Math History Summary By TopicSpring 2011

Bolded items are more important.

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Numeration/Notation

Numeration

Egypt

• 3200BC – 200

• decimal; hieroglyphic, hieratic numerals

• fractions: unit fractions only

◦ Old Kingdom (before 2050 BC): Eye of Horus fractions; MiddleKingdom: hieroglyphic fractions

◦ Rhind Papyrus, c. 1650 BC, is our most important source forEgyptian mathematics

Babylonia

• tokens in Mesopotamia, 8000 BC – 2000 BC

• cuneiform, 2000 BC: sexagesimal

◦ dot for placeholder

• fractions: sexagesimal

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India

• Hindu (Brahmin) numerals, 3rd cent BC

• place value, 8th-9th cent

• fractions writtenababab (Muslims added the bar later)

al-Khwarizmi, 800: book on numeration taught Muslim world the Hindunumeration system

• worked in the House of Wisdom in Baghdad

10th century, decimal fractions in Muslim world

Fibonacci, 1200: taught Europe the Hindu numeration system (Liber Abaci)

Notation

Diophantus, 250: some algebraic notation, didn’t catch on

15-17th centuries, symbols for arithmetic develop

16th century, decimal fractions in Europe (Rudolff, Stevin)

Viéte, 1600: symbolism for algebra

Leibniz, early 18th century: notation for calculus

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Algebra

Babylonia, 2000 BC: some linear equations;solving some quadratics by completing the square

Egypt, Rhind Mathematical Papyrus, 1650 BC: some linearequations

Egyptian, Babylonian mathematics all examples using specific numbers

• no proofs

• no abstractions

Pythagorean theorem led to incommensurablesZeno’s paradoxes resulted in avoidance of study of infinity

(“horror infiniti”)

• wanted to show that change (specifically, motion) was impossible

Eudoxus, 400 BC: geometric algebra

• essentially algebraic problems recast as geometrical to avoid prob-lems with irrational numbers

• only objects with same dimension can be equated

• solutions are line segments, not numbers

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al-Khwarizmi, 800: book on algebra(“completion and balancing”)

• solutions to all quadratics with at least one positive real root

• classifies them into five types

Khayyam, 1100: geometric solutions to many cubics

Fibonacci, 1225: book on quadratics and problems leading to them (LiberQuadratorum)

Stifel, 15th century: allows negative coefficients but not neg solnsto equations

Solution of the cubic

• del Ferro, 1500: x3+ px = q

• Tartaglia, 1530: x3+ px2 = q

• Cardano, 1540: general cubic (Cardano’s formulas)

◦ used complex numbers, but didn’t understand them at all (casusirreducibilis)

◦ but they seemed unavoidable

Ferrari, 1548: solution of the quartic

Bombelli, 1572: complex numbers can be written a+b√−1a+b√−1a+b√−1, a,b ∈ Ra,b ∈ Ra,b ∈ R

Harriott, 1600: negative solutions to equations allowed;move all terms to one side to solve equations

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Viéte, 1600

• developed algebraic symbolism

• Viéte’s formulas: coefficients of a polynomial are symmetric functions ofthe roots

Descartes (Discourse on Method), 1637: invention of analyticgeometry, connecting algebra and geometry

• showed that Eudoxus’ dimensional restriction was unnecessary byshowing that all geometric computations could be considered to re-sult in lengths

• knew that a polynomial of degree n must have n roots (no proof)

Ruffini, 1799: almost proves that the general quintic and higherdegree polynomial cannot be solved by radicals

• invents lots of mathematics to do it, including stuff about permutationgroups

• ignored

Gauss

• 1799: Fundamental Theorem of Algebra

• 1801: modular arithmetic; amounts to much of abelian group theory

Abel, 1824: unsolvability of quintic

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Galois, 1832: solvability of polynomials by radicals linked toproperties of groups of permutations of their roots

• normal subgroups of those groups

Cauchy, 1834: studied permutation groups

• products, order of a perm, cycles, conjugacy, Cauchy’s theorem

W.R. Hamilton, 1843: quaternions, first “artificial” algebraicsystem

• development of such systems led to problems with negative numbersfading away

Kummer, 1844: ideal numbers (generalization of integers)

