Showing posts with label Year Of Mathematcs. Show all posts
Showing posts with label Year Of Mathematcs. Show all posts

Tuesday, 24 July 2012

IMPORTANCE OF MATHEMATICS IN EVERYDAY LIFE

Sumisha k k        
X.B        

                                    “Education should be started with mathematics. For it forms well designed brains that are able to reason right. It is even admitted that those who have studied mathematics during their childhood should be trusted, for they have acquired solid bases for arguing which become to them a sort of second nature”.
             Mathematics is around us. It is present in different forms whenever we pick up the phone, manage the money, travel to some place, play soccer, meet new friends; unintentionally in all these things mathematics is involved.

For Example

Cooking: the idea of proportion  Percentage
Medicine/Pharmacy
Bank: savings and credit
Chance to win in lottery: Probability
Area
Geometry in clothing
Geometry in house decoration
Geometry in art
Geometry in architecture
Symmetry in the nature

  • The physical sciences (chemistry, physics, engineering)
  • The life and heath sciences (biology,psychology,pharmacy,nursing,optometry)
  • The social science, including Anthropology, communications, Eco-comics, Linguists, Education, geography
  • The tech Sciences, like computer science, networking, software development
  • Business and commerce
  • Medicine
  • Actuarial science, used by insurance companies

Mathematics and applications

  • Every area of mathematics has its own unique applications to the different career options. For example
  • Algebra: computer sciences, cryptology, networking, study of symmetry in chemistry and physics
  • Calculus (differential equations): Chemistry, biology, physics engineering, the motion of water, rocket science, molecular structure, option
    price modeling in business and economics models, etc...
  • Students are encouraged to give serious attention to their future. The career world is competitive!
                      
               Most of university degree require mathematics. Students who choose not to take mathematics seriously or to ignore it in high school forfeit many future career opportunities that they could have. They essentially turn their backs on more than half the job market. The importance of mathematics for potential careers cannot be over emphasized. To get degrees in the following areas one need to have good knowledge of mathematics and statistics
                         It’s almost impossible to get through a day without using maths in some way, because our world is full of numbers to handle and problems to solve. Studying maths provides you with the tools to make sense of it all, making life that little bit easier.          

SO ,WE CONCLUDE THAT MATHEMATICS PLAY AN IMPORTANT ROLE IN OUR DAILY LIFE.

Sunday, 22 July 2012

Prizes in mathematics

There is no Nobel Prize in mathematics, though sometimes mathematicians have won the Nobel Prize in a different field, such as economics. Prominent prizes in mathematics are 


 Abel prize
Chern prize 
Field’s medal 
Gauss Prize 
Balzan prize 
Crafoord prize 
 Shaw prize 
 Wolf prize 
Schock prize 
  Nevanlinna prize.

Saturday, 21 July 2012

Why Year Of Mathematics.....?


Prime Minister Manmohan Singh has declared the year 2012 as the 'national year of mathematics' for the country. This is to pay a tribute and mark the birth anniversary of mathematical genius Srinivasa Ramanujan. 
The great mathematician, Ramanujan , came from an economically weak background in Tamil Nadu and had minimal training in mathematics. Till few years of his initial childhood , he could not even speak, read or write. Yet he overcame all the formidable difficulties and contributed some of the greatest theories of maths.

His birthday, December 22, has been declared as the 'national mathematics day'

Courtsey: Times Of India

Tuesday, 17 July 2012

Ramanujan



SRINIVASA RAMANUJAM
                               (1887-1920)

 Srinivas Ramanujam was a natural genius in mathematics. During a very short life span of 32 years, he did a vast amount of work in mathematics.
    He was born on 22nd December 1887 at Erode in a middle class family. He did his primary schooling at Kumbhakonam. His brilliance in Mathematics was apparent from his school days. In the high school he won many prizes for Mathematics. But he could not get admission in university due to his poor record in all other subjects. So he concentrated only on the study of pure Mathematics.
   In 1909 he got married to Janki Ammal. In 191 his first paper was published in the Indian Mathematical Society’s Journal. This paper was based on the study of Bernoulli’s numbers. He extensively studied G.S.Carr’s Synopsis of Elememtary Results in Pure Mathematics.
   He sent his papers to the Cambridge University in 1913. Srinivas did not have any formal college education. But he studied Mathematics from books which he borrowed from the libraries. In 1914 the University of Madras granted him a scholarship to study at the Trinity College, Cambridge. He produced many papers while studying there.
  Ramanujam was elected the fellow of the Royal Society of London. He was the first Indian to earn this honour. He got a degree of Bachelor of Science by Research.
  Although he was a genius, his outstanding work did not help him financially. But he was an acclaimed scientist world-wide.
  While in England, he fell ill in 1917. He did not recover completely  in spite of the best treatment. So he had to return back to India in 1919. This remarkable mathematician died on 26th April 1920.






