BR Maths Corner1
Re: BR Maths Corner1
I also found Beckmann's History of PI to be quite good  that is if you can get past of his extreme antagonism towards communism and militarism.
Re: BR Maths Corner1
Amber G. wrote:Also "The Man Who Loved Only Numbers." It's by Paul Hoffman.
^^^ Thanks about mentioning Dr Hobbs. Small world. He was a good friend of Erdos and liked to tell stories about him in his class ..there is interesting story told by him (I am sure you have heard it) of becoming Erods number of 1 (those who wrote a paper with him)
Here is the story..(Thanks to google I found a reference so I can just cut and paste)Dr. Hobbs got accepted to Waterloo, Paul Erdos also visited Waterloo since he dabbled in graph theory (as with everything) himself. Hobbs found out that Erdos was going to some conference and made sure to get on the same bus. He went up to Erdos and expressed his desire to publish a paper with him (he wanted an Erdos number of 1). So, Erdos asked him if he had any idea for what they could write about. Hobbs replied, "I was hoping you did." Erdos sighed and asked Hobbs what he did. "I did my thesis on Hamiltonian cycles." Okay... is every cubic graph with girth less than 4 Hamiltonian? Hobbs thought he would try the Dirac formula on the question, but if Erdos hadn't solved it yet then that probably didn't work. Sure enough, the Dirac formula didn't work. He woke up Erdos (Erdos tended to sleep when people bored him) and told him what he found. Erdos nodded, gave him an idea of a proof, and where the hard spots were. Hobbs went back to work, solved the problem, and got his Erdos number
I think I have heard this one when we visited his home. Dr.Hobbs is an interesting character...
Re: BR Maths Corner1
Reflections Around the Ramanujan Centenary
Reflections of Atle Selberg. A modern great talking about mathematics.
Andre Weil is another modern great.Theres a very interesting India connection between Andre Weil and India.Some may be interested in exploring it.A.Weil was born into an agnostic Jewish family and had a life long interest in India.He too was an agnostic but was inspired by the Karma Yoga(as he understood it) of Bhagavad Gita.
http://www.ams.org/notices/199904/memweil.pdf
Just thought I will share the reflections of these greats.
http://www.frontlineonnet.com]Beyond Ramanujan
http://en.wikipedia.org/wiki/List_of_Indian_mathematicians
http://en.wikipedia.org/wiki/HarishChandra
http://en.wikipedia.org/wiki/Subrahmanyan_Chandrasekhar
http://en.wikipedia.org/wiki/Calyampudi_Radhakrishna_Rao
http://en.wikipedia.org/wiki/Shreeram_Shankar_Abhyankar
http://en.wikipedia.org/wiki/Narendra_Karmarkar
http://en.wikipedia.org/wiki/Manindra_Agrawal
http://en.wikipedia.org/wiki/Manjul_Bhargava
http://en.wikipedia.org/wiki/Madhu_Sudan
Reflections of Atle Selberg. A modern great talking about mathematics.
Andre Weil is another modern great.Theres a very interesting India connection between Andre Weil and India.Some may be interested in exploring it.A.Weil was born into an agnostic Jewish family and had a life long interest in India.He too was an agnostic but was inspired by the Karma Yoga(as he understood it) of Bhagavad Gita.
http://www.ams.org/notices/199904/memweil.pdf
His precocious fascination for epic poetry began
with Homer’s Iliad, in Greek, which in turn quickened
his interest in Sanskrit, and led inevitably to
the great epic Maha¯bha¯rata. Its core, the Bhagavad
Gõ¯ta (the Song of God) stirred his blood as
nothing else did either before
or since. He acquired
sufficient Sanskrit to be
able to read the Gõ¯ta in the
original with the help of a
SanskritFrench dictionary
and an English translation.
He was taken in as much
by the beauty of the poem
as by the thought that inspired
it. The Gõ¯ta is perhaps
the most systematic
spiritual statement of the
“perennial philosophy”,
embodying those universal
truths to which no one
people or age can make exclusive
claim. Eminent Indologists
like A. K.