Cayley, 1849, 1878: abstract groups, group tables

Dedekind, 1871: ideals, prime ideals (in the Gaussian integers)

van Dyck, 1882-3: free groups, generators and relations

Hilbert, 1888: Hilbert Basis Theorem

Burnside, 1897: modern group theory

Fraenkel, 1914: first definition of an abstract ring

Noether, 1920: modern defn of ring; many theorems, esp. in ideal theory

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Number Theory

Before Greeks, just arithmetic

Pythagoreans, from 500 BC

• many results

• used figurative numbers

• all things held in common, including credit for mathematical results

• Pythagorean triples: could generate infinitely many, but not all

• Pythagorean theorem implied existence of incommensurables

◦ threatened Pythagorean idea that all quantities were ratios ofintegers

◦ Eudoxus’ “geometric algebra” avoided the problem by dealingonly with magnitudes, not with numbers

Euclid, 325 BC: Elements: two chapters on number theory

• formulas for all possible Pythagorean triples, but no proof

Eratosthenes, 200 BC: Sieve of Eratosthenes for finding primes

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Diophantus, 250 (Arithmetica): many number theory problems

Brahmagupta, 7th century: explains negative numbers by

“debt and fortune”

India, 9th century: zero is a numberFibonacci, 1225 (Liber Quadratorum): proof that Euclid’s formulas

give all Pythagorean triples

Fermat, first half of 17th century

• many problems, theorems, most without proof

• method of infinite descent

• Last Theorem: no solution to xn+ yn = znxn+ yn = znxn+ yn = zn in integers for n > 2n > 2n > 2

◦ not proved until 1995

◦ attempts to prove it generated much good mathematics

Pascal, first half of 17th century: Pascal’s triangle, connection tobinomial coefficients

• full development of mathematical induction from Maurolico’s firstuse

Euler, 18th century

• proofs of many of Fermat’s theorems

• conjectured the quadratic reciprocity law

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Lagrange, second half of 18th century into 19th

• Wilson’s theorem, solution to a Pell’s equation, proofs of many of Fer-mat’s theorems

Legendre, late 18th into 19th century

• conjectured a form of the quadratic reciprocity law

• contributed to proof of Fermat’s Last Theorem

Gauss

• 1801: Disquisitiones Arithmeticae: modular arithmetic

• proof of the quadratic reciprocity law

• conjectured the prime number theorem

Dirichlet, first half of 19th century

• Dirichlet series, the zeta function

• this is the beginning of analytic number theory

Riemann, 1859: Riemann zeta function, theorems on thedistribution of primes

Dedekind, second half of 19th into 20th century

• algebraic number fields, ideals, zeta function of a number field

• Dedekind cuts construct RRR from QQQ

◦ but require the use of completed infinities

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Hilbert

• 1893: synthesis of algebraic number theory based on Dedekind’swork

◦ formed algebraic number theory into a field with its own meth-ods and results

• some of his 23 problems were number-theoretic and were very influential

Hardy/Littlewood/Ramanujan, first half of 20th century: manyresults in number theory

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Geometry

Before the Greeks, some formulas for areas of plane figures,volumes of solids

Thales, 600 BC: first proofs, a few theorems (Thales’ theorem)

Pythagoreans, from 500 BC

• many theorems (Pythagorean theorem)

• Platonic solids

• believed both that lines were made up of points and that they wereinfinitely divisible

Eudoxus, 400 BC: method of exhaustion

• no records left; Archimedes says he invented it

Three big classical problems of geometry

• squaring the circle

• doubling the cube

• trisecting the angle

Hippocrates of Chios, 400 BC: worked on all three of the bigproblems

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Euclid

• 325 BC: Elements of Geometry

◦ axiomatized geometry; all results derived from a few axioms

· five Common Notions: assumptions about quantity, espe-cially equality

· five Postulates: specifically geometric assumptions

◦ first axiomatic system

◦ Parallel Postulate (P5) controversial from the beginning

◦ geometric algebra

Archimedes, 3rd century BC

• The Measurement of a Circle

◦ ratio of circumference to diameter, approximation of π

◦ method of exhaustion: approximate circumference more andmore closely by polygons the perimeters of which can be cal-culated

• On the Sphere and the Cylinder

◦ surface area of sphere, other results

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• The Sand-Reckoner

◦ shows how to extend the Greek numeration system to describe arbi-trarily large numbers

• On Spirals

◦ spiral of Archimedes

• Quadrature of the Parabola

◦ by a different use of the method of exhaustion

• The Method of Mechanical Theorems

◦ think of surfaces as “made up of” lines, volumes of revolution as“made up of” circles

