards will provide books and monetary grants to offset the costs of online resources.
DPatrick
2021-12-02 20:17:02
The Templeton-Ramanujan Fellowships will provide monetary grants to offset the costs of enrolling in approved enrichment programs, such as summer camps.
The Templeton-Ramanujan Fellowships will provide monetary grants to offset the costs of enrolling in approved enrichment programs, such as summer camps.
KenOno
2021-12-02 20:17:06
Such as AoPS courses!
Such as AoPS courses!
DPatrick
2021-12-02 20:17:25
In case you're wondering...the "Templeton" in these names is the Templeton World Charity Foundation, created by the late Sir John Templeton, and which is providing the funding for SoR.
In case you're wondering...the "Templeton" in these names is the Templeton World Charity Foundation, created by the late Sir John Templeton, and which is providing the funding for SoR.
KenOno
2021-12-02 20:17:36
We have now supported 66 people worldwide...from 14 countries.
We have now supported 66 people worldwide...from 14 countries.
DPatrick
2021-12-02 20:18:07
Indeed, 2021-22 is the 6th year of the program. About half of the students supported are from the U.S. and half from elsewhere in the world.
Indeed, 2021-22 is the 6th year of the program. About half of the students supported are from the U.S. and half from elsewhere in the world.
KenOno
2021-12-02 20:18:14
Two of our former winners this year won the Regeneron Science Talent Search...and also a Rhodes Scholarship.
Two of our former winners this year won the Regeneron Science Talent Search...and also a Rhodes Scholarship.
DPatrick
2021-12-02 20:18:18
Basically, SoR is hoping to play to role for future Ramanujans that G. H. Hardy did for the actual Ramanujan: access to resources and mentorship to nurture future mathematical and scientific talent.
Basically, SoR is hoping to play to role for future Ramanujans that G. H. Hardy did for the actual Ramanujan: access to resources and mentorship to nurture future mathematical and scientific talent.
DPatrick
2021-12-02 20:18:41
To learn more about SoR, please visit the website spiritoframanujan.com. (Should be easy to remember, no?) The SoR website includes a link to the application page. You can also read brief bios of the previous winners on the website.
To learn more about SoR, please visit the website spiritoframanujan.com. (Should be easy to remember, no?) The SoR website includes a link to the application page. You can also read brief bios of the previous winners on the website.
DPatrick
2021-12-02 20:19:02
Please note that although awards for the 2021-22 school year will be made on a rolling basis, you should apply by April 30, 2022 in order to guarantee full consideration.
Please note that although awards for the 2021-22 school year will be made on a rolling basis, you should apply by April 30, 2022 in order to guarantee full consideration.
KenOno
2021-12-02 20:19:23
In recent years we have offered scholarships to things like AoPS courses...Canada/USA MathCamp, PROMYS, etc...
In recent years we have offered scholarships to things like AoPS courses...Canada/USA MathCamp, PROMYS, etc...
DPatrick
2021-12-02 20:19:56
Yes, I believe that generally Ken and the foundation works with the award winners to tailor a program that's best for that person.
Yes, I believe that generally Ken and the foundation works with the award winners to tailor a program that's best for that person.
KenOno
2021-12-02 20:20:14
David is on our board...along with Fields medalist Manjul Bhargava...and others....
David is on our board...along with Fields medalist Manjul Bhargava...and others....
KenOno
2021-12-02 20:20:51
Yes, please apply. We seek enthusiastic creative minds...Especially if our support can make participation possible.
Yes, please apply. We seek enthusiastic creative minds...Especially if our support can make participation possible.
DPatrick
2021-12-02 20:20:57
...and Po-Shen Loh I believe, is that right?
...and Po-Shen Loh I believe, is that right?
KenOno
2021-12-02 20:21:27
Ah yes...How can I forget Po-Shen...
Ah yes...How can I forget Po-Shen...
DPatrick
2021-12-02 20:21:39
Certainly a familiar name to many on AoPS!
Certainly a familiar name to many on AoPS!
P3Tan
2021-12-02 20:21:53
who is po shen?
who is po shen?
DPatrick
2021-12-02 20:22:04
He is the leader of the US Team for the International Math Olympiad.
He is the leader of the US Team for the International Math Olympiad.
DPatrick
2021-12-02 20:22:08
(Among other things!)
(Among other things!)
DPatrick
2021-12-02 20:23:18
That's all we have for tonight...we can try to answer some questions for a little while if you have any. There's also lots of info on the SoR website, along with contact info if you'd like to follow up.