SRINIVASA Ramanujan was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series and continued fractions. Ramanujan was said to be a natural genius by the English mathematician G.H. Hardy, in the same league as mathematicians like Euler and Gauss. Born in a poor Brahmin family, Ramanujan's introduction to formal mathematics began at age 10. He demonstrated a natural ability, and was given books on advanced trigonometry written by S. L. Loney that he mastered by the age of 12; he even discovered theorems of his own, and re- discovered Euler's identity independently. He demonstrated unusual mathematical skills at school, winning accolades and awards. By 17, Ramanujan had conducted his own mathematical research on Bernoulli numbers and the Euler–Mascheroni constant.Ramanujan received a scholarship to study at Government College in
Kumbakonam, but lost it when he failed his non-mathematical coursework. He joined another college to pursue independent mathematical research, working as a clerk in the Accountant-General's office at the Madras Port Trust Office to support himself. In 1912– 1913, he sent samples of his theorems to three academics at the University of Cambridge. G. H. Hardy, recognizing the brilliance of his work, invited Ramanujan to visit and work with him at Cambridge. He became a Fellow of the Royal Society and a Fellow of Trinity College, Cambridge. Srinivasa died of illness, malnutrition, and possibly liver infection in 1920 at the age of 32.During his short lifetime, Ramanujan independently compiled
nearly 3900 results (mostly identities and equations). Most of his claims have now been proven correct, although a small number of these results were actually false and some were already known. He stated results that were both original and highly unconventional,
such as the Ramanujan prime and the Ramanujan theta function, and these have inspired a vast amount of further research. However, the mathematical mainstream has been rather slow in absorbing some of his major discoveries. The Ramanujan Journal, an international publication, was launched to publish work in all areas of mathematics influenced by his work. In Dec 2011, in recognition of his contribution to mathematics, the Government of India declared that Ramanujan's birthday (22 December) should be
celebrated every year as National Mathematics Day, and also declared 2012 the National Mathematical Year. Ramanujan has been described as a person with a somewhat shy and quiet disposition, a dignified man with pleasant manners. He lived a rather Spartan life while at Cambridge. Ramanujan's first Indian biographers describe him as rigorously orthodox. Ramanujan credited his acumen to his family Goddess, Namagiri of Namakkal.
He looked to her for inspiration in his work, and claimed to dream of blood drops that symbolized her male consort, Narasimha, after which he would receive visions of scrolls of complex mathematical content unfolding before his eyes. He often said, "An equation
for me has no meaning, unless it represents a thought of God." Hardy cites Ramanujan as remarking that all religions seemed equally true to him. Hardy further argued that Ramanujan's religiousness had been romanticised by Westerners and overstated—in reference to his belief, not practice—by Indian biographers. At the same time, he remarked on Ramanujan's strict observance of vegetarianism. In mathematics, there is a distinction between having an insight and having a proof. Ramanujan's talent
suggested a plethora of formulae that could then be investigated in depth later. It is said that Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets the eye. As a by-product, new directions of research were opened up.
Examples of the most interesting of these formulae include the intriguing infinite series for π. A common anecdote about Ramanujan relates to the number 1729. Hardy arrived
at Ramanujan's residence in a cab numbered 1729. Hardy commented that the number 1729 seemed to be uninteresting. Ramanujan is said to have stated on the spot that it was
actually a very interesting number mathematically, being the smallest natural number representable in two different ways as a sum of two positive cubes SRINIVASA Ramanujan was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to
