Wednesday, January 9, 2019

Compilation of Mathematicians and Their Contributions

I. neoclassic Mathematicians Thales of Miletus Birthdate 624 B. C. Died 547-546 B. C. Nationality classic epithet Regarded as sustain of Science Contri completely whenions * He is credited with the prototypal implement of deductive reasoning apply to geometry. * husking that a pass rough isbisectedby its diameter, that the stand angles of an isosceles triplicity atomic deem 18 check and thatvertical anglesargon live. * recognise with put ination of the Ionian enlighten of maths that was a centre of learning and question. * Thales theorems use up in Geometry . The pairs of opposite enounce angles formed by 2 intersecting situations atomic subr outine 18 commensupace. 2. The base angles of an isosceles tri afterwardsal ar mates. 3. The append of the angles in a trilateral is star hundred eighty. 4. An angle inscribed in a semicircle is a effective angle. Pythagoras Birthdate 569 B. C. Died 475 B. C. Nationality classic Contri justions * Pythagorea n Theorem. In a unspoilt angled triangle the squ atomic derive 18 of the hypotenuse is extend to to the make sense of the squ atomic weigh 18s on the separate 2 si diethylstil tirepassrol. Note A by rights triangle is a triangle that contains single right (90) angle.The longest side of a right triangle, c solelyed the hypotenuse, is the side opposite the right angle. The Pythagorean Theorem is big in maths, physics, and astronomy and has unimaginative coats in surveying. * Developed a in advance(p) numerology in which odd depends annunciated priapic and even female 1 is the gen timetor of tote ups and is the tot up of reason 2 is the twist of opinion 3 is the flake of harmony 4 is the sum up of umpire and retri simplyion (opinion squ atomic write up forth 18d) 5 is the ph iodine scrap of conjugation (union of the ? rst male and the ? st female total) 6 is the public figure of creation 10 is the holiest of every last(predicate), and was the number of the universe, because 1+2+3+4 = 10. * Discoin truth of incommensurate ratios, what we would c any right a focus anomalous verse. * do the ? rst inroads into the branch of mathematics which would at unmatchable timeadays be c in altogethered Number conjecture. * Setting up a secret mystical society, cognize as the Pythagoreans that taught math and Physics. Anaxagoras Birthdate 500 B. C. Died 428 B. C. Nationality Grecian Contributions * He was the laidoff to explain that the laze shines due to reflected light from the sun. Theory of flake constituents of things and his emphasis on mechanical processes in the formation of slump out that paved the itinerary for the atomic attainable action. * Advocated that matter is composed of place elements. * Introduced the nonion of head t apieceer (Greek, mind or reason) into the philosophy of origins. The imagination of nous (mind), an unconditi unitaryd and unchanging perfume that enters into and controls every li ving object. He regarded subjective substance as an endless wad of imperishable primary elements, referring all genesis and disappearance to mixture and separation, respectively.Euclid Birthdate c. 335 B. C. E. Died c. 270 B. C. E. Nationality Greek gentle Father of Geometry Contributions * produce a reserve called the Elements serving as the main text for t individuallyingmathematics(e special(prenominal)lygeometry) from the time of its emergence until the late 19th or archetypical 20th century. The Elements. unrivalled of the ol stilboestrolt surviving fragments of EuclidsElements, prime atOxyrhynchus and dated to circa AD 100. * Wrote kit and caboodle on perspective, conical sections, rattlingism(a) geometry,number hypothesisand gracelessness. In addition to theElements, at least quintuple gui stilbesterol of Euclid wealthy or sobody survived to the present day. They follow the alike synthetic twist asElements, with explanations and resurrectd proposit ions. Those are the chase 1. Datadeals with the nature and implications of disposed(p) information in geo metric serve upal worrys the eccentric matter is closely related to the off vex printing quartette controls of theElements. 2. On Divisions of Figures, which survives save part inArabictranslation, concerns the division of geometrical figures into ii or to a spaciouser extent than(prenominal) equal split or into parts in donratios.It is similar to a triad century AD maneuver byHeron of Alexandria. 3. Catoptrics, which concerns the numeric possibleness of mirrors, peculiarly the encounters formed in flavorless and spherical concave mirrors. The attribution is held to be anachronic however by J J OConnor and E F Robertson who nameTheon of Alexandriaas a more likely source. 4. Phaenomena, a treatise onspherical astronomy, survives in Greek it is quite similar toOn the lamentable SpherebyAutolycus of Pitane, who flourished slightly 310 BC. * noteworthy five inquires of Euclid as menti unmatchabled in his day phonograph recording Elements . Point is that which has no part. 2. Line is a b eng seasonthless aloofness. 3. The extremities of musical notes are gratuitys. 4. A bully bend lies equally with respect to the draw a dip ons on itself. 5. single(a) raft draw a straight short garner from any(prenominal) point to any point. * TheElements in any con order include the following five common notions 1. Things that are equal to the same thing are as comfortably equal to one new(prenominal) (Transitive property of equality). 2. If equals are added to equals, consequently the safe and sounds are equal. 3. If equals are subtracted from equals, and thenly the last outders are equal. 4.Things that coincide with one an separate(prenominal) equal one another (Reflexive Property). 5. The whole is greater than the part. Plato Birthdate 424/423 B. C. Died 348/347 B. C. Nationality Greek Contributions * He helped to div ert mingled withpureand utilize mathematicsby widening the gap amongst arithmetic, now callednumber possible actionand logistic, now calledarithmetic. * Founder of the academyinAthens, the start-off institution of higher(prenominal)(prenominal) learning in theWestern adult male. It provided a comprehensive curriculum, including much(prenominal) subjects as astronomy, biology, mathematics, political possible action, and philosophy. Helped to lay the beations ofWestern philosophyandscience. * Platonic unbendables Platonic solid is a regular, bulging polyhedron. The faces are congruent, regular polygons, with the same number of faces marching at to each one vertex. at that place are exactly five solids which meet those criteria each is named according to its number of faces. * Polyhedron Vertices Edges FacesVertex chassis 1. tetrahedron4643. 3. 3 2. cube / hexahedron81264. 4. 4 3. octahedron61283. 3. 3. 3 4. dodecahedron2030125. 5. 5 5. icosahedron1230203. 3. 3. 3. 3 Ar istotleBirthdate 384 B. C. Died 322 BC (aged 61 or 62) Nationality Greek Contributions * Founded the Lyceum * His biggest share to the product line of mathematics was his ontogeny of the subscribe of logic, which he termed analyticals, as the buns for numeric study. He wrote extensively on this concept in his give-up the ghost Prior Analytics, which was create from Lyceum lecture notes some(prenominal)(prenominal) hundreds of days after his death. * Aristotles Physics, which contains a discussion of the innumerable that he believed existed in scheme only, sparked very much debate in later centuries.It is believed that Aristotle may hire been the origin philosopher to draw the sign amid actual and likely timeless populace. When considering twain actual and potential timeless existence, Aristotle states this 1. A carcass is settled as that which is bounded by a surface, therefore there micklenot be an innumerous body. 2. A Number, Numbers, by definitio n, is countable, so there is no number called infinity. 