August 3, 2015
Mathematics and Hindu Religion lecture series
An author with 60 books, 62 e- books; columnist with 700 article published in various newspaper and magazine, blogger with 500 + blogs.
India Book of Record Holder for writing 106 books on Mathematics.
My books are published by Rupa, Britannica, Pustak Mahal, Scholastic, Prabhat, Ocean books, Rising Publishers, Amar Ujala.
All my books are available on Amazon, Flipkart, Bookadda and google books.
My weekly articles get published on Every Wednesday in Amar Ujala Newspaper. I also write column for Dainik Jagaran, Navbharat Times, Vigyan Pragati, Teacher Plus, Aviskar, Vigyan Aapke liye .
Presently working at DIET Daryaganj as an Assistant Professor.
I am also the Resource person working with CBSE, NCERT, NIOS, VVM, SSUN, AIRMC, VIPNET,IISF, Sarayu Trust, RIT Chennai, etc.
July 24, 2015
Fermat
Pierre
De Fermat
Born: - August
17, 1601 Died: -
January 12, 1665
Fermat was one
of the leading mathematicians of early 17th century. Pierre Fermat's father was
a wealthy leather merchant and second consul of Beaumont- de- Lomagne. Although
there is little evidence concerning his school education it must have been at
the local Franciscan monastery. He was an amateur mathematician.
He attended the University of Toulouse
before moving to Bordeaux in the second half of the 1620s. In Bordeaux he began
his first serious mathematical researches and in 1629 he gave a copy of his
restoration of Apollonius Plane loci to one of the
mathematicians there. From Bordeaux Fermat went to Orléans where he studied law
at the University and spent his working life as a magistrate in
the small provincial town of Castres.
After he moved to Toulouse , he gained a new
mathematical friend Carcavi. In 1636 Carcavi went to Paris as royal librarian and made contact with Mersenne and his group. Mersenne's interest was aroused by Carcavi's descriptions of Fermat's
discoveries on falling bodies, and he wrote to Fermat. Fermat replied on 26
April 1636 and, in addition to telling Mersenne about errors which he believed
that Galileo had made in his description of
free fall. Fermat had little interest in physical applications of mathematics.
Even with his results on free fall he was much more interested in proving
geometrical theorems than in their relation to the real world. Fermat sent a
letter to Mersenne containing two problems on maxima which Fermat asked Mersenne to pass on to the Paris
mathematicians and this was to be the typical style of Fermat's letters, he
would challenge others to find results which he had already obtained. Roberval and Mersenne found that Fermat's problems
in this first, and subsequent, letters were extremely difficult and usually not
soluble using current techniques. They asked him to divulge his methods and
Fermat sent Method for
determining Maxima and Minima and Tangents to Curved Lines.
His reputation as one of the leading
mathematicians in the world came quickly but attempts to get his work published
failed mainly because Fermat never really wanted to put his work into a
polished form. However some of his methods were published in Cursus mathematicus a work by Herigone. With
Pascal, Fermat stands as one of the founder of mathematical theory of
probability. Pierre de Fermat independently founded the new branch of
mathematics called Analytical Geometry. This work led to violent controversies
over question of priority with Rene Descartes. Fermat's
pioneering work in analytic geometry was circulated in manuscript form in 1636,
predating the publication of Descartes' famous La géométrie. This manuscript
was published posthumously in 1679 in "Varia opera mathematica", as Ad Locos Planos et Solidos Isagoge,
("Introduction to Plane and Solid Loci").[
In Methodus ad disquirendam maximam et
minima and in De tangentibus linearum curvarum,
Fermat developed a method for determining maxima, minima, and tangents to
various curves that was equivalent to differentiation.
He is probably best known for
his work on number theory. He also left one of the famous unsolved problems in
maths called- Fermat’s last theorem. This theorem
states that xn + yn = zn
has no non-zero integer solutions for x, y and z when n > 2. in the margin of Bachet's
translation of Diophantus's Arithmetica
, Fermat worte “ I have discovered a truly remarkable proof which this margin
is too small to contain.” These marginal notes only became known after
Fermat's son Samuel published an edition of Bachet's translation of Diophantus's Arithmetica with his father's notes in 1670. Unsuccessful attempts to prove the theorem over a 300 year period led to
the discovery of commutative ring theory and a wealth of other mathematical
discoveries. The truth of
Fermat's assertion was proved in June 1993 by the British mathematician Andrew Wiles.
The second stamp was released after it was proved by Andrew Wiles.
In 1656 Fermat had started a
correspondence with Huygens. This grew out of Huygens interest in probability and
the correspondence was soon manipulated by Fermat onto topics of number theory.
