Saturday, 23 August 2014

Pythagorean triple

Pythagorean triple


                                                                                                                                                Manisha 
                                                                                                                                                        Lect. Maths,GSSS Town Hall, Asr
The Pythagorean theorem : 
                                                                          Perp2 + base2 = Hyp2
A triple is commonly written (a, b, c), and a well-known example is (3, 4, 5).

                                     a2 + b2 = c2

 A primitive Pythagorean triple is one in which a, b and c are coprime. A right triangle whose sides form a Pythagorean triple is called a Pythagorean triangle.

The name is derived from the Pythagorean theorem by famous mathematician  Pythagoras , stating that every right triangle has side lengths satisfying the formula a2 + b2 = c2; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides a = b = 1 and c = √2 is right, but (1, 1, √2) is not a Pythagorean triple because √2 is not an integer. Moreover, 1 and √2 do not have an integer common multiple because √2 is irrational.

There are 16 primitive Pythagorean triples with c ≤ 100:
(3, 4, 5 )
(5, 12, 13)
(8, 15, 17)
(7, 24, 25)
(20, 21, 29)
(12, 35, 37)
( 9, 40, 41)
(28, 45, 53)
(11, 60, 61)
(16, 63, 65)
(33, 56, 65)
(48, 55, 73)
(13, 84, 85)
(36, 77, 85)
(39, 80, 89)
(65, 72, 97)
Note, for example, that (6, 8, 10) is not a primitive Pythagorean triple, as it is a multiple of (3, 4, 5). Each one of these low-c points forms one of the more easily recognizable radiating lines in the scatter plot.
Additionally these are all the primitive Pythagorean triples with 100 < c ≤ 300:
(20, 99, 101)
(60, 91, 109)
(15, 112, 113)
(44, 117, 125)
(88, 105, 137)
(17, 144, 145)
(24, 143, 145)
(51, 140, 149)
(85, 132, 157)
(119, 120, 169)
(52, 165, 173)
(19, 180, 181)
(57, 176, 185)
(104, 153, 185)
(95, 168, 193)
(28, 195, 197)
(84, 187, 205)
(133, 156, 205)
(21, 220, 221)
(140, 171, 221)
(60, 221, 229)
(105, 208, 233)
(120, 209, 241)
(32, 255, 257)
(23, 264, 265)
(96, 247, 265)
(69, 260, 269)
(115, 252, 277)
(160, 231, 281)
(161, 240, 289)
(68, 285, 293)

                                                                                                                    

Complete mathematical formula guide for VIth to Xth class and Complete formula guide of Differentiation / Integration / Limit and continuity / Trigonometry

Complete mathematical formula guide for VIth to Xth class and Complete formula guide of Differentiation / Integration / Limit and continuity / Trigonometry  developed by Manisha  Lecturer Maths, GSSS Town Hall and Rajwinder Singh,   Math Master, GSSS ,Verka, Amritsar, Punjab, India and many more on http://mathsworldtlm.blogspot.in/

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Saturday, 16 August 2014

Indian-origin wizard wins 'Nobel Prize' of Mathematics

Congratulation to first Indian Mathematician Manjul Bhargava from Mathsworld.in
Indian-origin wizard wins Fields Medal 'Nobel Prize' of Mathematics
 Mathematicians of Indian- and Iranian-origin are among the four winners of the 2014 Fields Medal, widely considered the Nobel Prize for maths that has been broadly dominated by white males since it was instituted in 1936.

The award going to Princeton University's Manjul Bhargava, a Canadian-American maths wizard was no surprise; although he is the first person of Indian origin, he was the hot favorite in pre-award polls among peers.

The sensational co-winner is Maryam Mirzakhani, a female Iranian mathematician who teaches at Stanford University. It is the first time a female mathematician has won the Fields medal; all 52 previous winners have been men in a field traditionally dominated by the male of the species.

Expectedly, it created a ripple in the rarefied maths world.

Mirzakhani's success was "hugely symbolic and I hope it will encourage more women to get into mathematics because we need more women. I am very happy that now we can put to rest that particular 'it has never happened before'," Ingrid Daubechies, herself the first female president of the International mathematical Union (IMU), said while announcing the award.

The two other winners this year are Artur Avila from Brazil and Martin Hairer from Austria. Avila is also the first Brazilian and Latin American to win the medal.