3. Perceptible bodies exist somewhere, they do a place, so there scum bagnot be an in exhaustible body. But Aristotle says that we scum bagnot say that the in exhaustible does not exist for these reasons 1.If no infinite, magnitudes impart not be divisible into magnitudes, but magnitudes after part be divisible into magnitudes (potentially infinitely), therefore an infinite in some sense exists. 2. If no infinite, number would not be infinite, but number is infinite (potentially), therefore infinity does exist in some sense. * He was the open of baronial logic, pioneered the study ofzoology, and go forth every future scientist and philosopher in his debt done and through his plowshares to the scientific regularity. Erasthosthenes Birthdate 276 B. C. Died 194 B. C. Nationality Greek Contributions * strive of Eratosthenes Worked on roseola numbers.He is remembered for his prime number sieve, the Sieve of Eratosthenes which, in modified form, is take over an great tool innumber openingresearch. Sieve of Eratosthenes- It does so by iteratively bulls eye as composite (i. e. not prime) the multiples of each prime, starting with the multiples of 2. The multiples of a given prime are generated starting from that prime, as a sequence of numbers with the same variance, equal to that prime, among consecutive numbers. This is the Sieves key note of hand from information trial division to sequentially test each candidate number for divisibility by each prime. Made a surprisingly complete measurement of the circuit of the Earth * He was the branch someone to use the term geographics in Greek and he invented the discipline of geography as we understand it. * He invented a musical arrangement oflatitudeandlongitude. * He was the archetypal to calculate thetilt of the Earths axis( withal with remarkable accuracy). * He may in any case have accurately calculated thedistance from the populace t o the sunand invented theleap day. * He as rise as created the runnermap of the worldincorporating parallels and meridians in spite of appearance his cartographic depictions based on the usable geographical knowledge of the era. Founder of scientificchronology. Favourite Mathematician Euclid paves the way for what we cognize straightaway as Euclidian Geometry that is considered as an ingrained for everyone and should be studied not only by students but by everyone because of its great applications and relevance to everyones daily life. It is Euclid who is skilful with knowledge and therefore became the pillar of todays success in the ambit of geometry and mathematics as a whole. at that place were great mathematicians as there were legion(predicate) great numeral knowledge that god wants us to know.In consideration however, there were some(prenominal) sagacious Greek mathematicians that had imparted their great percentages and therefore they deserve to be appreciate d. But since my task is to declare my favourite mathematician, Euclid deserves roughly of my congratulations for laying down the institution of geometry. II. Mathematicians in the Medieval Ages Leonardo of Pisa Birthdate 1170 Died 1250 Nationality Italian Contributions * surmount know to the innovativee world for the cattle ranch of the HinduArabic numeral remains in Europe, primarily through the egress in 1202 of his Liber Abaci ( account earmark of Calculation). Fibonacci introduces the so-called Modus Indorum ( form of the Indians), today cognise as Arabic numerals. The cry of honor advocated enumeration with the digits 09 and place value. The entertain bear witnessed the practical importance of the new numeral system, utilize lattice multiplication and Egyptian engage outs, by applying it to commercial playscriptkeeping, conversion of weights and measures, the slowness of interest, money-changing, and other applications. * He introduced us to the hold back we use in dissevers, previous to this, the numerator has quotations around it. * The square root banknote is too a Fibonacci order. He wrote following books that deals mathematics teachings 1. Liber Abbaci (The Book of Calculation), 1202 (1228) 2. Practica Geometriae (The Practice of Geometry), 1220 3. Liber Quadratorum (The Book of square(a) Numbers), 1225 * Fibonacci sequence of numbers in which each number is the sum of the previous both numbers, starting with 0 and 1. This sequence begins 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 The higher up in the sequence, the closer cardinal consecutive Fibonacci numbers of the sequence split up by each other depart approach the golden ratio (approximately 1 1. 18 or 0. 618 1). Roger Bacon Birthdate 1214 Died 1294 Nationality English Contributions * spell Majus contains give-and-takes of mathematics and optics, alchemy, and the positions and sizes of the celestial bodies. * Advocated the experimental m ethod as the true strandation of scientific knowledge and who in addition did some cause in astronomy, chemistry, optics, and machine design. Nicole Oresme Birthdate 1323 Died July 11, 1382 Nationality cut Contributions * He also certain a phraseology of ratios, to relate speed to force and resistance, and utilize it to physical and cosmological head words. He make a careful study of musicology and utilise his determinations to develop the use of nonsensical exponents. * low gear to theorise that sound and light are a transfer of energy that does not displace matter. * His al near(prenominal) inviolable contributions to mathematics are contained in Tractatus de configuratione qualitatum et motuum. * Developed the premiere use of barons with half(prenominal) exponents, deliberateness with irrational proportions. * He heard the divergence of the benevolent serial publication, using the standard method pacify taught in narrow chalkstone classes today. Omar Kha yyam Birhtdate 18 whitethorn 1048Died 4 declination 1131 Nationality Arabian Contibutions * He derived ascendants to isometric equalitys using the crossover of conic sections with circles. * He is the author of one of the most big treatises on algebra written in front modern times, the Treatise on Demonstration of Problems of Algebra, which includes a geometric method for solving cube- wrought compares by intersecting a hyperbola with a circle. * He contributed to a calendar reform. * Created cardinal bailiwicks on geometry, specifically on the supposition of proportions. Omar Khayyams geometric etymon to troika-d equatings. Binomial theorem and extraction of grow. * He may have been runner to develop Pascals Triangle, on with the essential Binomial Theorem which is sometimes called Al-Khayyams command (x+y)n = n ? xkyn-k / k (n-k). * Wrote a book entitled Explanations of the delicateies in the postulates in Euclids Elements The treatise of Khayyam can be consid ered as the starting intervention of parallels axiom which is not based on petitio principii but on more intuitive postulate. Khayyam refutes the previous attempts by other Greek and Persian mathematicians to show up the proposition.In a sense he make the first attempt at legislationting a non-Euclidean postulate as an alternative to the parallel postulate. happy Mathematician As furthermost as mediaeval times is relate, people in this era were challenged with chaos, social turmoil, economic issues, and umpteen other disputes. Part of this era is tinted with so called regretful Ages that marked the history with unfavourable events. Therefore, mathematicians during this era-after they undergone the much(prenominal) toils-were deserving individuals for gratitude and praises for they had supplemented the following generations with numeric ideas that is very reclaimable and applicable.Leonardo Pisano or Leonardo Fibonacci caught my heed therefore he is my favourite mathem atician in the medieval times. His desire to spread divulge the Hindu-Arabic numerals in other countries thus signifies that he is a person of generosity, with his noble will, he deserves to be III. Mathematicians in the Renaissance stream Johann Muller Regiomontanus Birthdate 6 June 1436 Died 6 July 1476 Nationality German