in physics. He also worked out, but did not publish,
the principles of HYPERBOLIC GEOMETRY independently
of JÁNOS BOLYAI (1802–60) and NIKOLAI IVANOVICH
LOBACHEVSKY (1792–1856), and developed the theory
of COMPLEX NUMBERS and complex functions.
Gauss demonstrated a talent for mathematics at an
early age. (It is said that he astonished his elementary
school teachers by summing the integers from 1 to 100
in an instant by observing that the sum amounts to 50
pairs of numbers each summing to 101.) Before leaving
high school, Gauss had independently discovered the
PRIME NUMBER THEOREM, the BINOMIAL THEOREM,
important results in modular arithmetic, and the general arithmetic–geometric-mean inequality. He entered
the University of Göttingen in 1795 but, for reasons
that are not clear to historians, left before completing a
degree to return to his hometown of Brunswick. He
submitted a doctoral dissertation to the University of
Helmstedt on the topic of the fundamental theorem of
algebra, which was accepted in absentia. Gauss published his masterpiece Disquisitiones arithmeticae soon
afterward.
In 1801 Gauss created a sensation in the community of astronomers when he correctly predicted the
orbital positions of the asteroid Ceres, discovered January 1, 1801, by Italian astronomer Giuseppe Piazzi.
Piazzi had the opportunity to observe an extremely
small portion of its orbit before it disappeared behind
the sun. Other astronomers published predictions
about where it would reappear several months later, as
did Gauss, offering a prediction that differed greatly
from common opinion. When the comet was observed
again December 7, 1801, it was almost exactly where
Gauss had predicted. Although Gauss did not reveal
his method of prediction at the time, it is known that
he used his method of least squares to make the computation. In 1807 Gauss left Brunswick to head the
Göttingen observatory, and in 1809 he published his
general text on the mathematics of astronomy, Theoria
motus corporum coelestium (Theory of the motion of
heavenly bodies).
While at the observatory, Gauss continued work on
mathematics. He began developing a theory of surface
curvature and GEODESICs, and studied the convergence
of SERIES, integration techniques, STATISTICS, potential
theory, and more. In 1818 he was asked to conduct a
geodesic survey of the state of Hanover, and 14 years
later, he assisted in a project to map the magnetic field
of the Earth. His mathematical work on the theory of
differential geometry allowed Gauss to prove a number
of properties that the Earth’s field must possess, which
allowed him to correctly predict the value of the field at
different locations, as well as the location of the magnetic South Pole. (Moreover, he proved mathematically
that there can only be two magnetic poles.)
Gauss remained at Göttingen for the latter part of
his career. He was awarded many honors throughout
his life, including election as a foreign member of the
prestigious ROYAL SOCIETY of London in 1804 and
receiving the Copley Medal from the society in 1838.
He also won the Copenhagen University Prize in 1822.
Gauss published over 300 significant pieces of work,
mostly written in Latin, and kept a large number of
unpublished notebooks and correspondences that
proved to be as mathematically rich as much of his
published work.
Gauss died in Göttingen on February 23, 1855. It is
impossible to exaggerate the influence Gauss has had on
almost every branch of mathematics and mathematical
physics. He was a master at solving difficult problems
that lay at the heart of complex mathematical ideas, and
the very completeness and thoroughness of his work
paved the way for significant advances in mathematics.
A number of fundamental concepts in number theory,
differential geometry, and statistics (such as Gaussian
reciprocity, Gaussian curvature, and the Gaussian distribution) are today named in his honor.
Gaussian elimination (pivoting) Named in honor
of CARL FRIEDRICH GAUSS (1777–1855), the process
of Gaussian elimination provides the means to find
the solution (if one exists) for a system of n SIMULTANEOUS LINEAR EQUATIONS in n unknowns by multiplying selected equations with carefully chosen constants
and subtracting equations to eliminate variables. The
method is best explained with an example. Consider
the following system of three linear equations in three
unknowns:
y + 3z = 0
2x + 4y – 2z = 18
x + 5y + 3z = 14
Interchange the first two equations so that the variable
x appears in the first row:
Gaussian elimination 219
the principles of HYPERBOLIC GEOMETRY independently
of JÁNOS BOLYAI (1802–60) and NIKOLAI IVANOVICH
LOBACHEVSKY (1792–1856), and developed the theory
of COMPLEX NUMBERS and complex functions.
