He started his journey on 17th March
April 14th is important in many ways. One of
which that goes unreported is the great son of India whose journey into
recognition as one of the greatest genius happened with his arrival in London
on this Day.
We need to be reminded that top mathematicians have mined
many theorems from his works and worked up their minds to find out how so many
mind boggling theorems have popped out of a single mind.
Two great people were born on 22nd December one
is this great genius and another is great soul, my mother.
My God, I could not stop reading the link given below at a stretch
yesterday night. I am always amazed at this great genius, ordinary human being
with an extraordinary brain. He is a reminder of people who can be labeled as a
category of 'the greater the thinking the simpler the living'. The world is
witness to many unschooled geniuses. But seldom on subjects as scientific and
complicated as this, especially a wide range of contribution in mathematical
analysis, number theory, infinite series, continued fractions etc.
I am very poor even in the basic Mathematics but I know its
immense importance in terms of its application in almost all spheres of life. I
have read many interesting books on mathematics and its relevance. I have also
listened to many lectures about how India has been a great pioneer in some of
the most advanced Mathematics which were even embedded in many ancient
scriptures etc. Equally I have also read some interesting books on advanced
number theories by Arabic scholars.
This PDF gives brief accounts of this great genius whose
life continues to be greater puzzle than the most complicated theorems he came
up with, the complex mathematical problems he solved so easily, the great
solutions he provided with ease to many unsolvable problems.
He was a man immersed in number who understood and
interpreted everything connected with life through numbers.
French
historian of Mathematics and author of the book, The Universal History of
Numbers Georges Ifrah says,
"The
Indian mind has always had for calculations and the handling of numbers an
extraordinary inclination, ease and power, such as no other civilization in
history ever possessed to the same degree. So much so that Indian culture
regarded the science of numbers as the noblest of its arts...A thousand years
ahead of Europeans, Indian savants knew that the zero and infinity were
mutually inverse notions."
(source: Histoire Universelle des Chiffres - By Georges Ifrah
Paris - Robert Laffont, 1994, volume 2. p. 3 ).
Claiming
India to be the true birthplace
of our numerals, Ifrah salutes the Indian researchers saying that the
"...real inventors of this fundamental discovery, which is no less
important than such feats as the mastery of fire, the development of
agriculture, or the invention of the wheel, writing or the steam engine, were
the mathematicians and astronomers of the Indian civilization: scholars who,
unlike the Greeks, were concerned with practical applications and who were
motivated by a kind of passion for both numbers and numerical
calculations."
He
refers to 24 evidences from scriptures from India, whose dates range from 1150
BC until 458 BC. Of particular interest is the work by Indian mathematician Bhaskaracharya known as Bhaskara (1150 BC) where he makes
a reference to zero and the place-value system were invented by the god Brahma. In
other words, these notions were so well established in Indian thought and
tradition that at this time they were considered to have always been used by
humans, and thus to have constituted a "revelation"
of the divinities.
"It
was only after the eighth century BC, and doubtless due to the influence of the
Indian Buddhist missionaries, that Chinese mathematicians introduced the use of
zero in the form of a little circle or dot (signs that originated in
India),...".
The
early passion which Indian civilization had for high numbers was a significant
factor contributing to the discovery of the place-value system, and not only
offered the Indians the incentive to go beyond the "calculable"
physical world, but also led to an understanding (much earlier than in our
civilization) of the notion of mathematical infinity itself.
Sanskrit notation had
an excellent conceptual quality. It was easy to use and moreover it facilitated
the conception of the highest imaginable numbers. This is why it was so well
suited to the most exuberant numerical or arithmetical-cosmogonic speculations
of Indian culture."
"The Indian people
were the only civilization to take the decisive step towards the perfection of
numerical notation. We owe the discovery of modern numeration and the
elaboration of the very foundations of written calculations to India
alone."
"It is clear how much we owe to this brilliant
civilization, and not only in the field of arithmetic; by opening the way to
the generalization of the concept of the number, the Indian scholars enabled
the rapid development of mathematics and exact sciences. The discoveries of
these men doubtless required much time and imagination, and above all a great
ability for abstract thinking. These major discoveries took place within an environment which
was at once mystical, philosophical, religious, cosmological, mythological and
metaphysical."
"In India, an
aptitude for the study of numbers and arithmetical research was often combined
with a surprising tendency towards metaphysical abstractions; in fact, the latter is so deeply ingrained in Indian thought and
tradition that one meets it in all fields of study, from the most advanced
mathematical ideas to disciplines completely unrelated to 'exact sciences.
In short, Indian
science was born out of a mystical and religious culture and the etymology of
the Sanskrit words used to describe numbers and the science of numbers bears
witness to this fact. "
"Sanskrit means
“complete”, “perfect” and “definitive”. In fact, this language is extremely elaborate, almost artificial, and is capable of
describing multiple levels of
meditation, states of consciousness and psychic, spiritual and even intellectual
processes. As for vocabulary, its
richness is considerable and highly diversified. Sanskrit has for centuries
lent itself admirably to the diverse rules of prosody and versification. Thus
we can see why poetry has played such a preponderant role in all of Indian
culture and Sanskrit literature. "
1729 = 13 + 123 = 93 + 103.