Contributions * He completed De Triangulis omnimodus. De Triangulis (On Triangles) was one of the first textbooks presenting the current state of trigonometry. His work on arithmetic and algebra, Algorithmus Demonstratus, was among the first containing emblematical algebra. * De triangulis is in five books, the first of which gives the rudimentary definitions mensuration, ratio, equality, circles, arcs, chords, and the sine answer. * The crater Regiomontanus on the Moon is named after him. Scipione del Ferro Birthdate 6 February 1465 Died 5 November 1526 Nationality Italian Contributions * Was the first to solve the brick-shaped equation. * Contributions t o the rationalization of fractions with denominators containing sums of cube grow. Investigated geometry problems with a range of a melt down set at a icy angle. Niccolo Fontana Tartaglia Birthdate 1499/1500 Died 13 December 1557 Nationality Italian Contributions He make galore(postnominal) a(prenominal) books, including the first Italian translations of Archimedes and Euclid, and an acclaimed compilation of mathematics. Tartaglia was the first to apply mathematics to the investigation of the paths of cannonballs his work was later validate by Galileos studies on falling bodies. He also promulgated a treatise on retrieving sunken ships. Cardano-Tartaglia Formula. He makes dissolvers to solid equations. Formula for solving all types of boxy equations, involving first sure use of Gordian numbers (combinations of real and imaginary numbers). Tartaglias Triangle (earlier version of Pascals Triangle) A trilateral pattern of numbers in which each number is equal to the sum of the devil numbers promptly above it. He gives an grammatical construction for the good deal of a tetrahedron Girolamo Cardano Birthdate 24 kinsfolk 1501 Died 21 September 1576 Nationality Italian Contributions * He wrote more than 200 works on medicine, mathematics, physics, philosophy, religion, and music. Was the first mathematician to make opinionated use of numbers less than zero. * He published the solutions to the cubic and bi quadratic equation polynomial equations in his 1545 book Ars Magna. * musical composition novum de proportionibus he introduced the binominal coefficients and the binomial theorem. * His book about games of chance, Liber de ludo aleae (Book on Games of disaster), written in 1526, but not published until 1663, contains the first systematic treatment of hazard. * He studied hypocycloids, published in de proportionibus 1570. The generating circles of these hypocycloids were later named Cardano circles or cardanic ircles and were utilize for th e reflection of the first high-speed printing presses. * His book, Liber de ludo aleae (Book on Games of Chance), contains the first systematic treatment of opportunity. * Cardanos Ring Puzzle also cognise as Chinese Rings, still manufactured today and related to the towboat of Hanoi puzzle. * He introduced binomial coefficients and the binomial theorem, and introduced and solved the geometric hypocyloid problem, as salutary as other geometric theorems (e. g. the theorem implicit in(p) the 21 spur bike which converts circular to reciprocal recti additive motion).Binomial theorem- legality of nature for breeding devil-part mirror image a numeral shape utilise to calculate the value of a deuce-part numerical expression that is squared, cubed, or raised to another power or exponent, e. g. (x+y)n, without explicitly multiplying the parts themselves. Lodovico Ferrari Birthdate February 2, 1522 Died October 5, 1565 Nationality Italian Contributions * Was mainly responsible f or the solution of quartic equations. * Ferrari aided Cardano on his solutions for quadratic polynomial equations and cubic equations, and was mainly responsible for the solution of quartic equations that Cardano published.As a result, mathematicians for the next some(prenominal)(prenominal) centuries tried to find a formula for the grow of equations of degree five and higher. favorite(a) Mathematician Indeed, this period is supplemented with great mathematician as it move on from the Dark Ages and undergone a re support. Enumerated mathematician were all astounding with their performances and contributions. But for me, Niccolo Fontana Tartaglia is my favourite mathematician not only because of his undisputed contributions but on the way he keep himself alleviate despite of conflicts between him and other mathematicians in this period. IV. Mathematicians in the 16th ampere-secondFrancois Viete Birthdate 1540 Died 23 February 1603 Nationality cut Contributions * He unquestion able the first infinite-product formula for ?. * Vieta is most noted for his systematic use of ten-fold notation and variable earns, for which he is sometimes called the Father of Modern Algebra. (Used A,E,I,O,U for un cognises and consonants for parameters. ) * Worked on geometry and trigonometry, and in number system. * Introduced the polar triangle into spherical trigonometry, and stated the multiple-angle formulas for sin (nq) and cos (nq) in terms of the powers of sin(q) and cos(q). * Published Francisci Viet? universalium inspectionum ad canonem mathematicum liber singularis a book of trigonometry, in brief fiaten mathematicum, where there are umteen formulas on the sine and cosine. It is unusual in using decimal numbers. * In 1600, numbers potestatum ad exegesim resolutioner, a work that provided the means for extracting grow and solutions of equations of degree at most 6. commode Napier Birthdate 1550 Birthplace Merchiston Tower, Edinburgh Death 4 April 1617 Contri butions * credi cardinalrthy for advancing the notion of the decimal fraction by introducing the use of the decimal point. His prompting that a naive point could be utilise to eparate whole number and fragmentary parts of a number presently became accepted practice throughout massive Britain. * Invention of the Napiers Bone, a oil hand calculator which could be used for division and root extraction, as well as multiplication. * Written Works 1. A Plain husking of the Whole manifestation of St. John. (1593) 2. A Description of the Wonderful Canon of Logarithms. (1614) Johannes Kepler Born December 27, 1571 Died November 15, 1630 (aged 58) Nationality German Title Founder of Modern Optics Contributions * He reason Alhazens Billiard Problem, develop the notion of curvature. He was first to notice that the set of Platonic regular solids was incomplete if concave solids are admitted, and first to establish that there were only 13 Archimedean solids. * He proved theorems of solid geometry later spy on the historied palimpsest of Archimedes. * He reascertained the Fibonacci series, utilize it to botany, and noted that the ratio of Fibonacci numbers converges to the Golden Mean. * He was a key early pioneer in concretion, and embraced the concept of continuity (which others avoided due to Zenos paradoxes) his work was a direct inspiration for Cavalieri and others. He authentic mensuration methods and anticipated Fermats theorem (df(x)/dx = 0 at function extrema). * Keplers Wine lay Problem, he used his rudimentary calculus to deduce which barrel shape would be the best bargain. * Keplers Conjecture- is a numerical conjecture about sphere fisticuffs in terce-dimensional Euclidean space. It says that no arrangement of equally sized spheres alter space has a greater scrap-rate density than that of the cubic close packing (face- pertained cubic) and hexagonal close packing arrangements.Marin Mersenne Birthdate 8 September 1588 Died 1 September 164 8 Nationality cut Contributions * Mersenne primes. * Introduced several innovating concepts that can be considered as the basis of modern reflecting crushs 1. Instead of using an eyepiece, Mersenne introduced the revolutionary idea of a second mirror that would reflect the light feeler from the first mirror. This allows one to focus the image behind the primary mirror in which a hole is drilled at the centre to unblock the rays. 2.Mersenne invented the afocal telescope and the spread compressor that is useful in some multiple-mirrors telescope designs. 