Gauss demonstrated a talent for mathematics at an
early age. (It is said that he astonished his elementary
school teachers by summing the integers from 1 to 100
in an instant by observing that the sum amounts to 50
pairs of numbers each summing to 101.) Before leaving
high school, Gauss had independently discovered the
PRIME NUMBER THEOREM, the BINOMIAL THEOREM,
important results in modular arithmetic, and the general arithmetic–geometric-mean inequality. He entered
the University of Göttingen in 1795 but, for reasons
that are not clear to historians, left before completing a
degree to return to his hometown of Brunswick. He
submitted a doctoral dissertation to the University of
Helmstedt on the topic of the fundamental theorem of
algebra, which was accepted in absentia. Gauss published his masterpiece Disquisitiones arithmeticae soon
afterward.
In 1801 Gauss created a sensation in the community of astronomers when he correctly predicted the
orbital positions of the asteroid Ceres, discovered January 1, 1801, by Italian astronomer Giuseppe Piazzi.
Piazzi had the opportunity to observe an extremely
small portion of its orbit before it disappeared behind
the sun. Other astronomers published predictions
about where it would reappear several months later, as
did Gauss, offering a prediction that differed greatly
from common opinion. When the comet was observed
again December 7, 1801, it was almost exactly where
Gauss had predicted. Although Gauss did not reveal
his method of prediction at the time, it is known that
he used his method of least squares to make the computation. In 1807 Gauss left Brunswick to head the
Göttingen observatory, and in 1809 he published his
general text on the mathematics of astronomy, Theoria
motus corporum coelestium (Theory of the motion of
heavenly bodies).
While at the observatory, Gauss continued work on
mathematics. He began developing a theory of surface
curvature and GEODESICs, and studied the convergence
of SERIES, integration techniques, STATISTICS, potential
theory, and more. In 1818 he was asked to conduct a
geodesic survey of the state of Hanover, and 14 years
later, he assisted in a project to map the magnetic field
of the Earth. His mathematical work on the theory of
differential geometry allowed Gauss to prove a number
of properties that the Earth’s field must possess, which
allowed him to correctly predict the value of the field at
different locations, as well as the location of the magnetic South Pole. (Moreover, he proved mathematically
that there can only be two magnetic poles.)
Gauss remained at Göttingen for the latter part of
his career. He was awarded many honors throughout
his life, including election as a foreign member of the
prestigious ROYAL SOCIETY of London in 1804 and
receiving the Copley Medal from the society in 1838.
He also won the Copenhagen University Prize in 1822.
Gauss published over 300 significant pieces of work,
mostly written in Latin, and kept a large number of
unpublished notebooks and correspondences that
proved to be as mathematically rich as much of his
published work.
Gauss died in Göttingen on February 23, 1855. It is
impossible to exaggerate the influence Gauss has had on
almost every branch of mathematics and mathematical
physics. He was a master at solving difficult problems
that lay at the heart of complex mathematical ideas, and
the very completeness and thoroughness of his work
paved the way for significant advances in mathematics.
A number of fundamental concepts in number theory,
differential geometry, and statistics (such as Gaussian
reciprocity, Gaussian curvature, and the Gaussian distribution) are today named in his honor.
Gaussian elimination (pivoting) Named in honor
of CARL FRIEDRICH GAUSS (1777–1855), the process
of Gaussian elimination provides the means to find
the solution (if one exists) for a system of n SIMULTANEOUS LINEAR EQUATIONS in n unknowns by multiplying selected equations with carefully chosen constants
and subtracting equations to eliminate variables. The
method is best explained with an example. Consider
the following system of three linear equations in three
unknowns:
y + 3z = 0
2x + 4y – 2z = 18
x + 5y + 3z = 14
Interchange the first two equations so that the variable
x appears in the first row:
Gaussian elimination 219