This topic did not interest Huygens but Fermat tried hard and in New Account of Discoveries in the Science of Numbers sent to Huygens via Carcavi in 1659, he revealed more of
his methods than he had done to others. Fermat described his method of infinite
descent and gave an example on how it could be used to prove that every prime of the form 4k + 1 could be written as the sum of two squares.
He died at Castres, Tarn on January 12, 1655. The oldest and most
prestigious high school in Toulouse is named after him:
Send your comments to
rkthakur1974@gmail.com
An author with 60 books, 62 e- books; columnist with 700 article published in various newspaper and magazine, blogger with 500 + blogs.
India Book of Record Holder for writing 106 books on Mathematics.
My books are published by Rupa, Britannica, Pustak Mahal, Scholastic, Prabhat, Ocean books, Rising Publishers, Amar Ujala.
All my books are available on Amazon, Flipkart, Bookadda and google books.
My weekly articles get published on Every Wednesday in Amar Ujala Newspaper. I also write column for Dainik Jagaran, Navbharat Times, Vigyan Pragati, Teacher Plus, Aviskar, Vigyan Aapke liye .
Presently working at DIET Daryaganj as an Assistant Professor.
I am also the Resource person working with CBSE, NCERT, NIOS, VVM, SSUN, AIRMC, VIPNET,IISF, Sarayu Trust, RIT Chennai, etc.
Galois
Evariste Galois
Evariste
Galois was a French mathematician who made significant contribution to the
theory of functions, the theory of equations and number theory. His work became
the basis for Galois Theory and Group theory. He was the first to use the word
Group as a technical term in mathematics to represent a group of
permutations.
The mathematical genius Evariste Galois was
born on 25th October 1811 in France. He was the second son of
Nicholas Gabriel Galois and Adelaide Marie. His father was the director of a
boarding school and later mayor of Bourg – la- Reive. Evariste had a happy
childhood. He seemed to have a phenomenal memory. While still in teens, he was able to determine
a necessary and sufficient condition for a polynomial to be solvable by
radicals, thereby solving a long standing problem. Up to the age of 12, he was
educated by his mother who instilled in him knowledge of classics and a
sceptical attitude towards religion.
He
entered the College Royale de Louis le Grand in Paris in 1823 where his
precocious mathematical genius first emerged. He studied the Legendre’s text book
Elements de Geometrie and Lagrange’s
work on Differential Equation and Analytic Function. By the age of 16, he had
published many papers. He appeared for the examination of Ecole Polytechnic but
failed. The next year Galois had the good fortune to be studying maths under a
distinguished teacher Louis Paul Emile Richard who recognised his exceptional
gifts. Richard was enthusiastic about Galois mathematical work and this led to
his publication of first paper on continued fraction. It was at around the same time that he began making
fundamental discoveries in the theory of polynomial
equations. He submitted two papers on this topic to
the Academy of Sciences but the Academy refused to accept them for publication.
On 28 July 1829 Galois's father committed
suicide after a bitter political dispute with the village priest. A
couple of days later, Galois made his second and last attempt to enter the
Polytechnique, and failed yet again. Having
been denied admission to the Polytechnique, Galois took the Baccalaureate
examinations in order to enter the École
Normale. He passed, receiving his degree on 29 December 1829. His examiner in
mathematics reported, "This pupil is sometimes obscure in expressing his
ideas, but he is intelligent and shows a remarkable spirit of research."
Galois lived during a time of political turmoil
in France. Charles X had succeeded
Louis XVIII in 1824, but in 1827 his party suffered a major electoral setback
and by 1830 the opposition liberal party became the majority and Louis Philippe
became the king. His anti monarchist view led him to imprisonment leading his
expulsion from the college.
Galois quit school immediately and joined the
staunchly Republican artillery unit of the National Guard. He divided his time
between his mathematical work and his political affiliations. Due to
controversy surrounding the unit, soon after Galois became a member, on 31
December 1830, the artillery of the National Guard was disbanded out of fear
that they might destabilize the government. At around the same time, nineteen
officers of Galois' former unit were arrested and charged with conspiracy to
overthrow the government. They were later acquitted of all charges.
During his imprisonment, he continued developing
his mathematical ideas. On the advice of Simon Poisson he submitted his work on
the Theory of Equation for publication but Poisson decleared his work
incomprehensible. Galois reacted violently to the rejection and decided to
abandon publishing his papers through the Academy and instead publish them
privately through his friend Auguste Chevalier.
During 1830s Galois published several papers in Bulletin des Sciences mathematiques of Baren
de Ferussace, a review that normally only published work by established
scientist. These articles contained most of his work on Theory of Equation which is now known as Galois Theory.