3. Mersenne recognized also that he could objurgate the spherical aberration of the telescope by using nonspherical mirrors and that in the grumpy case of the afocal arrangement he could do this correction by using dickens parabolic mirrors. * He also performed extensive experiments to determine the acceleration of falling objects by comparing them with the swing of pendulums, reported in his Cogitata Physico-Mathematica in 16 44.He was the first to measure the length of the seconds pendulum, that is a pendulum whose swing takes one second, and the first to observe that a pendulums swings are not isochronous as Galileo thought, but that braggy swings take longer than small swings. Gerard Desargues Birthdate February 21, 1591 Died September 1661 Nationality cut Contributions * Founder of the system of conic sections. Desargues offered a unified approach to the several types of conics through pick upion and section. * Perspective Theorem that when deuce triangles are in perspective the meets of determineing sides are col unidimensional. * Founder of projective geometry. Desarguess theorem The theorem states that if two triangles ABC and A? B? C? , situated in ternary-dimensional space, are related to each other in much(prenominal) a way that they can be imagen perspectively from one point (i. e. , the lines AA? , BB? , and CC? all intersect in one point), then the points of intersection of corresp onding sides all lie on one line provided that no two corresponding sides are * Desargues introduced the notions of the opposite ends of a straight line be regarded as coincident, parallel lines concussion at a point of infinity and regarding a straight line as circle whose center is at infinity. Desargues most important work Brouillon projet dune atteinte aux evenemens des rencontres d? une cone avec un plan (Proposed selective service for an essay on the results of taking woodworking matte sections of a cone) was printed in 1639. In it Desargues presented innovations in projective geometry apply to the surmisal of conic sections. dearie Mathematician Mathematicians in this period has its own distinct, and strange knowledge in the bailiwick of mathematics.They tackled the more complex world of mathematics, this complex world of maths had at times stirred up their lives, ignited some conflicts between them, unfolded their f impartial toneitys and weaknesses but at the en d, they build harmonious world through the unity of their formulas and much has benefited from it, they and so reflected the beauty of Mathematics. They were all excellent mathematicians, and no doubt in it. But I admire John Napier for giving birth to Logarithms in the world of Mathematics. V. Mathematicians in the seventeenth Century Rene Descartes Birthdate 31 March 1596 Died 11 February 1650Nationality French Contributions * Accredited with the invention of unionise geometry, the standard x,y co-ordinate system as the Cartesian plane. He developed the ordinate system as a spin to locate points on a plane. The machinate system includes two right lines. These lines are called axes. The vertical axis is designated as y axis while the level axis is designated as the x axis. The intersection point of the two axes is called the origin or point zero. The position of any point on the plane can be located by locating how far perpendicularly from each axis the point lays.The positi on of the point in the coordinate system is specified by its two coordinates x and y. This is written as (x,y). * He is credited as the father of uninflected geometry, the bridge between algebra and geometry, crucial to the uncovering of narrow calculus and analysis. * Descartes was also one of the key figures in the Scientific revolution and has been set forth as an example of genius. * He also pioneered the standard notation that uses superscripts to show the powers or exponents for example, the 4 used in x4 to indicate squaring of squaring. He invented the convention of doing un cognises in equations by x, y, and z, and cognizes by a, b, and c. * He was first to assign a fundamental place for algebra in our system of knowledge, and believed that algebra was a method to automate or fit out reasoning, occurrencely about pilfer, un cognize quantities. * Rene Descartes created analytic geometry, and discovered an early form of the law of conservation of nerve impulse (the term momentum refers to the momentum of a force). * He developed a rule for determining the number of arbitrary and negative roots in an equation.The Rule of Descartes as it is know states An equation can have as many true positive roots as it contains changes of sign, from + to or from to + and as many false negative roots as the number of times two + signs or two signs are found in succession. Bonaventura Francesco Cavalieri Birthdate 1598 Died November 30, 1647 Nationality Italian Contributions * He is known for his work on the problems of optics and motion. * Work on the precursors of infinitesimal calculus. * Introduction of logs to Italy. First book was Lo Specchio Ustorio, overo, Trattato delle settioni coniche, or The Burning Mirror, or a Treatise on Conic Sections. In this book he developed the opening of mirrors shaped into parabolas, hyperbolas, and ellipses, and various combinations of these mirrors. * Cavalieri developed a geometrical approach to calculus and publ ished a treatise on the topic, Geometria indivisibilibus continuorum nova quadam ratione promota (Geometry, developed by a new method through the indivisibles of the continua, 1635).In this work, an land is considered as constituted by an unfixed number of parallel segments and a mint as constituted by an indistinct number of parallel planar cranial orbits. * Cavalieris linguistic rule, which states that the volumes of two objects are equal if the areas of their corresponding cross-sections are in all cases equal. Two cross-sections correspond if they are intersections of the body with planes equidistant from a chosen base plane. * Published tables of logarithms, accenting their practical use in the field of astronomy and geography.capital of South Dakota de Fermat Birthdate 1601 or 1607/8 Died 1665 Jan 12 Nationality French Contributions * Early teachings that led to infinitesimal calculus, including his proficiency of adequality. * He is recognized for his discovery of an overlord method of purpose the great and the smallest ordinates of reduced lines, which is analogous to that of the divergential gear calculus, then unknown, and his research into number system. * He make famous contributions to analytic geometry, hazard, and optics. * He is best known for Fermats Last Theorem. Fermat was the first person known to have evaluated the built-in of commonplace power functions. Using an ingenious trick, he was able to reduce this evaluation to the sum of geometric series. * He invented a factoring methodFermats factorization methodas well as the conclusion technique of infinite descent, which he used to prove Fermats Last Theorem for the case n = 4. * Fermat developed the two-square theorem, and the polygonal number theorem, which states that each number is a sum of common chord triangular numbers, foursome square numbers, five pentagonal numbers, and so on. With his gift for number traffic and his dexterity to find produces for many o f his theorems, Fermat fundamentally created the modern theory of numbers. Blaise Pascal Birthdate 19 June 1623 Died 19 August 1662 Nationality French Contributions * Pascals toy * Famous contribution of Pascal was his Traite du triangle arithmetique (Treatise on the Arithmetical Triangle), commonly known today as Pascals triangle, which demonstrates many mathematical properties like binomial coefficients. Pascals Triangle At the age of 16, he formulated a basic theorem of projective geometry, known today as