i hate mathematicians
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| Origins of probability theory | in gambling and the collection of statistical data for insurance and mortality tables
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| John Graunt | first to collect data and make statistical conclusions, used Bills of Mortality
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| Gods and randomness | casting dice had origins in religion, because the gods controlled the outcome
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| Cardan | first to point out that outcomes of a die toss had equal probability
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| Blaise Pascal | (1623-1662) founder of modern probability theory, developed mystic hexagon theorem, invented calculator for addition and multiplication, became religious, studied cycloid curve
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| Charles Babbage | (1792-1871) invented a calculator to generate logarithm tables, the Difference Engine and the Analytical Engine
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| Pierre de Fermat | (1601-1665) founder of modern probability theory, wrote most discoveries in margins of Arithmetica, leaving no proof, invented analytic or coordinate geometry, worked in number theory
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| Pascal’s Triangle | the arithmetic triangle where each entry is the sum of the two numbers above it, nth row gives the coefficients in the binomial expansion (x+y)^n
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| First to discover Pascal’s triangle | Chinese and Arabs in about 1050
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| Proof by induction | invented by Francesco Maurolico (1494-1575) , first rigorously used by Pascal
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| Steps in Mathematical Induction | 1. Show P(k_o) is true for some integer n=k_o. (Usually k_o=1) 2. Show P(k+1) is true if P(k) is true. Then P(n) is true for all n≥k_o.
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| James Bernouilli | (1654-1705) one of the first mathematicians to understand Leibniz’s calculus, wrote Ars Conjectandi , tutored brother John, started differential equations
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| John Bernouilli | (1667-1748) was tutored by brother James and then tutored L’Hopital, started differential equations
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| Ars Conjectandi | contained fundamental theorems of probability, permutations, combinations, games of chance, and Bernouilli’s Theorem on the Law of Large numbers
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| Abraham De Moivre | (1667-1754) published Doctrines of Chances , estimated Stirling’s formula n!≈√2πn n^n e^(-n), and also created own formula 〖(cosθ+isinθ)〗^n=cos(nθ)+i sin(nθ)
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| Pierre Simon Laplce | (1749-1827) called the Newton of France, broadened probability beyond games of chance, wrote Mecanique Celeste, on committee that established metric system
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| Mecanique Celeste | showed all members of the solar system acted under gravity
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| Mary Fairfax Somerville | (1780-1872) from Scotland translated Mecanique Celeste into English
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| Laplace’s main probability formulas | P[event], P[A or B], P[A and B]
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| Benouilli Trials Experiment | series of n independent trials with constant probability that are repeated and are either a success or a failure, p= P[success], q=1-p=P[failure]
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| Cristiaan Huygens | (1629-1895) developed the concept of expectation, the average amount gained or lost after playing several games, if E=0, the game is fair
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| Father Marin Mersenne | (1588-1648) worked with perfect numbers
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| Academie Royale des Sciences | formed in 1666 in France after multiple meetings started by Mersenne, members were well off
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| Royal Society of London | formed in 1660, but did not have the financial backing of the throne like in France
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| Perfect numbers | if n= the sum of the divisors less than n, ex. 6= 1+2+3 and 28=1+2+4+7+14, unknown if they are infinite, all known so far are even and end in 6 or 8
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| Saint Augustine | believed God created the universe in 6 days since 6 is a perfect number
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| First 5 perfect numbers | 6, 28, 496, 8128. 33550336
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| Euclid’s Theorem | if 2^k-1 is prime for k>1, then 2^(k-1) (2^k-1) is perfect
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| Leonhard Euler | (1707-1783) showed an even number n is perfect if and only 〖n=2〗^(k-1) (2^k-1) where 2^k-1 is prime, Swiss, blind in one eye and eventually the other, too many manuscripts to publish
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| Euler’s accomplishments | discouraged Diderot’s views on atheism, wrote essays on heat, tides, and calculating longitude at sea
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| Mersenne Numbers | have the form M_n=2^n-1 for n≥1, if M_nis prime it is a Mersenne prime
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| Amicable (or friendly) numbers | the proper divisors of one number sum to the other number and vice versa ex. 220 and 284, Thabit ibn Kurrah had a rule for finding some amicable pairs
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| Fermat’s Little Theorem | For any integers a and prime p, p divides a^p-a