In 1831, he was arrested again for keeping
illegal weapon. He was released on April 29, 1832. On May 25 he wrote to
Chevalier, a close friend of his from the Ecole Normale, expressing his
complete disenchantment with life, and hinting that a broken love affair was
the reason. The woman in question was Stephanie Dumotel, the daughter of the
resident physician at the hostel where Galois stayed during last months of his
life.
On May 30, Galois fought a duel with pistol and
was shot in the abdomen. Galois remained unattended for hours until a passerby
took him to the hospital. He refused the services of a priest and died of
peritonitis the following day, at the age of 20. A night before the duel was
scheduled he recorded his mathematical idea in a later to his former school
master Auguste Chevalier where he had out lined his work on elliptical
integrals and permutations of groups.
Hermann Weyl, a mathematician, said
of this testament, "This letter, if
judged by the novelty and profundity of ideas it contains, is perhaps the most
substantial piece of writing in the whole literature of mankind."
Galois' mathematical contributions were published
in full in 1843 when Liouville reviewed his
manuscript and declared it sound. It was finally published in the October–November
1846 issue of the Journal de
Mathématiques Pures et Appliquées. The
most famous contribution of this manuscript was a novel proof that there is no quintic formula, that is, that fifth and higher degree equations are not
solvable by radicals.
Unsurprisingly, Galois' collected works amount
to only some 60 pages, but within them are many important ideas that have had
far-reaching consequences for nearly all branches of mathematics. His work has been compared to that of Niels
Abel, another mathematician who died at a very young
age, and much of their work had significant overlap.
Send your comments to
rkthakur1974@gmail.com
An author with 60 books, 62 e- books; columnist with 700 article published in various newspaper and magazine, blogger with 500 + blogs.
India Book of Record Holder for writing 106 books on Mathematics.
My books are published by Rupa, Britannica, Pustak Mahal, Scholastic, Prabhat, Ocean books, Rising Publishers, Amar Ujala.
All my books are available on Amazon, Flipkart, Bookadda and google books.
My weekly articles get published on Every Wednesday in Amar Ujala Newspaper. I also write column for Dainik Jagaran, Navbharat Times, Vigyan Pragati, Teacher Plus, Aviskar, Vigyan Aapke liye .
Presently working at DIET Daryaganj as an Assistant Professor.
I am also the Resource person working with CBSE, NCERT, NIOS, VVM, SSUN, AIRMC, VIPNET,IISF, Sarayu Trust, RIT Chennai, etc.
Euclid
Euclid
Euclid of
Alexandria is better known as the Father of Geometry. He is one of the most
prominent mathematicians of antiquity best known for his treatise The Elements.
Little is known about Euclid's life, as there
are only a handful of references to him. The date and place of Euclid's birth
and the date and circumstances of his death are unknown. But whatever
speculated data are available shows that he was born about 325 BC in
Alexandria and died about 265 BC.
An
Arabian author, al-Qifti (d. 1248), recorded that Euclid's father was Naucrates
and his grandfather was Zenarchus, that he was a Greek, born in Tyre and lived
in Damascus. But there is no real proof that this is the same Euclid. During
the reign of Ptolemy I he taught at Alexandria. Ptolemy had created the great library at Alexandria,
which was known as the Museum, because it was considered a house of the muses
for the arts and sciences. Many scholars worked and taught there, and that is
where Euclid wrote The
Elements.
There is a very
interesting story about Euclid---
A student who
had begun to learn geometry with Euclid asked him “What shall I get by learning
all these things in Geometry?” Euclid called his slave and told him to give
three pence since he must make gain out of what he learns.
Euclid said to his students that --- There is no royal road
to Geometry.
A page from Euclid’s Elements
Euclid's most famous for his treatise on mathematics called The Elements. The book is a compilation of
knowledge that became the centre of mathematical teaching for 2000 years.
Probably no results in The Elements were first proved by Euclid. The Elements is divided into
thirteen books which cover plane geometry, arithmetic and number theory,
irrational numbers, and solid geometry.
Euclid
organized the known geometrical ideas, starting with simple definitions, axioms;
formed statements called theorems, and set forth methods for logical proofs. He
began with accepted mathematical truths, axioms and postulates, and
demonstrated logically 467 propositions in plane and solid geometry. One of the
proofs was for the theorem of Pythagoras, proving that the equation is always
true for every right triangle. The Elements was the most widely used textbook
of all time. It has appeared in more than 1,000 editions since it was first
printed in 1482 and is thought to have sold more copies than any book other
than the Bible. Original of Euclid's Elements have not been
preserved, but Arabic
mathematicians obviously had a
full copy as an Arabic version of The Elements appeared at the end of the 8th
century BC.
