About mathematiciansAbout mathematicians

EUCLIDEUCLID

Euclid may have been a student of Aristotle. He founded the school of mathematics at the great university of Alexandria. He was the first to prove that

there are infinitely many prime numbers; he stated and proved the Unique Factorization Theorem; and he devised Euclid's algorithm for computing gcd. He introduced the Mersenne primes and observed that (M2+M)/2 is always perfect

(in the sense of Pythagoras) if M is Mersenne. (The converse, that any even perfect number has such a corresponding Mersenne prime, was tackled by Alhazen and proven by Euler.) He proved that there are only five "Platonic

solids," as well as theorems of geometry far too numerous to summarize; among many with special historical interest is the proof that rigid-compass constructions can be implemented with collapsing-compass constructions. Although notions of trigonometry were not in use, Euclid's theorems include some closely related to

the Laws of Sines and Cosines. Among several books attributed to Euclid are The Division of the Scale (a mathematical discussion of music), The Optics, The

Cartoptrics (a treatise on the theory of mirrors), a book on spherical geometry, a book on logic fallacies, and his comprehensive math textbook The Elements.

Several of his masterpieces have been lost, including works on conic sections and other advanced geometric topics. Apparently Desargues' Homology Theorem (a pair of triangles is coaxial if and only if it is copolar) was proved in one of these

lost works; this is the fundamental theorem which initiated the study of projective geometry

Euclid ranks #14 on Michael Hart's famous list of Euclid ranks #14 on Michael Hart's famous list of the Most Influential Persons in History. the Most Influential Persons in History. The The ElementsElements introduced the notions of axiom and introduced the notions of axiom and theorem; was used as a textbook for 2000 years; theorem; was used as a textbook for 2000 years; and in fact is still the basis for high school and in fact is still the basis for high school geometry, making Euclid the leading mathematics geometry, making Euclid the leading mathematics teacher of all time. Some think his best inspiration teacher of all time. Some think his best inspiration was recognizing that the Parallel Postulate must was recognizing that the Parallel Postulate must be an axiom rather than a theorem. There are be an axiom rather than a theorem. There are many famous quotations about Euclid and his many famous quotations about Euclid and his books. Abraham Lincoln abandoned his law studies books. Abraham Lincoln abandoned his law studies when he didn't know what "demonstrate" meant when he didn't know what "demonstrate" meant and "went home to my father's house [to read and "went home to my father's house [to read Euclid], and stayed there till I could give any Euclid], and stayed there till I could give any proposition in the six books of Euclid at sight. I proposition in the six books of Euclid at sight. I then found out what demonstrate means, and then found out what demonstrate means, and went back to my law studies." went back to my law studies."

Indian mathematicians excelled for thousands of years, and eventually even developed advanced

techniques like Taylor series before Europeans did, but they are denied credit because of Western

ascendancy. Among the Hindu mathematicians, Aryabhatta (called Arjehir by Arabs) may be most

famous. While Europe was in its early "Dark Age," Aryabhatta advanced arithmetic, algebra, elementary analysis,

and especially trigonometry, using the decimal system. Aryabhatta is sometimes called the "Father of Algebra" instead of al-Khowârizmi (who himself cites the work of Aryabhatta). His most famous accomplishment in

mathematics was the Aryabhatta Algorithm (connected to continued fractions) for solving Diophantine

equations.

Aryabhatta made several important discoveries in astronomy; for example, his estimate of the Earth's circumference was more accurate than any achieved in ancient Greece. He was among the ancient scholars who realized the Earth rotated daily on an axis; claims that he also espoused heliocentric orbits are controversial, but may be confirmed by the writings of al-Biruni. Aryabhatta is said to have introduced the constant e. He used π ≈ 3.1416; it is unclear whether he discovered this independently or borrowed it from Liu Hui of China. Among theorems first discovered by Aryabhatta is the famous identity Σ (k3) = (Σ k)2

Archimedes is universally acknowledged to be the greatest of ancient mathematicians. He studied at Euclid's school (probably after Euclid's death), but his work far surpassed the works of Euclid. His achievements are particularly impressive given the lack of good mathematical notation in his day. His proofs are noted not only for brilliance but for unequalled clarity, with a modern biographer (Heath) describing Archimedes' treatises as "without exception monuments of mathematical exposition ... so impressive in their perfection as to create a feeling akin to awe in the mind of the reader." Archimedes made advances in number theory, algebra, and analysis, but is most renowned for his many theorems of plane and solid geometry

He was first to prove Heron's formula for the area of a triangle. His excellent approximation to √3 indicates that he'd partially anticipated the method of continued fractions. He found a method to trisect an arbitrary angle (using a markable straightedge — the construction is impossible using strictly Platonic rules). One of his most remarkable and famous geometric results was determining the area of a parabolic section, for which he offered two independent proofs, one using his Principle of the Lever, the other using a geometric series. Many of Archimedes' discoveries are known only second-hand: Pappus reports that he discovered the Archimedean solids; Thabit bin Qurra reports his method to construct a regular heptagon; Alberuni credits the Broken-Chord Theorem to him; etc.

