Transcript A Cartoon-Assisted Proof of The Fundamental Theorem of Algebra

Fundamental Theorem of Algebra
A Cartoon-Assisted Proof
• Frank Wang
• LaGuardia
Community College
Topics
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Motivation
What Happens to Complex Numbers
Complex Functions
Fundamental Theorem of Algebra
Abel’s Theorem
New York Times
Elusive Proof, Elusive
Prover: A New
Mathematical Mystery
(Aug 15, 2006)
Letter to the New York Times
• In my modest attempt to understand Poincaré’s
Conjecture discussed in the Times, I thought of
how one of the greatest play ever written,
Sophocles’s Oedipus Rex, enacts the notion of
the sphere that can be expanded, reshaped, or
contracted to a point. In teaching this play, I
have used the image of the net to describe
human fate: you can push at the boundaries of
the net so that your individual choices allow you
a personal identity, but you can never get
outside the net—or the sphere… van Slyck
Popular Books
The most beautiful equation
i
e  cos   i sin 
i
e  1
Julia set c=0.33+0.45 i
Mandelbrot set and bifurcation
• Mandelbrot set
M  {c  C || Qc (0) | }
n
Qc : z  z 2  c
Newton’s Method
in the Complex Plane
Differential Equations
Ly  Ry  E cos t
Ly  Ry  Ee
i t
 cos(t )e

e
e
dt

Rt / L
i t Rt / L
dt
Isaac Newton
•
•
•
•
Invented Calculus
Laws of Motion
F=m a
Differential Equations
Galileo Galilei
• Pendulum Motion
g
   
l
Johannes Kepler
• Planetary Motion
GM
2

(r  r )  2
r
r  2r  0
(False) Conclusion
• Most systems of interest are periodic.
• Most differential equations are analytically
solvable.
• Receive an A in Differential Equations,
then one can solve any problem!
Henri Poincaré
• Three body problem
can NOT be
analytically solved
• The trajectory could
be chaotic!
Sonya Kovalevskaya
Rigid body problems
are NOT integrable
except for
• Euler Case
• Lagrange Case
• Kovalevskaya Case
Kovalevskaya to Mittag-Leffler
Dear Sir,
… It is a question of integrating
Ap  ( B  C )qr  z0   y0 
Bq  (C  A)rp  x0   z0
Cr  ( A  B) pq  y0  x0 
…, and I can show that these 3 cases
are the only ones [integrable]…
• Roger Cooke Trans.
Fundamental Quantum Condition
h
pq  qp 
2i
In order to understand the Origin of the
universe, we need to combine the
General Theory of Relativity, with
quantum theory. The best way of doing
so, seems to be to use Feinman's idea of
a sum over histories.
Richard
Feynman
“The universe has
every possible history.”
Richard Feinman was a colorful
character, who played the bongo drums
in a strip joint in Pasadena, and was a
brilliant physicist at the California
Institute of Technology. He proposed
that a system got from a state A, to a
state B, by every possible path or
history.
Feynman Sum Over Histories
A ~  eiS[g]/ћ
Sum over all metrics consistent with given
boundary conditions
Each path or history, has a certain
amplitude or intensity, and the probability
of the system going from A- to B, is given
by adding up the amplitudes for each
path. There will be a history in which the
moon is made of blue cheese, but the
amplitude is low, which is bad news for
mice.
Fundamental Theorem of Algebra
Let p(z) be a polynomial
over C, then there
exists at least one z
such that p(z)=0.
(Euler, Lagrange)
Gauss (1799)
Geometrical interpretation of
complex numbers
z  x  iy
z1  2  3i
z1  2  3i
Complex Functions
f ( z)  z
2
f ( z )  ( x  iy )  x  y  2ixy
2
2
f ( z )  u ( x, y )  iv ( x, y )
u ( x, y )  x  y
v( x, y )  2 xy
2
2
2
Complex Functions
f ( z)  z 3  i
f ( z )  ( x  iy )3  i  x 3  3xy2  i (3x 2 y  y 3  1)
f ( z )  u ( x, y )  iv ( x, y )
u ( x, y )  x 3  3xy2
v ( x, y )  3 x 2 y  y 3  1
z plane and w plane
z  w  f ( z)  z
2
z plane and w plane
z  w  f ( z)  z  i
3
z plane and w plane
(Riemann’s Hypothesis)
z  w  f ( z)   ( z)
Complex Numbers
Cartesian vs Polar Forms
z  x  iy
z  re
i
(r  x  y ,  tan
2
2
1
y
)
x
Parametrized curve
z (t )  cos( 2t )  i sin( 2t )
Map from z to w
z (t )  cos( 2t )  i sin( 2t )
w  z  z i
3
De Moivre’s formula
z  x  iy  r (cos   i sin  )
z  r (cos n  i sin n )
n
n
Winding number (2)
z (t )  cos( 2t )  i sin( 2t )
w  z  0.01z
2
Winding number (3)
z (t )  cos( 2t )  i sin( 2t )
w  z  0.05 z
3
Property
• When a loop is large enough, the winding
number is the degree of the polynomial.
w( )  n
Expanding r
z (t )  r cos( 2t )  ir sin( 2t )
w  z  z  z i
3
2
Expanding r
z (t )  r cos( 2t )  ir sin( 2t )
w  z  z  z i
4
2
Index 1, 2, 3
dz
 f (z )
dt
Index
dz
 z n  t (an 1 z n 1    a0 )
dt
0  t 1
Abel’s Theorem
If n>4 and f(x) is the
general polynomial of
degree n, then f(x) is
not solvable by
radicals.
Galois Theory
• The Galois group of
the general
polynomial is Sn, the
symmetric group on n
symbols.
• For n>4, Sn is not
solvable.
Riemann surface
w  f ( z)
x  Re( z )
y  Im( z )
h  Re( w)
(w  z )
5
Riemann surface
w  f ( z)
x  Re( z )
y  Im( z )
h  Re( w)
(w  z  3 z )
Dodecahedron
• Permutation of the
Riemann sheets is a
symmetric group
• The rotation group of
the dodecahedron is
isomorphic to the
alternating group A5
• A5 is not solvable
Integrated Calculus and Physics
at LaGuardia C C
• Frank Y. Wang
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[email protected]
http://faculty.lagcc.cuny.edu/fwang
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http://www.wiley.com