Euclid
believed that we can't be sure of any axioms without proof, so he devised
logical steps to prove them. There are 5 axioms and 5 postulates found in the
book of Euclid. He called Axiom "Common Notions," because they were
common to all sciences.
Axioms
1. Things which are equal to the same
thing are also equal to one another.
2. If equals are added to equals, the
sums are equal.
3. If equals are subtracted from
equals, the remainders are equal.
4. Things which coincide with one
another are equal to one another.
5. The whole is greater than the part.
Postulates
1. You can draw a straight line between
any two points.
2. You can extend the line
indefinitely.
3. You can draw a circle using any line
segment as the radius and one end point as the center.
4. All right angles are equal.
5. Given a line and a point, you can
draw only one line through the point that is parallel to the first line.
The fifth postulates later became the cause of discovery of
new branch of geometry known as Non Euclidian Geometry.
Euclid's Elements is remarkable for the clarity with which the theorems are
stated and proved. This
wonderful book, with all its imperfections, which are indeed slight enough when
account is taken of the date it appeared, is and will doubtless remain the
greatest mathematical textbook of all time.
Euclid
proved that it is impossible to find the "largest prime number,"
because if you take the largest known prime number, add 1 to the product of all
the primes up to and including it; you will get another prime number. Euclid's
proof for this theorem is generally accepted as one of the "classic" proofs
because of its conciseness and clarity. Millions of prime numbers are known to
exist, and more are being added by mathematicians and computer scientists.
Mathematicians since Euclid have attempted without success to find a pattern to
the sequence of prime numbers.
Although
best known for its geometric results, the Elements also include number theory. It considers the
connection between perfect
numbers and Mersenne primes, the infinitude of prime numbers, Euclid's lemma on factorization (which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations), and the Euclidean algorithm for finding the greatest common divisor of two numbers.
Euclid
also wrote Data, which
contains 94 propositions, Phaenomena,
concerning spherical astronomy, Caloptrics, about mirrors, Optics, the theory of
perspective, and a work of music theory. In his works about optics, Euclid made
light rays part of geometry, working with them as if they were straight lines.
Many of the works ascribed to Euclid are no longer in existence or are
incomplete.
Send your comments to
rkthakur1974@gmail.com
An author with 60 books, 62 e- books; columnist with 700 article published in various newspaper and magazine, blogger with 500 + blogs.
India Book of Record Holder for writing 106 books on Mathematics.
My books are published by Rupa, Britannica, Pustak Mahal, Scholastic, Prabhat, Ocean books, Rising Publishers, Amar Ujala.
All my books are available on Amazon, Flipkart, Bookadda and google books.
My weekly articles get published on Every Wednesday in Amar Ujala Newspaper. I also write column for Dainik Jagaran, Navbharat Times, Vigyan Pragati, Teacher Plus, Aviskar, Vigyan Aapke liye .
Presently working at DIET Daryaganj as an Assistant Professor.
I am also the Resource person working with CBSE, NCERT, NIOS, VVM, SSUN, AIRMC, VIPNET,IISF, Sarayu Trust, RIT Chennai, etc.
Emmy Noether
Emmy Noether
Emmy Noether was an
influential German mathematician known for her groundbreaking contributions to
abstract algebra and theoretical physics. Albert Einstein described her as the
most important woman in mathematical history. She lived and worked in Germany
during the same period that Ramanujan worked in England and India, but whereas
Ramanujan’s contributions were in analysis and number theory, hers were in
abstract algebra and in the application of algebra to theoretical physics. She
revolutionized the theories of rings, fields and algebras. In physics Noether’s
theorem explains the fundamental connection between symmetry and conservation
laws. The innovative approach to modern abstract algebra of Emmy Noether not
only produced major new results, but also inspired highly productive work by
students and colleagues who emulated her techniques.
Amalie Emmy Noether to
give her the full name was born on March 23, 1882 in Erlangen in Germany. Her
father Max Noether was professor of mathematics at the University of Erlangen
while her mother Ida Amalia was the daughter of a wealthy Jewish family of
Cologne. She was better known as Emmy. She loved to dance and enjoyed music. She attended the Municipal school for the
Higher Education of Daughters until she was 18. Emmy Noether showed early
proficiency in French and English. In the spring of 1900 she took the
examination for teachers of these languages and received an overall score of very good. Her performance qualified
her to teach languages at schools reserved for girls, but she chose instead to
continue her studies at the University
of Erlangen. This was an unconventional decision because as late as 1900, women
were not allowed to enroll in Universities in Germany. Professors frequently
refused permission for women event to attend their lectures, and only very
rarely women allowed taking university examination. The obstacle in a way could
not deter her to enroll in a university, she had
to get permission of the professors to take an entrance exam -- she did and she
passed, after sitting in on mathematics lectures at the University of Erlangen.
