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
devenware
2020-11-10 19:30:08
Welcome to the 2020-21 Spirit of Ramanujan Math Jam!
Welcome to the 2020-21 Spirit of Ramanujan Math Jam!
devenware
2020-11-10 19:30:09
Adul
2020-11-10 19:30:31
woohoo
woohoo
Jianning
2020-11-10 19:30:31
whoo hoo! so excited
whoo hoo! so excited
devenware
2020-11-10 19:30:37
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 16 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 16 years, and I've written or co-written a few of our textbooks.
ryanzhao
2020-11-10 19:30:43
um hi
um hi
devenware
2020-11-10 19:30:45
Wait a second...
Wait a second...
leafwhisker
2020-11-10 19:30:49
OMG HI DAVE
OMG HI DAVE
Adul
2020-11-10 19:31:00
DAVID PATRICK!!!!?
DAVID PATRICK!!!!?
Countcountcount
2020-11-10 19:31:00
You're not Dave Patrick...
You're not Dave Patrick...
Mathbee2020
2020-11-10 19:31:00
rrllly
rrllly
NehaSeban
2020-11-10 19:31:00
Ur not Dave Patrick tho
Ur not Dave Patrick tho
devenware
2020-11-10 19:31:04
I think they prepared this for someone else.
I think they prepared this for someone else.
devenware
2020-11-10 19:31:21
We are pleased to have a very special guest here with us tonight. Ken Ono (KenOno) is the Thomas Jefferson Professor of Mathematics at the University of Virginia and is also Vice President of the American Mathematical Society (AMS). The AMS promotes mathematical research, fosters excellence in mathematics education, and increases the awareness of the value of mathematics to society.
We are pleased to have a very special guest here with us tonight. Ken Ono (KenOno) is the Thomas Jefferson Professor of Mathematics at the University of Virginia and is also Vice President of the American Mathematical Society (AMS). The AMS promotes mathematical research, fosters excellence in mathematics education, and increases the awareness of the value of mathematics to society.
devenware
2020-11-10 19:31:29
Say hi, Ken!
Say hi, Ken!
KenOno
2020-11-10 19:31:47
Hi everyone! Pleased to be here.
Hi everyone! Pleased to be here.
vmr123
2020-11-10 19:31:56
Hi Ken
Hi Ken
leafwhisker
2020-11-10 19:31:56
Hi Ken!
Hi Ken!
NehaSeban
2020-11-10 19:31:56
Hello Ken!
Hello Ken!
Pandayue
2020-11-10 19:31:56
Now, THAT's right! Hi!
Now, THAT's right! Hi!
Mathbee2020
2020-11-10 19:31:56
Hello Ken!
Hello Ken!
AOPqghj
2020-11-10 19:31:56
hi, Ken!
hi, Ken!
SmartGroot
2020-11-10 19:31:56
hello ken!
hello ken!
Andrew_Dou
2020-11-10 19:31:56
hi
hi
nathanj
2020-11-10 19:31:56
Hi!
Hi!
devenware
2020-11-10 19:32:05
Ken is also the Director of the "Spirit of Ramanujan" project, which AoPS is also a partner with, and which we'll be talking about tonight.
Ken is also the Director of the "Spirit of Ramanujan" project, which AoPS is also a partner with, and which we'll be talking about tonight.
Anthony0512
2020-11-10 19:32:50
THANKS IN ADVANCE FOR ALL THE HARD WORK THE MODS ARE DOING
THANKS IN ADVANCE FOR ALL THE HARD WORK THE MODS ARE DOING
devenware
2020-11-10 19:32:57
Speaking of which, today we have a teaching assistant: Victoria Li (mag1c).
Speaking of which, today we have a teaching assistant: Victoria Li (mag1c).
bsu1
2020-11-10 19:33:16
HI!
HI!
yesufsa
2020-11-10 19:33:16
Hi
Hi
Skyhigh1218
2020-11-10 19:33:16
Hi mag1c!
Hi mag1c!
GXboy12345
2020-11-10 19:33:16
Hello!
Hello!
Adul
2020-11-10 19:33:16
hello!!
hello!!
Anthony0512
2020-11-10 19:33:16
hello!
hello!
vmr123
2020-11-10 19:33:16
Hi Victoria
Hi Victoria
interactivemath
2020-11-10 19:33:16
Hello mag1c!
Hello mag1c!
yesufsa
2020-11-10 19:33:16
Hi!
Hi!
bsu1
2020-11-10 19:33:16
Hi Victoria!
Hi Victoria!
GXboy12345
2020-11-10 19:33:16
Hello Victoria!
Hello Victoria!
Countcountcount
2020-11-10 19:33:16
Hi Victoria!
Hi Victoria!
KenOno
2020-11-10 19:33:19
Hi Victoria.
Hi Victoria.
mag1c
2020-11-10 19:33:29
Hi!
Hi!
devenware
2020-11-10 19:33:34
Victoria studied mathemagics at HCSSiM and Colorado Math Circle, did research fellowships on AMO physics and genome modification off-target prediction at RSI and BioFrontiers Institute, studied mathematics at four universities in two countries, has an MS Applied Mathematics from the University of Colorado Boulder, and has been teaching math/programming/Chinese since 2010. As a learner and educator, she strives for:
1. Compassion (everyone can become a successful mathematician when taught by the right teacher)
2. Open-mindedness (there are many ways to solve a problem and many different learning styles)
3. Constant Improvement (learning never stops...)
In her free time she likes to ponder questions such as "What happened before the Big Bang (13.8 billion years ago)?", "Why do Fibonacci numbers occur in pinecones?", and "What are the implications of Gödel's Incompleteness Theorem..."
Victoria studied mathemagics at HCSSiM and Colorado Math Circle, did research fellowships on AMO physics and genome modification off-target prediction at RSI and BioFrontiers Institute, studied mathematics at four universities in two countries, has an MS Applied Mathematics from the University of Colorado Boulder, and has been teaching math/programming/Chinese since 2010. As a learner and educator, she strives for:
1. Compassion (everyone can become a successful mathematician when taught by the right teacher)
2. Open-mindedness (there are many ways to solve a problem and many different learning styles)
3. Constant Improvement (learning never stops...)
In her free time she likes to ponder questions such as "What happened before the Big Bang (13.8 billion years ago)?", "Why do Fibonacci numbers occur in pinecones?", and "What are the implications of Gödel's Incompleteness Theorem..."
devenware
2020-11-10 19:33:43
Victoria 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.
Victoria 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.
Rishab-AOPS
2020-11-10 19:34:03
Wat is this?
Wat is this?
Senguamar
2020-11-10 19:34:03
what will this be about If not relevant I gotta go
what will this be about If not relevant I gotta go
devenware
2020-11-10 19:34:04
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.
devenware
2020-11-10 19:34:09
GXboy12345
2020-11-10 19:34:28
Mathematics or Mathemagics?
Mathematics or Mathemagics?
Anthony0512
2020-11-10 19:34:28
so history?
so history?
devenware
2020-11-10 19:34:33
All of the above.
All of the above.
devenware
2020-11-10 19:34:46
But I'll let Dr. Patrick show you all that. I think his internet kicked back on.
But I'll let Dr. Patrick show you all that. I think his internet kicked back on.
KenOno
2020-11-10 19:35:14
I have a first day issue of that stamp.
I have a first day issue of that stamp.
DPatrick
2020-11-10 19:35:23
Hopefully I shooed all the gremlins away that were gnawing at my internet cable...
Hopefully I shooed all the gremlins away that were gnawing at my internet cable...
DPatrick
2020-11-10 19:35:51
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
2020-11-10 19:36:12
Did anybody here see the movie?