Archimedes anticipated integral calculus, most notably by determining the centers of mass of hemisphere and cylindrical wedge, and the volume of two cylinders' intersection. Although Archimedes made little use of differential calculus, Chasles credits him (along with Kepler, Cavalieri, and Fermat) as one of the four who developed calculus before Newton and Leibniz. He was similar to Newton in that he used his (non-rigorous) calculus to discover results, but then devised rigorous geometric proofs for publication. His original achievements in physics include the principles of leverage, the first law of hydrostatics, and inventions like the compound pulley, the hydraulic screw, and war machines. His books include Floating Bodies, Spirals, The Sand Reckoner, Measurement of the Circle, and Sphere and Cylinder.

He developed the Stomachion puzzle (and solved a difficult enumeration problem involving it).

Archimedes proved that the volume of a sphere is two-thirds the volume of a circumscribing cylinder.

He requested that a representation of such a sphere and cylinder be inscribed on his tomb.

Archimedes discovered formulae for the volume and surface area of a sphere, and may even have been

first to notice and prove the simple relationship between a circle's circumference and area. For these reasons, π is often called Archimedes' constant. His approximation 223/71 < π < 22/7 was the best of his day. (Apollonius soon surpassed it, but by using Archimedes' method.) That Archimedes shared the

attitude of later mathematicians like Hardy and Brouwer is suggested by Plutarch's comment that

Archimedes regarded applied mathematics "as ignoble and sordid ... and did not deign to [write

about his mechanical inventions; instead] he placed his whole ambition in those speculations the beauty

and subtlety of which are untainted by any admixture of the common needs of life."

Some of Archimedes' greatest writings are preserved on a palimpsest which has been rediscovered and properly studied only since 1998. Ideas unique to that work are calculating the volume of a cylindrical wedge (previously first attributed to Kepler), and perhaps an implication that Archimedes understood the distinction between countable and uncountable infinities (a distinction which wasn't resolved until Georg Cantor, who lived 2300 years after the time of Archimedes .

Some of Archimedes' greatest writings are preserved on a palimpsest which has been rediscovered and properly studied only since 1998. Ideas unique to that work are calculating the volume of a cylindrical wedge (previously first attributed to Kepler), and perhaps an implication that Archimedes understood the distinction between countable and uncountable infinities (a distinction which wasn't resolved until Georg Cantor, who lived 2300 years after the time of Archimedes

Like Abel, Ramanujan was a self-taught prodigy who lived in a country distant from his mathematical peers, and suffered from poverty: childhood dysentery and vitamin deficiencies probably led to his early death. Yet he produced 4000 theorems or conjectures in number theory, algebra, and combinatorics. He might be almost unknown today, except that his letter caught the eye of Godfrey Hardy, who saw remarkable, almost inexplicable formulae which "must be true, because if they were not true, no one would have had the imagination to invent them.

." Ramanujan's specialties included infinite series, elliptic functions, continued fractions, partition enumeration, definite integrals, modular equations, gamma functions, "mock theta" functions, hypergeometric series, and "highly composite" numbers. Much of his best work was done in collaboration with Hardy, for example a proof that almost all numbers n have about log log n prime factors (a result which developed into probabilistic number theory). Much of his methodology, including unusual ideas about divergent series, was his own invention.

(As a young man he made the absurd claim that 1+2+3+4+... = -1/12. Later it was noticed that this claim translates to a true statement

about the Riemann zeta function, with which Ramanujan was

unfamiliar.) Ramanujan's innate ability for algebraic manipulations equaled or surpassed that of Euler

and Jacobi. Ramanujan's most famous work

was with the partition enumeration function p(), Hardy guessing that some of these discoveries would

have been delayed at least a century without Ramanujan.

Together ,Hardy and Ramanujan developed an analytic approximation to p(). (Rademacher and Selberg later discovered an exact expression to replace the Hardy-Ramanujan formula; when Ramanujan's notebooks were studied it was found he had anticipated their technique, but had deferred to his friend and mentor.)

Many of Ramanujan's other results would also probably never have been discovered without him, and are so inspirational that there is a periodical dedicated to them. The theories of strings and crystals have benefited from Ramanujan's work. (Today some professors achieve fame just by finding a new proof for one of Ramanujan's many results.) Unlike Abel, who insisted on rigorous proofs, Ramanujan often omitted proofs. (Ramanujan may have had unrecorded proofs, poverty leading him to use chalk and erasable slate rather than paper.)

Unlike Abel, much of whose work depended on the complex numbers, most of Ramanujan's work focused

on real numbers. Despite these limitations, Ramanujan is considered one of the greatest geniuses ever. Because of its fast convergence, an odd-looking formula of Ramanujan is

sometimes used to calculate π: 992 / π = √8 ∑k=0,∞ ((4k)!

