modern period (17th-19th century) on mathematics
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Modern PerioMathematic7
TH
19
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Century
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17thCentury Mathematics
John Napier
Marin MersenneRene Descartes
Pierre de Fermat
Blaise PascalGerard (Girard) Desargues
Isaac Newton
Gottfried Wilhelm Leibniz
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17thCentury Mathematics
John Napier (1550 - 4 April 1617)
Born in Edinburgh, Scotland,into the Scottish nobility
Studied in Europe
Interested in Mathematics,Astronomy, Religion, andPolitics
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17thCentury Mathematics
Logarithms Description
Significant contributions
Used by Johannes Kepler for his Third Law of PlanetaryThe ratio of the squares of the revolutionary periods f
planets is equal to the ratio of the cubes of their semi- Simplified large-number calculations
Biomathematics(Modeling Population Growth)
Physics(Radioactive Decay, Astronomy)
Chemistry(pH)
Jo
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17thCentury Mathematics Napiers
Rods/Bones Made of ivory so
that it looked likebones
Aided inmultiplication anddivision
Jo
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Napiers Rods/Bones
Example:
J
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17thCentury Mathematics
Marin Mersenne(8 Sept. 1588 - 1 Sept. 1648) French theologian, natural
philosopher, andmathematician
Generated a formula tofind prime numbers of theform, = 2 1, -prime
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Published works on Music TheoryMathematics, Physics, and Astro
Uses of Mersenne primes:Number Theory
Cryptography
Marin
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Rene Descartes (31 Mar, 159611 Feb, 1650) French Philosopher,
Mathematician, and Writer
Father of Modern Philosophy
Made an importantcontribution in AnalyticalGeometry by developing theCartesian Plane.
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The Cartesian Plane Translates Algebrato Geometry; thus,making newinnovations inAnalytic Geometry
Rene
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17thCentury Mathematics
Pierre de Fermat(August 17, 1601January 12, 16 Small town amateur mathematician Inspired by Arithmeticaby Diophantus
Contributions on Number Theory, Modern Cand Probability
Despite showing interest in Mathematstudied law at Orlans and received thecouncillor at the High Court of JudicaToulouse in 1631, which he held for the re
life.
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17thCentury Mathematics Sworn in by the Grand Chambre in May 16
hes entitled to change his name fromFermat to Pierre de Fermat.
Fluent in Latin, Greek, Italian and Spanish, apraised for his written verse in several lang
and eagerly sought for advice on the emeof Greek texts.
Dominique Fermat (father) wealthy merchant and Claire, ne de Long (m
daughter of a prominent family.
Pierre
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Most of Fermats work was written in lett
friends, which often provided little or noof his theorems. Although he himself clato have proved all his arithmetic theorefew records of his proofs have survived,
many mathematicians have doubted sof his claims, especially given the difficusome of the problems and the limitedmathematical tools available to Ferma
Pierre
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17thCentury MathematicsFermats Contributions
Methodus ad disquirendam maximam et minin De tangentibus linearum curvarum, Fermat dea method for determining maxima, minimtangents to various curves that was eqto differential calculus.
In these works, Fermat obtained a technique fothe centers of gravity of various plane and solidwhich led to his further work in quadrature (nintegration).
Pierre
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17thCentury MathematicsPierre
Probability (gambling)
Infinite Descent
Two Square Theorem
Fermats FactorizaMethod
Fermats Prime Nu
Fermats Little The
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Fermats LastTheorem Puzzled mathematicians for 350
years.
Found by his son in his copy of anedition of Diophantus and itincludes the statement that themargin was too small to include the
proof.
Pierre d
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17thCentury Mathematics
Blaise Pascal(June 19, 1623August 19, 1662) Known as a child prodigy tienne Pascal (father) and Antoinette B
(mother, whom which died in 1626).
French mathematician, physicist, inventwriter and Christian philosopher.
First education was confined to languagand not included any mathematics.