She was then allowed to audit courses -- first at the University of Erlangen
and then the University of Göttingen, neither of which would permit a woman to
attend classes for credit. Finally, in 1904, the University of Erlangen decided
to permit women to enroll as regular students, and Emmy Noether returned there. She began to focus solely on mathematics. Under
the supervision of Paul Gordan she wrote her dissertation, Über die Bildung des Formensystems
der ternären biquadratischen Form (On
Complete Systems of Invariants for Ternary Biquadratic Forms). Her dissertation in algebraic math earned her a
doctorate summa cum laude in 1908.
Having
obtained her doctorate, Noether was well qualified for a position at a
university, but the persistent sexiest atmosphere in Germany prevented the
brilliant young woman from being able to even apply for a job. This depressed
her greatly, but she began helping her father in his research. She also started
publishing papers on her own, which were so well received that she was invited
to join a number of European Mathematical Societies, including the German
Mathematical Association, which had been founded by George Cantor, but still
she could not obtain a paying position at a University in Germany. By 1915
Noether was a famous mathematician in her own right and her papers were read
with interest throughout the world. They dealt primarily with algebra.
In
1915, Emmy Noether's mentors, Felix Klein and David Hilbert, invited her to
join them at the Mathematical Institute in Gottingen, again without
compensation. There, she pursued important mathematical work that confirmed key
parts of the general theory of relativity. Noether arrived at Gottingen and
began her work on invariance in mathematical physics. Meanwhile, Klein rallied
to get her appointed a professor at Gottingen, but he had to struggle with the
administration until 1919, when his request was finally granted. She became a privatdozent allowing her to teach
students and students would pay her directly. In 1922, the University gave her
a position as an adjunct professor with a small salary and no tenure or
benefits. Soon after arriving at Göttingen, however, she
demonstrated her capabilities by proving the theorem now known as Noether's theorem, which shows
that a conservation law is associated with any differentiable symmetry of a physical system.
American physicists Leon M. Lederman and Christopher
T. Hill argue in their book Symmetry and the Beautiful Universe that Noether's theorem is
"certainly one of the most important mathematical theorems ever proved in
guiding the development of modern physics, possibly on a par with the Pythagorean Theorem". Her work went far beyond mathematical physics. She
made important contributions to Galois Theory, to many other areas of abstract
algebra and to topology. Noether was, in fact the greatest algebraist of her
time.
Noether’s
groundbreaking work in algebra began in 1920 with a paper on non commutative
fields. Her work earned her enough recognition that she was invited as a
visiting professor in 1928-1929 at the University of Moscow and in 1930 at the
University of Frankfurt. In 1932 Emmy Noether and Emil Artin received the
Ackermann Teubner Memorial Award for their contributions to mathematics. Though she was never able to gain a regular
faculty position at Göttingen, she was one of many Jewish faculty members who
were purged by the Nazis in 1933.
In America,
the Emergency Committee to Aid Displaced German Scholars obtained for Emmy
Noether an offer of a professorship at Bryn Mawr College in America, and they
paid, with the Rockefeller Foundation, her first year's salary. The grant was
renewed for two more years in 1934. This was the first time that Emmy Noether
was paid a full professor's salary and accepted as a full faculty member. In
1934, Noether began lecturing at the Institute for Advanced Study in Princeton
upon the invitation of Abraham
Flexner and Oswald Veblen. She also worked with
and supervised Abraham Albert and Harry Vandiver. Her
time in the United States was pleasant, surrounded as she was by supportive
colleagues and absorbed in her favorite subjects.
In April 1935 doctors
discovered a tumor in Noether's pelvis. She was admitted to hospital for surgery to
remove a uterine tumor. Although the operation was successful but on April 14,
she fell unconscious and she developed a high fever resulting to her death.
Weyl said to her
funeral – The memory of her work in science and of her personality among her
fellows will not soon pass away. She was a great mathematician, the greatest, I
firmly believe, her sex has ever produced, and a great woman.
An author with 60 books, 62 e- books; columnist with 700 article published in various newspaper and magazine, blogger with 500 + blogs.
India Book of Record Holder for writing 106 books on Mathematics.
My books are published by Rupa, Britannica, Pustak Mahal, Scholastic, Prabhat, Ocean books, Rising Publishers, Amar Ujala.
All my books are available on Amazon, Flipkart, Bookadda and google books.
