Bernhard Riemann
Revolutionized geometry and complex analysis
Quotes by Bernhard Riemann
It is known that the results of geometry rest upon certain assumptions concerning the nature of space.
The purpose of this work is to investigate the number of primes less than a given magnitude.
The function ζ(s) can be analytically continued to the entire complex plane except for a pole at s=1.
Non-Euclidean geometries arise when the curvature of space is not zero.
The metric of space determines its geometry.
In the theory of functions, the concept of analytic continuation is fundamental.
The sum over the reciprocals of primes diverges, as established by Euler.
The zeros of the zeta function lie on the critical line Re(s)=1/2.
Geometry is not absolute, but relative to the properties of space.
The differential geometry of surfaces reveals the intrinsic properties independent of embedding.
Complex analysis provides tools to study real functions through their extensions.
The Riemann integral generalizes the notion of integration for bounded functions.
Abel's theorem on integrals finds a natural place in the theory of elliptic functions.
The conformal mapping preserves angles and is essential in geometry.
Space itself, and time, and all their contents, may be only shadowy illusions.
Mathematics is the language in which God has written the universe.
The value of a problem is not in its solution, but in the path to it.
In the vastness of mathematics, we glimpse the infinite.
The beauty of a proof lies in its simplicity and generality.
Curiosity is the engine of mathematical discovery.