Pascals theorem. * Pascals law (a hydrostatics principle). * He invented the mechanical calculator. He built 20 of these machines (called Pascals calculator and later Pascaline) in the following ten historic period. * Corresponded with Pierre de Fermat on probability theory, strongly influencing the development of modern economics and social science. * Pascals theorem. It states that if a hexagon is inscribed in a circle (or conic) then the three intersection points of opposi te sides lie on a line (called the Pascal line).Christiaan Huygens Birthdate April 14, 1629 Died July 8, 1695 Nationality Dutch Contributions * His work include early telescopic studies elucidating the nature of the peal of Saturn and the discovery of its moon Titan. * The invention of the pendulum clock. chute driven pendulum clock, designed by Huygens. * Discovery of the centrifugal force, the laws for collision of bodies, for his role in the development of modern calculus and his original observations on sound perception. Wrote the first book on probability theory, De ratiociniis in ludo aleae (On Reasoning in Games of Chance). * He also designed more accurate clocks than were available at the time, suitable for sea soaring. * In 1673 he published his mathematical analysis of pendulums, Horologium Oscillatorium sive de motu pendulorum, his greatest work on horology. Isaac atomic number 7 Birthdate 4 Jan 1643 Died 31 March 1727 Nationality English Contributions * He set(p) the foundations for differential and organic calculus. compression-branch of mathematics concerned with the study of such concepts as the rate of change of one variable quantity with respect to another, the slope of a curve at a prescribed point, the numeration of the maximum and minimum values of functions, and the calculation of the area bounded by curves. Evolved from algebra, arithmetic, and geometry, it is the basis of that part of mathematics called analysis. * Produced simple analytical methods that unified many separate techniques antecedently developed to solve apparently misrelated problems such as determination areas, tangents, the lengths of curves and the maxima and minima of functions. Investigated the theory of light, explained gravity and hence the motion of the planets. * He is also famed for inventing nitrogenian mechanism and explicating his famous three laws of motion. * The first to use fractional indices and to employ coordinate geometry to derive solutions to Diophantine equations * He discovered Newtons identities, Newtons method, classified cubic plane curves (polynomials of degree three in two variables) Newtons identities, also known as the NewtonGirard formulae, give relations between two types of symmetric polynomials, namely between power sums and easy symmetric polynomials.Evaluated at the roots of a monic polynomial P in one variable, they allow expressing the sums of the k-th powers of all roots of P (counted with their multiplicity) in terms of the coefficients of P, without real finding those roots * Newtons method (also known as the NewtonRaphson method), named after Isaac Newton and Joseph Raphson, is a method for finding in turn better approximations to the roots (or zeroes) of a real-valued function. Gottfried Wilhelm Von Leibniz Birthdate July 1, 1646 Died November 14, 1716 Nationality GermanContributions * Leibniz invented a mechanical reason machine which would multiply as well as add, the chemical mechanism of which were still being used as late as 1940. * Developed the infinitesimal calculus. * He became one of the most fecund inventors in the field of mechanical calculators. * He was the first to describe a pinwheel wind collector calculator in 16856 and invented the Leibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator. * He also refined the double star number system, which is at the foundation of virtually all digital ready reckoners. Leibniz was the first, in 1692 and 1694, to employ it explicitly, to cite any of several geometric concepts derived from a curve, such as abscissa, ordinate, tangent, chord, and the perpendicular. * Leibniz was the first to put one across that the coefficients of a system of linear equations could be arranged into an array, now called a matrix, which can be manipulated to find the solution of the system. * He introduced several notations used to this day, for instance the integral sign ? representing an elongated S, from the Latin word summa and the d used for differentials, from the Latin word differentia.This cleverly suggestive notation for the calculus is probably his most enduring mathematical legacy. * He was the ? rst to use the notation f(x). * The notation used today in Calculus df/dx and ? f x dx are Leibniz notation. * He also did work in decided mathematics and the foundations of logic. Favorite Mathematician Selecting favourite mathematician from these single persons in mathematics is a severe task, but as I read the contributions of these Mathematicians, I found Sir Isaac Newton to be the greatest mathematician of this period.He invented the useful but rough subject in mathematics- the calculus. I found him cooperative with different mathematician to derive useful formulas despite the fact that he is graphic enough. Open-mindedness towards others opinion is what I discerned in him. VI. Mathematicians in the 18th Century Jacob Bernoulli Birthdate 6 January 1655 Died 16 August 1705 Nationality Swiss Contributions * Founded a school for mathematics and the sciences. * Best known for the work Ars Conjectandi (The Art of Conjecture), published eight years after his death in 1713 by his nephew Nicholas. Jacob Bernoullis first important contributions were a booklet on the parallels of logic and algebra published in 1685, work on probability in 1685 and geometry in 1687. * Introduction of the theorem known as the law of large numbers. * By 1689 he had published important work on infinite series and published his law of large numbers in probability theory. * Published five treatises on infinite series between 1682 and 1704. * Bernoulli equation, y = p(x)y + q(x)yn. * Jacob Bernoullis paper of 1690 is important for the history of calculus, since the term integral appears for the first time with its integration meaning. discovered a general method to determine evolutes of a curve as the envelope of its circles of curvature. He also investigated caustic curves an d in particular he studied these associated curves of the parabola, the logarithmic verticillated and epicycloids around 1692. * Theory of electrical switchs and combinations the so-called Bernoulli numbers, by which he derived the exponential series. * He was the first to think about the convergence of an infinite series and proved that the series is convergent. * He was also the first to propose incessantly compounded interest, which led him to investigate Johan Bernoulli Birthdate 27 July 1667Died 1 January 1748 Nationality Swiss Contributions * He was a brilliant mathematician who make important discoveries in the field of calculus. * He is known for his contributions to infinitesimal calculus and educated Leonhard Euler in his youth. * Discovered fundamental principles of mechanics, and the laws of optics. * He discovered the Bernoulli series and make advances in theory of navigation and ship sailing. * Johann Bernoulli proposed the brachistochrone problem, which asks what s hape a telegraph must be for a bead to slide from one end to the other in the shortest possible time, as a challenge to other mathematicians in June 1696.For this, he is regarded as one of the founders of the calculus of variations. Daniel Bernoulli Birthdate 8 February 1700 Died 17 March 1782 Nationality Swiss Contributions * He is particularly remembered for his applications of mathematics to mechanics. * His pioneering work in probability and statistics. Nicolaus Bernoulli Birthdate February 6, 1695 Died July 31, 1726 Nationality Swiss Contributions Worked mostly on