(1103+26390 k) / (k!4 3964k))

Newton was an industrious lad who built marvelous toys (e.g. a model windmill powered by a mouse on treadmill). At about age 22, on leave from University, this genius began revolutionary advances in mathematics, optics, dynamics, thermodynamics, acoustics and celestial mechanics. He is famous for his Three Laws of Motion (inertia, force, reciprocal action) but, as Newton himself acknowledged, these Laws weren't fully novel: Hipparchus, Ibn al-Haytham, Galileo and Huygens had all developed much basic mechanics already, and Newton credits the First Law itself to Aristotle.

However Newton was also apparently the first person to conclude that the ordinary gravity we observe on Earth is the very same force that keeps the planets in orbit. His Law of Universal Gravitation was revolutionary and due to Newton alone. (Christiaan Huygens, the other great mechanist of the era, had independently deduced that Kepler's laws imply inverse-square gravitation, but he considered the action at a distance in Newton's theory to be "absurd.") Newton's other intellectual interests included chemistry, theology, astrology and alchemy.

Although this list is concerned only with mathematics, Newton's greatness is indicated by the wide range of his physics: even without his revolutionary Laws of Motion and his Cooling Law of thermodynamics, he'd be famous just for his work in optics, where he explained diffraction and observed that white light is a mixture of all the rainbow's colors. (Although his corpuscular theory competed with Huygen's wave theory, Newton understood that his theory was incomplete without waves.) Newton also designed the first reflecting telescope, first reflecting microscope, and the sextant.

Although others also developed the techniques independently, Newton is regarded as the Father of Calculus (which he called "fluxions"); he shares credit with Leibniz for the Fundamental Theorem of Calculus (that integration and differentiation are each other's inverse operation). He applied calculus for several purposes: finding areas, tangents, the lengths of curves and the maxima and minima of functions. In addition to several other important advances in analytic geometry, his mathematical works include the Binomial Theorem, his eponymous numeric method, the idea of polar coordinates, and power series for exponential and trigonometric functions. (His equation ex = ∑ xk / k! has been called the "most important series in mathematics.")

He contributed to algebra and the theory of equations; he was first to state Bézout's Theorem; he generalized Déscartes' rule of signs. (The generalized rule of signs was incomplete and finally resolved two centuries later by Sturm and Sylvester.) He developed a series for the arcsin function. He developed facts about cubic equations (just as the "shadows of a cone" yield all quadratic curves, Newton found a curve whose "shadows" yield all cubic curves). He proved that same-mass spheres of any radius have equal gravitational attraction: this fact is key to celestial motions. He discovered Puiseux series almost two centuries before they were re-invented by Puiseux. (Like some of the greatest ancient mathematicians, Newton took the time to compute an approximation to π; his was better than Vieta's, though still not as accurate as al-Kashi's.)

Newton is so famous for his calculus, optics and laws of motion, it is easy to overlook that he was also one of the greatest geometers. He solved the Delian cube-doubling problem. Even before the invention of the calculus of variations, Newton was doing difficult work in that field, e.g. his calculation of the "optimal bullet shape." Among many marvelous theorems, he proved several about quadrilaterals and their in- or circum-scribing ellipses, and constructed the parabola defined by four given points. He anticipated Poncelet's Principle of Continuity. An anecdote often cited to demonstrate his brilliance is the problem of the brachistochrone, which had baffled the best mathematicians in Europe, and came to Newton's attention late in life. He solved it in a few hours and published the answer anonymously. But on seeing the solution Jacob Bernoulli immediately exclaimed "I recognize the lion by his footprint.

In 1687 Newton published Philosophiae Naturalis Principia Mathematica, surely the greatest scientific book ever written. The motion of the planets was not understood before Newton, although the heliocentric system allowed Kepler to describe the orbits. In Principia Newton analyzed the consequences of his Laws of Motion and introduced the Law of Universal Gravitation. With the key mystery of celestial motions finally resolved, the Great Scientific Revolution began. (In his work Newton also proved important theorems about inverse-cube forces, work largely unappreciated until Chandrasekhar's modern-day work.) Newton once wrote "Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things." Sir Isaac Newton was buried at Westminster Abbey in a tomb inscribed "Let mortals rejoice that so great an ornament to the human race has existed."

Newton ranks #2 on Michael Hart's famous list of the Most Influential Persons in History. (Muhammed the Prophet of Allah is #1.) Whatever the criteria, Newton would certainly rank first or second on any list of physicists, or scientists in general, but some listmakers would demote him slightly on a list of pure mathematicians: his emphasis was physics not mathematics, and the contribution of Leibniz (Newton's rival for the title Inventor of Calculus) lessens the historical importance of Newton's calculus. One reason I've ranked him at #1 is a comment by Gottfried Leibniz himself: "Taking mathematics from the beginning of the world to the time when Newton lived, what he has done is much the better part."

AcknowledgementAcknowledgement

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om mishraom mishra

Prepared by-Prepared by- VIPUL SINGHVIPUL SINGH AKSHAT KATIYARAKSHAT KATIYAR MAYANK RAWATMAYANK RAWAT ANSHUL CHAUHANANSHUL CHAUHAN ISHU KASHYAPISHU KASHYAP

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