Transcript Leonhard Euler - UT Mathematics

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
eix  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