curves, differential equations, and probability. He also contributed to nomadic dynamics. Abraham de Moivre Birthdate 26 may 1667 Died 27 November 1754 Nationality French Contributions Produced the second textbook on probability theory, The Doctrine of Chances a method of calculating the probabilities of events in play. * Pioneered the development of analytic geometry and the theory of probability. * Gives the fir st statement of the formula for the usual dispersal curve, the first method of finding the probability of the occurrence of an error of a given size when that error is show in terms of the variability of the diffusion as a unit, and the first denomination of the probable error calculation. Additionally, he applied these theories to gambling problems and actuarial tables. In 1733 he proposed the formula for estimating a factorial as n = cnn+1/2e? n. * Published an clause called Annuities upon Lives, in which he revealed the normal scattering of the mortality rate over a persons age. * De Moivres formula which he was able to prove for all positive integral values of n. * In 1722 he suggested it in the more well-known(a) form of de Moivres Formula Colin Maclaurin Birthdate February, 1698 Died 14 June 1746 Nationality economical Contributions * Maclaurin used Taylor series to characterize maxima, minima, and points of inflexion for infinitely differentiable functions in his Tre atise of Fluxions. Made significant contributions to the gravitation attraction of roundeds. * Maclaurin discovered the EulerMaclaurin formula. He used it to sum powers of arithmetic progressions, derive Stirlings formula, and to derive the Newton-Cotes numerical integration formulas which includes Simpsons rule as a special case. * Maclaurin contributed to the study of ovoid integrals, reducing many intractable integrals to problems of finding arcs for hyperbolas. * Maclaurin proved a rule for solving square linear systems in the cases of 2 and 3 unknowns, and discussed the case of 4 unknowns. Some of his important works are Geometria Organica 1720 * De Linearum Geometricarum Proprietatibus 1720 * Treatise on Fluxions 1742 (763 pages in two volumes. The first systematic exposition of Newtons methods. ) * Treatise on Algebra 1748 (two years after his death. ) * Account of Newtons Discoveries partial upon his death and published in 1750 or 1748 (sources disagree) * Colin Mac laurin was the name used for the new Mathematics and Actuarial Mathematics and Statistics Building at Heriot-Watt University, Edinburgh. Lenard Euler Birthdate 15 April 1707 Died 18 September 1783 Nationality Swiss Contributions He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. * He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. * He is also renowned for his work in mechanics, fluid dynamics, optics, and astronomy. * Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a function 2 and was the first to write f(x) to denote the function f applied to the argument x. He also introduced the modern notation for the trigonometric functions, the letter e for the base of the natural logarithm (now also known as Eulers number), t he Greek letter ? for summations and the letter i to denote the imaginary unit. * The use of the Greek letter ? to denote the ratio of a circles lap to its diameter was also popularized by Euler. * swell up known in analysis for his browse use and development of power series, the expression of functions as sums of infinitely many terms, such as * Euler introduced the use of the exponential function and logarithms in analytic proofs. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and complex numbers, thus greatly expanding the scope of mathematical applications of logarithms. * He also defined the exponential function for complex numbers, and discovered its relation to the trigonometric functions. * Elaborated the theory of higher mysterious functions by introducing the gamma function and introduced a new method for solving quartic equations. He also found a way to calculate integrals with comple x fixates, predict the development of modern complex analysis.He also invented the calculus of variations including its best-known result, the EulerLagrange equation. * Pioneered the use of analytic methods to solve number theory problems. * Euler created the theory of hypergeometric series, q-series, hyperbolic trigonometric functions and the analytic theory of continued fractions. For example, he proved the limitlessness of primes using the divergence of the harmonic series, and he used analytic methods to gain some understanding of the way prime numbers are distributed. Eulers work in this area led to the development of the prime number theorem. He proved that the sum of the reciprocals of the primes diverges. In doing so, he discovered the connection between the Riemann zeta function and the prime numbers this is known as the Euler product formula for the Riemann zeta function. * He also invented the totient function ? (n) which is the number of positive whole numbers less th an or equal to the integer n that are coprime to n. * Euler also conjectured the law of quadratic reciprocality. The concept is regarded as a fundamental theorem of number theory, and his ideas paved the way for the work of Carl Friedrich Gauss. * Discovered the formula V ?E + F = 2 relating the number of vertices, edges, and faces of a convex polyhedron. * He made great strides in improving the numerical approximation of integrals, inventing what are now known as the Euler approximations. Jean Le Rond De Alembert Birthdate 16 November 1717 Died 29 October 1783 Nationality French Contributions * DAlemberts formula for obtaining solutions to the beckon equation is named after him. * In 1743 he published his most famous work, Traite de dynamique, in which he developed his own laws of motion. * He created his ratio test, a test to see if a series converges. The DAlembert operator, which first arose in DAlemberts analysis of vibrating strings, plays an important role in modern theoret ical physics. * He made several contributions to mathematics, including a suggestion for a theory of limits. * He was one of the first to appreciate the importance of functions, and defined the first derivative of a function as the limit of a quotient of increments. Joseph Louise Lagrange Birthdate 25 January 1736 Died 10 April 1813 Nationality Italian French Contributions * Published the Mecanique Analytique which is considered to be his monolithic work in the pure maths. His most prominent influence was his contribution to the the metric system and his addition of a decimal base. * Some refer to Lagrange as the founder of the Metric System. * He was responsible for developing the groundwork for an alternate method of write Newtons Equations of Motion. This is referred to as Lagrangian Mechanics. * In 1772, he described the Langrangian points, the points in the plane of two objects in orbit around their common center of gravity at which the combined gravitational forces are zero, and where a third member of negligible mass can remain at rest. He made significant contributions to all fields of analysis, number theory, and classical and celestial mechanics. * Was one of the creators of the calculus of variations, lineage the EulerLagrange equations for extrema of functionals. * He also extended the method to take into account possible constraints, arriving at the method of Lagrange multipliers. * Lagrange invented the method of solving differential equations known as variation of parameters, applied differential calculus to the theory of probabilities and come through notable work on the solution of equations. * He proved that every natural number is a sum of four squares. Several of his early papers also deal with questions of number theory. 1. Lagrange (17661769) was the first to prove that Pells equation has a nontrivial solution in the integers for any non-square natural number n. 7 2. He proved the theorem, stated by Bachet without justification, that every positive integer is the sum of four squares, 1770. 