My weekly articles get published on Every Wednesday in Amar Ujala Newspaper. I also write column for Dainik Jagaran, Navbharat Times, Vigyan Pragati, Teacher Plus, Aviskar, Vigyan Aapke liye .
Presently working at DIET Daryaganj as an Assistant Professor.
I am also the Resource person working with CBSE, NCERT, NIOS, VVM, SSUN, AIRMC, VIPNET,IISF, Sarayu Trust, RIT Chennai, etc.
Diophantus
Diophantus
Diophantus of
Alexandria was a Greek mathematician. He was the
first Greek mathematician who recognized fractions as numbers. He
allowed positive rational
numbers for the coefficients and solutions. He played a major role in the development of
algebra and was a considerable influence on later number theorists. Diophantine
analysis, which is closely related to algebraic geometry, has experienced a
resurgence of interest in the past half century is due to him.
Little is known
about Diophantus except the fact that he was born around 200 AD in Alexandria
and died at the age of 84 in 284 AD. The claims that Diophantus lived from
about 200 to 284 and spent time at Alexandria are based on detective work in
finding clues to the times he flourished in his and others’ writing. Theon of
Alexandra quoted Diophantus in 365 and his work was the subject of a commentary
written by Theon’s daughter Hypatia at the beginning of 5th century,
which unfortunately is lost. Michael Psellus, an 11th century
Byzantine scholar mentioned that Diophantus dealt with Egyptian arithmetic
which later led to other people working and developing the same branch of
mathematics. The most details we have of Diophantus’s life come from the Greek
Anthology which estimates his life span to be around 84 years. This anthology
says----
His boyhood lasted 1/6th of
his life; he married after 1/7th
more; his beard grew after 1/12th
more, and his son was born 5 years later; the son
lived to half his father's age, and the father died 4 years after the son.
Diophantus is best
known for his famous work Arithmetica, written on the solution
of algebraic equations and on the theory of numbers. The Arithmetica is a
collection of 130 problems giving numerical solutions of determinate equations
(those with a unique solution), and indeterminate equations. The
method of solving indeterminate equation is known as Diophantine analysis. An
equation is indeterminate if it has more than one variable. The Pythagorean
Triplets containing the integers x, y and z satisfying the equation x2
+ y2 = z2 is the example of Diophantine analysis.
Historically, the problem of solving Diophantine
equation has been to find expressions that show the relationship of the
integral values of x and y that satisfy an indeterminate equation ax + by = c, where the coefficients a, b
and c are integers. Such equations have infinite number of solutions.
The
Arithmetica had 13 volumes but only six of the original 13 books were thought to
have survived. However, an Arabic manuscript in the library Astan-i-Quds (The
Holy Shrine Library) in Meshed Iran has a title claiming it is a translation of
Arithmetica and was discovered in 1968. In Arithmetica he considered the
solutions of linear and quadratic equation. Diophantus looked at three types of
quadratic equations ax2 + bx = c, ax2 = bx + c and ax2 + c = bx. Since there was no
symbol of zero when the book was written so Diophantus took all coefficients to
be positive. Diophantus considered negative or
irrational square root solutions "useless", "meaningless",
and even "absurd". There
is no evidence that suggests Diophantus even realized that there could be two
solutions to a quadratic equation. Besides that, he also solved problem from
Simultaneous equations.
Looking at the work of Diophantus it was quite
clear that he was fully aware of the fact that every number can be written as
the sum of four squares which was later proved by Lagrange. Here are two
examples of the problems solved in Arithmetica.
1.
Find four numbers, the sum of every arrangement
three at a time being given: say 22, 24 , 27 and 20. (Answer:- 9, 7 , 4 and 11)
2.
Divide a number, such as 13 which is the sum of
two squares 4 and 9, into two other squares. (Answer:- 324/25 and 1/25)
In the Arithmetica, Diophantus stated
certain theorem without proofs. It is believed that his proofs were found in
his lost book The Porism, a
collection of lemmas. One lemma states
that the difference of the cubes of two rational numbers is equal to the sum of
the cubes of two other rational numbers. For any
given numbers a, b there
exist numbers c, d such that a3 - b3= c3 + d3.
Diophantus'
work has had a large influence in history. Editions of Arithmetica exerted a
profound influence on the development of algebra in Europe in the late
sixteenth and through the seventeenth and eighteenth centuries. Diophantus and
his works have also influenced Arab
mathematics and were of great fame among Arab mathematicians.
Diophantus' work created a foundation for work on algebra and in fact much of
advanced mathematics is based on algebra. European mathematicians did not learn
of the gems in Diophantus's Arithmetica until Regiomontanus wrote in 1463:-No one has yet
translated from the Greek into Latin the thirteen Books of Diophantus, in which
the very flower of the whole of arithmetic lies hid...