Did anybody here see the movie?
ananyasharma
2020-11-10 19:36:34
I did!
I did!
SuperJJ
2020-11-10 19:36:34
Meee!
Meee!
Askumar
2020-11-10 19:36:34
Yes
Yes
devenware
2020-11-10 19:36:34
That was a very entertaining movie! I saw it in theaters!
That was a very entertaining movie! I saw it in theaters!
BlazingSun1200
2020-11-10 19:36:34
YES!
YES!
hwdaniel
2020-11-10 19:36:34
Me!
Me!
lilac613
2020-11-10 19:36:34
I saw it a week ago because of this math jam!
I saw it a week ago because of this math jam!
BrightMonkey
2020-11-10 19:36:34
yes!
yes!
SuperJJ
2020-11-10 19:36:34
I loved the movie!
I loved the movie!
NightFury101
2020-11-10 19:36:34
me!
me!
DPatrick
2020-11-10 19:36:44
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
2020-11-10 19:36:57
Unfortunately as of November 2020 it does not appear to be available on any streaming service in the U.S.
But it is available for rent for $2.99 on a bunch of different sites.
Unfortunately as of November 2020 it does not appear to be available on any streaming service in the U.S.
But it is available for rent for $2.99 on a bunch of different sites.
DPatrick
2020-11-10 19:37:14
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.
BlazingSun1200
2020-11-10 19:37:28
Srinivasa Ramanujan had his interest in mathematics unlocked by a book.
Srinivasa Ramanujan had his interest in mathematics unlocked by a book.
DPatrick
2020-11-10 19:37:38
You're right! 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. (You can find it in Google Books!) Ramanujan read and studied it in great detail and it formed the foundation of his mathematical thought.
You're right! 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. (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
2020-11-10 19:38:02
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
2020-11-10 19:38:25
(And don't worry if you don't know what "quartic equation" means -- I'll explain that in a little bit!)
(And don't worry if you don't know what "quartic equation" means -- I'll explain that in a little bit!)
DPatrick
2020-11-10 19:38:45
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.
GXboy12345
2020-11-10 19:39:15
Shoutout to G. H. Hardy for beginning Ramanujan's career! ;P
Shoutout to G. H. Hardy for beginning Ramanujan's career! ;P
DPatrick
2020-11-10 19:39:29
Indeed, we're coming to that!
Indeed, we're coming to that!
DPatrick
2020-11-10 19:39:38
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.
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.
BlazingSun1200
2020-11-10 19:39:49
He sent multiple letters to Cambridge!
He sent multiple letters to Cambridge!
DPatrick
2020-11-10 19:40:11
A major turning point in Ramanujan's life was his writing a letter to the prominent British mathematician G. H. Hardy in January 1913.
A major turning point in Ramanujan's life was his writing a letter to the prominent British mathematician G. H. Hardy in January 1913.
Noam2007
2020-11-10 19:40:48
Was there a reply?
Was there a reply?
Countcountcount
2020-11-10 19:40:48
Did G. H. Hardy reply to him?
Did G. H. Hardy reply to him?
devenware
2020-11-10 19:41:28
Good question, I think Dave left you with a cliff hanger.
Good question, I think Dave left you with a cliff hanger.
Skyhigh1218
2020-11-10 19:41:48
Noo, I hate those!
Noo, I hate those!
devenware
2020-11-10 19:41:54
Before telling you if he got a reply, here's how the letter began.
Before telling you if he got a reply, here's how the letter began.
devenware
2020-11-10 19:42:00
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'.
devenware
2020-11-10 19:42:06
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.
nemoatl
2020-11-10 19:42:27
Did G.H Hardy listen?
Did G.H Hardy listen?
Noam2007
2020-11-10 19:42:47
(drumroll, please!)
(drumroll, please!)
devenware
2020-11-10 19:42:48
Hardy took the letter seriously!
Hardy took the letter seriously!
Noam2007
2020-11-10 19:43:04
Yay!
Yay!
hipeeps1104
2020-11-10 19:43:04
yayayayay
yayayayay
ananyasharma
2020-11-10 19:43:04
Yay!!
Yay!!
NehaSeban
2020-11-10 19:43:04
YAAAY
YAAAY
devenware
2020-11-10 19:43:05
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. Indeed, on Hardy's recommendation, Ramanujan was finally able to get a scholarship to the University of Madras.
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. Indeed, on Hardy's recommendation, Ramanujan was finally able to get a scholarship to the University of Madras.
devenware
2020-11-10 19:43:13
Finally, in 1914, Hardy arranged for Ramanujan to be able to travel and visit Trinity College at Cambridge University in England.
Finally, in 1914, Hardy arranged for Ramanujan to be able to travel and visit Trinity College at Cambridge University in England.
devenware
2020-11-10 19:43:16
Do you know how long it took Ramanujan to travel from India to London? Remember, this was slightly over 100 years ago!
Do you know how long it took Ramanujan to travel from India to London? Remember, this was slightly over 100 years ago!
danielWang816
2020-11-10 19:43:49
2 years
2 years
Peter0527
2020-11-10 19:43:49
1 year?
1 year?
hipeeps1104
2020-11-10 19:43:49
4 years?
4 years?
stevecabbage
2020-11-10 19:43:49
A year?
A year?
devenware
2020-11-10 19:43:55
I said 100 years ago, not 500.
I said 100 years ago, not 500.
vmr123
2020-11-10 19:44:21
a month???
a month???
MinervaPi
2020-11-10 19:44:21
months?
months?
TheGeometer
2020-11-10 19:44:21
About a month?
About a month?
amazingxin777
2020-11-10 19:44:21
one month
one month
amazingxin777
2020-11-10 19:44:21
one month?
one month?
devenware
2020-11-10 19:44:24
27 days!
27 days!
SmartGroot
2020-11-10 19:44:42
oh, wow
oh, wow
KLBBC
2020-11-10 19:44:42
wow
wow
Yash1212
2020-11-10 19:44:42
wow?
wow?
BlazingSun1200
2020-11-10 19:44:42
Wow!
Wow!
stevecabbage
2020-11-10 19:44:42
That's a lot faster than I expected
That's a lot faster than I expected
Morrigan_Black
2020-11-10 19:44:42
wow it would only take like 9 hours now
wow it would only take like 9 hours now
devenware
2020-11-10 19:44:45
Ramanujan left India on March 17, 1914 by boat, and arrived in London on April 14. Then 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.
Ramanujan left India on March 17, 1914 by boat, and arrived in London on April 14. Then 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.
devenware
2020-11-10 19:44:54
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.
danielWang816
2020-11-10 19:45:07
yay!
yay!
devenware
2020-11-10 19:45:15
Unfortunately, for much of his time in England, Ramanujan was in very poor health. He spent much of his time in hospitals while in England, and nearly died in 1917. (The fact that World War I was going on, affecting supplies, probably didn't help.) There's a famous mathematical anecdote about Hardy visiting Ramanujan in the hospital that I'll share in a little bit.
Unfortunately, for much of his time in England, Ramanujan was in very poor health. He spent much of his time in hospitals while in England, and nearly died in 1917. (The fact that World War I was going on, affecting supplies, probably didn't help.) There's a famous mathematical anecdote about Hardy visiting Ramanujan in the hospital that I'll share in a little bit.
amazingxin777
2020-11-10 19:47:34
c'mon devenware
c'mon devenware
devenware
2020-11-10 19:47:37
Apparently Dave's Gremlins came to my house too.
Apparently Dave's Gremlins came to my house too.
miya0829
2020-11-10 19:47:50
lol
lol
Morrigan_Black
2020-11-10 19:47:50
Atoms789 hi
Atoms789 hi
stevecabbage
2020-11-10 19:47:50
Huh?
