Egyptians of the time held mathematics in high regard.
The copyist, whose name is rendered as Ahmes or
Ahmose, copied also the grandiose title of the papyrus:
Accurate Rendering: The Entrance into the Knowledge
of All Existing Things and All Obscure Secrets. The
purpose of the text seems to be that of a textbook—a
series of examples and exercises organized to illustrate
certain mathematical skills and techniques. Some of the
latter problems discussed, however, have no practical
application and clearly indicate a delight in studying
mathematics for its own sake.
See also EGYPTIAN FRACTIONS; EGYPTIAN MATHEMATICS; EGYPTIAN MULTIPLICATION.
Richter scale See LOGARITHMIC SCALE.
Riemann, Georg Friedrich Bernhard (1826–1866)
German Geometry, Analysis, Calculus, Number Theory
Born on September 17, 1826, in Breselenz, Hanover
(now Germany), Bernhard Riemann is remembered as
one of the greatest mathematicians of the 19th century.
His work in NON-EUCLIDEAN GEOMETRY revolutionized
modern thinking on the study of shape and space and
provided the tools ALBERT EINSTEIN needed for his
description of the universe in the general theory of relativity. Among his many significant contributions to
ANALYSIS, Riemann is remembered for his construction
of the Riemann integral, which, for the first time,
extended the theory of CALCULUS beyond the realm of
continuous functions only. Riemann also developed a
theory of complex functions (that is, functions whose
inputs and outputs are COMPLEX NUMBERS) and studied
applications of calculus in this setting. In 1859, while
searching for a better approximation for the number of
primes as given by the famous PRIME-NUMBER THEOREM,
Riemann discovered a generalized version of LEONHARD EULER’s important ZETA FUNCTION. He made a
deep and insightful conjecture about the properties of
this function, the Riemann hypothesis, which, if true,
would provide invaluable information about the distribution of prime numbers. Proving (or disproving) the
Riemann hypothesis is considered the most important
unsolved mathematical problem of today.
Riemann exhibited a keen interest in mathematics
at an early age and was encouraged by the director of
his high school to pursue this line of study. In 1846
Riemann enrolled at the University of Göttingen to
study under the mathematician CARL FRIEDRICH GAUSS
(1777–1855). Surprisingly, even with Gauss as a faculty
member, Göttingen did not offer a strong program in
mathematics, and so, a year later, Riemann transferred
to Berlin University, an institution recognized for its
strength in mathematics at the time. While there, Riemann developed his general theory of complex variables and functions that later formed the basis of much
of his important work.
After completing his undergraduate degree at
Berlin, Riemann returned to Göttingen in 1849 to complete a doctoral thesis under the supervision of Gauss.
He was awarded a Ph.D. 2 years later in mathematics.
Gauss described Riemann’s thesis work as having
“a gloriously fertile originality.” By viewing the range
of outputs of a complex function as a surface in threeor even four-dimensional space, Riemann managed to
tie the theory of calculus and analysis to the study of
shape, space, and geometry. This innovative approach
was immediately recognized as a significant turning
point in the advancement of mathematics. On Gauss’s
recommendation, Riemann was appointed a position at
Göttingen, and he commenced work on his habilitation
degree. During this time Riemann turned his attention
to the study of FOURIER SERIES and questions of integrability. It was during this period that Riemann formulated his famous definition of an integral.
As the final step to completing his degree Riemann
was required to give a public lecture. Gauss asked him
to speak on the theory of geometry, and on June 10,
1854, Riemann presented to the outside world his
groundbreaking work on shape and space.
Riemann devised a general notion of distance in ndimensional space and developed a theory of TENSORs
that allowed him to properly define geometric shape. In
particular, Riemann developed the notion of curvature
tensor to measure, or define, the warping of space. The
mathematics Riemann presented on that day proved to
be the precise framework Einstein needed for structuring his theory of general relativity. Riemann continued
to present and publish groundbreaking work in the
years that followed.