Coomaraswamy have expressed
the opinion that
it is “probably the most
important single work produced
in India.” The Gõ¯ta remained Weil’s close companion
all his life, through thick and thin, as it did
with Gandhi.
Ka¯lida¯sa’s lyric poem Meghadu¯ta (the Cloudmessenger)
enraptured him—as it has at least fifty generations
of Indians—with its delicacy and grace, its
mellifluous diction, its lyrical concision, and its suggestive
power. Ka¯lida¯sa’s deceptively simple Sanskrit
is the despair of translators. Weil was struck by
Ka¯lida¯sa’s mastery of the Sanskrit language, of its
grammar and rhetoric and dramatic theory, “subjects
which Hindu savants have treated with great, if sometimes
hairsplitting, ingenuity,” in the words of Arthur
Ryder (Berkeley, 1912). He came upon the fact that
in India, Pa¯n. ini’s invention of grammar (ca. fourth
century B.C.) had preceded that of the decimal notation
and negative numbers. Pa¯n. ini’s As.t.
a¯dhya¯yi
(eight chapters) consists of nearly four thousand
aphorisms, the su¯tras, enumerating the technical
terms used in grammar and the rules for their interpretation
and application. The Sanskrit term for
grammar is Vya¯karan.a, which literally means “undoing”,
implying linguistic analysis. Weil could very
well say that “nothing he later came across in the writings
of Chomsky and his disciples seemed unfamiliar
to him.”
Having delved that deep into Sanskrit studies, he
was ready to jump at any offer of a chair in India,
which eventually turned out to be mathematics
(1930–32). It was in Helsinki on the opening day of
the Congress (ICM 1978), as we emerged from a reception
given by our Finnish hosts, and the evening
was spread against the sky, that he suddenly asked
me to recite the first line of the first stanza of
Meghadu¯ta, which he so dearly loved. I had not then
known, as I did later, the intensity of the impact of
India on his personality. Joseph Brodsky has said: “A man is what he
loves. That is why
he loves it; because
he is a part
of it.” His colleagues
at Princeton,
Gödel and
Oppenheimer,
were ardent admirers
of the Gõ ¯ta.
Oppenheimer’s citation
of lines
(often misquoted)
from the Gõ ¯ta (Ch.
XI, verse 32) as he
witnessed the
first nuclear explosion
has entered
the history
of American science.
Neither of
them, however,
could visit India as they had wished
Just thought I will share the reflections of these greats.
http://www.frontlineonnet.com]Beyond Ramanujan
The aura of Srinivasa Ramanujan, perhaps one of the greatest mathematicians of all time, has virtually eclipsed other Indian mathematicians
http://en.wikipedia.org/wiki/List_of_Indian_mathematicians
http://en.wikipedia.org/wiki/HarishChandra
http://en.wikipedia.org/wiki/Subrahmanyan_Chandrasekhar
http://en.wikipedia.org/wiki/Calyampudi_Radhakrishna_Rao
http://en.wikipedia.org/wiki/Shreeram_Shankar_Abhyankar
http://en.wikipedia.org/wiki/Narendra_Karmarkar
http://en.wikipedia.org/wiki/Manindra_Agrawal
http://en.wikipedia.org/wiki/Manjul_Bhargava
http://en.wikipedia.org/wiki/Madhu_Sudan
Re: BR Maths Corner1
^^^ There were number of physicists, especially quantum theorists who were interested in vedanta (in particular, advaitha vedanta). Erwin Schrödinger is perhaps a notable figure. You may be interested in his article,
What Is Life?
It is publicly available.
What Is Life?
It is publicly available.
Re: BR Maths Corner1
A small note about the proof by contradiction posted by AmberG ji  we need the fact that pi^2 is irrational. Just pi being irrational is not sufficient.
Re: BR Maths Corner1
^^ That's true, but pi^2 can be proven to be irrational (in fact any power of pi  or any polynomial of pi is irrational  pi is transcendental  it is hard to prove but is true. (BTW one of the first proof a student learns happens to prove that pi^2 is irrational ) (and hence pi is irrational)..
