Modern Period (17th-19th Century) on Mathematics

8/10/2019 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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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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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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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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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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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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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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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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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

18th Century Mathematics


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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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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



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BernoulliNumbers



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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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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



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= 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.



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= + - Eulers Formula.- demonstrate the deep relationship

between trigonometry, exponentiacomplex numbers



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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



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SevenBridges ofKnigsbergProblem



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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



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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.

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.



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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



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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

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.

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.

http://en.wikipedia.org/wiki/Continued_fractionshttp://en.wikipedia.org/wiki/Polynomial_equationhttp://en.wikipedia.org/wiki/Academy_of_Scienceshttp://en.wikipedia.org/wiki/Augustin_Louis_Cauchyhttp://en.wikipedia.org/wiki/Augustin_Louis_Cauchyhttp://en.wikipedia.org/wiki/Augustin_Louis_Cauchyhttp://en.wikipedia.org/wiki/Augustin_Louis_Cauchyhttp://en.wikipedia.org/wiki/Academy_of_Scienceshttp://en.wikipedia.org/wiki/Polynomial_equationhttp://en.wikipedia.org/wiki/Polynomial_equationhttp://en.wikipedia.org/wiki/Continued_fractions


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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.

http://en.wikipedia.org/wiki/Joseph_Liouvillehttp://en.wikipedia.org/wiki/Journal_de_Math%C3%A9matiques_Pures_et_Appliqu%C3%A9eshttp://en.wikipedia.org/wiki/Journal_de_Math%C3%A9matiques_Pures_et_Appliqu%C3%A9eshttp://en.wikipedia.org/wiki/Quintic_equationhttp://en.wikipedia.org/wiki/Quintic_equationhttp://en.wikipedia.org/wiki/Quintic_equationhttp://en.wikipedia.org/wiki/Quintic_equationhttp://en.wikipedia.org/wiki/Journal_de_Math%C3%A9matiques_Pures_et_Appliqu%C3%A9eshttp://en.wikipedia.org/wiki/Journal_de_Math%C3%A9matiques_Pures_et_Appliqu%C3%A9eshttp://en.wikipedia.org/wiki/Journal_de_Math%C3%A9matiques_Pures_et_Appliqu%C3%A9eshttp://en.wikipedia.org/wiki/Journal_de_Math%C3%A9matiques_Pures_et_Appliqu%C3%A9eshttp://en.wikipedia.org/wiki/Journal_de_Math%C3%A9matiques_Pures_et_Appliqu%C3%A9eshttp://en.wikipedia.org/wiki/Journal_de_Math%C3%A9matiques_Pures_et_Appliqu%C3%A9eshttp://en.wikipedia.org/wiki/Journal_de_Math%C3%A9matiques_Pures_et_Appliqu%C3%A9eshttp://en.wikipedia.org/wiki/Joseph_Liouville


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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

http://en.wikipedia.org/wiki/Group_(algebra)http://en.wikipedia.org/wiki/Group_theoryhttp://en.wikipedia.org/wiki/Normal_subgrouphttp://en.wikipedia.org/wiki/Normal_subgrouphttp://en.wikipedia.org/wiki/Cosethttp://en.wikipedia.org/wiki/Finite_fieldhttp://en.wikipedia.org/wiki/Galois_fieldhttp://en.wikipedia.org/wiki/Galois_fieldhttp://en.wikipedia.org/wiki/Finite_fieldhttp://en.wikipedia.org/wiki/Cosethttp://en.wikipedia.org/wiki/Normal_subgrouphttp://en.wikipedia.org/wiki/Normal_subgrouphttp://en.wikipedia.org/wiki/Group_theoryhttp://en.wikipedia.org/wiki/Group_(algebra)


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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

http://en.wikipedia.org/wiki/General_linear_grouphttp://en.wikipedia.org/wiki/Projective_special_linear_grouphttp://en.wikipedia.org/wiki/Simple_grouphttp://en.wikipedia.org/wiki/Alternating_grouphttp://en.wikipedia.org/wiki/Exceptional_objecthttp://en.wikipedia.org/wiki/Projective_linear_grouphttp://en.wikipedia.org/wiki/Projective_linear_grouphttp://en.wikipedia.org/wiki/Projective_linear_grouphttp://en.wikipedia.org/wiki/Projective_linear_grouphttp://en.wikipedia.org/wiki/Projective_linear_grouphttp://en.wikipedia.org/wiki/Projective_linear_grouphttp://en.wikipedia.org/wiki/Exceptional_objecthttp://en.wikipedia.org/wiki/Alternating_grouphttp://en.wikipedia.org/wiki/Simple_grouphttp://en.wikipedia.org/wiki/Projective_special_linear_grouphttp://en.wikipedia.org/wiki/General_linear_group


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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.

http://en.wikipedia.org/wiki/Polynomialhttp://en.wikipedia.org/wiki/Permutationhttp://en.wikipedia.org/wiki/Galois_grouphttp://en.wikipedia.org/wiki/Nth_roothttp://en.wikipedia.org/wiki/Abelian_grouphttp://en.wikipedia.org/wiki/Solvable_grouphttp://en.wikipedia.org/wiki/Theory_of_equationshttp://en.wikipedia.org/wiki/Theory_of_equationshttp://en.wikipedia.org/wiki/Theory_of_equationshttp://en.wikipedia.org/wiki/Theory_of_equationshttp://en.wikipedia.org/wiki/Solvable_grouphttp://en.wikipedia.org/wiki/Abelian_grouphttp://en.wikipedia.org/wiki/Nth_roothttp://en.wikipedia.org/wiki/Galois_grouphttp://en.wikipedia.org/wiki/Permutationhttp://en.wikipedia.org/wiki/Polynomial


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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

19th Century Mathematics Geo


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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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Documents

Modern Period (17th-19th Century) on Mathematics