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17thCentury Mathematics
Blaises
Contributions Essay on conic
sections (16 years old)
First ArithmeticalMachine (Pascaline,18)
Bla
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Trait du triangle arithmtique("Trea
on the Arithmetical Triangle") of 165 Problem of Points and Gamblers Ru
Pascals Principle in Fluids
Roulette Machine (accidental inven Wrist Watch
Others are theology related
Bla
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Gerard (Girard) Desargues (February 21, 1591Sep
Born in aristocratic family
Mathematician and Engineer
Worked as tutor, engineer, arcand technical consultant
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Gerards Contribution
Desargues Perspective Theore
Epicycloidial Wheel
Gerard
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17thCentury MathematicsGerard
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17thCentury MathematicsGerard
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17thCentury Mathematics
Isaac Newton (1643-1727) A physicist, astronomer,
alchemist and a theologian
Made a book called thePhilosophi NaturalisPrincipia Mathematica orMathematical Principlesof Natural Philosophy or
also called as Principia
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Newton's Three Laws of Motion
Law of Inertia Every body persists in its state of being at rest or of moving uniformly straigh
forward, except insofar as it is compelled to change its state by forceimpressed
Force and Acceleration
The alteration of motion is ever proportional to the motive force impress'd;and is made in the direction of the right line in which that force is impress'd
F=ma
Action-Reaction
To every action there is always an equal and opposite reaction: or the forceof two bodies on each other are always equal and are directed in oppositedirections..
Isa
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During the Great Plague of 1665-6, he
developed a theory of light, discovereand quantified gravitation, and pionea revolutionary new approach tomathematics: infinitesimal calculus.
calculated a derivative function ()which gives the slope at any point of function()
Isa
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17thCentury MathematicsIsa
Differe(derivaapprothe slocurve intervaappro
zero
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17thCentury Mathematics This process of calculating the slope or der
of a curve or function is called differentialcalculus or diffrentitation (in Newtonsterminology, the method of fluxions)
The opposite of differentiation is integratintegral calculus (or, in Newtons terminolomethod of fluents), and together differeand integration are the two main operatiocalculus.
Newtons Fundamental Theorem of Calcu
Isa
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17thCentury MathematicsIsa
Integrationapproximates thearea under acurve as the sizeof the samplesapproaches zero.
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Gottfried Wilhelm Leibniz(1646-1716)
German polymath
one of the three great 17thCentury rationalists
along with Descartes andSpinoza
politician and representativeof the royal house of Hanover
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17thCentury Mathematics He was perhaps the first to explicitly employ t
mathematicalnotion of a function to denote geometric conderivedfrom a curve, and he developed a system ofinfinitesimal calculus, independently of his
contemporary Sir Isaac Newton. Also revived the ancient method of solving
equations usingmatrices
Gottfried
th
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17thCentury Mathematics invented a practical calculating machine called S
Reckoner
pioneered the use of the binary system.
Gottfried
th i
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Also developed a very similar theo
calculus compared to Newton.
Within the short period of about tw
months he had developed a comtheory of differential calculus andintegral calculus
Gottfried
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17thCentury Mathematics Also often considered the most importa
logician between Aristotle in AncientGreece and George Boole and AugustuMorgan in the 19thCentury.
Even though he actually published noth
formal logic in his lifetime, he enunciateworking drafts the principal properties owe now call conjunction, disjunction,negation, identity, set inclusion and the
set
Gottfried
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Newton and Leibniz
Unlike Newton, however, he was more than happy to puwork, and so Europe first heard about calculus from Leibniz inand not from Newton (who published nothing on the subjec1693). When the Royal Society was asked to adjudicate betwrival claims of the two men over the development of the thecalculus, they gave credit for the first discovery to Newton, a
for the first publication to Leibniz. However, the Royal Societyunder the rather biased presidency of Newton himself, lateaccused Leibniz of plagiarism, a slur from which Leibniz neverecovered.
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Ironically, it wasLeibnizs mathematicsthat eventuallytriumphed, and hisnotation and his way of
writing calculus, notNewtons more clumsynotation, is the one stillused in mathematicstoday.