(See Hermites proof in wiki http://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational
****
(In any case, the proof I put was, as I have said, more of a "try to impress" or show/off type.. too complicated while much simpler ones exist like Euclid.. ) .. (The proofs that pi^2 is irrational, and zeta(2) can be reduced to that kind of product is not elementary math..)
(See Hermites proof in wiki http://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational
****
(In any case, the proof I put was, as I have said, more of a "try to impress" or show/off type.. too complicated while much simpler ones exist like Euclid.. ) .. (The proofs that pi^2 is irrational, and zeta(2) can be reduced to that kind of product is not elementary math..)
Re: BR Maths Corner1
There is a striking similarity between irrational numbers and transcendental numbers. Irrational numbers are nonrepeating when expressed in decimals similarly transcendental numbers are nonrecurring when written as continued fraction. However, some irrational numbers are algebraic. One remarkable fact about transcendental numbers are that they are uncountable, although they seem to be only few (e, pi etc.).
Qn. Prove algebraic numbers are countably infinite. {Hint: Think polynomials}
Qn. Prove algebraic numbers are countably infinite. {Hint: Think polynomials}
Last edited by kasthuri on 07 Jun 2012 17:01, edited 1 time in total.
Re: BR Maths Corner1
One remarkable fact about transcendental numbers are that they are uncountable, although they seem to be only few (e, pi etc.).
What you want to say that there are only a few which have popularly known name(s)....(most can't name others except may be gamma, or logs (which are sort or related to e)..):)
As to concept of "countable infinity" .. being explained to a 8 year old is one of the best I have seen. The letter is from Bethe's (Noble prize winner Physicist from Cornell) son (who, of course is also Bethe..).. The letter was written when Feynman passed away.. (One of the best proof that all integers or rational numbers or algebraic numbers are alephzero is in Gamow's 123 infinity)
Dear Mrs. Feynman,
We have not met, I believe, frequently enough for either of us to have taken root in the other's conscious memory. So please forgive any impertinence, but I could not let Richard's death pass unnoticed, or to take the opportunity to add my own sense of loss to yours.
Dick was the best and favorite of several “uncles” who encircled my childhood. During his time at Cornell he was a frequent and always welcome visitor at our house, one who could be counted on to take time out from conversations with my parents and other adults to lavish attention on the children. He was at once a great player of games with us and a teacher even then who opened our eyes to the world around us.
My favorite memory of all is of sitting as an eightornineyearold between Dick and my mother, waiting for the distinguished naturalist Konrad Lorenz to give a lecture. I was itchy and impatient, as all young are when asked to sit still, when Dick turned to me and said, “Did you know that there are twice as many numbers as numbers?”
“No, there are not!” I was defensive as all young of my knowledge.
“Yes there are; I'll show you. Name a number.”
“One million.” A big number to start.
“Two million.”
“Twentyseven.”
“Fiftyfour.”
I named about ten more numbers and each time Dick named the number twice as big. Light dawned. {75}
“I see; so there are three times as many numbers as numbers.”
“Prove it,” said Uncle Dick. He named a number. I named one three times as big. He tried another. I did it again. Again.
He named a number too complicated for me to multiply in my head. “Three times that,” I said.
“So, is there a biggest number?” he asked.
“No,” I replied. “Because for every number, there is one twice as big, one three times as big. There is even one a million times as big.”
“Right, and that concept of increase without limit, of no biggest number, is called ‘infinity.’ “
At that point Lorenz arrived, so we stopped to listen to him.
I did not see Dick often after he left Cornell. But he left me with bright memories, infinity, and new ways of learning about the world. I loved him dearly.
Excerpt from "What Do You Care What Other People Think?”
Last edited by Amber G. on 07 Jun 2012 09:16, edited 1 time in total.
Re: BR Maths Corner1
^^^ Nice...really wonderful. May be I will use this when our baby grows up.