Huh?
Mathbee2020
2020-11-10 19:47:50
nuu
nuu
devenware
2020-11-10 19:47:52
I'm now on a mobile hotspot, we got this.
I'm now on a mobile hotspot, we got this.
SmartGroot
2020-11-10 19:48:26
lol I thought I lost connection
lol I thought I lost connection
devenware
2020-11-10 19:48:30
Nope, just me.
Nope, just me.
devenware
2020-11-10 19:48:55
In early 1919, the war was over and Ramaujan's health had improved, 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.
In early 1919, the war was over and Ramaujan's health had improved, 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.
vmr123
2020-11-10 19:49:41
But sadly, he died.
But sadly, he died.
BobDBuilder321
2020-11-10 19:49:41
hope...
hope...
Infinite_May
2020-11-10 19:49:41
he got sick again
he got sick again
devenware
2020-11-10 19:49:44
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.
SmartGroot
2020-11-10 19:50:31
oh.
oh.
micyang
2020-11-10 19:50:31
rjeagle2019
2020-11-10 19:50:31
ouyangc
2020-11-10 19:50:31
devenware
2020-11-10 19:50:35
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.
devenware
2020-11-10 19:50:47
As an aside, this website is a great resource: it has biographies of nearly 3,000 mathematicians spanning over 35 centuries, from Ahmes (1680-1620 BC) to Mirzakhani (1977-2017), and includes dozens of living mathematicians as well.
As an aside, this website is a great resource: it has biographies of nearly 3,000 mathematicians spanning over 35 centuries, from Ahmes (1680-1620 BC) to Mirzakhani (1977-2017), and includes dozens of living mathematicians as well.
amazingxin777
2020-11-10 19:50:57
is that the whole story of his lfe?
is that the whole story of his lfe?
devenware
2020-11-10 19:51:06
Well, we gave a brief summary! You can find much more detail on that page.
Well, we gave a brief summary! You can find much more detail on that page.
EpicAarn
2020-11-10 19:51:15
is this it? or is there more?
is this it? or is there more?
Anthony0512
2020-11-10 19:51:15
We still got math right?
We still got math right?
devenware
2020-11-10 19:51:20
Now let's talk about some the mathematics that Ramanujan studied.
Now let's talk about some the mathematics that Ramanujan studied.
URcurious2
2020-11-10 19:51:33
Yay math!!!!
Yay math!!!!
Mathbee2020
2020-11-10 19:51:33
yessss
yessss
rjeagle2019
2020-11-10 19:51:33
Yay Math
Yay Math
devenware
2020-11-10 19:51:38
Ramanujan did a lot of mathematics in his relatively short life, and a great deal of it was quite advanced. 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. But there are some objects of his study that we can discuss tonight.
BlazingSun1200
2020-11-10 19:51:42
MATH IS AS INTERESTING AS IT IS BECAUSE OF PEOPLE LIKE HIM!!
MATH IS AS INTERESTING AS IT IS BECAUSE OF PEOPLE LIKE HIM!!
devenware
2020-11-10 19:51:48
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.
devenware
2020-11-10 19:52:01
To start, many of you already know how to solve quadratic equations, which is an equation involving a polynomial of degree 2.
To start, many of you already know how to solve quadratic equations, which is an equation involving a polynomial of degree 2.
devenware
2020-11-10 19:52:03
In particular, how can we solve $ax^2 + bx + c = 0$?
In particular, how can we solve $ax^2 + bx + c = 0$?
Andrew_Dou
2020-11-10 19:53:03
we can use the quadratic formula
we can use the quadratic formula
umikakv2013
2020-11-10 19:53:03
quardratic equation
quardratic equation
Jianning
2020-11-10 19:53:03
factoring quadratic
factoring quadratic
Spaced_Out
2020-11-10 19:53:03
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
dolphin7
2020-11-10 19:53:03
quadratic formula
quadratic formula
InsaneMoosen
2020-11-10 19:53:03
quadratic formula
quadratic formula
devenware
2020-11-10 19:53:06
We use the well-known quadratic formula: $$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$
We use the well-known quadratic formula: $$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$
devenware
2020-11-10 19:53:11
This is not too hard to prove using basic algebra. (We prove it in AoPS's Introduction to Algebra textbook and classes.)
This is not too hard to prove using basic algebra. (We prove it in AoPS's Introduction to Algebra textbook and classes.)
BlazingSun1200
2020-11-10 19:53:14
complete the square
complete the square
amazingxin777
2020-11-10 19:53:20
or complete the square
or complete the square
devenware
2020-11-10 19:53:24
Yeah, that's how you do it!
Yeah, that's how you do it!
devenware
2020-11-10 19:53:28
The next step up is solving the cubic equation $ax^3 + bx^2 + cx + d = 0$. There is a formula for solving this as well, but it's a lot more complicated.
The next step up is solving the cubic equation $ax^3 + bx^2 + cx + d = 0$. There is a formula for solving this as well, but it's a lot more complicated.
devenware
2020-11-10 19:53:45
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$$\begin{array}{rcl}
p &=& -\frac{b}{3a} \\
q &=& p^3 + \frac{bc-3ad}{6a^2} \\
r &=& \frac{c}{3a}
\end{array}$$
Anthony0512
2020-11-10 19:53:50
cubic formula?
cubic formula?
devenware
2020-11-10 19:53:52
then
$$x = \sqrt[3]{q + \sqrt{q^2 + (r-p^2)^3}} + \sqrt[3]{q - \sqrt{q^2 + (r-p^2)^3}} + p.$$
then
$$x = \sqrt[3]{q + \sqrt{q^2 + (r-p^2)^3}} + \sqrt[3]{q - \sqrt{q^2 + (r-p^2)^3}} + p.$$
Peter0527
2020-11-10 19:54:05
wow
wow
rjeagle2019
2020-11-10 19:54:05
Wow that's big
Wow that's big
Mathbee2020
2020-11-10 19:54:05
wow long formula
wow long formula
Countcountcount
2020-11-10 19:54:07
Wow. That's complex
Wow. That's complex
devenware
2020-11-10 19:54:14
And we simplified it using our new variables...
And we simplified it using our new variables...
asengo
2020-11-10 19:54:23
holy cow that's complicated
holy cow that's complicated
Skyhigh1218
2020-11-10 19:54:23
cOMPLICATED
cOMPLICATED
CosmicIQ
2020-11-10 19:54:23
It's confusing!
It's confusing!
rjeagle2019
2020-11-10 19:54:26
Oh it gets even worse
Oh it gets even worse
devenware
2020-11-10 19:54:33
Exactly!
Exactly!
devenware
2020-11-10 19:54:48
But, you know who loves cubics?
But, you know who loves cubics?
riverteng
2020-11-10 19:55:01
me
me
ezRobots
2020-11-10 19:55:01
me?
me?
devenware
2020-11-10 19:55:08
Then you and DPatrick have something in common!
Then you and DPatrick have something in common!
devenware
2020-11-10 19:55:16
The mention of cubics woke him up from his slumber.
The mention of cubics woke him up from his slumber.
DPatrick
2020-11-10 19:55:18
I do love cubics.
I do love cubics.
DPatrick
2020-11-10 19:55:34
And I hope the gremlins are gone for good this time.
And I hope the gremlins are gone for good this time.
DPatrick
2020-11-10 19:55:55
But...I don't love cubics enough to have memorized that formula for solving them.
But...I don't love cubics enough to have memorized that formula for solving them.
DPatrick
2020-11-10 19:56:15
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. I don't even guarantee that I typed it correctly above!)