3. He proved Wilsons theorem that n is a prime if and only if (n ? 1) + 1 is always a multiple of n, 1771. 4. His papers of 1773, 1775, and 1777 gave demonstrations of several results enunciated by Fermat, and not previously proved. 5.His Recherches dArithmetique of 1775 developed a general theory of binary quadratic forms to handle the general problem of when an integer is representable by the form. Gaspard Monge Birthdate may 9, 1746 Died July 28, 1818 Nationality French Contributions * Inventor of descriptive geometry, the mathematical basis on which practiced drafting is based. * Published the following books in mathematics 1. The Art of Manufacturing Cannon (1793)3 2. Geometrie descriptive. Lecons donnees aux ecoles normales (Descriptive Geometry) a arranging of Monges lectures. (1799) Pierre Simon Laplace Birthdate 23 March 1749Died 5 March 1827 Nationality French Contributions * Formulated Laplaces equation, and pi oneered the Laplace understand which appears in many branches of mathematical physics. * Laplacian differential operator, widely used in mathematics, is also named after him. * He restated and developed the cloudlike hypothesis of the origin of the solar system * Was one of the first scientists to postulate the existence of black holes and the notion of gravitational collapse. * Laplace made the non-trivial extension of the result to three dimensions to coming back a more general set of functions, the spherical harmonics or Laplace coefficients. Issued his Theorie analytique des probabilites in which he laid down many fundamental results in statistics. * Laplaces most important work was his Celestial Mechanics published in 5 volumes between 1798-1827. In it he sought to give a complete mathematical interpretation of the solar system. * In Inductive probability, Laplace set out a mathematical system of inductive reasoning based on probability, which we would today recognise as Bay esian. He begins the text with a series of principles of probability, the first six being 1.Probability is the ratio of the elevate events to the total possible events. 2. The first principle assumes equal probabilities for all events. When this is not true, we must first determine the probabilities of each event. accordingly, the probability is the sum of the probabilities of all possible favored events. 3. For independent events, the probability of the occurrence of all is the probability of each work out together. 4. For events not independent, the probability of event B following event A (or event A causing B) is the probability of A multiplied by the probability that A and B both occur. 5.The probability that A will occur, given that B has occurred, is the probability of A and B occurring divided by the probability of B. 6. Three corollaries are given for the sixth principle, which amount to Bayesian probability. Where event Ai ? A1, A2, An exhausts the list of possible ca uses for event B, Pr(B) = Pr(A1, A2, An). Then * Amongst the other discoveries of Laplace in pure and applied mathematics are 1. Discussion, contemporaneously with Alexandre-Theophile Vandermonde, of the general theory of determinants, (1772) 2. Proof that every equation of an even degree must have at least one real quadratic factor 3.Solution of the linear partial differential equation of the second holy order 4. He was the first to consider the difficult problems involved in equations of mixed unlikenesss, and to prove that the solution of an equation in finite differences of the first degree and the second order might always be obtained in the form of a continued fraction and 5. In his theory of probabilities 6. Evaluation of several common definite integrals and 7. General proof of the Lagrange reversion theorem. Adrian Marie Legendere Birthdate 18 September 1752 Died 10 January 1833 Nationality French Contributions Well-known and important concepts such as the Legendre polyn omials. * He developed the least squares method, which has broad application in linear regression, signal processing, statistics, and curve fitting this was published in 1806. * He made substantial contributions to statistics, number theory, abstract algebra, and mathematical analysis. * In number theory, he conjectured the quadratic reciprocity law, subsequently proved by Gauss in connection to this, the Legendre image is named after him. * He also did pioneering work on the distribution of primes, and on the application of analysis to number theory. Best known as the author of Elements de geometrie, which was published in 1794 and was the leading elementary text on the topic for around 100 years. * He introduced what are now known as Legendre functions, solutions to Legendres differential equation, used to determine, via power series, the attraction of an ellipsoid at any exterior point. * Published books 1. Elements de geometrie, textbook 1794 2. Essai sur la Theorie des Nombres 1798 3. Nouvelles Methodes pour la Determination des Orbites des Cometes, 1806 4. Exercices de Calcul Integral, book in three volumes 1811, 1817, and 1819 5.Traite des Fonctions Elliptiques, book in three volumes 1825, 1826, and 1830 Simon Dennis Poison Birthdate 21 June 1781 Died 25 April 1840 Nationality French Contributions * He published two memoirs, one on Etienne Bezouts method of elimination, the other on the number of integrals of a finite difference equation. * Poissons well-known correction of Laplaces second order partial differential equation for potential today named after him Poissons equation or the potential theory equation, was first published in the Bulletin de la societe philomatique (1813). Poissons equation for the divergence of the gradient of a scalar field, ? in 3-dimensional space Charles Babbage Birthdate 26 December 1791 Death 18 October 1871 Nationality English Contributions * mechanically skillful engineer who originated the concept of a programmable co mputer. * assign with inventing the first mechanical computer that ultimately led to more complex designs. * He invented the Difference Engine that could compute simple calculations, like multiplication or addition, but its most important trait was its ability create tables of the results of up to seven-degree polynomial functions. Invented the analytic Engine, and it was the first machine ever designed with the idea of programming a computer that could understand commands and could be programmed much like a modern-day computer. * He produced a Table of logarithms of the natural numbers from 1 to 108000 which was a standard reference from 1827 through the end of the century. Favorite Mathematician Noticeably, Leonard Euler made a mark in the field of Mathematics as he contributed several concepts and formulas that encompasses many areas of Mathematics-Geometry, Calculus, Trigonometry and etc.He deserves to be praised for doing such great things in Mathematics, indeed, his work la id foundation to make the lives of the following generation sublime, ergo, He is my favourite mathematician. VII. Mathematicians in the 19th Century Carl Friedrich Gauss Birthdate 30 April 1777 Died 23 February 1855 Nationality German Contributions * He became the first to prove the quadratic reciprocity law. * Gauss also made important contributions to number theory with his 1801 book Disquisitiones Arithmeticae (Latin, Arithmetical Investigations), which, among things, introduced the symbol ? or congruence and used it in a clean presentation of modular arithmetic, contained the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary quadratic forms, stated the class number problem for them, and showed that a regular heptadecagon (17-sided polygon) can be constructed with straightedge and compass. * He developed a method of measuring the horizontal intensity of the magnetised field which was in use well into the second half of the 20th ce ntury, and worked out the