Along these lines...today we had a group meeting in which I asked my fellow physicians (cancer specialists) a question. I drew an interval [0,1] and asked how many points where there. "Infinity" came the reply. Next I drew a unit square and asked how many points where there. Also, came the reply "infinity". Then I asked them which is "more" infinite. Almost everybody said square has more points and so more infinite. It was time I thought and started to explain spacefilling curves and Peano's insight. At that point everybody grew tired
Along these lines...today we had a group meeting in which I asked my fellow physicians (cancer specialists) a question. I drew an interval [0,1] and asked how many points where there. "Infinity" came the reply. Next I drew a unit square and asked how many points where there. Also, came the reply "infinity". Then I asked them which is "more" infinite. Almost everybody said square has more points and so more infinite. It was time I thought and started to explain spacefilling curves and Peano's insight. At that point everybody grew tired
Re: BR Maths Corner1
Historically, did indians excel more than the chinese in mathematics (discoveries, styles, etc)? Going through k12 and college in the US, the chinese kids seem like they are a bit stronger overall and seem to win more competitions. At the same time indians seemed a tad lazier and parents didn't really seem to care as much and were less assertive about their civilizational identity. So i'm sure this also plays a role...Moreover, i've developed an interest on the effects of board games on mathematical ability. The two that come to my mind are Go (chinese) and Chess (indian). IMO, Go seems more complicated and takes a bit more creativity to beat the opponent vs chess.
Re: BR Maths Corner1
RoyG wrote:Historically, did indians excel more than the chinese in mathematics (discoveries, styles, etc)? ...
...Go seems more complicated and takes a bit more creativity to beat the opponent vs chess.
RoyG ji
IMHO, the answer to the first question is "it depends". Godel's theorems depend on what is called Chinese remainder theorem discovered during 35 century CE. It is also useful in RSA algorithm. Around the same time we had Aryabhatta who developed Mathematics and its applications to Astronomy. One things with Chinese mathematics is that they have documented lot more than the Indian Mathematics that is lost (putative) where in the lies the problem  has it really been lost or is it not there to start with?
Regards
Re: BR Maths Corner1
Good luck to India's official team in IMO..
<link>
Also, there is an official Facebook group for contestants of IMO 2012.. If you have friends in IMO team.. please let them know
Link: https://www.facebook.com/groups/409088145791398
<link>
Also, there is an official Facebook group for contestants of IMO 2012.. If you have friends in IMO team.. please let them know
Link: https://www.facebook.com/groups/409088145791398
Re: BR Maths Corner1
Okay ... some may find this interesting ..
(If you wish, do not google before seeing how much you can guess)
These pictures are from a mathematics professor, writing about history of math in India..
1. This following fort is famous for Historic reasons.. Can you guess why?
2. Inside of this fort there is a much older temple.. (It is named चतुर्भुज मंदिर)
3. In this temple, just near चतुर्भुज's right hand, there is a tablet:
Why this is of interest to a history of mathematics professor?
(The inscription says  rough translation  Vaillabhatta's son caused this temple to be built on a piece of land 270 hastas in length. at one particular date  given in vikram samvat...)
(If you wish, do not google before seeing how much you can guess)
These pictures are from a mathematics professor, writing about history of math in India..
1. This following fort is famous for Historic reasons.. Can you guess why?
2. Inside of this fort there is a much older temple.. (It is named चतुर्भुज मंदिर)
3. In this temple, just near चतुर्भुज's right hand, there is a tablet:
Why this is of interest to a history of mathematics professor?
(The inscription says  rough translation  Vaillabhatta's son caused this temple to be built on a piece of land 270 hastas in length. at one particular date  given in vikram samvat...)
Re: BR Maths Corner1
^^^ Historic reason for the fort is captured in this excerpt
रानी बढ़ी कालपी आई, कर सौ मील निरंतर पार,
घोड़ा थक कर गिरा भूमि पर गया स्वर्ग तत्काल सिधार,
यमुना तट पर अंग्रेज़ों ने फिर खाई रानी से हार,
विजयी रानी आगे चल दी, किया ग्वालियर पर अधिकार।
अंग्रेज़ों के मित्र सिंधिया ने छोड़ी रजधानी थी,
बुंदेले हरबोलों के मुँह हमने सुनी कहानी थी,
खूब लड़ी मर्दानी वह तो झाँसी वाली रानी थी।।
Re: BR Maths Corner1
Amber G. wrote:^^^ Historic reason for the fort is captured in this
Is it Gwalior fort? Haven't seen it but guessing from the poetry, it is somehow related to Jhansi ki Rani and Madhaorao Scindia.
