leonhard euler: his life and work michael p. saclolo, ph.d. st. edward’s university austin, texas
TRANSCRIPT
![Page 1: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/1.jpg)
Leonhard Euler: His Life and Work
Michael P. Saclolo, Ph.D.
St. Edward’s University
Austin, Texas
![Page 2: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/2.jpg)
Pronunciation
Euler = “Oiler”
![Page 3: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/3.jpg)
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.
![Page 4: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/4.jpg)
Images of Euler
![Page 5: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/5.jpg)
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)
![Page 6: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/6.jpg)
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
![Page 7: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/7.jpg)
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
![Page 8: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/8.jpg)
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
![Page 9: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/9.jpg)
Mathematical Predecessors
• Isaac Newton
• Pierre de Fermat
• René Descartes
• Blaise Pascal
• Gottfried Wilhelm Leibniz
![Page 10: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/10.jpg)
Mathematical Successors
• Pierre-Simon Laplace
• Johann Carl Friedrich Gauss
• Augustin Louis Cauchy
• Bernhard Riemann
![Page 11: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/11.jpg)
Mathematical Contemporaries
• Bernoullis-Johann, Jakob, Daniel
• Alexis Clairaut
• Jean le Rond D’Alembert
• Joseph-Louis Lagrange
• Christian Goldbach
![Page 12: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/12.jpg)
Contemporaries: Non-mathematical
• Voltaire– Candide– Academy of Sciences, Berlin
• Benjamin Franklin
• George Washington
![Page 13: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/13.jpg)
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
![Page 14: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/14.jpg)
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)
![Page 15: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/15.jpg)
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
![Page 16: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/16.jpg)
Other Contributions--Some highlights
• Mechanics
• Motion of celestial bodies
• Motion of rigid bodies
• Propulsion of Ships
• Optics
• Fluid mechanics
• Theory of Machines
![Page 17: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/17.jpg)
Named after Euler
• Over 50 mathematically related items (own estimate)
![Page 18: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/18.jpg)
Euler Polyhedral Formula (Euler Characteristic)
• Applies to convex polyhedra
![Page 19: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/19.jpg)
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)
![Page 20: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/20.jpg)
Euler Polyhedral Formula (Euler Characteristic)
• Five Platonic Solids– Tetrahedron
– Hexahedron (Cube)
– Octahedron
– Dodecahedron
– Icosahedron
• #Vertices - #Edges+ #Faces = 2
![Page 21: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/21.jpg)
Euler Polyhedral Formula (Euler Characteristic)
• What would be the Euler characteristic of– a triangular prism?
– a square pyramid?
![Page 22: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/22.jpg)
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
![Page 23: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/23.jpg)
The Bridges of Königsberg—The Birth of Graph Theory
![Page 24: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/24.jpg)
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)
![Page 25: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/25.jpg)
The Bridges of Königsberg—The Birth of Graph Theory
![Page 26: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/26.jpg)
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
![Page 27: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/27.jpg)
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
![Page 28: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/28.jpg)
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)
![Page 29: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/29.jpg)
The Bridges of Königsberg—The Birth of Graph Theory
• Nowadays we use graph theory to solve problem (see ACTIVITIES)
![Page 30: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/30.jpg)
Knight’s Tour (on a Chessboard)
![Page 31: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/31.jpg)
Knight’s Tour (on a Chessboard)
• Problem proposed to Euler during a chess game
![Page 32: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/32.jpg)
Knight’s Tour (on a Chessboard)
![Page 33: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/33.jpg)
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
![Page 34: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/34.jpg)
Knight’s Tour (on a Chessboard)
![Page 35: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/35.jpg)
Basel Problem
• First posed in 1644 (Mengoli)
• An example of an INFINITE SERIES (infinite sum) that CONVERGES (has a particular sum)
6...
1...
3
1
2
1
1
1 2
2222
k
![Page 36: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/36.jpg)
Euler and Primes
• If
• Then
• In a unique way• Example
22 bap
14 np
22 121)1(45
![Page 37: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/37.jpg)
Euler and Primes
• This infinite series has no sum
• Infinitely many primes
...1
...11
1
7
1
5
1
3
1
2
11
p
![Page 38: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/38.jpg)
Euler and Complex Numbers
• Recall
1i
![Page 39: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/39.jpg)
Euler and Complex Numbers
p
Euler’s Formula:
![Page 40: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/40.jpg)
Euler and Complex Numbers
• Euler offered several proofs
• Cotes proved a similar result earlier
• One of Euler’s proofs uses infinite series
![Page 41: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/41.jpg)
Euler and Complex Numbers
...54321432132121
15432
xxxx
xex
...54321
)(
4321
)(
321
)(
21
)(1
5432
ixixixix
ixeix
...54321432132121
15432
ixxixx
ixeix
![Page 42: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/42.jpg)
Euler and Complex Numbers
...432121
1cos42
xx
...54321321
sin53
xx
xx
...54321321
sin53
ixix
ixxi
![Page 43: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/43.jpg)
Euler and Complex Numbers
...54321432132121
15432
ixxixx
ixeix
...54321321
...432121
15342
ixix
ixxx
eix
![Page 44: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/44.jpg)
Euler and Complex Numbers
Euler’s Identity:
01ie1)sin(cos1 iei
1011 iei
01ie
![Page 45: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/45.jpg)
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/
![Page 46: Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward’s University Austin, Texas](https://reader036.vdocuments.mx/reader036/viewer/2022062516/56649de55503460f94adcd4f/html5/thumbnails/46.jpg)
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