mathematical analysis, number theory, infinite series and continued fractions. Ramanujan was said to be a natural genius by the English mathematician G.H. Hardy, in the same league as mathematicians like Euler and Gauss. Born in a poor Brahmin family, Ramanujan's introduction to formal mathematics began at age 10. He demonstrated a natural ability, and was given books on advanced trigonometry written by S. L. Loney that he mastered by the age of 12; he even discovered theorems of his own, and re- discovered Euler's identity independently. He demonstrated unusual mathematical skills at school, winning accolades and awards. By 17, Ramanujan had conducted his own mathematical research on Bernoulli numbers and the Euler–Mascheroni constant.Ramanujan received a scholarship to study at Government College in
Kumbakonam, but lost it when he failed his non-mathematical coursework. He joined another college to pursue independent mathematical research, working as a clerk in the
Accountant-General's office at the Madras Port Trust Office to support himself. In 1912– 1913, he sent samples of his theorems to three academics at the University of Cambridge. G. H. Hardy, recognizing the brilliance of his work, invited Ramanujan to visit and work with him at Cambridge. He became a Fellow of the Royal Society and a Fellow of Trinity College, Cambridge. Srinivasa died of illness, malnutrition, and possibly liver infection in 1920 at the age of 32.During his short lifetime, Ramanujan independently compiled nearly 3900 results (mostly identities and equations). Most of his claims have now been proven correct, although a small number of these results were actually false and some were already known. He stated results that were both original and highly unconventional, such as the Ramanujan prime and the Ramanujan theta function, and these have inspired a vast amount of further research. However, the mathematical mainstream has been rather
slow in absorbing some of his major discoveries. The Ramanujan Journal, an international publication, was launched to publish work in all areas of mathematics influenced by his work. 
        In Dec 2011, in recognition of his contribution to mathematics,
the Government of India declared that Ramanujan's birthday (22 December) should be celebrated every year as National Mathematics Day, and also declared 2012 the National
Mathematical Year. Ramanujan has been described as a person with a somewhat shy and quiet disposition, a dignified man with pleasant manners. He lived a rather Spartan lifestyle at Cambridge. Ramanujan's first Indian biographers describe him as rigorouslyvorthodox. Ramanujan credited his acumen to his family Goddess, Namagiri of Namakkal.vHe looked to her for inspiration in his work, and claimed to dream of blood drops thatvsymbolised her male consort, Narasimha, after which he would receive visions of scrollsvof complex mathematical content unfolding before his eyes. He often said, "An equation for me has no meaning, unless it represents a thought of God." Hardy cites Ramanujan as remarking that all religions seemed equally true to him. Hardy further argued that Ramanujan's religiousness had been romanticized by Westerners and overstated—in reference to his belief, not practice—by Indian biographers. At the same time, he remarked on Ramanujan's strict observance of vegetarianism. In mathematics,
there is a distinction between having an insight and having a proof. Ramanujan's talent suggested a plethora of formula that could then be investigated in depth later. It is said that Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets the eye. As a by-product, new directions of research were opened up. Examples of the most interesting of these formula include the intriguing infinite series for π. A common anecdote about Ramanujan relates to the number 1729. Hardy arrived
at Ramanujan's residence in a cab numbered 1729. Hardy commented that the number 1729 seemed to be uninteresting. Ramanujan is said to have stated on the spot that it was actually a very interesting number mathematically, being the smallest natural number re presentable in two different ways as a sum of two positive cubes.

