# Hilbert's Problems

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```					HILBERT’S PROBLEMS
By: Bree Murdock and Mary Michelli
DAVID HILBERT
 Born: January 23, 1862 in Prussia
 Attended Konigsberg University

 Taught at Gottingen University

 Married in 1892 to his second cousin

 He had a son who was mentally disabled so Hilbert did
not claim him
 Published two books

 Forced to retire in 1932 by the Nazis
“DO YOU KNOW HILBERT? NO. THEN
WHAT ARE YOU DOING IN HIS SPACE?”

 Hilbert space: a vector space, H, with an inner
product (f,g) such that the norm is defined by
              turns H into a complete metric
space.

 http://mathworld.wolfram.com/HilbertSpace.html
 Example:
HILBERT’S HOTEL
 There   is a hotel with an infinite number of
rooms
 When a new guest arrives, all the current
guests must move to the room whose number
is one greater than the one they are currently
in
 But there is a way to accommodate new
guests so that the current guests only have to
move once
 Each guest must move to the room whose
number is twice their current room number
 So infinitely many odd numbered rooms will
be open for an infinite number of guests
HILBERT’S 23 PROBLEMS
 In the summer of 1900 Hilbert presented a set of
problems at the Second International Congress of
Mathematics in Paris
 “Who of us would not be glad to lift the veil
behind which the future lies hidden; to cast a
glance at the next advances of our science in the
secret of it’s development during the future
centuries?”
THE UNSOLVED PROBLEMS
 #6: Mathematical Treatment of the Axioms of
physics
 # 8: Problems of Prime Numbers

 #12: Extension of Kroneker’s theorem of abelian
fields to any algebraic realm of reationality
 # 16: Problems of topology of algebraic curves and
surfaces
PARTIALLY SOLVED OR IMPOSSIBLE TO
PROVE
   #1: Impossible – Cantor’s problem of the cardinal number of a
continuum
   # 2: Impossible – Compatibility of arithmetical axioms
   # 4: Too vague – Problem of the straight line as the shortest
distance between two points
   #5: Partially Solved – Lie’s concept of a continuous group of
transformations without the assumption of the differentiability of
the functions defining the group
   #9: Partially Solved – Proof of the most general law of reciprocity
in any number field
   # 11: Partially Solved – Quadratic Forms with any algebraic
numerical coefficients
   #15: Partially Solved – Rigorous foundation of Schubert’s
enumerative calculus
THE SOLVED PROBLEMS
 # 3: The equality of two volumes of two tetrahedra of
equal bases and equal altitudes
 # 7: Irrationality in transcendence of certain numbers

 # 10: Determination of the solvability of a diophantine
equation
 # 13: Impossibility of the solution of the general
equation of the seventh degree by means of functions
of only two arguments
 # 14: Proof of the finiteness of certain complete
systems of functions
 #17: Expression of definite forms by squares

 # 18: Building up of space from congruent polyhedra
MORE SOLVED PROBLEMS
#  19: Are the solutions of regular problems in
the calculus of variations always necessarily
analytic
 # 20: The general problem of boundary values

 # 21: The proof of the existence of linear
differential equations having a prescribed
monodromic group
 #22: Uniformization of analytic relations by
means of automorphic functions
 #23: Further development of the method of
calculus of variations
PROBLEM NUMBER THREE: THE EQUALITY OF TWO
VOLUMES OF TWO TETRAHEDRA OF EQUAL BASES AND
EQUAL ALTITUDES

 Solved by Max Dehn
 Used a counterexample to prove that this cannot
be done
 Considered the easiest of the twenty three
problems
 D(P) = Dehn invariant

 D(P) = D(P1) + D(P2)

 D(P) = D(P1)+…+D(Pn)
PROBLEM NUMBER SEVEN:
IRRATIONALITY IN TRANSCENDENCE OF
CERTAIN NUMBERS

 Proven by the Gelfond-Schneider Theorem
 Solved in 1934

 A and b are algebraic numbers

 A^b is transcendental if b is irrational

 Not restricted to the real numbers
4 PROBLEMS REMAIN UNSOLVED…
…Could you be the next to solve one????

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