One to four Fields Medals are awarded once every four years to mathematicians under the age of 40 years at the International Congress of the International Mathematical Union (IMU), which meets every four years. The presentation will take place in Seoul on Wednesday at the quadrennial IMU Congress (Hyderabad hosted it in 2010).

Although the prize money ($15,000) is chump change (approximately 1/100th) compared to the Nobel Prize, the award, long dominated by Americans, Russians, French, and Britons (38 medals between them), is the highest recognition in the world of mathematics.

Canadian mathematician John Charles Fields instituted it at a time mathematicians felt short-changed that they had no Nobel recognition. The Nobel Prize is awarded for literature, peace, economics, physiology or medicine, chemistry and physics — but not for mathematics.

Legend — or the apocryphal story — goes that Swedish chemist Alfred Nobel, who instituted the Nobel Prize, disdained maths after someone he loved cheated on him — with a mathematician. But there is no historical basis to the story. More likely fact is that Nobel didn't care much for maths because it was not considered a practical science from which humanity could benefit (a chief purpose for creating the Nobel Foundation).

All that has changed, of course. Mathematics offers solutions to everyday issues from airline scheduling to Internet security, even though many practitioners pursue esoteric problems described in dense language incomprehensible to the layman. Bhargava's PhD thesis, for instance, is said to have helped in the "determination of the asymptotic density of discriminants of quartic and quintic number fields."

Although a Canadian-American who was born in Hamilton, Ontario, Bhargava is no stranger to India or to Indian mathematicians. Indeed, his mother, Mira Bhargava, is herself a rare female mathematician, teaching at Hofstra University (another well-known female Indian-American mathematician is Bhama Srinivasan at the University of Chicago).


Canadian-American maths wizard
Manjul Bhargava.

Manjul has also collaborated with many Indian mathematicians, and his work with fellow Princeton scholar Arul Shankar, his PhD student, won them the Fermat Prize in 2011. Manjul's own PhD advisor was Andrew Wiles, famous for proving Fermat's last theorem.

Bhargava was awarded the 2012 Infosys Prize in mathematics for his "extraordinarily original work in algebraic number theory, which has revolutionized the way in which number fields and elliptic curves are counted." That came on top of almost every other top prize in maths, from the SASTRA Ramanujan Prize in 2005 to the American mathematical Society's Cole Prize in 2008. So the Fields Medal comes as no great surprise to the mathematical community in the US or in India.

Last week, as speculation heated up about possible 2014 winners of Fields Medal, an online poll put Bhargava on top with 516 votes, with Avila coming second with 486 votes. Apparently, his peers pretty much expected it. Which is not surprising for someone who became a tenured full professor within two years of finishing graduate school, an Ivy League record, and the second youngest full professor in Princeton's history.

That's not all. Before you think all he does is crunch numbers, Bhargava is also an accomplished tabla player (tutored by Zakir Hussain) and has the number on Sanskrit, which he learned from his grandfather Purushottam Lal Bhargava, was the head of the Sanskrit department of the University of Rajasthan, during family visits to Jaipur. He sees close links between his three loves noting how beats of tabla and rhythms of Sanskrit poetry are highly mathematical.

Such recognition came to him early. In past interviews, he has often recounted how in Grade 3, he became curious about how many oranges it takes to make a pyramid. Just as well his mathematician mother and chemist father were well-to-do: they indulged him with oranges till he figured out the answer, which was not long coming. Now he's at the pinnacle of his calling.

 TOI dated 17th aug 2014

Saturday, 22 February 2014

VEDIC MATHEMATICS,Simplest way to Mathematics

VEDIC MATHEMATICS
(Simplest way to Mathematics)
What is vedic maths?
Ø  Vedic mathematics is ancient system of Indian mathematics ,which help in calculations simpler and faster
Ø  This  technique of calculations is simple. Simple principles and rules help in solving any  mathematical problem mentally of any branch ( arithmetic, algebra, geometry or trigonometry) .
Why Vedic Mathematics is ?
Vedic Mathematics is becoming popular all over the world as It helps to solve problems 10-15 times faster , reduces burden (Need to learn tables up to nine only), provides one line answer save student’s time in competition exams and  increases his concentration.

http://mathsworld-vedicmath.blogspot.in/

Monday, 6 January 2014

Mathematics Made simple

Math Park,  TLM photographs by DRP Math Fazilka


many more Mathematical TLM on
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Measure Circumference of Circle