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ENDof the
17thCENTURY
18th C t M th ti
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18thCentury Mathematics
Bernoulli Brothers
(Jacob Bernoulli & Johann Bernoulli)Leonhard Euler
Christian Goldbach
Abraham de Moivre
Joseph Louis Lagrange
Pierre-Simon Laplace
Adrien-Marie Legendre
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Bernoulli Brothers Jacob (1654-1705)
Johann Bernoulli (1667-1748)
Bernoulli family - prosp
of traders and scholar- Basel in Switzerlaat that time was the gcommercial hub of ceEurope
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Jacob Bernoulli(1654-1705)
professor at Basel University
helped to consolidate infinitesimal c
developed a technique for solving sedifferential equations
18th Century Mathematics Jac
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18thCentury Mathematics
The Art of Conjecture - published in
- consolidated existing knowledge on protheory and expected values
- theory of permutations and combination
- Bernoulli trials and Bernoulli distribution- Bernoulli Numbers sequence
18th Century Mathematics Jac
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BernoulliNumbers
18th Century Mathematics Jac
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published papers on transcendental c
invented polar coordinates the first to use the word integral to re
the area under a curve.
discovered the approximate value of tirrational number .
died from tuberculosis at the age of 54
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Johann Bernoulli(1667-1748 ) took over his brother's position further developed infinitesimal calculus inc
the calculus of variation, functions for cufastest descent (brachistochrone) and cat
curve calculus of variations - useful in fields as d
as engineering, financial investment, archiand construction, and even space travel
18th Century Mathematics Johan
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first derivebrachistrocurve, usi
calculus ovariation
18th Century Mathematics Johan
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18thCentury Mathematics
Guillaume de l'Hpital- published a bo
his own name consisting almost entirelyJohann's lectures de l'Hpital's Rule - famous rule about
0 0
his sons Nicolaus, Daniel and Johann IIgrandchildren Jacob II and Johann III, all accomplished mathematicians andteachers
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18 Century Mathematics
Leonhard Euler (1707-1783 )
18th Century Mathematics Leon
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18 Century Mathematics born in Basel, Switzerland, and he studied fo
while under Johann Bernoulli at Basel Univer
spent his academic life in Russia and Germaespecially in the burgeoning St. Petersburg othe Great and Catherine the Great.
collected works comprise nearly 900 books
produced on average one mathematical pevery week
much of the notation used by mathematicitoday was either created, popularized or
standardi ed b E ler
18th Century Mathematics Leon
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18th Century Mathematics Leon
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18 Century Mathematics
= 1-sometimes known as Eulers Identity- combines arithmetic, calculus, trigonand complex analysis into what has be
called "the most remarkable formula inmathematics", "uncanny and sublime""filled with cosmic beauty", among othdescriptions.
18th Century Mathematics Leon
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18 Century Mathematics
= + - Eulers Formula.- demonstrate the deep relationship
between trigonometry, exponentiacomplex numbers
18th Century Mathematics Leon
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18 Century Mathematics
Basel problem
- calculation of infinite sums- Bernoullis had tried and failed to solve it
- what was the precise sum of the reciprocals of thsquares of all the natural numbers to infinity i.e. 112
132 + 142 ... (a zeta function using a zeta constant- showed that the infinite series was equivalent to aproduct of prime numbers, an identity which wouldinspire Riemanns investigation of complex zeta fun
18th Century Mathematics Leon
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18 Century Mathematics
SevenBridges ofKnigsbergProblem
18th Century Mathematics Leon
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18 Century Mathematics
Some list of theorems and methods
pioneered by Euler demonstration of geometrical proper
such as Eulers Line and Eulers Circle;
definition of the Euler Characteristicfor the surfaces of polyhedra
new method for solving quartic equa
18th Century Mathematics Leon
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18 Century Mathematics
the Prime Number Theorem
proofs (and in some cases disproosome of Fermats theorems and
conjectures
discovery of over 60 amicable num method of calculating integrals w
complex limits
18thCentury Mathematics Leon
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8 Ce u y a e a cs
18thCentury Mathematics Leon
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y
1766he accepted an invitation from
Catherine the Great to return to the SPetersburg Academy in Russia.