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. I don't even guarantee that I typed it correctly above!)
mathworm04
2020-11-10 19:56:24
can't there be up to 3 solutions?
can't there be up to 3 solutions?
DPatrick
2020-11-10 19:56:29
Indeed, good observation!
Indeed, good observation!
DPatrick
2020-11-10 19:56:38
That formula only give us one solution...
That formula only give us one solution...
DPatrick
2020-11-10 19:56:40
...or so it seems.
...or so it seems.
DPatrick
2020-11-10 19:57:03
But there are three cube roots if you consider complex numbers!
But there are three cube roots if you consider complex numbers!
TheGeometer
2020-11-10 19:57:11
Use polynomial division and the quadratic formula to find the other two.
Use polynomial division and the quadratic formula to find the other two.
Heavytoothpaste
2020-11-10 19:57:11
we can then use quadratic formula
we can then use quadratic formula
DPatrick
2020-11-10 19:57:31
Or you can do that: once you have one root, you can divide it down to a quadratic.
Or you can do that: once you have one root, you can divide it down to a quadratic.
DPatrick
2020-11-10 19:57:45
You'll learn about all this stuff in algebra classes down the line if you don't know it already.
You'll learn about all this stuff in algebra classes down the line if you don't know it already.
DPatrick
2020-11-10 19:58:00
This formula, and other similar formulas for solving cubics, were discovered in the 1500s. Ramanujan would have learned this in his early teens.
This formula, and other similar formulas for solving cubics, were discovered in the 1500s. Ramanujan would have learned this in his early teens.
amazingxin777
2020-11-10 19:58:20
is this mentioned because he worked with cubics?
is this mentioned because he worked with cubics?
DPatrick
2020-11-10 19:58:29
Yes---and he took it one step further!
Yes---and he took it one step further!
rjeagle2019
2020-11-10 19:58:37
And he found the quartic?
And he found the quartic?
Heavytoothpaste
2020-11-10 19:58:41
quartics
quartics
DPatrick
2020-11-10 19:58:43
What Ramanujan then undertook, and eventually successfully solved, was solving the quartic equation $ax^4 + bx^3 + cx^2 + dx + e = 0$.
What Ramanujan then undertook, and eventually successfully solved, was solving the quartic equation $ax^4 + bx^3 + cx^2 + dx + e = 0$.
DPatrick
2020-11-10 19:59:07
You saw how complicated the cubic formula was...imagine how complicated the quartic formula is!
You saw how complicated the cubic formula was...imagine how complicated the quartic formula is!
DPatrick
2020-11-10 19:59:23
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.
samrocksnature
2020-11-10 19:59:35
Can we see it?
Can we see it?
yiz76
2020-11-10 19:59:35
$\boxed{\text{show me it}}$
$\boxed{\text{show me it}}$
os31415
2020-11-10 19:59:35
I wish I could see it
I wish I could see it
DPatrick
2020-11-10 19:59:39
Oh, OK...
Oh, OK...
DPatrick
2020-11-10 19:59:44
amazingxin777
2020-11-10 20:00:02
yikes
yikes
A1K2C3
2020-11-10 20:00:02
aaaah
aaaah
mathworm04
2020-11-10 20:00:02
!!
!!
amazingxin777
2020-11-10 20:00:02
that is horrifying
that is horrifying
Allen31415
2020-11-10 20:00:06
a wolfram alpha screenshot?
a wolfram alpha screenshot?
DPatrick
2020-11-10 20:00:15
It's from http://pjk.scripts.mit.edu/pkj/wp-content/uploads/2018/11/quartic_formula.jpg
It's from http://pjk.scripts.mit.edu/pkj/wp-content/uploads/2018/11/quartic_formula.jpg
Countcountcount
2020-11-10 20:00:28
that is soooo long
that is soooo long
sturdycrab
2020-11-10 20:00:28
He derived that himself??
He derived that himself??
DPatrick
2020-11-10 20:00:49
Yes. It might now have been this equal formula, but it was something equivalent.
Yes. It might now have been this equal formula, but it was something equivalent.
DPatrick
2020-11-10 20:00:56
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
2020-11-10 20:01:11
But it is an extremely complicated formula, and it's amazing that Ramanujan was able to discover it, essentially 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 on his own, at age 15.
sanickode
2020-11-10 20:01:27
w==but you cant solve a plynomial to the 5thdegree right?
w==but you cant solve a plynomial to the 5thdegree right?
DPatrick
2020-11-10 20:01:37
Good question! Ramanujan attempted to solve the quintic equation:
$$ ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0 $$
Good question! Ramanujan attempted to solve the quintic equation:
$$ ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0 $$
DPatrick
2020-11-10 20:01:50
He did not have success. Does anyone know why?
He did not have success. Does anyone know why?
rjeagle2019
2020-11-10 20:01:57
Yes Quintics are impossible
Yes Quintics are impossible
boing123
2020-11-10 20:02:10
IT'S IMPOSSIBLE
IT'S IMPOSSIBLE
NCEE
2020-11-10 20:02:10
impossible?
impossible?
RaymondOuyang
2020-11-10 20:02:10
Is it not possible?
Is it not possible?
Pandayue
2020-11-10 20:02:10
Cuz it's impossible?
Cuz it's impossible?
ezRobots
2020-11-10 20:02:10
Quintics are impossible?
Quintics are impossible?
TheGeometer
2020-11-10 20:02:18
Abel-Ruffini I believe...
Abel-Ruffini I believe...
Afo
2020-11-10 20:02:18
Abel–Ruffini theorem
Abel–Ruffini theorem
DPatrick
2020-11-10 20:02:25
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.)
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.)
UntamedMathBeast
2020-11-10 20:02:48
Wow
Wow
URcurious2
2020-11-10 20:02:48
Cool
Cool
Quacker88
2020-11-10 20:02:48
does that mean degree 5 and up are all impossible?
does that mean degree 5 and up are all impossible?
DPatrick
2020-11-10 20:03:00
That's right. Degree 5 and up are all impossible.
That's right. Degree 5 and up are all impossible.
DPatrick
2020-11-10 20:03:07
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.
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.
A1K2C3
2020-11-10 20:03:22
Ooh can we see the proof?
Ooh can we see the proof?
DPatrick
2020-11-10 20:03:44
I used to teach this proof! But it came at the year of a year-long abstract algebra class taught at university.
I used to teach this proof! But it came at the year of a year-long abstract algebra class taught at university.
DPatrick
2020-11-10 20:03:48
So maybe we'll skip it today.
So maybe we'll skip it today.
DPatrick
2020-11-10 20:04:10
Again, Ramanujan probably didn't know any of this at the time. But more importantly, he didn't let his lack of success in this particular problem deter him from his continued pursuit of mathematics.
Again, Ramanujan probably didn't know any of this at the time. But more importantly, he didn't let his lack of success in this particular problem deter him from his continued pursuit of mathematics.
rama1728
2020-11-10 20:04:28
Also, sir what is ramanujan's biggest achievement
Also, sir what is ramanujan's biggest achievement
DPatrick
2020-11-10 20:04:39
There are so many to choose from! But the next one I'll talk about might be the most famous.
There are so many to choose from! But the next one I'll talk about might be the most famous.
DPatrick
2020-11-10 20:04:52
One of the most famous stories regarding Ramanujan occurred when Hardy was visiting Ramanujan in the hospital during one of his many illnesses.
One of the most famous stories regarding Ramanujan occurred when Hardy was visiting Ramanujan in the hospital during one of his many illnesses.
URcurious2
2020-11-10 20:05:03
1729?
1729?
BobDBuilder321
2020-11-10 20:05:03
1729!!!