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| Fermat’s favorite method of proof | the method of infinite descent
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| Fermat’s last theorem | if the integer n>2, then x^n+y^n=z^n has no nonzero solution for (x, y, z)
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| Andrew Wiles | proved Fermat’s last Theorem in 1993 using a new theory of elliptic curves
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| 18th century | known as Age of Enlightenment, researchers quit publishing in Latin, universities stifled by the church, so knowledge was spread in salons where learning was a social event
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| Denis Diderot and Jean d’Alembert | (1717-1783) French, compiled the first Encyclopedia
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| Introductio in Analysin Infinitorum | written by Euler, solidified foundations of modern analysis, popularized notation of e,π,and i, discovered equation e^ix+1=0, expanded infinite sum e
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| First paper on graph theory | written by Euler to solve Konigsberg Bridges Problem, showed it was impossible to cross each bridge only once
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| Goldbach’s Conjecture | currently unsolved problem, every even number >2 is a sum of 2 primes
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| Joseph Louis Lagrange | (1736-1813), Italian, works led to Galois’s group theory, known for the theorem that every positive integer can be written as sum of 4 squares
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| Sophie Germain | (1776-1831) tutored by Lagrange since she couldn’t enroll
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| Mecanique Analytique | written by Lagrange, reconstructed mechanics analytically, freeing mechanics from geometry
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| Theorie des Fonctions Analytiques | written by Lagrange, laid down the theory of analytic functions (can be represented by Taylor series)
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| Carl Friedrich Gauss | (1777-1855) German, called Prince of Mathematicians, last mathematician to impact all the branches of mathematics, worked in astronomy
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| Gauss’s accomplished | developed method of least squares, created polygon of 17 sides with straight edge and compass, proved Fundamental Theorem of Algebra, introduced idea of congruence
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| Wilhelm Weber | laid down the first telegraph wire with Gauss
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| Euclid’s Parallel Postulate | If a line falling on two lines makes interior angles on the same side less than two right angles, then the two lines will eventually meet on that side
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| Equivalents of the Parallel postulate | Playfair’s axiom, and seven others
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| Playfair’s axiom | Through a given point not on a given line, only one parallel can be drawn
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| Fundamental Theorem of Algebra | a polynomial of degree n has n roots counting multiplicity
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| Proclus | attempted to prove the parallel postulate, made an incorrect assumption
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| John Wallis | made the same mistake as Proclus while trying to prove the Parallel Postulate
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| Girolama Saccheri | (1667-1733) in Italy, first to study the results of denying the parallel postulate, tried to prove postulate by contradiction, but failed
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| Johann Lambert | (1728-1777) tried to prove parallel postulate by contradiction, but failed, proved pi is irrational
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| Proof pi is irrational | if tan x is rational, then x is irrational, since tan pi/4=1 is rational, then pi/4 is irrational
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| Adrien-Marie Legendre | (1752-1833) French, competed with Gauss, determined the length of a meter, wrote Elements de Geometrie where he rearranged Euclids Elements for clarity
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| Non-Euclidean Geometry | a whole new geometry if the Parallel postulate is denied, invented simultaneously by Gauss, Bolyai, and Lobachevsky,
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| Gauss’s non Euclidean geometry | first to develop consistent geometry, but did not publish because of Kant’s popularity, also they never reached his level of perfection
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| Immanual Kant | (1724-1804) popular philosopher who proposed no other system of geometry could exist because no other system of geometry was imaginable
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| John Bolyai | (1802-1860) invented non-Euclidean geometry, he published an Appendix in father’s Tentamen explaining “the absolutely true science of space”. Disheartened hearing Gauss had already done it
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| Nicolai Lobachevsky | (1793-1856) first to publish non-Euclidean geometry in On the Foundations of Geometry calling it imaginary geometry, received no recognition
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| Lobachevskian Parallel Postulate | There exist two lines parallel to a given line through a point not on a line, proves the angles sum of a triangle is less than 180 degrees
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| Bernhard Riemann | (1826-1866) studied under Gauss, developed his own Non-Euclidean Geometry, results of this show that lines are finite, perpendiculars meet in 2 pts and angle sum of a triangle is > 180