Bombelli translated much of the work in 1570
but it was never published. Bombelli did borrow many of Diophantus's
problems for his own Algebra. The most famous Latin translation of
the Diophantus's Arithmetica is
due to Bachet in 1621. It is that edition which Fermat studied. Certainly Fermat was inspired by this work which has
become famous in recent years due to its connection with Fermat's Last Theorem. Diophantus earliest
work was probably his short essay on polygonal numbers, containing ten
propositions in which he employed the classical method in which numbers are
represented by line segments. The topic
on polygonal numbers was of great interest to Pythagoreans.
Diophantus worked before the
introduction of modern algebraic notation, but he moved from rhetorical algebra
to syncopated algebra, where abbreviations are used. Diophantus made important advancement in mathematical notation.
He
was the first person to use algebraic notation and symbolism. Before him
everyone wrote out equations completely. Diophantus introduced an algebraic
symbolism that used an abridged notation for frequently occurring operations,
and an abbreviation for the unknown and for the powers of the unknown.
Mathematical historian Kurt Vogel states: “The
symbolism that Diophantus introduced for the first time, and undoubtedly
devised himself, provided a short and readily comprehensible means of
expressing an equation... Since an abbreviation is also employed for the word
‘equals’, Diophantus took a fundamental step from verbal algebra towards
symbolic algebra.”
Send your comments at
rkthakur1974@gmail.com
Rajesh Thakur
An author with 60 books, 62 e- books; columnist with 700 article published in various newspaper and magazine, blogger with 500 + blogs.
India Book of Record Holder for writing 106 books on Mathematics.
My books are published by Rupa, Britannica, Pustak Mahal, Scholastic, Prabhat, Ocean books, Rising Publishers, Amar Ujala.
All my books are available on Amazon, Flipkart, Bookadda and google books.
My weekly articles get published on Every Wednesday in Amar Ujala Newspaper. I also write column for Dainik Jagaran, Navbharat Times, Vigyan Pragati, Teacher Plus, Aviskar, Vigyan Aapke liye .
Presently working at DIET Daryaganj as an Assistant Professor.
I am also the Resource person working with CBSE, NCERT, NIOS, VVM, SSUN, AIRMC, VIPNET,IISF, Sarayu Trust, RIT Chennai, etc.
July 9, 2015
Augustin Louis Cauchy
Augustin
Louis Cauchy
The
greatest French Mathematician who has been described as the first revolution of
rigour in mathematics is undoubtedly Augustin Louis Cauchy. He contributed in a
major way to almost every branch of mathematics. It is humorously said—Cauchy
did not so much master mathematics, mathematics mastered Cauchy.
Augustin
Louis Cauchy was born in Paris on August 21, 1789. He was christened Augustin
because he was born in August and Louis after his father’s name Louis Francois.
His father Louis Francois was a parliamentary lawyer and a lieutenant of police
in Paris. His mother Marie- Madeleine came from a very reputed Parisian family.
Cauchy’s childhood fell in the bloodiest period of the revolution. In 1794, the
Cauchy family fled to Arcueil to escape the terror. Louis Francois could hardly
manage to feed his family and as a consequence Cauchy grew up delicate and
underdeveloped physically.
Two
years later the Cauchy family returned to Paris where Louis Francois began to
rebuild his carrier under the new regime and was appointed as secretary general
to the newly constituted Senate of which Laplace was chancellor. Louis had a
great friendship with Lagrange’s and Laplace and they used to visit Cauchy’s
house every now and then. Lagrange was impressed by the ability of Augustin and
on his advice he was enrolled at the Ecole Centrale de Pantheon to study
humanities. Pointing to Louis, Lagrange once said--- he will supplement all of us so far as we are mathematicians. Don’t
let him touch a mathematical book till he is 17.
At
the age of 13, Augustin Cauchy entered the Central school of Pantheon. He won
the grand prize for the best student instituted by Napoleon. On leaving the
school in 1804, Cauchy won the sweepstakes and a special prize in mathematics.
For the next ten months he studied mathematics intensively with a good tutor
and at the age of 16, he entered the Ecole Polytechnic. By 1810, he was a
qualified Junior Engineer.
In
March 1810, he left Paris to work on the construction of a naval base at Cherbourg,
the Port Napoleon. Cauchy took 4 books
with him to Cherbourg-- The Mecanique
celeste of Laplace, The Traite des fonctions analytiques of Lagrange, The
Imitation of Christ by Thomas a Kempls and the work of Virgil. He remained
there for 3 years where he used to do some mathematical research in part time.