1771he was marred by tragedy
1773his dear wife Katharina died.He later married Katharina's half-sister, Sa
Abigail
1783he died from a brain hemorrhage
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y
Christian Goldbach (1690-1764)
18thCentury Mathematics Christian
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y
Goldbach Conjecture
- every even integer greater than 2 can bexpressed as the sum of two primes- every integer greater than 5 can be exp
as the sum of three primes Goldbach-Euler Theorem
- the sum of 1/(p 1) over the set of perfepowers p, excluding 1 and omittingrepetitions, converges to 1
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y
Abraham de Moivre (1667-1754)
18thCentury Mathematics Abraham
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y
de Moivre'sformula:
( + ) = cos() + ( generalized Newtons famous binomia
theorem into the multinomial theorem pioneered the development of analyt
geometry work on the normal distribution probability theory
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Joseph Louis Lagrange (1736-1813 )
18thCentury Mathematics Joseph Lou
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joint work on the calculus of variatio contributed to differential equation
number theory originate the theory of groups
four-square theoremAny natural number can berepresented as the sum of foursquar
18thCentury MathematicsJoseph Lou
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Lagranges Theorem or Lagranges
Value Theorem
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Pierre-Simon Laplace (1749-1827 )
18thCentury Mathematics Pierre Sim
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the French Newton
Celestial Mechanics- translated the geometric study of classical me
to one based on calculus.
work on differential equations and finitdifferences
he developed his own version of the soBayesian interpretation of probabilityindependently of Thomas Bayes.
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Adrien-Marie Legendre (1752-1833 )
18thCentury Mathematics Adrien Ma
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contributions to statistics, number th
abstract algebra and mathematicaanalysis Least squares method for curve-fittin
linear regression, the quadratic reci
law, the prime number theorem andwork on elliptic functions Elements of Geometry
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ENDof the
18th
CENTURY
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Jean Robert Argand (1768-1822)
19th Century MathematicsJean Ro
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Born in Switzerland on 1768. He was a Parisiabookkeeper and an amateur mathematici
His background and education are mostlyunknown. Since his knowledge of mathemawas self-taught and he did not belong to amathematical organizations, he likely pursumathematics as a hobby rather than a pro
In 1806, he published his own invention andelaboration of a geometric representationof complex numbers and the operations upthem (his major contribution to mathematic
19th Century Mathematics Jean Ro
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Essai sur une maniere de reprenter les quaimaginaires daps les constructions gomriqu
(Essay on a method of representing imaginaryquantities)- discussion of models for generating negative n
by repeated subtraction; one used weights removepan of a beam balance, the other subtracted fransum of money.
- concluded that distance may be considered adirection, and that whether a negative quantity isconsidered real or imaginary depends upon the quantity measured.
19th Century Mathematics Jean Ro
http://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Complex_number -
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- This initial use of the word imaginary for anegative number is related to the
mathematical-philosophical debates ofthe time as to whether negative numberswere numbers, or even existed.
- In 1813, it was republished in the Frenchjournal Annales de Mathmatiques. TheEssay discussed a method ofgraphing complex numbers via analyticalgeometry. It proposed the interpretation ofthe value i as a rotation of 90 degrees inthe Argand plane.
19th Century Mathematics Jean Ro
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He included in the book, the first use of the wordabsolute in the sense of the absolute value of apositive, negative, or complex number; of the bar pair of letters to indicate what is today called a ve
Later in the Essay, Argand used the term modulus(module) for the absolute value or the length of a representing a complex number.