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| Felix Klein | (1849-1925) first to show Riemann’s Non-euclidean geometry has a physical interpretation on the sphere
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| Klein’s definition of geometry | study of properties of figures remaining the same under a particular group of transformations
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| Grace Chisholm Young | English, studied under Klein, and was the first women in Germany to receive a PhD in any subject
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| Beltrami | used a pseudosphere similar to Klein to show Lobachevsky’s Non-Euclidean Geometry is valid
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| Augustin Louis Cauchy | (1789-1857) cleaned up definitions of Taylor series, limits, continuity, differentiability, definite integrals, and infinite series
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| Cauchy Convergence Criterion | states that a sequence Sn converges to a limit if and only if |Sm-Sn| can be made as small as possible for m and n large enough
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| Karl Weierstrass | (1815-1897) father of modern analysis, reduced all calculus concepts to equations and inequalities and our current notation for limits, found a continuous function that isn’t differentiable
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| Sonia Kovalevsky | (1850-1891) Weierstrass’s student, won Prix Bordin for anonymous paper “On the Rotation of a Solid Body about a Fixed Point”
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| David Hilbert | (1862-1943) restructured Euclid’s geometry to have 21 axioms and 6 undefined terms in Grundlagen der Geometrie and in 1900 stated 23 unsolved problems for the next century
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| Biggest revolution in mathematics in this century | “computer revolution”
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| Ada Lovelace | wrote Sketch of the Analytic Engine and is considered by some the inventor of computer programming
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| Caspar Wessel of Norway | found a geometric representation of complex number a+bi as a point in the plane treated a complex number as a vector from the origin to the point
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| Jean Robert Argand of Switzerland | found a geometric representation of complex number a+bi as a point in the plane treated a complex number as a vector from the origin to the point
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| Gauss of Germany | found a geometric representation of complex number a+bi as a point in the plane and represented a+bi by the point (a,b)
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| William Rowan Hamilton | (1805-1865) expanded on the idea of a complex number as an ordered pair, defined addition and multiplication preserving associative, commutative and distributive laws
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| Quaternion numbers | also preserve associative and distributive laws but are not commutative for multiplication
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| Arthur Cayley | (1821-1895) English, worked with James Sylvester to develop a system of algebra involving arrays of matrices
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| Matrix addition and multiplication | addition is commutative and associative, but multiplication is associative and not commutative
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| George Boole and Augustus DeMorgan | developed the field of symbolic logic where the emphasis is on the rules not the symbols
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| DeMorgan and Boole logic | symbol x stands for a set or proposition, 1= entire universe, 0=empty set, x*y=intersection of x and y, x+y= union of x and y, 1-x=complement of x
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| American Universities | evolved from an emphasis on theology to a classical curriculum, then in 19th century American mathematicians traveled to Germany who then relayed their research
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| First research university in the U.S. | Johns Hopkins University, patterned after the University of Berlin in 1876
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| James Sylvester | headed the mathematics department at Johns Hopkins after short stint at UVA
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| University of Chicago | founded in 1892 had reputable mathematics dept whose faculty and graduates elevated American mathematics to world prominence
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| First doctorates in mathematics in US | awarded by Yale in 1862 and Harvard in 1873
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| Winifred Edgerton | first American woman to receive a PhD in 1886 from Columbia University
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| Elbert Cox | in 1925 first African-American to complete a PhD degree at Cornell
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| Marjorie Lee Brown and Evelyn Boyd Granville | first Arican-American women to receive PhDs from Yale in 1949
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| Charlotte Angas Scott | from Girton College wrote her doctorate under Cayley’s direction from the U. of London in 1885, became mathematics dept chair at Bryn Mawr College, 1st woman living in US with PhD
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| Thomas Fiske | in 1888 founded the New York Mathematical Society later becoming American Mathematical Society with research journal Bulletin of the American Mathematical Society
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| American Mathematical Monthly | founded by Benjamin Finkel in 1894, first journal on mathematics education