Generalizations of this idea
have created the notion of "taxicab
numbers". Coincidentally, 1729 is also a Carmichael
number.
1729 = 13 + 123 = 93 + 103.
The
following passages are from http://www.believermag.com/issues/201501/?read=article_schneider_phelan
by ROBERT SCHNEIDER WITH BENJAMIN PHELAN
Robert Schneider is the lead singer
of The Apples in stereo, a record producer (Neutral Milk Hotel, Olivia Tremor
Control), and cofounder of the Elephant 6 collective of musicians and artists.
He is currently pursuing a PhD in number theory at Emory University, in
Atlanta, Georgia, where he lives with his wife and son.
Benjamin Phelan is a writer and musician who lives in Louisville, Kentucky, and is a multi-instrumentalist in Apples in Stereo..
Benjamin Phelan is a writer and musician who lives in Louisville, Kentucky, and is a multi-instrumentalist in Apples in Stereo..
“ENCOUNTER
WITH THE INFINITE
HOW
DID THE MINIMALLY TRAINED, ISOLATED SRINIVASA RAMANUJAN, WITH LITTLE MORE THAN
AN OUT-OF-DATE ELEMENTARY TEXTBOOK, ANTICIPATE SOME OF THE DEEPEST THEORETICAL
PROBLEMS OF MATHEMATICS—INCLUDING CONCEPTS DISCOVERED ONLY AFTER HIS DEATH?
There is a form of Buddhism so potent, adherents say, that to hear its
name spoken is to receive a promise of premature enlightenment, of early
freedom from the wheel of incarnations. Something similar is true of Srinivasa
Ramanujan, the super-genius who was born into deep poverty in an obscure part
of southern India, who taught himself mathematics from a standard textbook, and
in total isolation became a mathematician of such power that a hundred years
after his death, at the age of thirty-two, the meaning of much of his work is
still a mystery. In the middle of what I thought would be my life’s work,
writing and producing music, I heard his story; now I find myself in
graduate school studying number theory.
That even as he approached the infinite, Ramanujan found a wormhole
through, and beyond. Even on his deathbed, mathematics was an act of worship.
Worship of a single infinity, in infinite forms, all of them knowable.
Ramanujan was unfashionable. His
body of work consisted of notebooks filled with short formulae, so there was no
overarching theory to study, and formula writing had been out of style in
serious mathematics for more than a century.2 The formulists had had their time. They were the
sorcerers of math’s prehistory who had discovered the deep connections among
the key concepts and encoded them in mathematical haiku. Modern
mathematicians-in-training studied modern theorists, technicians who labored
over proofs of narrowly defined conjectures, mastered this or that technique,
and polished the gleaming apparatus free of fingerprints.
When Ono began to dig a little more
deeply into Ramanujan’s formulae, he was surprised at the tangle of roots he
encountered below the surface. Ramanujan’s crazy tricks linked up with some of
the deepest concepts in math. They could not exist unless they concealed
massive theoretical edifices.
Take the tau function, an oddity
that Ramanujan discovered and studied during his five years at Cambridge. A
function is a mathematical expression that, when fed with a number, produces
another number. It’s a machine that takes some raw material and then stretches,
compresses, reshapes, or transforms it into something else. Functions embody
the relationships between numbers; they are central objects of study in number
theory. Ramanujan found the tau function important enough to spend upward of
thirty pages in his notebook exploring it, but it was hard for other
mathematicians to see why he’d been so interested. On its face, there was
nothing special about the tau function. Hardy, Ramanujan’s chief collaborator
at Cambridge, worried that the tau function’s homeliness might lead future
mathematicians to see it as a mathematical “backwater.” For decades after
Ramanujan’s death, it was treated as one.
Then, in the 1960s, a French
mathematician named Jean-Pierre Serre realized that the tau function was an
unassuming front for a powerful force. Its existence could be explained only if
there was a brand-new theory of functions encoded in it. Serre called this
theory, suspected but not proven, the Galois representations. Not long after,
the Belgian researcher Pierre Deligne proved that the Galois representations
actually existed, and in the process clarified that the tau function was deeply
connected to algebraic geometry and algebraic number theory. For proving the
Galois representations, Deligne won a Fields Medal, the ne plus ultra of
mathematical achievement, awarded every four years to a mathematician under the
age of forty. In 1995, the Galois representations appeared as the key component
of Andrew Wiles’s epochal proof of Fermat’s Last Theorem, the largest, most
notorious open problem in mathematics, which had gone unproved for over three hundred
years and was suspected of being unprovable. Wiles, forty-one when he published
the final version of his proof, was ineligible for a Fields, which only seems
unjust: no prize, not even a Fields medal, could be adequate to the mastery in
his proof. When the International Mathematical Union convened to hand out
Fields Medals that year, it created a special award for Wiles and, for the
ceremony, built two stages: one for the Fields Medalists and one above it,
where Wiles stood alone.