His last article appeared in the volume ofAnnales18151816 and dealt with a problem in combinatioit Argand devised the notation (m, n) for thecombinations of mthings taken nat a time and thnotation Z(m, n) for the number of such combinatio
19th Century Mathematics Jean Ro
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Argand is also renowned for delivering a proof ofthe Fundamental Theorem of Algebrain his 1814
work Rflexions sur la nouvelle thoried'analyse (Reflections on the new theory ofanalysis). It was the first complete and rigorousproofof the theorem, and was also the first proofto generalize the fundamental theorem ofalgebra to include polynomialswith complex
coefficients. In 1978, it was called by TheMathematical Intelligencer both ingenious andprofound, and was later referencedin Cauchy'sCours dAnalyseand Chrystal'sinfluential textbook Algebra.
19th Century MathematicsJean Ro
http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebrahttp://en.wikipedia.org/wiki/Rigorous_proofhttp://en.wikipedia.org/wiki/Rigorous_proofhttp://en.wikipedia.org/wiki/Polynomialshttp://en.wikipedia.org/wiki/Cauchyhttp://en.wikipedia.org/wiki/George_Chrystalhttp://en.wikipedia.org/wiki/George_Chrystalhttp://en.wikipedia.org/wiki/Cauchyhttp://en.wikipedia.org/wiki/Polynomialshttp://en.wikipedia.org/wiki/Rigorous_proofhttp://en.wikipedia.org/wiki/Rigorous_proofhttp://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra -
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Argand recognized the nonrigorous nature of hisreasoning, but he defined his goals as clarifying
thinking about imaginaries by setting up a new viethem and providing a new tool for research ingeometry.
He used complex numbers to derive severaltrigonometric identities, to prove Ptolemys theore
and to give a proof of the fundamental theorem algebra.
He died on 1822 in Paris.
19th Century Mathematics va
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variste Galois (1811-183
Galois was born on 25 OctobNicolas-Gabriel Galois and AMarie (born Demante).
His mother, the daughter of aa fluent reader of Latinand cliteratureand was responsibleson's education for his first twe
19th Century Mathematics va
http://en.wikipedia.org/wiki/Latinhttp://en.wikipedia.org/wiki/Classical_literaturehttp://en.wikipedia.org/wiki/Classical_literaturehttp://en.wikipedia.org/wiki/Classical_literaturehttp://en.wikipedia.org/wiki/Classical_literaturehttp://en.wikipedia.org/wiki/Latin -
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In October 1823, he entered the Lyce Louis-
le-Grand, and despite some turmoil in theschool at the beginning of the term (whenabout a hundred students were expelled),Galois managed to perform well for the firsttwo years, obtaining the first prize in Latin.
He soon became bored with his studies and,at the age of 14, he began to take a seriousinterest in mathematics.
va19th Century Mathematics
http://en.wikipedia.org/wiki/Lyc%C3%A9e_Louis-le-Grandhttp://en.wikipedia.org/wiki/Lyc%C3%A9e_Louis-le-Grandhttp://en.wikipedia.org/wiki/Lyc%C3%A9e_Louis-le-Grandhttp://en.wikipedia.org/wiki/Lyc%C3%A9e_Louis-le-Grandhttp://en.wikipedia.org/wiki/Lyc%C3%A9e_Louis-le-Grandhttp://en.wikipedia.org/wiki/Lyc%C3%A9e_Louis-le-Grandhttp://en.wikipedia.org/wiki/Lyc%C3%A9e_Louis-le-Grandhttp://en.wikipedia.org/wiki/Lyc%C3%A9e_Louis-le-Grandhttp://en.wikipedia.org/wiki/Lyc%C3%A9e_Louis-le-Grand -
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In 1828, he attempted the entrance examination fthe cole Polytechnique, the most prestigiousinstitution for mathematics in France at the time.Without the usual preparation in mathematics, andfailed for lack of explanations on the oral examina
In that same year, he entered the cole Normale(
known as l'cole prparatoire), a far inferior institutfor mathematical studies at that time, where he fosome professors sympathetic to him.