Added later
I verified my guess that it is the first dateable record of zero usage c. 875 CE
Re: BR Maths Corner1
^^^ Above ("first dateable record of zero usage") is correct .. The information, apparently is in wikl too.. but a good collection of photographs are at "Shunya's (sanskrit word for zero) collection"
http://www.shunya.net/Pictures/NorthIndia/Gwalior/Gwalior.htm
From:http://en.wikipedia.org/wiki/Gwalior_Fort
"0" has been used much before 9th century, but in this tablet, definite date (in vikram samvat) can be seen clearly.
The fort is It is one of the biggest forts in India, and has a reputation that it was virtually never (rarely?) conquered by an enemy..Rani Laximbai died here and it also contains Tansen's tomb.
The source of the first post is AMS article, "All for nought"
http://www.ams.org/samplings/featurecolumn/fcarcindiazero
Gwalior's most famous son of recent times, is of course, Atal Bihari Vajpayee.
http://www.shunya.net/Pictures/NorthIndia/Gwalior/Gwalior.htm
From:http://en.wikipedia.org/wiki/Gwalior_Fort
Gwalior Fort also occupies a unique place in the human civilization as the place which has the first ever recorded use of zero. Also referred as 'Shunya' in sanskrit, this site is of mathematical interest because of what is written on a tablet recording the establishment of a small 9th century Hindu temple on the eastern side of the plateau. By accident, it records the oldest "0" { see "270 in picture above} in India for which a definite date can be assigned
"0" has been used much before 9th century, but in this tablet, definite date (in vikram samvat) can be seen clearly.
The fort is It is one of the biggest forts in India, and has a reputation that it was virtually never (rarely?) conquered by an enemy..Rani Laximbai died here and it also contains Tansen's tomb.
The source of the first post is AMS article, "All for nought"
http://www.ams.org/samplings/featurecolumn/fcarcindiazero
Gwalior's most famous son of recent times, is of course, Atal Bihari Vajpayee.
Proofs from the Book
I was thinking of reading this book for a long time (after reading The man who love only numbers). Just recently gone to the library and checked it out.
AmberG ji would be thrilled to know that there are six proofs for infinity of primes one of them being Euclid's of course. I found two proofs especially interesting. Will post them soon. One I could not understand fully as this involves Topology which I am not familiar with.
AmberG ji would be thrilled to know that there are six proofs for infinity of primes one of them being Euclid's of course. I found two proofs especially interesting. Will post them soon. One I could not understand fully as this involves Topology which I am not familiar with.
Re: BR Maths Corner1
^^^ There are literally dozens of proof (and we are counting only "interesting ones") for that. ( Ribenboim's book about prime numbers, lists about dozen interesting ones)
Here is one fun proof many may find very interesting and simple...(but not that well known) (There are quite a few proofs based on similar theme)
(n is positive integer >1)
N2 = n(n + 1) must have at least two different prime factors.
( Because n and n+1 are coprime )
Similarly since N2 and N2+1 are coprime, so
N3 = N2(N2+1) = n(n+1)(n(n+1)+1) must have at lest three different prime factors
This can be continued indefinitely.
(Topological proof(s) eg like Fürstenberg's (wiki can give details..) are interesting too)
Here is one fun proof many may find very interesting and simple...(but not that well known) (There are quite a few proofs based on similar theme)
(n is positive integer >1)
N2 = n(n + 1) must have at least two different prime factors.
( Because n and n+1 are coprime )
Similarly since N2 and N2+1 are coprime, so
N3 = N2(N2+1) = n(n+1)(n(n+1)+1) must have at lest three different prime factors
This can be continued indefinitely.