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| Georg Ferdinand Cantor | Russian, paper began the notion of set theory, defined two sets to be equivalent if there is a 1-1 correspondence between the two sets, suggested some infinite sets are larger
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| Cardinal number | represents the number of elements
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| Which is larger set of real number or set of integers? | real numbers, since there is no 1-1 correspondence
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| Leopold Kronecker | (1823-1891) reproached Cantor causing mental illness, claimed “God created the natural numbers, and the rest is the work of man”
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| Cantor’s theorem | the union of a countable number of countable sets is countable
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| Cantor’s corollary | the union of a finite number of countable sets is countable
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| Set equivalencies | N=(natural numbers), Z=(integers) and Q=(rational numbers) are all equivalent, R is not since it is uncountable, I is also uncountable
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| A transcendental number | a real number that is not algebraic, ex. Pi
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| Lindemann | proved in 1882 that squaring the circle is impossible since pi is transcendental and lengths by straight edge and compass are roots of special algebraic equations
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| Continuum hypothesis | there is no cardinal number between X0 and c, cannot be proven or disproven, so is undecidable
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| Cantor’s Theorem | for any set A the cardinal number o(A)
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| Betrand Russell | (1872-1970) found a paradox in set theory, mayor example of this, allows for a vicious-circle definition
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| Ernst Zermelo | (1871-1953) decided to supply set theory with axioms to avoid the paradoxes, called it the Axiom of specification
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| Logistic | logic creates mathematics advocated by Bertrand Russell and Alfred North Whitehead, create hierarchy of levels of elements, awkward and impossible not to overlap levels
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| Formalist | advocated by David Hilbert, mathematics should by axiomized into formal symbolic system, Godel showed this idea could not work due to lack of consistency
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| Intuitionist | L.E.J. Brouwer, mathematics should be developed constructing everything from the positive integers in a finite number of steps, Euclid’s proof of the infinite number of primes fails here
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| Godfrey Harold Hardy | (1877-1947) English, worked in analytic number theory with the Riemann zeta function
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| Riemann Hypothesis | conjecture is till unproven, Hardy came closest to proving it in 1914, ensured his safety across the North Sea Channel
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| John Endsor Littlewood | (1885-1977) collaborated with Hardy on many problems in number theory and analysis, showed if Riemann holds then every odd integer n≥7 can be written as sum of 3 primes
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| Srinivasa Raanujan | India’s greatest mathematician, devised many infinite series to approximate pi
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| Maurice Frechet | (1878-1973) considered concept of distance in abstract spaces
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| Distance of metric d(x,y) between 2 pts in a set X satisfies | d(x,y)≥0 is a real number, d(x,y)=0 if and only if x=y, d(x,y)=d(y,x), and d(x,y)≤d(x,z)+d(z,y) (triangle inequality)
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| A metric space (X,d) | a set X together with a metric d
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| A neighborhood of point x | the set S(x) of points whose distances are within E units of x
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| Felix Hausdorff | (1868-1942) sued the concept of a neighborhood to define topology, committed suicide to avoid concentration camp
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| The Polish School | group of mathematicians in Poland, specialized in and developed topology, including Kuratowski, Banach, Sierpinski, many killed by Nazis
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| Emmy Noether | (1882-1935) became one of the founders of modern (abstract) algebra with her work on rings and ideals
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| Nazi expulsion | Noether, Curant, Zorn, Einstein, Mrose, von Neumann, Goedel were all expelled for being Jewish, US mathematics was strengthened by these refugees in the 1930s and 40s
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| Computers | used to help solve Goldbach’s conjecture, Riemann zeta function, pi>50 billion decimal places, and their ability to handle large and tedious calculations
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| Four-color conjecture | proved in 1976 by computer, any map drawn on a plane or sphere can be colored using only four colors so that adjacent countries have different colors, impossible w/o computer
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| Kenneth Appel and Wolfgang Haken | after 100 years, used a computer to prove four-color conjecture, by generating all possible map configurations,
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| In the future | the definition of a proof may have to change to include computers
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