Hadamard,
Jacques (1865–1963) French Number
theory, Analysis, Mathematical physics Born on December 8, 1865, in Versailles, France, Jacques Hadamard
is remembered as one of the two mathematicians who
independently proved the famous PRIME NUMBER THEOREM first conjectured by the German mathematician
CARL FRIEDRICH GAUSS. Belgian scholar CHARLES-JEAN
DE LA VALLÉE-POUSSIN was the second mathematician to
prove the result.
Hadamard received a doctoral degree in 1892 from
the École Normale Supérieure after completing a dissertation on complex functions and TAYLOR SERIES. The
same year he also completed work on the Riemann ZETA
FUNCTION and the Riemann hypothesis, earning him the
Grand Prix des Sciences Mathématique from the institution. Hadamard soon came to also realize that, with the
recent developments in the field of complex functions,
all the necessary pieces were now in place to develop a
proof of the famous outstanding prime number conjecture made by Gauss. (Vallée-Poussin independently
made the same realization.) Hadamard presented his
proof of the famous result in 1896. Later that same year
Hadamard was appointed as professor of astronomy
and rational mechanics at the University of Bordeaux.
Throughout his career Hadamard also made significant contributions to the field of matrix theory by identifying a class of matrices that can be used in
COMBINATORICS to create “block designs” and have
applications to PROBABILITY theory. He also studied VECTOR SPACE theory (defining the term functional for a linear function on a vector space) and contributed to the
study of mathematics education. Hadamard wrote over
300 scientific papers. In 1906 he was elected president of
the French Mathematical Society, and 3 years later was
appointed chair of the Collège de France. He stayed in
that position for only 3 years, before accepting the position as professor of analysis at the École Polytechnique.
He was elected to the Paris Academy of Sciences in 1912.
Hadamard died in Paris, France, on October 17,
1963. His proof of the prime number theorem is considered his most outstanding achievement.
half-plane The plane that lies on one side of a given
line is called a half-plane. If the points on the line itself
are considered part of the region, then we say that the
half-plane is “closed.” An “open” half-plane excludes
the points on the line.
If the equation of the given line is ax + by = c, then
the set of points (x,y) that satisfy ax + by > c form an
open half-plane on one side of the line, and those
points (x,y) satisfying ax + by < c form an open halfplane on the other side. If c is positive, then this second
inequality represents the half-plane that contains the
origin (we have a · 0 + b · 0 = 0 < c), and if c is negative, then the first inequality contains the origin. If c
equals zero, one must substitute in different values for
x and y to determine which inequality represents which
half-plane. The inequalities ax + by ≥ c and ax + by ≤ c
represent closed half-planes.
In three-dimensional space, a half-space is the
region of space that lies on one side of a plane. The
243
H
Jacques (1865–1963) French Number
theory, Analysis, Mathematical physics Born on December 8, 1865, in Versailles, France, Jacques Hadamard
is remembered as one of the two mathematicians who
independently proved the famous PRIME NUMBER THEOREM first conjectured by the German mathematician
CARL FRIEDRICH GAUSS. Belgian scholar CHARLES-JEAN
DE LA VALLÉE-POUSSIN was the second mathematician to
prove the result.
Hadamard received a doctoral degree in 1892 from
the École Normale Supérieure after completing a dissertation on complex functions and TAYLOR SERIES. The
same year he also completed work on the Riemann ZETA
FUNCTION and the Riemann hypothesis, earning him the
Grand Prix des Sciences Mathématique from the institution. Hadamard soon came to also realize that, with the
recent developments in the field of complex functions,
all the necessary pieces were now in place to develop a
proof of the famous outstanding prime number conjecture made by Gauss. (Vallée-Poussin independently
made the same realization.) Hadamard presented his
proof of the famous result in 1896. Later that same year
Hadamard was appointed as professor of astronomy
and rational mechanics at the University of Bordeaux.
Throughout his career Hadamard also made significant contributions to the field of matrix theory by identifying a class of matrices that can be used in
COMBINATORICS to create “block designs” and have
applications to PROBABILITY theory. He also studied VECTOR SPACE theory (defining the term functional for a linear function on a vector space) and contributed to the
study of mathematics education. Hadamard wrote over
300 scientific papers. In 1906 he was elected president of
the French Mathematical Society, and 3 years later was
appointed chair of the Collège de France. He stayed in
that position for only 3 years, before accepting the position as professor of analysis at the École Polytechnique.
He was elected to the Paris Academy of Sciences in 1912.
Hadamard died in Paris, France, on October 17,
1963. His proof of the prime number theorem is considered his most outstanding achievement.
half-plane The plane that lies on one side of a given
line is called a half-plane. If the points on the line itself
are considered part of the region, then we say that the
half-plane is “closed.” An “open” half-plane excludes
the points on the line.
If the equation of the given line is ax + by = c, then
the set of points (x,y) that satisfy ax + by > c form an
open half-plane on one side of the line, and those
points (x,y) satisfying ax + by < c form an open halfplane on the other side. If c is positive, then this second
inequality represents the half-plane that contains the
origin (we have a · 0 + b · 0 = 0 < c), and if c is negative, then the first inequality contains the origin. If c
equals zero, one must substitute in different values for
x and y to determine which inequality represents which
half-plane. The inequalities ax + by ≥ c and ax + by ≤ c
represent closed half-planes.
In three-dimensional space, a half-space is the
region of space that lies on one side of a plane. The
243
H