(Topological proof(s) eg like Fürstenberg's (wiki can give details..) are interesting too)
Re: BR Maths Corner1
^^^ Here is another cute proof .. (Think attributed to Goldbach)
Take F_n = 2^(2^n)+1 (Known as Fermat's numbers)
Easy to prove, there are no common factors between any two F_n's
Hence, each F_n has (at least) one new prime number as its factor ..
Since there are infinite F_n's (n can become as large as one wants).. there are infinite prime numbers..
Take F_n = 2^(2^n)+1 (Known as Fermat's numbers)
Easy to prove, there are no common factors between any two F_n's
Hence, each F_n has (at least) one new prime number as its factor ..
Since there are infinite F_n's (n can become as large as one wants).. there are infinite prime numbers..
Re: BR Maths Corner1
Okay, let me add one more proof, because it is cute, nontrivial, and may encourage someone to learn new fun facts in number theory ...
See my earlier post about an old problem first time asked to me by a Jain guru.
(See discussion about that problem)
http://forums.bharatrakshak.com/viewtopic.php?f=2&t=4201&p=791120&hilit=11111111111#p791120
So here is the proof (that there are infinite primes..)
Proof by contradiction:
If the number is finite ... Let p be the largest prime.
Take the number 11111... (p times)
Any factor of the above is larger than p.. (why?  Can you prove this)
Contradiction...
QED
(Example, suppose one tells you 11 is the largest prime, you take the factors of 11111111111 ("1" written 11 times).. they are larger than 11 hence the contradiction )
****
If fact, any factor of 1111.. p times has to be of the form 2p+1
Can you prove this... (This was mentioned in this form earlier)
for example factors of 11111111111 (11 times) are 21649 and 513239
and both are of the form 11n+1 ..
(Of course, both factors are larger than 11  that was the crux of the above proof..)
See my earlier post about an old problem first time asked to me by a Jain guru.
(See discussion about that problem)
http://forums.bharatrakshak.com/viewtopic.php?f=2&t=4201&p=791120&hilit=11111111111#p791120
So here is the proof (that there are infinite primes..)
Proof by contradiction:
If the number is finite ... Let p be the largest prime.
Take the number 11111... (p times)
Any factor of the above is larger than p.. (why?  Can you prove this)
Contradiction...
QED
(Example, suppose one tells you 11 is the largest prime, you take the factors of 11111111111 ("1" written 11 times).. they are larger than 11 hence the contradiction )
****
If fact, any factor of 1111.. p times has to be of the form 2p+1
Can you prove this... (This was mentioned in this form earlier)
for example factors of 11111111111 (11 times) are 21649 and 513239
and both are of the form 11n+1 ..
(Of course, both factors are larger than 11  that was the crux of the above proof..)
Re: BR Maths Corner1
Amber G. wrote:^^^ Here is another cute proof .. (Think attributed to Goldbach)
Take F_n = 2^(2^n)+1 (Known as Fermat's numbers)
Easy to prove, there are no common factors between any two F_n's
Hence, each F_n has (at least) one new prime number as its factor ..
Since there are infinite F_n's (n can become as large as one wants).. there are infinite prime numbers..
Yeah. That one I found to be one of the interesting ones.
Re: BR Maths Corner1
Amber G. wrote:Proof by contradiction:
If the number is finite ... Let p be the largest prime.
Take the number 11111... (p times)
Any factor of the above is larger than p.. (why?  Can you prove this)
Contradiction...
QED
OK. The other one using Lagrange's Theorem is similar to this. It uses coset counting.
Amber Ji have you read "Proofs from the book"? Lives up to the name. For others who have not read "The man who only loved numbers" (Bio of Erdos), Erdos believed that god keeps a book in which all the elegant proofs are written. Paraphrasing here  "Even if one doesn't believe in god, one should believe in the book". In that spirit the book "Proofs from the book" is written collecting some of the most elegant proofs.
Re: BR Maths Corner1
^^^ Don't know exactly what you mean by Lagrange's Theorem (and/or) coset counting... the proof of above was (at least outline was given) hinted in earlier posts...
Interestingly the above mention proof is essentially based on Fermat't little theorem..