In
December 1810, he had begun to go over again all branches of mathematics,
beginning with Arithmetic and finishing with Astronomy. Some of his discoveries
were sufficiently significant to attract the attention of learned society in
Paris particularly the memoir on polyhedral and that on symmetric functions,
which included the germs of the fundamental ideas that eventually blossomed
into Group Theory. He also extended the formula of Euler connecting the
numbers(E), faces(F) and vertices(V) of a polyhedron given as E+2 =F+V
He
also developed the theory of determinants but he got his due recognition as a
brilliant mathematician when he submitted a work on Calculation of Definite
Integrals to the Academy in 1814. In 1815, Cauchy created a sensation by
proving one of the great theorems which Fermat has bequeathed to a baffled
posteiry.
Every
positive integers is a sum of three triangles, four squares, five pentagons,
six hexagons and so on---
In
1816, he received the Grand Prize offered by the Academy for a theory of the propagation of waves on the
surface of a heavy fluid of indefinite depth. In 1816 itself he was elected
a member of the Paris Academy. This success made him the enemies of those great
mathematician likes Monge and Carnot who were displaced from Academy. In the
same year, he was appointed Assosiate Professor of Analysis at the Ecole
Polytechnic.
The discovery with
which Cauchy’s name is most firmly associated is his fundamental theorems in
Complex Analysis. The first pivotal theorem proved by
Cauchy, now known as Cauchy's integral theorem, was the following:
Where f(z) is a
complex-valued function holomorphic on and within the non-self-intersecting closed curve C (contour)
lying in the complex plane. The contour
integral is taken along the contour C.
He is therefore known as the father of Complex Analysis.
In
1818, Cauchy married with Aloise de Bure. She was a close relative of the publisher who published most of Cauchy's
works. They had two daughters Marie Francoise Alicia and Marie Mathilde
born in 1819 and 1823 respectively. Encouraged by Laplace, Cauchy in 1821 wrote
up for publication of his lecture on Analysis given in Ecole Polytechnic. This
book contains the definition of limits and continuity and convergence of
infinite series which we still read in the book of Calculus.
In
1825, Cauchy started his own monthly journal Exercises de mathemaliques. Its pages were exclusively filled by
Cauchy himself. Cauchy's writings
covered notable topics including: the theory of series, where he developed the
notion of convergence and
discovered many of the basic formulas for q-series. The theory of numbers and complex quantities; he was the
first to define complex numbers as pairs of real numbers.
The
turmoil in 1830 in Paris had a great impact on the mind of Cauchy. Cauchy lost most of his positions at the
institutes because he refused to take the oath of allegiance to the new king,
Louis-Philippe. He left for
Switzerland and remain there in self imposed exile for eight years where he was
offered the Post of Professor in Mahematical Physics by Charles Albert, the King
of Sardinia. He taught in Turin during 1832-1833. In 1831, he had been
elected a foreign member of the Royal Swedish Academy of Sciences. In 1833, Cauchy was appointed tutor of
Charles’ grandson.
Cauchy went back to Paris in 1838 when he
finished his work with Charles X in Prague, and resumed his involvement with
the Academy. At the time, because Cauchy was a mathematician, he was exempted
from the oath of allegiance. After the establishment of the Second Republique
in 1848, Cauchy resumed his position at the Sorbonne. Cauchy continued with his
writings and publications through the remainder of his life. The
long hour of work made him ill. On medical advice he left for Paris on May 12,
1857 but his health suddenly detorieted and he died on 23 rd May 1857 at the
age of 67. His name is one of the 72
names inscribed on the Eiffel Tower.
Read 51 Greatest Mathematician of World by Rajesh Kumar Thakur
Rajesh Thakur
rkthakur1974@gmail.com
An author with 60 books, 62 e- books; columnist with 700 article published in various newspaper and magazine, blogger with 500 + blogs.
India Book of Record Holder for writing 106 books on Mathematics.
My books are published by Rupa, Britannica, Pustak Mahal, Scholastic, Prabhat, Ocean books, Rising Publishers, Amar Ujala.
All my books are available on Amazon, Flipkart, Bookadda and google books.
My weekly articles get published on Every Wednesday in Amar Ujala Newspaper. I also write column for Dainik Jagaran, Navbharat Times, Vigyan Pragati, Teacher Plus, Aviskar, Vigyan Aapke liye .
Presently working at DIET Daryaganj as an Assistant Professor.
I am also the Resource person working with CBSE, NCERT, NIOS, VVM, SSUN, AIRMC, VIPNET,IISF, Sarayu Trust, RIT Chennai, etc.
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