1729!!!
hwdaniel
2020-11-10 20:05:03
Because of Ramanujan, 1729 is now my favourite number
Because of Ramanujan, 1729 is now my favourite number
rama1728
2020-11-10 20:05:09
oh you mean 1729?
oh you mean 1729?
audio-on
2020-11-10 20:05:09
$1729$
$1729$
DPatrick
2020-11-10 20:05:15
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.
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.
DPatrick
2020-11-10 20:05:17
Do you know why?
Do you know why?
BobDBuilder321
2020-11-10 20:05:46
Sum of 2 cubes, 2 ways
Sum of 2 cubes, 2 ways
aopsuser305
2020-11-10 20:05:46
smallest number that can be represented as the sum of 2 cubes in 2 ways
smallest number that can be represented as the sum of 2 cubes in 2 ways
amazingxin777
2020-11-10 20:05:46
it is the least one that is the sum of two cubes in two different ways
it is the least one that is the sum of two cubes in two different ways
audio-on
2020-11-10 20:05:46
$1729$ is the smallest positive integer than is the sum of $2$ positive pairs of $2$ positive cubes.
$1729$ is the smallest positive integer than is the sum of $2$ positive pairs of $2$ positive cubes.
DPatrick
2020-11-10 20:05:57
It is the smallest number that can be written as a sum of two positive cubes in two different ways! Can you find them?
It is the smallest number that can be written as a sum of two positive cubes in two different ways! Can you find them?
KenOno
2020-11-10 20:06:04
Nice...
Nice...
URcurious2
2020-11-10 20:06:34
12^3 + 1^3 = 10^3 + 9^3 = 1729
12^3 + 1^3 = 10^3 + 9^3 = 1729
LikMCAMC
2020-11-10 20:06:34
12^3+1=10^3+9^3
12^3+1=10^3+9^3
rama1728
2020-11-10 20:06:34
12^3+1^3=10^3+9^3
12^3+1^3=10^3+9^3
Aaryabhatta1
2020-11-10 20:06:34
${10}^3 + 9^3, 1^3 + {12}^3$
${10}^3 + 9^3, 1^3 + {12}^3$
aopsuser305
2020-11-10 20:06:34
$12^3+1^3$ and $10^3+9^3$
$12^3+1^3$ and $10^3+9^3$
os31415
2020-11-10 20:06:34
$1^3+12^3=9^3+10^3=1729$
$1^3+12^3=9^3+10^3=1729$
audio-on
2020-11-10 20:06:34
$1^3+12^3$ and $9^3+10^3$
$1^3+12^3$ and $9^3+10^3$
tennisplayer2007
2020-11-10 20:06:34
1^3 + 12^3 and 9^3 + 10^3?
1^3 + 12^3 and 9^3 + 10^3?
BobDBuilder321
2020-11-10 20:06:34
$12^{3}+1^{3}$ and $10^{3}+9^{3}$
$12^{3}+1^{3}$ and $10^{3}+9^{3}$
DPatrick
2020-11-10 20:06:41
$1729 = 1^3 + 12^3 = 1 + 1728$
$1729 = 9^3 + 10^3 = 729 + 1000$
$1729 = 1^3 + 12^3 = 1 + 1728$
$1729 = 9^3 + 10^3 = 729 + 1000$
DPatrick
2020-11-10 20:06:53
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.
Bluedolphingogo
2020-11-10 20:07:04
is there a proof?
is there a proof?
dolphin7
2020-11-10 20:07:20
bashing is the proof
bashing is the proof
DPatrick
2020-11-10 20:07:23
Right. The proof is essentially: you check that all of 1 through 1728 don't work.
Right. The proof is essentially: you check that all of 1 through 1728 don't work.
pandapo
2020-11-10 20:07:31
why is it the second and not the first?
why is it the second and not the first?
Champ1
2020-11-10 20:07:31
what is the first taxicab number?
what is the first taxicab number?
DPatrick
2020-11-10 20:07:40
Great question with a slightly silly answer.
Great question with a slightly silly answer.
DPatrick
2020-11-10 20:07:53
The first is the smallest number that can be written as a sum of two positive cubes in at least one way.
The first is the smallest number that can be written as a sum of two positive cubes in at least one way.
DPatrick
2020-11-10 20:08:09
Which is...?
Which is...?
URcurious2
2020-11-10 20:08:26
2
2
NCEE
2020-11-10 20:08:26
wow so it is 2
wow so it is 2
pandapo
2020-11-10 20:08:26
2?
2?
NCEE
2020-11-10 20:08:26
2
2
aopsuser305
2020-11-10 20:08:26
2
2
dolphin7
2020-11-10 20:08:26
2
2
asengo
2020-11-10 20:08:26
2
2
NCEE
2020-11-10 20:08:26
2?
2?
BobDBuilder321
2020-11-10 20:08:26
2
2
aopsuser305
2020-11-10 20:08:26
$2$
$2$
URcurious2
2020-11-10 20:08:26
2?
2?
KingJerryX
2020-11-10 20:08:26
2
2
DPatrick
2020-11-10 20:08:41
Right. The first taxicab number is $2 = 1^3 + 1^3$.
Right. The first taxicab number is $2 = 1^3 + 1^3$.
DPatrick
2020-11-10 20:08:56
And continuing, 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?)
And continuing, 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?)
atl3236
2020-11-10 20:09:22
8?
8?
truffle
2020-11-10 20:09:22
8
8
DPatrick
2020-11-10 20:09:28
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
2020-11-10 20:09:39
It's likely that this was first discovered as recently as 1957.
It's likely that this was first discovered as recently as 1957.
DPatrick
2020-11-10 20:09:50
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.
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.
tokumi
2020-11-10 20:10:04
what's the largest known taxicab number?
what's the largest known taxicab number?
Countcountcount
2020-11-10 20:10:04
Have we discovered the 4th taxicab number yet?
Have we discovered the 4th taxicab number yet?
DPatrick
2020-11-10 20:10:10
Only the first 6 taxicab numbers are known:
$2$
$1729$
$87539319$
$6963472309248$
$48988659276962496$
$24153319581254312065344$
Only the first 6 taxicab numbers are known:
$2$
$1729$
$87539319$
$6963472309248$
$48988659276962496$
$24153319581254312065344$
DPatrick
2020-11-10 20:10:20
This is all pretty new! The last of these was discovered by computer search in 2008.
This is all pretty new! The last of these was discovered by computer search in 2008.
Arr0w
2020-11-10 20:10:28
What is the practicality of the the taxicab number though? Does it translate to an applied purpose?
What is the practicality of the the taxicab number though? Does it translate to an applied purpose?
DPatrick
2020-11-10 20:10:36
I am so glad you asked!
I am so glad you asked!
DPatrick
2020-11-10 20:10:46
This is not just a fun game to find these numbers. Just five years ago 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.
This is not just a fun game to find these numbers. Just five years ago 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.
Countcountcount
2020-11-10 20:11:19
wow!
wow!
KenOno
2020-11-10 20:11:20
Thanks David...And this appears prominently in black hole physics.
Thanks David...And this appears prominently in black hole physics.
sanickode
2020-11-10 20:11:29
I searched it up and they found the 7th taxicab number a month ago at MIT
I searched it up and they found the 7th taxicab number a month ago at MIT
DPatrick
2020-11-10 20:11:47
Really??? I didn't know that! I'll have to look it up afterwards.
Really??? I didn't know that! I'll have to look it up afterwards.
yofro
2020-11-10 20:11:57
Is there a proof that there are infinitely many taxicab numbers?
Is there a proof that there are infinitely many taxicab numbers?
DPatrick
2020-11-10 20:12:15
Indeed there is.