On July 30, 1859, Riemann was appointed chair
of mathematics at Göttingen, and a few days later he
was honored with election to the Berlin Academy of
Sciences. By this time Riemann had developed an interest in NUMBER THEORY and the distribution of prime
Riemann, Georg Friedrich Bernhard 447
The copyist, whose name is rendered as Ahmes or
Ahmose, copied also the grandiose title of the papyrus:
Accurate Rendering: The Entrance into the Knowledge
of All Existing Things and All Obscure Secrets. The
purpose of the text seems to be that of a textbook—a
series of examples and exercises organized to illustrate
certain mathematical skills and techniques. Some of the
latter problems discussed, however, have no practical
application and clearly indicate a delight in studying
mathematics for its own sake.
See also EGYPTIAN FRACTIONS; EGYPTIAN MATHEMATICS; EGYPTIAN MULTIPLICATION.
Richter scale See LOGARITHMIC SCALE.
Riemann, Georg Friedrich Bernhard (1826–1866)
German Geometry, Analysis, Calculus, Number Theory
Born on September 17, 1826, in Breselenz, Hanover
(now Germany), Bernhard Riemann is remembered as
one of the greatest mathematicians of the 19th century.
His work in NON-EUCLIDEAN GEOMETRY revolutionized
modern thinking on the study of shape and space and
provided the tools ALBERT EINSTEIN needed for his
description of the universe in the general theory of relativity. Among his many significant contributions to
ANALYSIS, Riemann is remembered for his construction
of the Riemann integral, which, for the first time,
extended the theory of CALCULUS beyond the realm of
continuous functions only. Riemann also developed a
theory of complex functions (that is, functions whose
inputs and outputs are COMPLEX NUMBERS) and studied
applications of calculus in this setting. In 1859, while
searching for a better approximation for the number of
primes as given by the famous PRIME-NUMBER THEOREM,
Riemann discovered a generalized version of LEONHARD EULER’s important ZETA FUNCTION. He made a
deep and insightful conjecture about the properties of
this function, the Riemann hypothesis, which, if true,
would provide invaluable information about the distribution of prime numbers. Proving (or disproving) the
Riemann hypothesis is considered the most important
unsolved mathematical problem of today.
Riemann exhibited a keen interest in mathematics
at an early age and was encouraged by the director of
his high school to pursue this line of study. In 1846
Riemann enrolled at the University of Göttingen to
study under the mathematician CARL FRIEDRICH GAUSS
(1777–1855). Surprisingly, even with Gauss as a faculty
member, Göttingen did not offer a strong program in
mathematics, and so, a year later, Riemann transferred
to Berlin University, an institution recognized for its
strength in mathematics at the time. While there, Riemann developed his general theory of complex variables and functions that later formed the basis of much
of his important work.
After completing his undergraduate degree at
Berlin, Riemann returned to Göttingen in 1849 to complete a doctoral thesis under the supervision of Gauss.
He was awarded a Ph.D. 2 years later in mathematics.
Gauss described Riemann’s thesis work as having
“a gloriously fertile originality.” By viewing the range
of outputs of a complex function as a surface in threeor even four-dimensional space, Riemann managed to
tie the theory of calculus and analysis to the study of
shape, space, and geometry. This innovative approach
was immediately recognized as a significant turning
point in the advancement of mathematics. On Gauss’s
recommendation, Riemann was appointed a position at
Göttingen, and he commenced work on his habilitation
degree. During this time Riemann turned his attention
to the study of FOURIER SERIES and questions of integrability. It was during this period that Riemann formulated his famous definition of an integral.
As the final step to completing his degree Riemann
was required to give a public lecture. Gauss asked him
to speak on the theory of geometry, and on June 10,
1854, Riemann presented to the outside world his
groundbreaking work on shape and space.
Riemann devised a general notion of distance in ndimensional space and developed a theory of TENSORs
that allowed him to properly define geometric shape. In
particular, Riemann developed the notion of curvature
tensor to measure, or define, the warping of space. The
mathematics Riemann presented on that day proved to
be the precise framework Einstein needed for structuring his theory of general relativity. Riemann continued
to present and publish groundbreaking work in the
years that followed.
On July 30, 1859, Riemann was appointed chair
of mathematics at Göttingen, and a few days later he
was honored with election to the Berlin Academy of
Sciences. By this time Riemann had developed an interest in NUMBER THEORY and the distribution of prime
Riemann, Georg Friedrich Bernhard 447