Interestingly the above mention proof is essentially based on Fermat't little theorem..
Re: BR Maths Corner1
Lagrange's Theorem of Algebra states that the order of a subgroup of a finite group divides the order of the group it is a subgroup of. Langrange's Theorem is proved by counting the cosets (right or left doesn't matter as long as one always takes right or left) of a subgroup of a group.
Using the above theorem and Mersenne numbers, there is a proof for infinitude of primes.
Using the above theorem and Mersenne numbers, there is a proof for infinitude of primes.
Re: BR Maths Corner1
matrimc wrote:OK. The other one using Lagrange's Theorem is similar to this. It uses coset counting.
I thought Lagrange's theorem is about the order of subgroups and the order of a group  that it essentially divides. I vaguely remember Herstein using normal subgroups (and the cosets) to arrive at the result.
Re: BR Maths Corner1
^^^ ok...matrimcji got ahead of me there...
Re: BR Maths Corner1
kasthuri wrote:matrimc wrote:OK. The other one using Lagrange's Theorem is similar to this. It uses coset counting.
I thought Lagrange's theorem is about the order of subgroups and the order of a group  that it essentially divides. I vaguely remember Herstein using normal subgroups (and the cosets) to arrive at the result.
Okay .. I see using it on multiplicative group(s)..
But one can achieve the same with using elementary school math using Fermat's little theorem..
Re: BR Maths Corner1
matrimc wrote:Amber Ji have you read "Proofs from the book"? Lives up to the name. For others who have not read "The man who only loved numbers" (Bio of Erdos), Erdos believed that god keeps a book in which all the elegant proofs are written. Paraphrasing here  "Even if one doesn't believe in god, one should believe in the book". In that spirit the book "Proofs from the book" is written collecting some of the most elegant proofs.
Erdos is a legend. There are few spots on the shoulders of the giants, and Erdos occupies one. Wiki compares him with Euler, and rightfully so!
Quotable quote: "A mathematician is a machine that turns coffee into theorems"  Erdos.
Re: BR Maths Corner1
Amber G. wrote:Okay .. I see using it on multiplicative group(s)..
But one can achieve the same with using elementary school math using Fermat's little theorem..
Yeah. Showoff proof
On a side note, Rahul M ji note that memorization of definitions and theorems is important in certain contexts in that one doesn't prove everything starting from fundamentals. Of course, it helps to read the proof of a theorem which once internalized helps to remember the theorem itself.
Re: BR Maths Corner1
I am not a mathematician. My mathematical education extends up to multivariate calculus and linear algebra. In my college days, I also spend quite a bit of time reading up on elementary number theory (I loved Apostol's book, but don't remember much from it).
Given all that, I am wondering if I should buy and read Proofs from The Book.
What do the math people here think? Would it be accessible to me?
Given all that, I am wondering if I should buy and read Proofs from The Book.
What do the math people here think? Would it be accessible to me?
Re: BR Maths Corner1
Nandu ji, I am still in the process of reading it (among other things, unfortunately). Will let you know  hopefully in the near future.
Re: BR Maths Corner1
matrimc wrote:On a side note, Rahul M ji note that memorization of definitions and theorems is important in certain contexts in that one doesn't prove everything starting from fundamentals. Of course, it helps to read the proof of a theorem which once internalized helps to remember the theorem itself.
This is a nice subject to think about. I think memorization is not only important but it is essential. I want to put 'memorization' as a subclass of 'becoming familiar'.
The world is quite simple for me  the more one becomes familiar on a subject, he becomes an expert in it. This *includes* math, physics, chemistry and what not!
Can you imagine anybody researching group theory without mastering a basic multiplication table? I don't know if I can find any examples around. Memorization is another name for "internalizing" a concept. It is truly a meditation.
Last edited by kasthuri on 28 Jun 2012 07:35, edited 1 time in total.
Re: BR Maths Corner1
Nandu wrote:My mathematical education extends up to multivariate calculus and linear algebra.