19th Century Mathematics va
http://en.wikipedia.org/wiki/%C3%89cole_Polytechniquehttp://en.wikipedia.org/wiki/%C3%89cole_Normalehttp://en.wikipedia.org/wiki/%C3%89cole_Normalehttp://en.wikipedia.org/wiki/%C3%89cole_Normalehttp://en.wikipedia.org/wiki/%C3%89cole_Normalehttp://en.wikipedia.org/wiki/%C3%89cole_Polytechniquehttp://en.wikipedia.org/wiki/%C3%89cole_Polytechniquehttp://en.wikipedia.org/wiki/%C3%89cole_Polytechnique -
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In 1829, Galois's first paper, on continued fractwas published. It was at around the same timehe began making fundamental discoveries in theory of polynomial equations.
He submitted two papers on this topic tothe Academy of Sciences. Augustin Louis
Cauchyreferred these papers, but refused to them for publication for reasons that still remaiunclear.
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Galois's mathematical contributions were publisfull in 1843 when Liouvillereviewed his manuscrideclared it sound. It was finally published in theOctoberNovember 1846 issue of the Journal dMathmatiques Pures et Appliques. The most fcontribution of this manuscript was a novel proo
there is no quintic formulathat is, that fifth anddegree equations are not generally solvable byradicals.
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While many mathematicians before Galois gave considwhat are now known as groups, it was Galois who was t
use the word group (in French groupe) in a sense close technical sense that is understood today, making him afounders of the branch of algebra known as group theo
He developed the concept that is today known as a nosubgroup. He called the decomposition of a group intoright cosetsa proper decomposition if the left and rightcoincide, which is what today is known as a normal subalso introduced the concept of a finite field(also knowna Galois fieldin his honor), in essentially the same form aunderstood today.
I hi l t l tt t Ch li d tt h d i t
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In his last letter to Chevalier, and attached manuscripts, second of three, he made basic studies of linear groups fields:
He constructed the general linear group over a prime GL(,p) and computed its order, in studying the Galoithe general equation of degree p.
He constructed the projective special linear groupPSLGalois constructed them as fractional linear transformsobserved that they were simple except if p was 2 or 3.were the second family of finite simple groups, afterthe alternating groups.
He noted the exceptional factthat PSL(2,p) is simple aon p pointsif and only if p is 5, 7, or 11.
G l i t i ifi t t ib ti t th ti b f
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Galois most significant contribution to mathematics by far development of Galois theory which make hailed him as Fof Modern Algebra
He realized that the algebraic solution to a polynomialequrelated to the structure of a group of permutationsassociawith the roots of the polynomial, the Galois groupof thepolynomial.
He found that an equation could be solved in radicalsif on
find a series of subgroups of its Galois group, each one normits successor with abelianquotient, or its Galois group is solvThis proved to be a fertile approach, which later mathemaadapted to many other fields of mathematics besides the tof equationsto which Galois originally applied it to.
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Georg Ferdinand Cantor(18451918)
Georg Ferdinand Ludwig Philipp Cantor is aGerman mathematician who was born in1845 in Russia.
His first ten papers were on number theory,after which he turned to calculus (analysis at
that time). He is best known as the firstmathematician to really understand theconcept of infinity and to give itmathematical precision.
C t t ti i t t th t it ht t b
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Cantors starting point was to say that it ought to be poto add infinity and infinity. He realized that it was actua
possible to add and subtract infinities, and that beyonwas normally thought of as infinity existed another, larginfinity, and then other infinities beyond that. In fact, heshowed that there may be infinitely many sets of infinitnumbers - an infinity of infinities - some bigger than othconcept which clearly has philosophical, as well as jusmathematical, significance. The sheer audacity of Catheory set off a quiet revolution in the mathematicalcommunity, and changed forever the way mathematapproached.
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In the early 1870s, he realized that the set of natural nuand any infinite subset of the natural numbers have the
number of elements. Same is true with the set of naturanumbers paired to the set of integers and the set of ratnumbers. However, when Cantor considered an infiniteof decimal numbers, he then proved that the infinity odecimal numbers is bigger than the infinity of natural n
He coined the word transfinite to distinguish these infnumbers from the absolute infinity (which he equated He used the Hebrew letter aleph to describe the sizes osets and developed transfinite arithmetic.
