Leonhard Euler: His Life and Work

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Leonhard Euler His Life and Work

• Michael P. Saclolo, Ph.D.
• St. Edwards University
• Austin, Texas

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Pronunciation

• Euler Oiler

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Leonhard Euler

• Lisez Euler, lisez Euler, c'est notre maître à
  tous.
• -- Pierre-Simon Laplace
• Read Euler, read Euler, hes the master (teacher)
  of us all.

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Images of Euler
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Eulers Life in Bullets

• Born April 15, 1707, Basel, Switzerland
• Died 1783, St. Petersburg, Russia
• Father Paul Euler, Calvinist pastor
• Mother Marguerite Brucker, daughter of a pastor
• Married-Twice 1)Katharina Gsell, 2)her half
  sister
• Children-Thirteen (three outlived him)

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Academic Biography

• Enrolled at University of Basel at age 14
• Mentored by Johann Bernoulli
• Studied mathematics, history, philosophy
  (masters degree)
• Entered divinity school, but left to pursue more
  mathematics

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Academic Biography

• Joined Johann Bernoullis sons in St. Russia (St.
  Petersburg Academy-1727)
• Lured into Berlin Academy (1741)
• Went back to St. Petersburg in 1766 where he
  remained until his death

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Other facts about Eulers life

• Loss of vision in his right eye 1738
• By 1771 virtually blind in both eyes
• (productivity did not suffer-still averaged 1
  mathematical publication per week)
• Religious

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Mathematical Predecessors

• Isaac Newton
• Pierre de Fermat
• René Descartes
• Blaise Pascal
• Gottfried Wilhelm Leibniz

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Mathematical Successors

• Pierre-Simon Laplace
• Johann Carl Friedrich Gauss
• Augustin Louis Cauchy
• Bernhard Riemann

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Mathematical Contemporaries

• Bernoullis-Johann, Jakob, Daniel
• Alexis Clairaut
• Jean le Rond DAlembert
• Joseph-Louis Lagrange
• Christian Goldbach

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Contemporaries Non-mathematical

• Voltaire
• Candide
• Academy of Sciences, Berlin
• Benjamin Franklin
• George Washington

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Great Volume of Works

• 856 publications550 before his death
• Works catalogued by Enestrom in 1904 (E-numbers)
• Thousands of letters to friends and colleagues
• 12 major books
• Precalculus, Algebra, Calculus, Popular Science

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Contributions to Mathematics

• Calculus (Analysis)
• Number Theoryproperties of the natural numbers,
  primes.
• Logarithms
• Infinite Seriesinfinite sums of numbers
• Analytic Number Theoryusing infinite series,
  limits, calculus, to study properties of
  numbers (such as primes)

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Contributions to Mathematics

• Complex Numbers
• Algebraroots of polynomials, factorizations of
  polynomials
• Geometryproperties of circles, triangles,
  circles inscribed in triangles.
• Combinatoricscounting methods
• Graph Theorynetworks

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Other Contributions--Some highlights

• Mechanics
• Motion of celestial bodies
• Motion of rigid bodies
• Propulsion of Ships
• Optics
• Fluid mechanics
• Theory of Machines

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Named after Euler

• Over 50 mathematically related items (own
  estimate)

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Euler Polyhedral Formula (Euler Characteristic)

• Applies to convex polyhedra

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Euler Polyhedral Formula (Euler Characteristic)

• Vertex (plural Vertices)corner points
• Faceflat outside surface of the polyhedron
• Edgewhere two faces meet
• V-EFEuler characteristic
• Descartes showed something similar (earlier)

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Euler Polyhedral Formula (Euler Characteristic)

• Five Platonic Solids
• Tetrahedron
• Hexahedron (Cube)
• Octahedron
• Dodecahedron
• Icosahedron
• Vertices - Edges Faces 2

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Euler Polyhedral Formula (Euler Characteristic)

• What would be the Euler characteristic of
• a triangular prism?
• a square pyramid?

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The Bridges of KönigsbergThe Birth of Graph
Theory

• Present day Kaliningrad (part of but not
  physically connected to mainland Russia)
• Königsberg was the name of the city when it
  belonged to Prussia

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The Bridges of KönigsbergThe Birth of Graph
Theory
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The Bridges of KönigsbergThe Birth of Graph
Theory

• Question 1Is there a way to visit each land mass
  using a bridge only once? (Eulerian path)
• Question 2Is there a way to visit each land mass
  using a bridge only once and beginning and
  arriving at the same point? (Eulerian circuit)

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The Bridges of KönigsbergThe Birth of Graph
Theory
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The Bridges of KönigsbergThe Birth of Graph
Theory

• One can go from A to B via b (AaB).
• Using sequences of these letters to indicate a
  path, Euler counts how many times a A (or B)
  occurs in the sequence

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The Bridges of KönigsbergThe Birth of Graph
Theory

• If there are an odd number of bridges connected
  to A, then A must appear n times where n is half
  of 1 more than number of bridges connected to A

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The Bridges of KönigsbergThe Birth of Graph
Theory

• Determined that the sequence of bridges (small
  letters) necessary was bigger than the current
  seven bridges (keeping their locations)

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The Bridges of KönigsbergThe Birth of Graph
Theory

• Nowadays we use graph theory to solve problem
  (see ACTIVITIES)

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Knights Tour (on a Chessboard)
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Knights Tour (on a Chessboard)

• Problem proposed to Euler during a chess game

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Knights Tour (on a Chessboard)
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Knights Tour (on a Chessboard)

• Euler proposed ways to complete a knights tour
• Showed ways to close an open tour
• Showed ways to make new tours out of old

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Knights Tour (on a Chessboard)
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Basel Problem

• First posed in 1644 (Mengoli)
• An example of an INFINITE SERIES (infinite sum)
  that CONVERGES (has a particular sum)

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Euler and Primes

• If
• Then
• In a unique way
• Example

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Euler and Primes

• This infinite series has no sum
• Infinitely many primes

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Euler and Complex Numbers

• Recall

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Euler and Complex Numbers

Eulers Formula
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Euler and Complex Numbers

• Euler offered several proofs
• Cotes proved a similar result earlier
• One of Eulers proofs uses infinite series

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Euler and Complex Numbers
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Euler and Complex Numbers
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Euler and Complex Numbers
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Euler and Complex Numbers

• Eulers Identity

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How to learn more about Euler

• How Euler did it. by Ed Sandifer
• http//www.maa.org/news/howeulerdidit.html
• Monthly online column
• Euler Archive
• http//www.math.dartmouth.edu/euler/
• Eulers works in the original language (and some
  translations)
• The Euler Society
• http//www.eulersociety.org/

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How to learn more about Euler

• Books
• Dunhamm, W., Euler the Master of Us All,
  Dolciani Mathematical Expositions, the
  Mathematical Association of America, 1999
• Dunhamm, W (Ed.), The Genius of Euler
  Reflections on His Life and Work, Spectrum, the
  Mathematical Association of America, 2007
• Sandifer, C. E., The Early Mathematics of
  Leonhard Euler, Spectrum, the Mathematical
  Associatin of America, 2007