Indeed there is.
KenOno
2020-11-10 20:12:21
The buzz words are: K3 surfaces, 3d quantum gravity...
The buzz words are: K3 surfaces, 3d quantum gravity...
DPatrick
2020-11-10 20:12:28
It's been proven that for any $n$, there is some number that can be written as a sum of two positive cubes in at least $n$ different ways. Hardy proved this in 1938 along with his colleague E. M. Wright.
It's been proven that for any $n$, there is some number that can be written as a sum of two positive cubes in at least $n$ different ways. Hardy proved this in 1938 along with his colleague E. M. Wright.
DPatrick
2020-11-10 20:12:54
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?
DPatrick
2020-11-10 20:13:26
It hasn't been on in a while, but it's still a favorite.
It hasn't been on in a while, but it's still a favorite.
DPatrick
2020-11-10 20:13:39
Apparently the creators of Futurama are big fans of taxicab numbers.
Apparently the creators of Futurama are big fans of taxicab numbers.
DPatrick
2020-11-10 20:13:50
In particular, the number $1729$ is hidden in several episodes. It's the registration number of the Planet Express ship, for one thing.
In particular, the number $1729$ is hidden in several episodes. It's the registration number of the Planet Express ship, for one thing.
DPatrick
2020-11-10 20:13:54
DPatrick
2020-11-10 20:14:01
Image from mathworld.wolfram.com
Image from mathworld.wolfram.com
amazingxin777
2020-11-10 20:14:14
i never realized this
i never realized this
MinervaPi
2020-11-10 20:14:14
Woah!
Woah!
DPatrick
2020-11-10 20:14:17
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
2020-11-10 20:14:21
DPatrick
2020-11-10 20:14:27
Image from Gawker Media
Image from Gawker Media
fasterthanlight
2020-11-10 20:14:33
87539319
87539319
A1K2C3
2020-11-10 20:14:40
87539319
87539319
happyhari
2020-11-10 20:14:40
the 3rd taxicab number
the 3rd taxicab number
amazingxin777
2020-11-10 20:14:40
87539319
87539319
DPatrick
2020-11-10 20:14:44
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
2020-11-10 20:14:54
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!
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!
DPatrick
2020-11-10 20:15:17
One more bit of math tonight before we talk about the Spirit of Ramanujan program.
One more bit of math tonight before we talk about the Spirit of Ramanujan program.
DPatrick
2020-11-10 20:15:37
We showed you the Spirit of Ramanujan Project logo at the start of the Math Jam:
We showed you the Spirit of Ramanujan Project logo at the start of the Math Jam:
DPatrick
2020-11-10 20:15:42
DPatrick
2020-11-10 20:15:55
Does anybody know the significance of this logo?
Does anybody know the significance of this logo?
Bluedolphingogo
2020-11-10 20:16:14
ahh the nested radical looking thingy
ahh the nested radical looking thingy
Allen31415
2020-11-10 20:16:14
Has lots of square roots
Has lots of square roots
swan11
2020-11-10 20:16:14
sqrt rt?
sqrt rt?
KingJerryX
2020-11-10 20:16:14
square roots?
square roots?
williamxiao
2020-11-10 20:16:14
idk infinite square root
idk infinite square root
DPatrick
2020-11-10 20:16:18
That's right!
That's right!
DPatrick
2020-11-10 20:16:28
The logo was inspired by the following famous problem that was posed by Ramanujan:
The logo was inspired by the following famous problem that was posed by Ramanujan:
DPatrick
2020-11-10 20:16:33
Evaluate the following expression:
$$\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}$$
Evaluate the following expression:
$$\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}$$
DPatrick
2020-11-10 20:16:46
This is an example of a nested radical.
This is an example of a nested radical.
samrocksnature
2020-11-10 20:16:56
OH THAT ONE
OH THAT ONE
Arr0w
2020-11-10 20:16:56
Oh I've seen that before!
Oh I've seen that before!
Skyhigh1218
2020-11-10 20:16:56
Ohh
Ohh
DPatrick
2020-11-10 20:17:13
Yes, it's a pretty famous item in Ramanujan's history.
Yes, it's a pretty famous item in Ramanujan's history.
DPatrick
2020-11-10 20:17:21
He submitted this problem to the Journal of the Indian Mathematical Society as a challenge for others to solve.
He submitted this problem to the Journal of the Indian Mathematical Society as a challenge for others to solve.
DPatrick
2020-11-10 20:17:31
After 6 months passed and no one had solved it(!), he provided the solution himself.
After 6 months passed and no one had solved it(!), he provided the solution himself.
DPatrick
2020-11-10 20:17:43
Any idea how we might solve it?
Any idea how we might solve it?
DPatrick
2020-11-10 20:17:52
(If you've seen this before and know how it works, please don't spoil it for others!)
(If you've seen this before and know how it works, please don't spoil it for others!)
KingJerryX
2020-11-10 20:18:12
set one part as a variable
set one part as a variable
Peter0527
2020-11-10 20:18:16
rewrite the things within the radicals
rewrite the things within the radicals
DPatrick
2020-11-10 20:18:47
Good ideas...let's try a simpler problem first to see how these ideas work.
Good ideas...let's try a simpler problem first to see how these ideas work.
gladIasked
2020-11-10 20:18:56
A similar problem is in the intro to algebra textbook
A similar problem is in the intro to algebra textbook
DPatrick
2020-11-10 20:19:05
Absolutely! I think it may be this one in fact:
Absolutely! I think it may be this one in fact:
DPatrick
2020-11-10 20:19:11
Suppose we want to evaluate $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}}$.
Suppose we want to evaluate $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}}$.
Bluedolphingogo
2020-11-10 20:19:31
let x = sqrt(2 + sqrt2..)
let x = sqrt(2 + sqrt2..)
fasterthanlight
2020-11-10 20:19:31
Set that value to a variable, $x$.
Set that value to a variable, $x$.
DPatrick
2020-11-10 20:19:38
Good idea. We can set it to be equal to a variable $x$, so that
$$x = \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}}.$$
Good idea. We can set it to be equal to a variable $x$, so that
$$x = \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}}.$$
yesufsa
2020-11-10 20:19:53
Square both sides of the equation
Square both sides of the equation
mathwiz03
2020-11-10 20:19:53
square it
square it
suvamkonar
2020-11-10 20:19:53
Square it
Square it
DPatrick
2020-11-10 20:19:59
Sounds like a good thing to try.
Sounds like a good thing to try.
DPatrick
2020-11-10 20:20:03
We square both sides:
$$x^2 = 2 + \sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}.$$
We square both sides:
$$x^2 = 2 + \sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}.$$
DPatrick
2020-11-10 20:20:13
But now what do you notice?
But now what do you notice?
Bluedolphingogo
2020-11-10 20:20:35
x^2 = 2 + x
x^2 = 2 + x
mathwiz03
2020-11-10 20:20:35
let the other square root parts be x
let the other square root parts be x
mathchampion1
2020-11-10 20:20:35
We have $x$ again
We have $x$ again
PhysKid11
2020-11-10 20:20:35
get x again
get x again
fasterthanlight
2020-11-10 20:20:35
$x^2=2+x$
$x^2=2+x$
buckystark
2020-11-10 20:20:35
This means that $x^2 = 2+x$
This means that $x^2 = 2+x$
boing123
2020-11-10 20:20:35
hey, the thing on the right looks like $x$
hey, the thing on the right looks like $x$
gladIasked
2020-11-10 20:20:35
There's the same thing inside
There's the same thing inside
dolphin7
2020-11-10 20:20:35
right side is 2+x
right side is 2+x
LikMCAMC
2020-11-10 20:20:35
it is just x^2=2+x
it is just x^2=2+x
URcurious2
2020-11-10 20:20:35
x^2 = 2 + x
x^2 = 2 + x
happyhari
2020-11-10 20:20:35
the secopnd part is the same as the original
the secopnd part is the same as the original
DPatrick
2020-11-10 20:20:43
Aha! That big square root on the right side is just another copy of $x$!