◦ discovery technique, not a proof technique, for Archimedes

Apollonius of Perga, 200 BC: Conics

Proclus, 450: our source for much ancient work

• tried to revive Greek geometry (unsuccessfully)

Saccheri

• 1733: tried to prove that adding the negation of the Parallel Postulateto the other postulates of Euclidean geometry resulted in a contradic-tion

• Saccheri quadrilaterals

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Pascal, 1639: Mystic Hexagon Theorem

Legendre, 1794: famous geometry text, first to displace Euclid

Gauss

• 1796: construction of regular 17-gon

• 1816–1824: non-Euclidean geometry

◦ first to develop it

◦ hyperbolic geometry

◦ told only a few people at first

• 1827: differential geometry (Theorema Egregium, Gauss-Bonnet Theo-rem)

Bolyai, 1823: independently developed hyperbolic geometry

Lobachevsky, 1826: independently developed hyperbolic geometry

Riemann, 1854: elliptic geometry

Beltrami, 1868:

• Parallel Postulate is independent of the other four

◦ proved that geometry of geodesics on the pseudosphere was hy-perbolic geometry

◦ found a map from the plan to the pseudosphere that preservedangles

◦ this sent Postulates 1–4 to true statements

◦ but Postulate 5 is not true on the pseudosphere, so it cannot beproved from the other four

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• hyperbolic geometry consistent iff Euclidean geometry is so

Klein, 1872: Erlanger Programm: general defn of geometry interms of symmetry groups

Hilbert

• 1892: Nullstellensatz (algebraic geometry)

• 1899: first completely rigorous axiomatization of Euclidean geome-try

Poincaré, 1895: invented algebraic topology

• Poincaré conjecture (surfaces with same fundamental group as Sn arehomeomorphic to Sn)

◦ finally proved by Perelman, 2003

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Calculus/Analysis

Archimedes

• used method of exhaustion two different ways to approximate ratio ofcircumference of circle to diameter and to do quadrature of parabola

• Method of Mechanical Theorems

Napier, 1614: logarithms

Descartes, 1637: analytic geometry, solution of tangent problem

Fermat, first half of 17th century

• independent invention of analytic geometry

• quadrature of y = xpqy = xpqy = xpq (by ad hoc method)

• method for finding extrema of some curves

Pascal, first half of 17th century: quadrature of sine curve

Cavalieri, first half of 17th century

• quadrature of y = xny = xny = xn for small n

• Cavalieri’s Principle

Wallis, 1655: quadrature of y = xny = xny = xn

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Barrow, mid 17th century: finding tangents using the differential triangle

• explicitly let quantities→ 0

Newton

• 1665: General Binomial Theorem

◦ allows infinite-series expansion of some functions

• 1666: method of fluxions (differential calculus)

◦ manuscript De Analysi, 1669

◦ curve generated by moving point

· curve is a fluent, velocity of generation is its fluxion

◦ algebraic approach based on binomial theorem

• 1687: Principia (mathematical physics)

Leibniz, 1670s: developed much calculus

• geometric approach

• product rule, Fundamental Theorem of Calculus

• great notation; we use it today

Bernoullis

• Jacob and Johann, late 17th into 18th cent

• development and applications of calculus and DEs

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• Jacob: beginnings of calc of variations

• Jacob: book on probability, left unfinished

Berkeley, 1734: The Analyst: criticism of infinitesimals

• in calculations, people first divided by these (so they can’t be zero)and then threw them away and treated the results as exact (so themust be zero)

• both Newton and Leibniz were concerned about them

Taylor, 1715: finite differences, Taylor series, Taylor’s Theorem

Maclaurin

• Maclaurin series

• 1742: Treatise on Fluxions

◦ convinced English mathematicians that calculus could be foundedon geometry

d’Alembert, mid-18th century

• mechanics, calculus/DEs (esp. PDE—wave equation)

• idea of limit, but too vague to be useful

◦ thought calculus should somehow be based on limits

• ratio test

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Laplace, late 18th–early 19th cent

• analysis/DEs

• celestial mechanics

• determinants

• full development of probability theory using calculus

Euler, 18th century

• calculus/DEs, esp. infinite series

• complex analysis

• definitions of function—first in terms of formulas, then in terms offunctional dependency

◦ sine, cosine are functions of a real variable

• invented graph theory for solution of Seven Bridges problem

Lagrange, second half of 18th into 19th cent

• theoretical mechanics (Lagrangian mechanics)

◦ mechanics as pure mathematics

◦ special solution to the three-body problem (Lagrange points)

• calculus/DEs (variation of parameters)

