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Leonhard Euler: His Life and Work

Leonhard Euler: His Life and Work

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Leonhard Euler: His Life and Work

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  1. Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas

  2. Pronunciation Euler = “Oiler”

  3. Leonhard Euler Lisez Euler, lisez Euler, c'est notre maître à tous.” -- Pierre-Simon Laplace Read Euler, read Euler, he’s the master (teacher) of us all.

  4. Images of Euler

  5. Euler’s 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)

  6. Academic Biography • Enrolled at University of Basel at age 14 • Mentored by Johann Bernoulli • Studied mathematics, history, philosophy (master’s degree) • Entered divinity school, but left to pursue more mathematics

  7. Academic Biography • Joined Johann Bernoulli’s 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

  8. Other facts about Euler’s 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

  9. Mathematical Predecessors • Isaac Newton • Pierre de Fermat • René Descartes • Blaise Pascal • Gottfried Wilhelm Leibniz

  10. Mathematical Successors • Pierre-Simon Laplace • Johann Carl Friedrich Gauss • Augustin Louis Cauchy • Bernhard Riemann

  11. Mathematical Contemporaries • Bernoullis-Johann, Jakob, Daniel • Alexis Clairaut • Jean le Rond D’Alembert • Joseph-Louis Lagrange • Christian Goldbach

  12. Contemporaries: Non-mathematical • Voltaire • Candide • Academy of Sciences, Berlin • Benjamin Franklin • George Washington

  13. Great Volume of Works • 856 publications—550 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

  14. Contributions to Mathematics • Calculus (Analysis) • Number Theory—properties of the natural numbers, primes. • Logarithms • Infinite Series—infinite sums of numbers • Analytic Number Theory—using infinite series, “limits”, “calculus, to study properties of numbers (such as primes)

  15. Contributions to Mathematics • Complex Numbers • Algebra—roots of polynomials, factorizations of polynomials • Geometry—properties of circles, triangles, circles inscribed in triangles. • Combinatorics—counting methods • Graph Theory—networks

  16. Other Contributions--Some highlights • Mechanics • Motion of celestial bodies • Motion of rigid bodies • Propulsion of Ships • Optics • Fluid mechanics • Theory of Machines

  17. Named after Euler • Over 50 mathematically related items (own estimate)

  18. Euler Polyhedral Formula (Euler Characteristic) • Applies to convex polyhedra

  19. Euler Polyhedral Formula (Euler Characteristic) • Vertex (plural Vertices)—corner points • Face—flat outside surface of the polyhedron • Edge—where two faces meet • V-E+F=Euler characteristic • Descartes showed something similar (earlier)

  20. Euler Polyhedral Formula (Euler Characteristic) • Five Platonic Solids • Tetrahedron • Hexahedron (Cube) • Octahedron • Dodecahedron • Icosahedron • #Vertices - #Edges+ #Faces = 2

  21. Euler Polyhedral Formula (Euler Characteristic) • What would be the Euler characteristic of • a triangular prism? • a square pyramid?

  22. The Bridges of Königsberg—The 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

  23. The Bridges of Königsberg—The Birth of Graph Theory

  24. The Bridges of Königsberg—The Birth of Graph Theory • Question 1—Is there a way to visit each land mass using a bridge only once? (Eulerian path) • Question 2—Is there a way to visit each land mass using a bridge only once and beginning and arriving at the same point? (Eulerian circuit)

  25. The Bridges of Königsberg—The Birth of Graph Theory

  26. The Bridges of Königsberg—The 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

  27. The Bridges of Königsberg—The 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

  28. The Bridges of Königsberg—The Birth of Graph Theory • Determined that the sequence of bridges (small letters) necessary was bigger than the current seven bridges (keeping their locations)

  29. The Bridges of Königsberg—The Birth of Graph Theory • Nowadays we use graph theory to solve problem (see ACTIVITIES)

  30. Knight’s Tour (on a Chessboard)

  31. Knight’s Tour (on a Chessboard) • Problem proposed to Euler during a chess game

  32. Knight’s Tour (on a Chessboard)

  33. Knight’s Tour (on a Chessboard) • Euler proposed ways to complete a knight’s tour • Showed ways to close an open tour • Showed ways to make new tours out of old

  34. Knight’s Tour (on a Chessboard)

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

  36. Euler and Primes • If • Then • In a unique way • Example

  37. Euler and Primes • This infinite series has no sum • Infinitely many primes

  38. Euler and Complex Numbers • Recall

  39. Euler and Complex Numbers Euler’s Formula:

  40. Euler and Complex Numbers • Euler offered several proofs • Cotes proved a similar result earlier • One of Euler’s proofs uses infinite series

  41. Euler and Complex Numbers

  42. Euler and Complex Numbers

  43. Euler and Complex Numbers

  44. Euler and Complex Numbers Euler’s Identity:

  45. 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/ • Euler’s works in the original language (and some translations) • The Euler Society • http://www.eulersociety.org/

  46. 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