That's all we have for tonight...we can try to answer some questions for a little while if you have any. There's also lots of info on the SoR website, along with contact info if you'd like to follow up.
DPatrick
2021-12-02 20:23:26
spiritoframanujan.com
spiritoframanujan.com
KenOno
2021-12-02 20:23:30
To repeat...please visit spiritoframanujan.com
To repeat...please visit spiritoframanujan.com
DPatrick
2021-12-02 20:23:37
Can't repeat it enough!
Can't repeat it enough!
DofL
2021-12-02 20:24:06
when is the optimal time to apply for SoR? what qualifications do we need for it?
when is the optimal time to apply for SoR? what qualifications do we need for it?
DPatrick
2021-12-02 20:24:18
Anytime between now and April 30 will guarantee full consideration.
Anytime between now and April 30 will guarantee full consideration.
DPatrick
2021-12-02 20:24:25
The info you'll need to apply is on the website.
The info you'll need to apply is on the website.
KenOno
2021-12-02 20:24:27
We have made awards to students as young as 8-9....
We have made awards to students as young as 8-9....
SmartGroot
2021-12-02 20:24:31
Oh KevinOno you face revealed!
Oh KevinOno you face revealed!
DPatrick
2021-12-02 20:24:41
Yes, if you want to see what Ken looks like, go to the SoR website.
Yes, if you want to see what Ken looks like, go to the SoR website.
DPatrick
2021-12-02 20:24:59
(If you want to see what I look like, you can find my picture on the AoPS website somewhere I think.)
(If you want to see what I look like, you can find my picture on the AoPS website somewhere I think.)
KenOno
2021-12-02 20:24:59
Except I am going bald...The photo doesn't reveal...
Except I am going bald...The photo doesn't reveal...
KenOno
2021-12-02 20:25:44
Thanks for having me tonight...Super fun...
Thanks for having me tonight...Super fun...
Avemaytas
2021-12-02 20:25:49
This was one of the best Math Jams ever!
This was one of the best Math Jams ever!
DPatrick
2021-12-02 20:25:59
Thanks...glad you liked it!
Thanks...glad you liked it!
dBIT
2021-12-02 20:26:07
Also this month is National Mathematics Day in India December 22 on Ramanujan's Bday
Also this month is National Mathematics Day in India December 22 on Ramanujan's Bday
DPatrick
2021-12-02 20:26:16
That's right! Thanks for mentioning that.
That's right! Thanks for mentioning that.
DPatrick
2021-12-02 20:26:38
I think Ken is doing an event with the Museum of Math in New York which will be online -- is that correct?
I think Ken is doing an event with the Museum of Math in New York which will be online -- is that correct?
DPatrick
2021-12-02 20:26:44
(I may be misremembering)
(I may be misremembering)
DPatrick
2021-12-02 20:27:01
(On or around Dec 22)
(On or around Dec 22)
KenOno
2021-12-02 20:27:31
Yes, Steven Strogatz is hosting a virtual event on December 20th called "Starring Math". I will be speaking with Manjul Bhargava and director Matt Brown. Look for info online at the Museum of Math website (MoMath).
Yes, Steven Strogatz is hosting a virtual event on December 20th called "Starring Math". I will be speaking with Manjul Bhargava and director Matt Brown. Look for info online at the Museum of Math website (MoMath).
QueenRaven337
2021-12-02 20:27:51
Thank you so much for an amazing evening!
Thank you so much for an amazing evening!
DPatrick
2021-12-02 20:28:05
Thanks everyone for coming! Have a nice evening!
Thanks everyone for coming! Have a nice evening!
KenOno
2021-12-02 20:28:14
Good night!
Good night!
HonestCat
2021-12-02 20:29:16
Thank you!
Thank you!
DofL
2021-12-02 20:29:16
thank you so much for the wondeful oppurtunities Dr. Ono!
thank you so much for the wondeful oppurtunities Dr. Ono!
peterguo
2021-12-02 20:29:16
good math jam
good math jam
Kai_Zhao
2021-12-02 20:29:16
bye
.
bye
.
P3Tan
2021-12-02 20:29:16
Have GREAT night ya'll!
Have GREAT night ya'll!
horse869
2021-12-02 20:29:16
Thank you
Thank you
HappyNarwhal
2021-12-02 20:29:16
Bye!
Bye!
yashraj31
2021-12-02 20:29:16
bye!
bye!
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