“All that, from Serre to the Fields
medal to Wiles, is from only about ten or fifteen pages from Ramanujan’s
notebooks, out of the hundreds that he wrote,” Ono says. “Which is typical! And
in fact, studying the tau function, the British mathematician Louis Mordell
proved some properties that were later developed into Hecke algebras and the
Langlands program, among the two or three most important developments in
twentieth-century math. And that’s from a different five pages of Ramanujan’s
work on tau that have no intersection with the previous fifteen. In fact, it
might be as short as a page. One page from Ramanujan’s work may have given
birth to all that.”
There’s a subtlety here that needs
to be made explicit. It’s not remarkable that Ramanujan’s work on the tau
function led to interesting new mathematics. That kind of thing happens all the
time; it’s how the subject advances.
With Ramanujan there is a seeming
reversal of cause and effect. No one can write down a formula with deep, hidden
properties unless they first know what the deep properties are that they are
trying to encode. This is the way mathematicians understand math to work; it is
the only way they—we—know to approach the subject. But the significance of the
tau function—the reason to write it down—wasn’t discovered until Ramanujan had
been dead for sixty years.
“There’s no way Ramanujan knew all
these intermediate things,” says Ono. “The concepts [encoded in the tau
function] didn’t exist when he was alive. That’s the mind-boggling part:
Ramanujan anticipated the work of people who would live long after him. He had
visions that said there were going to be some theories in the future. Somehow.
He didn’t need any intermediate steps for him to anticipate that there would be
all these subjects, and that he would find the first examples of them, and that
they would go on to be the prototypes that we desperately needed to build our
subjects. Whether he’s in fashion or out of fashion has more to do with us,
with where we are in coming to grips with him.”
When Ono started looking into the
mock-theta functions, there were a few hints as to what they might mean. They
seemed to help describe the spread of cancer tumors, and physicists had begun
to find them useful in understanding how black holes unravel space and time and
how string theory knits them together. This was peculiar, since the concept of
string theory didn’t exist in 1920, when Ramanujan wrote his letter, and black
holes were brand-new objects of speculation among a handful of physicists. But
still—when modern astrophysicists peer inside their black-hole models, they
find they are looking at mock-theta functions.
Despite a few research applications,
the mathematical understanding of mock-theta functions was in a bizarre state.
Dozens of papers had been written on them, but no one could explain in the most
basic sense what a mock-theta functionwas.
When Ramanujan died, there were no clues anywhere in the mathematical
literature to explain why he found the mock-theta functions interesting. It’s
probably not going too far to say that, in fact, they weren’t interesting. All they did, Ramanujan
wrote, was imitate a class of functions called the theta functions, which had
been around for a century or so. In that time, the theta functions had been
working perfectly on their own. No one had needed to imitate them. Ramanujan
had produced a solution to a nonexistent problem. Who
cares?would not have been an unreasonable response.
In the summer of 2012, Ono found
that the only way he could understand the mock-theta functions was via Serre,
Deligne, and others’ work on the tau function. This made no sense. It meant
that it was not Ramanujan’s own work on tau that had led Ramanujan from tau to
the mock thetas, but the work of others, of Serre and Deligne, that would not
be carried out until he’d been dead for decades.
Ono had the sensation of Ramanujan
walking in his footsteps, but from the wrong direction in time.
“Whatever Ramanujan was thinking
about between the tau function and the deathbed letter somehow must have been
parallel to what I was doing, without him knowing I was doing it,
ninety years later,” he says.
With Ramanujan looking over his
shoulder like a “chubby guardian angel,” Ono found that, as the numbers being
spit out by the theta functions started to grow at an unimaginable speed,
approaching and then far exceeding the number of atoms in the universe, the
mock-theta functions began to imitate them with eerie precision. In the lower
reaches of the number line, the behavior of the function and its doppelgänger
was unlovely and chaotic. But out here, in the immense realms that had driven
Cantor insane and enraged the European mathematics establishment, their
relationship became clear. You could take the ludicrous, unmanageable output of
a theta function, then subtract the ludicrous, unmanageable output of a
mock-theta function, and the answer was shocking in its simplicity. The answer
was 4.
With pencil and paper and pages of
calculations in front of you, to see these titanic quantities consume each
other so precisely bends the mind.
“It doesn’t take any imagination,”
says Ono, “to recognize that four is a beautiful number.”
As Ramanujan lies dying, racing
toward infinity, a dot of light appears in the great wall. The gleaming
apparatus is about to crash, but the mock theta function does its crazy trick,
and the infinite dissolves, just a little. A portal the size of an atom
appears. The apparatus threads the hole. And then it keeps going, and going,
and going…”
Maths
and this blog has lot of interesting stuff on Maths
some useful sites for mathematics
.
Also read a very interesting research work 'Alex's Adventures in Numberland' by Alex Bellos
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