Aha! That big square root on the right side is just another copy of $x$!
DPatrick
2020-11-10 20:20:52
So our equation becomes just $x^2 = 2 + x$.
So our equation becomes just $x^2 = 2 + x$.
audio-on
2020-11-10 20:21:07
Quadratic!
Quadratic!
fasterthanlight
2020-11-10 20:21:07
Turn it into a quadratic: $x^2-x-2=0$
Turn it into a quadratic: $x^2-x-2=0$
Bob.The.Builder
2020-11-10 20:21:07
quadratic!
quadratic!
happyhari
2020-11-10 20:21:16
it'S A QUADRATIC!
it'S A QUADRATIC!
gladIasked
2020-11-10 20:21:16
Then we have a quadratic
Then we have a quadratic
DPatrick
2020-11-10 20:21:20
This becomes $x^2 - x + 2 = 0$, which factors as $(x-2)(x+1) = 0$.
This becomes $x^2 - x + 2 = 0$, which factors as $(x-2)(x+1) = 0$.
tennisplayer2007
2020-11-10 20:21:27
Then solve the quadratic and use the positive root
Then solve the quadratic and use the positive root
LikMCAMC
2020-11-10 20:21:32
x^2-x-2=(x-2)(x+1) x has to be positive so x=2!!
x^2-x-2=(x-2)(x+1) x has to be positive so x=2!!
boing123
2020-11-10 20:21:36
and x is positive
and x is positive
Bluedolphingogo
2020-11-10 20:21:36
its equal to 2
its equal to 2
URcurious2
2020-11-10 20:21:36
2
2
KingJerryX
2020-11-10 20:21:36
answer is 2
answer is 2
DPatrick
2020-11-10 20:21:45
Right: either $x=2$ or $x=-1$. And since square roots are positive, we must have $x=2$.
Right: either $x=2$ or $x=-1$. And since square roots are positive, we must have $x=2$.
DPatrick
2020-11-10 20:22:03
So we conclude that
$$2 = \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}}.$$
So we conclude that
$$2 = \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}}.$$
DPatrick
2020-11-10 20:22:20
So now back to Ramanujan's problem:
$$\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}$$
So now back to Ramanujan's problem:
$$\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}$$
DPatrick
2020-11-10 20:22:41
Ramanujan did the same sort of clever substitution, but he had to be a little more clever.
Ramanujan did the same sort of clever substitution, but he had to be a little more clever.
tokumi
2020-11-10 20:22:46
we don't have complete repetition here though, it changes a little bit each time
we don't have complete repetition here though, it changes a little bit each time
DPatrick
2020-11-10 20:22:54
Right. His nested radical doesn't exactly repeat, but there's definitely a pattern.
Right. His nested radical doesn't exactly repeat, but there's definitely a pattern.
DPatrick
2020-11-10 20:23:16
So he couldn't just use a single variable to represent a repeating part.
So he couldn't just use a single variable to represent a repeating part.
DPatrick
2020-11-10 20:23:40
Instead he used a function to capture the pattern.
Instead he used a function to capture the pattern.
DPatrick
2020-11-10 20:23:46
Specifically, he let $$f(x) = \sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+\cdots}}}}.$$
Specifically, he let $$f(x) = \sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+\cdots}}}}.$$
DARTHTATOR0112
2020-11-10 20:24:09
Wow that was good thinking
Wow that was good thinking
DPatrick
2020-11-10 20:24:12
Notice that if we plug in $x=2$ into this function, we get the value that we want! So our goal is to determine $f(2)$.
Notice that if we plug in $x=2$ into this function, we get the value that we want! So our goal is to determine $f(2)$.
DPatrick
2020-11-10 20:24:19
And so now what should we do?
And so now what should we do?
DPatrick
2020-11-10 20:24:48
What did we do in the easier problem at this stage?
What did we do in the easier problem at this stage?
tokumi
2020-11-10 20:25:07
squared
squared
Zhaom
2020-11-10 20:25:07
square
square
happyhari
2020-11-10 20:25:07
square both sides
square both sides
Countcountcount
2020-11-10 20:25:07
square it
square it
connorlonnorl
2020-11-10 20:25:07
square it
square it
mathcounting
2020-11-10 20:25:07
Square it?
Square it?
rjeagle2019
2020-11-10 20:25:07
square it?
square it?
Arr0w
2020-11-10 20:25:07
We squared it I think...
We squared it I think...
DPatrick
2020-11-10 20:25:13
Yeah! Let's square it!
Yeah! Let's square it!
DPatrick
2020-11-10 20:25:19
When we square this, we get $$(f(x))^2 = 1 + x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+\cdots}}}.$$
When we square this, we get $$(f(x))^2 = 1 + x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+\cdots}}}.$$
DPatrick
2020-11-10 20:25:27
And now, what is the big square root term?
And now, what is the big square root term?
happyhari
2020-11-10 20:25:49
f(x+1)
f(x+1)
boing123
2020-11-10 20:25:49
$f(x+1)$?
$f(x+1)$?
fasterthanlight
2020-11-10 20:25:49
f(x+1)
f(x+1)
Hamroldt
2020-11-10 20:25:49
f(x plus 1)
f(x plus 1)
aopsuser305
2020-11-10 20:25:49
f(x+1)
f(x+1)
Concerto-in-A-minor
2020-11-10 20:25:49
f(x+1)
f(x+1)
First_Mathematician
2020-11-10 20:25:49
f(x+1)
f(x+1)
fasterthanlight
2020-11-10 20:25:49
$f(x+1)$
$f(x+1)$
DPatrick
2020-11-10 20:26:10
It's $f(x+1)$. Everywhere there was an $x$ in the original function, we replace it by $x+1$. So the old $x+1$ becomes $(x+1)+1$ or $x+2$, and so on.
It's $f(x+1)$. Everywhere there was an $x$ in the original function, we replace it by $x+1$. So the old $x+1$ becomes $(x+1)+1$ or $x+2$, and so on.
DPatrick
2020-11-10 20:26:17
And now we have the functional equation $$(f(x))^2 = 1 + xf(x+1).$$
And now we have the functional equation $$(f(x))^2 = 1 + xf(x+1).$$
DPatrick
2020-11-10 20:26:29
And then because Ramanujan was very good with algebra, he was essentially able to "guess and check" what $f(x)$ must be. Can you?
And then because Ramanujan was very good with algebra, he was essentially able to "guess and check" what $f(x)$ must be. Can you?
hwdaniel
2020-11-10 20:26:52
x+1
x+1
DPatrick
2020-11-10 20:27:00
Right! It's $f(x) = x+1$. Let's check.
Right! It's $f(x) = x+1$. Let's check.
DPatrick
2020-11-10 20:27:09
If we use $f(x) = x+1$, then our functional equation become $$(x+1)^2 = 1 + x((x+1)+1).$$
If we use $f(x) = x+1$, then our functional equation become $$(x+1)^2 = 1 + x((x+1)+1).$$
DPatrick
2020-11-10 20:27:16
Does this work?
Does this work?
yofro
2020-11-10 20:27:30
yes
yes
tonyme.2
2020-11-10 20:27:30
Yes?
Yes?
Countcountcount
2020-11-10 20:27:30
YES!