mathematical theory for separating the inner and outermost (magnetospheric) sources of Earths magnetic field.Agustin Cauchy Birthdate 21 August 1789 Died 23 May 1857 Nationality French Contributions * His most notable research was in the theory of residues, the question of convergence, differential equations, theory of functions, the legitimate use of imaginary numbers, operations with determinants, the theory of equations, the theory of probability, and the applications of mathematics to physics. * His writings introduced new standards of rigor in calculus from which grew the modern field of analysis.In Cours danalyse de lEcole Polytechnique (1821), by developing the concepts of limits and continuity, he provided the foundation for calculus basically as it is today. * He introduced the epsilon-delta definition for limits (epsilon for error and delta for difference). * He modify the theory of complex functions by discovering integral theorems and introdu cing the calculus of residues. * Cauchy founded the modern theory of snapshot by applying the notion of pressure on a plane, and assuming that this pressure was no longer perpendicular to the plane upon which it acts in an elastic body.In this way, he introduced the concept of latent hostility into the theory of elasticity. * He also examined the possible deformations of an elastic body and introduced the notion of strain. * One of the most prolific mathematicians of all time, he produced 789 mathematics papers, including 500 after the age of fifty. * He had sixteen concepts and theorems named for him, including the Cauchy integral theorem, the Cauchy-Schwartz inequality, Cauchy sequence and Cauchy-Riemann equations. He defined continuity in terms of infinitesimals and gave several important theorems in complex analysis and initiated the study of permutation radicals in abstract algebra. * He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner. * He was the first to define complex numbers as pairs of real numbers. * Most famous for his single-handed development of complex function theory.The first important theorem proved by Cauchy, now known as Cauchys integral theorem, was the following where f(z) is a complex-valued function holomorphic on and within the non-self-intersecting closed in(p) curve C (contour) lying in the complex plane. * He was the first to prove Taylors theorem rigorously. * His greatest contributions to mathematical science are enveloped in the rigorous methods which he introduced these are mainly embodied in his three great treatises 1. Cours danalyse de lEcole royale polytechnique (1821) 2. Le Calcul infinitesimal (1823) 3.Lecons sur les applications de calcul infinitesimal La geometrie (18261828) Nicolai Ivanovich Lobachevsky Birthdate December 1, 1792 Died February 24, 1856 Nationality Russian Contributions * Lobachevskys great contribution to the development of modern mathemati cs begins with the fifth postulate (sometimes referred to as axiom XI) in Euclids Elements. A modern version of this postulate reads Through a point lying outside a given line only one line can be displace parallel to the given line. * Lobachevskys geometry found application in the theory of complex numbers, the theory of vectors, and the theory of relativity. Lobachevskiis deductions produced a geometry, which he called imaginary, that was internally consistent and harmonious yet different from the traditional one of Euclid. In 1826, he presented the paper Brief Exposition of the Principles of Geometry with vigorous Proofs of the Theorem of Parallels. He refined his imaginary geometry in subsequent works, dating from 1835 to 1855, the last being Pangeometry. * He was well respected in the work he developed with the theory of infinite series especially trigonometric series, integral calculus, and probability. In 1834 he found a method for approximating the roots of an algebraic alal equation. * Lobachevsky also gave the definition of a function as a correspondence between two sets of real numbers. Johann Peter Gustav Le Jeune Dirichlet Birthdate 13 February 1805 Died 5 May 1859 Nationality German Contributions * German mathematician with deep contributions to number theory (including creating the field of analytic number theory) and to the theory of Fourier series and other topics in mathematical analysis. * He is credited with being one of the first mathematicians to give the modern formal definition of a function. Published important contributions to the biquadratic reciprocity law. * In 1837 he published Dirichlets theorem on arithmetic progressions, using mathematical analysis concepts to tackle an algebraic problem and thus creating the branch of analytic number theory. * He introduced the Dirichlet characters and L-functions. * In a couple of papers in 1838 and 1839 he proved the first class number formula, for quadratic forms. * Based on his researc h of the structure of the unit group of quadratic fields, he proved the Dirichlet unit theorem, a fundamental result in algebraic number theory. He first used the pigeonhole principle, a basic reckoning argument, in the proof of a theorem in diophantine approximation, later named after him Dirichlets approximation theorem. * In 1826, Dirichlet proved that in any arithmetic progression with first term coprime to the difference there are infinitely many primes. * Developed significant theorems in the areas of elliptic functions and applied analytic techniques to mathematical theory that resulted in the fundamental development of number theory. * His lectures on the equilibrium of systems and potential theory led to what is known as the Dirichlet problem.It involves finding solutions to differential equations for a given set of values of the boundary points of the region on which the equations are defined. The problem is also known as the first boundary-value problem of potential theor em. Evariste Galois Birthdate 25 October 1811 Death 31 May 1832 Nationality French Contributions * His work laid the foundations for Galois Theory and group theory, two major branches of abstract algebra, and the subfield of Galois connections. * He was the first to use the word group (French groupe) as a technical term in mathematics to represent a group of permutations. Galois published three papers, one of which laid the foundations for Galois Theory. The second one was about the numerical resolution of equations (root finding in modern terminology). The third was an important one in number theory, in which the concept of a finite field was first articulated. * Galois mathematical contributions were published in full in 1843 when Liouville reviewed his manuscript and stated it sound. It was finally published in the OctoberNovember 1846 issue of the Journal de Mathematiques Pures et Appliquees. 16 The most famous contribution of this manuscript was a novel proof that there is no q uintic formula that is, that fifth and higher degree equations are not generally solvable by radicals. * He also introduced the concept of a finite field (also known as a Galois field in his honor), in essentially the same form as it is still today. * One of the founders of the branch of algebra known as group theory. He developed the concept that is today known as a normal subgroup. * Galois most significant contribution to mathematics by far is his development of Galois Theory.He realized that the algebraic solution to a polynomial equation is related to the structure of a group of permutations associated with the roots of the polynomial, the Galois group of the polynomial. He found that an equation could be solved in radicals if one can find a series of subgroups of its Galois group, each one normal in its surrogate with abelian quotient, or its Galois group is solvable. This proved to be a fertile approach, which later mathematicians vary to many other fields of mathematics besides the theory of equations to which Galois orig

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