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Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas Pronunciation Euler = “Oiler” 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. Images of Euler 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) 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 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 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 Mathematical Predecessors • • • • • Isaac Newton Pierre de Fermat René Descartes Blaise Pascal Gottfried Wilhelm Leibniz Mathematical Successors • • • • Pierre-Simon Laplace Johann Carl Friedrich Gauss Augustin Louis Cauchy Bernhard Riemann Mathematical Contemporaries • • • • • Bernoullis-Johann, Jakob, Daniel Alexis Clairaut Jean le Rond D’Alembert Joseph-Louis Lagrange Christian Goldbach Contemporaries: Non-mathematical • Voltaire – Candide – Academy of Sciences, Berlin • Benjamin Franklin • George Washington 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 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) 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 Other Contributions--Some highlights • • • • • • • Mechanics Motion of celestial bodies Motion of rigid bodies Propulsion of Ships Optics Fluid mechanics Theory of Machines Named after Euler • Over 50 mathematically related items (own estimate) Euler Polyhedral Formula (Euler Characteristic) • Applies to convex polyhedra 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) Euler Polyhedral Formula (Euler Characteristic) • Five Platonic Solids – Tetrahedron – Hexahedron (Cube) – Octahedron – Dodecahedron – Icosahedron • #Vertices - #Edges+ #Faces = 2 Euler Polyhedral Formula (Euler Characteristic) • What would be the Euler characteristic of – a triangular prism? – a square pyramid? 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 The Bridges of Königsberg—The Birth of Graph Theory 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) The Bridges of Königsberg—The Birth of Graph Theory 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 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 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) The Bridges of Königsberg—The Birth of Graph Theory • Nowadays we use graph theory to solve problem (see ACTIVITIES) Knight’s Tour (on a Chessboard) Knight’s Tour (on a Chessboard) • Problem proposed to Euler during a chess game Knight’s Tour (on a Chessboard) 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 Knight’s Tour (on a Chessboard) Basel Problem • First posed in 1644 (Mengoli) • An example of an INFINITE SERIES (infinite sum) that CONVERGES (has a particular sum) 1 1 1 1 ... ... 2 2 2 2 1 2 3 k 6 2 Euler and Primes • If • Then p 4n 1 p a b 2 2 • In a unique way • Example 5 4(1) 1 2 1 2 2 Euler and Primes • This infinite series has no sum • Infinitely many primes 1 1 1 1 1 1 1 ... ... 2 3 5 7 11 p Euler and Complex Numbers • Recall i 1 Euler and Complex Numbers Euler’s Formula: p Euler and Complex Numbers • Euler offered several proofs • Cotes proved a similar result earlier • One of Euler’s proofs uses infinite series Euler and Complex Numbers 2 3 4 5 x x x x x e 1 x ... 1 2 1 2 3 1 2 3 4 1 2 3 4 5 2 3 4 5 ( ix ) ( ix ) ( ix ) ( ix ) ix e 1 ix ... 1 2 1 2 3 1 2 3 4 1 2 3 4 5 2 3 4 5 x ix x ix ix e 1 ix ... 1 2 1 2 3 1 2 3 4 1 2 3 4 5 Euler and Complex Numbers 2 4 x x cos 1 ... 1 2 1 2 3 4 3 5 x x sin x x ... 1 2 3 1 2 3 4 5 3 5 ix ix i sin x ix ... 1 2 3 1 2 3 4 5 Euler and Complex Numbers 2 3 4 5 x ix x ix e 1 ix ... 1 2 1 2 3 1 2 3 4 1 2 3 4 5 ix 2 4 3 5 x x ix ix eix 1 ... ix ... 1 2 1 2 3 4 1 2 3 1 2 3 4 5 Euler and Complex Numbers Euler’s Identity: i e 1 0 i e 1 (cos i sin ) 1 i e 1 1 i 0 1 i e 1 0 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/ 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