YES!
yesufsa
2020-11-10 20:27:30
Yes
Yes
CrazyVideoGamez
2020-11-10 20:27:30
yes
yes
jxc516
2020-11-10 20:27:30
yes
yes
Peter0527
2020-11-10 20:27:30
yes
yes
Allen31415
2020-11-10 20:27:30
Yes
Yes
DPatrick
2020-11-10 20:27:35
It does! Both sides are $x^2 + 2x + 1$. So it worked!
It does! Both sides are $x^2 + 2x + 1$. So it worked!
DPatrick
2020-11-10 20:27:41
And how does this answer the original problem posed by Ramanujan?
And how does this answer the original problem posed by Ramanujan?
Zhaom
2020-11-10 20:28:18
f(2)=3
f(2)=3
Bluedolphingogo
2020-11-10 20:28:18
plug in x = 2
plug in x = 2
KingJerryX
2020-11-10 20:28:18
so f(2) = 3
so f(2) = 3
AOPqghj
2020-11-10 20:28:18
x=2
x=2
tokumi
2020-11-10 20:28:18
since x=2, f(x)=3!
since x=2, f(x)=3!
PhysKid11
2020-11-10 20:28:18
THE ANSWER IS 3!!! WOW THIS IS INGENIOUS!!!
THE ANSWER IS 3!!! WOW THIS IS INGENIOUS!!!
buckystark
2020-11-10 20:28:18
$f(2) = 2+1 = 3$
$f(2) = 2+1 = 3$
Aaryabhatta1
2020-11-10 20:28:18
$f(2) = 2+1 = 3$
$f(2) = 2+1 = 3$
AOPqghj
2020-11-10 20:28:18
plug in for x = 2
plug in for x = 2
Wesey
2020-11-10 20:28:18
x+1 = f(2), so the answer is 3
x+1 = f(2), so the answer is 3
DPatrick
2020-11-10 20:28:30
Our original quantity is $f(2)$, so if $f(x) = x+1$, then $f(2) = 2+1 = \boxed{3}$.
Our original quantity is $f(2)$, so if $f(x) = x+1$, then $f(2) = 2+1 = \boxed{3}$.
DPatrick
2020-11-10 20:28:44
Ramanujan was pretty clever, eh?
Ramanujan was pretty clever, eh?
asbodke
2020-11-10 20:28:47
how do you show it is the only function
how do you show it is the only function
boing123
2020-11-10 20:28:47
What if there are other functions?
What if there are other functions?
TheGeometer
2020-11-10 20:28:47
How do you know that $x+1$ is the only solution to the functional equation?
How do you know that $x+1$ is the only solution to the functional equation?
DPatrick
2020-11-10 20:29:05
Well...that's a really good point. There are certainly some icky details that I'm leaving out.
Well...that's a really good point. There are certainly some icky details that I'm leaving out.
DPatrick
2020-11-10 20:29:19
What Ramanujan did is something a little more general.
What Ramanujan did is something a little more general.
DPatrick
2020-11-10 20:29:25
He proved that $$x+n+a = \sqrt{ax + (n+a)^2 + x\sqrt{a(x+n) + (n+a)^2 + (x+n)\sqrt{\cdots}}}$$ for any values of $x$, $n$, and $a$, using a very similar functional equation.
He proved that $$x+n+a = \sqrt{ax + (n+a)^2 + x\sqrt{a(x+n) + (n+a)^2 + (x+n)\sqrt{\cdots}}}$$ for any values of $x$, $n$, and $a$, using a very similar functional equation.
DPatrick
2020-11-10 20:29:43
The original problem is the special case where $a=0$, $n=1$, and $x=2$.
The original problem is the special case where $a=0$, $n=1$, and $x=2$.
DPatrick
2020-11-10 20:29:57
It's believed that Ramanujan proved this formula when he was about 16 years old!
It's believed that Ramanujan proved this formula when he was about 16 years old!
fasterthanlight
2020-11-10 20:30:23
Wow!
Wow!
S304564
2020-11-10 20:30:23
cool
cool
Countcountcount
2020-11-10 20:30:23
cool!
cool!
tonyme.2
2020-11-10 20:30:23
WOW! HOW?
WOW! HOW?
Awesomeness_in_a_bun
2020-11-10 20:30:23
Wow thats really young
Wow thats really young
DPatrick
2020-11-10 20:30:56
So now that we've talked a little about the life and math of Ramanujan, let's transition into discussing the Spirit of Ramanujan initiative.
So now that we've talked a little about the life and math of Ramanujan, let's transition into discussing the Spirit of Ramanujan initiative.
DPatrick
2020-11-10 20:31:08
Let me start by citing an excerpt from an article called "Why Ramanujan Matters" written by Ken Ono and Robert Schneider:
Let me start by citing an excerpt from an article called "Why Ramanujan Matters" written by Ken Ono and Robert Schneider:
DPatrick
2020-11-10 20:31:17
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.
PrinceRoyale
2020-11-10 20:31:49
its true
its true
URcurious2
2020-11-10 20:31:49
Nice speech
Nice speech
DPatrick
2020-11-10 20:31:53
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
2020-11-10 20:32:17
We are AoPS are pleased that Ken Ono 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 are AoPS are pleased that Ken Ono 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
2020-11-10 20:32:38
My pleasure.
My pleasure.
DPatrick
2020-11-10 20:32:46
Specifically, SoR is a grant program that provides two types of awards.
Specifically, SoR is a grant program that provides two types of awards.
DPatrick
2020-11-10 20:32:57
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
2020-11-10 20:33:09
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
2020-11-10 20:33:18
This can include AoPS books and online programs.
This can include AoPS books and online programs.
DPatrick
2020-11-10 20:33:23
By the way, 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.
By the way, 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.
DPatrick
2020-11-10 20:33:34
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
2020-11-10 20:33:50
2020-21 is the 5th year of the program. To date, the program has identified 46 award winners, with about half from the U.S. and half from elsewhere in the world.
2020-21 is the 5th year of the program. To date, the program has identified 46 award winners, with about half from the U.S. and half from elsewhere in the world.
Butterfly0707
2020-11-10 20:33:54
how can one qualify for the awards?
how can one qualify for the awards?
DPatrick
2020-11-10 20:34:01
To learn more about SoR, please visit the website spiritoframanujan.com.
To learn more about SoR, please visit the website spiritoframanujan.com.
DPatrick
2020-11-10 20:34:11
The SoR website includes a link to the application page. You can also read brief bios of the previous winners on the website.
The SoR website includes a link to the application page. You can also read brief bios of the previous winners on the website.
DPatrick
2020-11-10 20:34:22
Please note that although awards will be made on a rolling basis, you should apply by March 1, 2021 in order to guarantee full consideration.
Please note that although awards will be made on a rolling basis, you should apply by March 1, 2021 in order to guarantee full consideration.
KenOno
2020-11-10 20:35:10
We have made several awards to the AoPs community, and we hope to do so again in 2021.
We have made several awards to the AoPs community, and we hope to do so again in 2021.
DPatrick
2020-11-10 20:35:15
And with that, this concludes our Math Jam! Thanks for attending! Ken Ono and I will stay around for a little while to answer questions.
And with that, this concludes our Math Jam! Thanks for attending! Ken Ono and I will stay around for a little while to answer questions.
KenOno
2020-11-10 20:35:54
Great show David...
Great show David...
KenOno
2020-11-10 20:36:06
Lots of very talented folks online.
Lots of very talented folks online.
DPatrick
2020-11-10 20:36:11
Thanks Ken! And very special thanks to Deven for jumping in while the gremlins attacked.
Thanks Ken! And very special thanks to Deven for jumping in while the gremlins attacked.
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