Monday, 16 July 2012



HARDY-RAMANUJAN NUMBER
    1729 is the natural number following 1728 and preceding 1730. 1729 is known as the Hardy–Ramanujan number , named after the British mathematician G.H.Hardy and Srinivasa Ramanujan.
  Now, a little about these great mathematicians:

1) SRINIVASA RAMANUJAN
  Ramanujan was said to be a natural genius by the English mathematician G.H. Hardy, in the same league as mathematicians like Euler and Gauss.During his short lifetime, Ramanujan independently compiled nearly 3900 results. However, the mathematical mainstream has been rather slow in absorbing some of his major discoveries.In Dec 2011, in recognition of his contribution to mathematics, the Government of India declared that Ramanujan's birthday (22 December) should be celebrated every year as National Mathematics Day, and also declared 2012 the National year of mathematics.

2) G.H.HARDY
   Godfrey Harold "G. H." Hardy is usually known by those outside the field of mathematics for his essay from 1940 on the aesthetics of mathematics, A Mathematician's Apology, which is often considered one of the best insights into the mind of a working mathematician written for the layman. When Hardy was asked what his greatest contribution to mathematics was, Hardy unhesitatingly replied that it was the discovery of Ramanujan-it was in an interview by Paul Erdős.

HARDY-RAMANJUAN NUMBER
          It is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.
The two different ways are these:
1729 = 13 + 123 = 93 + 103
In Hardy's words:“I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."”
Numbers that are the smallest number that can be expressed as the sum of two cubes in "n" distinct ways have been dubbed "taxicab numbers".
    The same expression defines 1729 as the first in the sequence of "Fermat near misses" defined as numbers of the form 1 + z3 which are also expressible as the sum of two other cubes . 1729 has another mildly interesting property: the 1729th decimal place is the beginning of the first occurrence of all ten digits consecutively in the decimal representation of the transcendental number.
Masahiko Fujiwara showed that 1729 is one of four positive integers which, when its digits are added together, produces a sum which, when multiplied by its reversal, yields the original number:
1 + 7 + 2 + 9 = 19
19 × 91 = 1729

Wednesday, 11 July 2012

1729

HARDY-RAMANUJAN NUMBER 1729

1729 is the natural number following 1728 and preceding 1730. 1729 is known as the Hardy–Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a hospital visit to the Indian mathematician Srinivasa Ramanujan. In Hardy's words:[1]“ I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways." ”

The two different ways are these:
1729 = 13 + 123 = 93 + 103
The quotation is sometimes expressed using the term "positive cubes", since allowing negative perfect cubes (the cube of a negative integer) gives the smallest solution as 91 (which is a divisor of 1729):
91 = 63 + (−5)3 = 43 + 33
Of course, equating "smallest" with "most negative", as opposed to "closest to zero" gives rise to solutions like −91, −189, −1729, and further negative numbers. This ambiguity is eliminated by the
term "positive cubes". Numbers that are the smallest number that can be expressed as the sum of two cubes in n distinct ways[2] have been dubbed "taxicab numbers". The number was also found in one of Ramanujan's notebooks dated years before the incident, and was noted by Frénicle de Bessy in 1657.
The same expression defines 1729 as the first in the sequence of "Fermat near misses" (sequence A050794 in OEIS) defined as numbers of the form 1 + z3 which are also expressible as the sum of two other cubes.
1729 is also the third Carmichael number and the first absolute Euler pseudoprime. It is also a sphenic number.
1729 is a Zeisel number. It is a centered cube number, as well as a dodecagonal number, a 24-gonal and 84-gonal number. Investigating pairs of distinct integer-valued quadratic forms that represent every integer the same number of times, Schiemann found that such quadratic forms must be in four or more variables, and the
least possible discriminant of a four-variable pair is 1729 (Guy 2004).
Because in base 10 the number 1729 is divisible by the sum of its digits, it is a Harshad number. It also has this property in octal (1729 = 33018, 3 + 3 + 0 + 1 = 7) and hexadecimal (1729 = 6C116, 6 + C + 1 = 1910), but not in binary.
1729 has another mildly interesting property: the 1729th decimal place is the beginning of the first occurrence of all ten digits consecutively in the decimal representation of the transcendental number e.[3]
Masahiko Fujiwara showed that 1729 is one of four positive integers (with the others being 81, 1458, and the trivial case 1) which, when its digits are added together, produces a sum which, when multiplied by its reversal, yields the original number:
1 + 7 + 2 + 9 = 19
19 × 91 = 1729
It suffices only to check sums congruent to 0 or 1 (mod 9) up to 19.





Tuesday, 10 July 2012

Discovery Of 1729......!

Once when he was ill in England, Hardy visited Ramanujan in the hospital. Hardy mentioned casually that the number of the taxicab that had brought him was 1729, which, he thought was 'rather a dull one'. At this, Ramanujan replied: "No,it is a very interesting number! It is the smallest number expressible as a sum of two cubes in two different ways". That is 1729 = 13 + 123 = 93 + 103. This is one of the best known anecdotes in the history of mathematics , and after this incident, the number 1729is known as the Hardy - Ramanujan number!