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Cantor is also responsible for the real origin of set
theory. Cantor showed that there could be infinisets of different sizes just as there were different fsets. He introduced the concepts of ordinalityancardinalityand the arithmeticof infinite sets.
Cantor died in 1918.
Georg Ferdinand Frobenius19th Century Mathematics
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Georg Ferdinand Frobenius(18491917)
He was born on 26 October 1849in Charlottenburg, a suburb of Berlinfrom parents Christian FerdinandFrobenius, a Protestantparson, andChristine Elizabeth Friedrich.
He was a Germanmathematician, bestknown for his contributions to the theoryof elliptic functions, differentialequationsand to group theory.
He is known for the famous determinantal identities
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He is known for the famous determinantal identitiesknown as Frobenius-Stickelberger formulae,
governing elliptic functions, and for developing thetheory of biquadratic forms.
He was also the first to introduce the notion of ratioapproximations of functions (nowadays known asapproximants), and gave the first full proof for
the CayleyHamilton theorem. He also lent his namcertain differential-geometric objects inmodern mathematical physics, known as Frobeniumanifolds.
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In 1867, after graduating, he went to the University ofGttingenwhere he began his university studies but h
studied there for one semester before returning to Behe attended lectures by Kronecker, Kummerand KarWeierstrass. He received his doctorate (awarded withdistinction) in 1870 supervised by Weierstrass. His thesisupervised by Weierstrass, was on the solution of diffeequations.
In 1874, after having taught at secondary school levethe Joachimsthal Gymnasium then at the Sophienreahe was appointed to the University of Berlin as an extprofessor of mathematics.
Group theory was one of Frobenius' principal interests in th
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Group theorywas one of Frobenius principal interests in thhalf of his career. One of his first contributions was the proothe Sylow theoremsfor abstract groups. Earlier proofs had
for permutation groups. His proof of the first Sylow theoremexistence of Sylow groups) is one of those frequently used
Burnside's lemma, sometimes also called Burnside's countitheorem, the Cauchy-Frobenius lemma or the orbit-counttheorem, is a result in group theorywhich is often useful in
taking account of symmetrywhen counting mathematicaIts various eponyms include William Burnside, George PlyLouis Cauchy, and Ferdinand Georg Frobenius. The result to Burnside himself, who merely quoted it in his book 'On thGroups of Finite Order', attributing it instead to Frobenius (
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Frobenius also has proved the following fundamentheorem: If a positive integer n divides the order |
a finite groupG, then the number of solutions of thequation xn = 1 in G is equal to kn for some positivinteger k. He also posed the following problem: If,above theorem, k = 1, then the solutions of theequation xn = 1 in G form a subgroup. Many years
this problem was solved for solvable groups. Only 1991, after the classification of finite simple groupsthis problem solved in general.
More important was his creation of the theory of group
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More important was his creation of the theory of groupcharactersand group representations, which are fundamtools for studying the structure of groups. This work led to
of Frobenius reciprocityand the definition of what are nocalled Frobenius groups. A group G is said to be a Frobenif there is a subgroup H < G such that
All known proofs of that theorem make use of characters
paper about characters (1896), Frobenius constructed thcharacter table of the group PSL(2,p) of order (1/2)(p3 odd primes p (this group is simple provided p > 3). He alsofundamental contributions to the representation theory osymmetric and alternating groups.
Frobenius introduced a canonical way of turning primes
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Frobenius introduced a canonical way of turning primesinto conjugacy classesin Galois groupsover Q. Specificaif K/Q is a finite Galois extension then to each (positive)prime p which does not ramifyin K and to each primeideal P lying over p in K there is a unique element g of Gasatisfying the condition g(x) = xp (mod P) for all integers xVarying P over p changes g into a conjugate (and everyconjugate of g occurs in this way), so the conjugacy cla
the Galois group is canonically associated to p. This is called the Frobenius conjugacy class of pand any
element of the conjugacy class is called a Frobenius eleof p.
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