IMHO, Linear algebra is the most useful tool that mankind has ever invented. With a strong linear algebra one can solve almost all the problems in this world. Of course, there are problems that can't be solved using linear algebra, but they are as little as measure zero  yes, I mean in the Lebesgue sense!
Re: BR Maths Corner1
Okay, I thought I should ask this to the math junta here.
Few days back, I was deeply thinking about Riemann integral. Most often the argument to disparage Riemann integral is as follows:
"For a function to be Riemann integrable, the domain of integration should be bounded and the function should be continuous almost everywhere, which places a severe restriction on the integrability of certain functions."
Agreed that domain of integration could be a serious problem, which can be overcome by defining improper Riemann integral (as a limit of Riemann sums), I don't see any reason why we need to move beyond improper Riemann integral  other than for physics and "abstract nonsense" purposes.
In other words, has anybody ever "practically" dealt with discontinous functions on a set of finite measure? Please note I am saying Dirichlet's function is for "aesthetic" and "abstract nonsense"/generalization purposes. Please prove me wrong as I think I will definitely learn from such examples.
Few days back, I was deeply thinking about Riemann integral. Most often the argument to disparage Riemann integral is as follows:
"For a function to be Riemann integrable, the domain of integration should be bounded and the function should be continuous almost everywhere, which places a severe restriction on the integrability of certain functions."
Agreed that domain of integration could be a serious problem, which can be overcome by defining improper Riemann integral (as a limit of Riemann sums), I don't see any reason why we need to move beyond improper Riemann integral  other than for physics and "abstract nonsense" purposes.
In other words, has anybody ever "practically" dealt with discontinous functions on a set of finite measure? Please note I am saying Dirichlet's function is for "aesthetic" and "abstract nonsense"/generalization purposes. Please prove me wrong as I think I will definitely learn from such examples.
Re: BR Maths Corner1
Amber G. wrote:^^^ Here is another cute proof .. (Think attributed to Goldbach)
Take F_n = 2^(2^n)+1 (Known as Fermat's numbers)
Easy to prove, there are no common factors between any two F_n's
Hence, each F_n has (at least) one new prime number as its factor ..
Since there are infinite F_n's (n can become as large as one wants).. there are infinite prime numbers..
Details are left as exercise (cruelly so ). Here are the details.
$\Pi_i_0^{n1} F_i = F_n  2$ which can be proved by induction.
To prove any two Fermat's numbers are relatively prime, observe that if there is a divisor $m$ which divides both some $F_i$ and $F_n$, then $m$ has to be either 1 or 2 (since it has to divide 2), but all of the Fermat's numberS are odd leading to the GCD being 1. QED.
Re: BR Maths Corner1
^^ Nice
(Can we ask brf admins to make brf LaTex enabled? )
(Can we ask brf admins to make brf LaTex enabled? )
Re: BR Maths Corner1
Amber G. wrote:^^ Nice
(Can we ask brf admins to make brf LaTex enabled? )
LaTex or there is also a web standard called MathML. Can that be done?
Re: BR Maths Corner1
Amber G. wrote:I believer this is the Indian team for the 53rd International Mathematical Olympiad (2012). ( July 416 in Mar del Plata, Argentina)
Good luck to The Indian Team:
(1) Debdyuti Banerjee
(2) Akashdeep Dey
(3) Prafulla Sushil Dhariwal
(4) Shubham Bhakta
(5) Rijul Saini
(6) Mrudul Madhav Thatte
Did not see it posted here before...
Prafulla and Debdyuti got Gold
Akashdeep, Rijul and Mrudul got Silver
Shubham got HM
Excellent. 2 Golds and 3 Silvers... One of the best result in last 10 years.. congrats.

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Re: BR Maths Corner1
Prof. C K Raju talks about teaching calculus effecitvely in less than 2 weeks. Any math wiz here, familar with his approach and willing to share what is different of each approach in comparison to the method(s) most of us followed.
Re: BR Maths Corner1
amber ji, pooch. i am kanpoosed on subsets. I understood what is proper subset, and subset 99% of it, but not complete, because of the definition that says the difference is proper subset has other elements not in the subset, hence proper sub. so, does this mean A c B > A = B ?
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