◦ tried to base calculus on infinite series

• calculus of variations

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Legendre, 18th to 19th century: mechanics, elliptic functions

Bolzano (late 18th–19th century): much work on limits

• mostly ignored; Cauchy and Weierstrass had to rediscover it

Gauss, late 18th–19th century

• differential geometry (Theorema Egregium, Gauss-Bonnet Theorem)

• complex plane, 1799

◦ not the first: Wessel, 1797

◦ Argand also thought of it, 1806

Fourier, 1822: Fourier series, study of heatJacobi, first half of 19th century: elliptic functions, PDEs,

determinants (the Jacobian)

• Abel did similar work on elliptic functions at about the same time

Dirichlet, first half of 19th century: Dirichlet series, the zeta function

Cauchy

• precise defn of limit, derivative, continuity, sum of infinite series

• developed calculus from these; makes infinitesimals unnecessary

◦ this finally answered Berkeley’s criticism of calculus

• Cauchy criterion for convergence of a sequence

• complex analysis (Cauchy Integral Theorem, etc.)

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Riemann, mid-19th century

• Riemann integral

• elliptic functions

• analytic number theory (Riemann zeta function, the Riemann hy-pothesis)

Weierstrass, second half of 19th century

• “father of modern analysis”

• complete rigor

◦ we do and teach analysis in his way

• much real, complex analysis

Poincaré, 19th to early 20th cent

• DEs, dynamical systems, chaos (Poincaré-Bendixson theorem)

• complex analysis

Hilbert, 19th to early 20th cent

• functional analysis (Hilbert spaces)

• mathematical physics

• address in 1900 gave 23 problems which set course for much of 20thcentury mathematics

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Function concept

Aristotle, 350 BC: used line segment to indicate durationOresme, 1350: perpendicular lines, one for duration, one for a quantity

depending on itGalileo, 1638: a 1–1 mapping between concentric circles

Leibniz, 1692: “function” : tangent line as function of point on curve(and other geometric dependencies)

Euler, 18th cent: defn first in terms of algebraic formulæ, later as onequantity depending on another

Fourier, 1822: function is any relation between quantities

Dirichlet, 1837: pretty modern; like Fourier’s

Frege, late 19th cent: function = set of ordered pairs

Wiener, 1914: fully modern defn

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InfinityGreeks allowed only “potential infinities,” not completed ones

Zeno’s paradoxes resulted in avoidance of study of infinity

• wanted to show that change (specifically, motion) was impossible

Aristotle, 4th century BC

• allowed only “potential infinities” (processes that never have to stop,like counting), not “completed infinities” (infinite sets, like NNN)

Augustine (400) accepted the totality of the natural numbers as a real thing

Aquinas (1250) accepted the infinite divisibility of the line

Gauss (19th century) agreed with no completed infinity

Bolzano (early 19th century): paradoxes of infinite sets (mostly ignored)

Kronecker (19th century) begins constructivism

• mathematical objects exist only if an algorithm can be given to con-struct them

Cantor, late 19th century: consistent theory of infinite sets

• definition of set, equal cardinality of sets, ordinals

• proved QQQ countable, RRR uncountable (Cantor’s diagonal argument)

• proved card(Rn) = card(R)card(Rn) = card(R)card(Rn) = card(R)

• conjectured the well-ordering axiom and the continuum hypothesis

• theory met with much resistance (and some support)

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Matrices

appear in the Nine Chapters, 263

Vandermonde uses idea of determinants, 1772

used but not studied by Gauss, 1801

studied by Cauchy, 1812

• determinant theroems, eigenvalues, diagonalization, but none of thesein general

Jacobi, 1830: determinants

Sylvester: 1850: determinant theorems; 1884: rank-nullity theoremCayley, middle of 19th cent: more general theory; inverse of a matrix;

case of Cayley-Hamilton theoremFrobenius, 1878: general theory

• full proof of C-H theorem, rank, orthogonality, etc.

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Group theory

started with Euler and Gauss, 18th to first part of 19th cent — modulararithmetic

Lagrange, 1771: studied perms, but didn’t define a product

Ruffini, 1799: proved lots of stuff about perm groups, but was ignored

Cauchy, 1815: groups of perms of roots of polynomials; 1844: groups ofpermutations

Galois, 1831: normal subgroups

Cayley, 1849: abstract groups, group tables; 1878: much theory

van Dyck, 1882-3: free groups, generators and relations

Burnside, 1897: modern group theory

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