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# Bernhard Riemann

Without certain figures in history some subjects would still be elementary in content compared to our current knowledge. If Einstein would have decided not to pursue his studies, the world could very well have an entirely different political structure.

It can be said that each subject is dependent upon the accomplishments of its past pioneers. If it weren’t for the mathematician Antoine Augustin Cournot, John Forbes Nash, Jr. may not have made the advances in Game theory which nearly redefined a branch of Economics. This can also be said about Georg Friedrich Bernhard Riemann.

## His Life

Riemann grew up in poverty and received his early education by his father and later in his youth by a local teacher. At sixteen, after moving a few times, he realized that he had a great desire to study mathematical ideas beyond what his school offered. Riemann approached the director of the school with his request to learn more and was lent books on a variety of different mathematical subjects. It wasn’t long after that, he was formally enrolled in a university. Riemann began studies in theology and philology, more than likely by his father’s request, but never passing on attending classes in mathematics. After his father approved of shifting the focus of his studies to mathematics, Riemann began to shine.

Studying his life’s passion, Riemann was led only to discover the absolute deficiency in his current university’s math program. He was studying under “The Prince of Mathematics” Carl Gauss (pronounced Goss), who only taught applied mathematics at an elementary level. This compromised Riemann’s level of expertise, so he made the decision to leave for a new program in Berlin studying under Carl Jacobi, Jakob Steiner, and Johann Dirichlet who had a great influence on his education.

After making another transition back to Johanneum Lüneburg, where he originally started, he began to study under the experimental physicist W. (Wilhelm) Weber. During this time he documented his thoughts on a standardized mathematical physical view of nature and formed his dissertation: *On the Hypotheses which lie at the Bases of Geometry.* His work was ground breaking and resolved the issues his previous professor, Gauss, had concerning physical paradoxes in his extension of Leibniz’s calculus into the domain of complex numbers. Gauss solved a problem by finding several ways to measure the curvature of a surface by seeing that every point on any type of surface has one point that is the most curved and one that is the least curved. Moreover, Gauss recognized that the relationship between the most and least curved is a characteristic of functions in the domain of complex numbers.

Riemann expands on Gauss and shows that not only is there such a harmonic relationship, but from the complex functions a complete class of physical manifolds can be expressed. Riemann saw these complex functions as a way to change a physical manifold into another. These thoughts he expanded on in his dissertation have been an integral part to the works of many physicists including Albert Einstein.

Riemann had other works which advanced many other branches of mathematics. Some of his contributions can be seen in topology, calculus, and real analysis. His contributions to calculus are essential to what is currently being studied. Newton and Leibniz found that there was a connection between differentiation and integration. Thus the Fundamental Theorem of Calculus was born and is now the frame work for modern calculus. Integration had a long history, dating as far back as ancient Egypt (1800 B.C.). Throughout the years it was only used and studied and needed a more rigorous definition. It was Riemann who formalized integration by using limits to represent an infinite amount of approximating rectangles. Ultimately his formalization led to finding the area under a curve and defining a definite integral.

Although the amount of his work that has been published is relatively small compared to others in the field, Riemann’s work was seen as genius. His mentors have been known to say that they have learned more from him than he from them. It wasn’t even that he was a child prodigy or even that he studied under someone with an impeccable ability to teach a vast amount of knowledge that defined his life. It was centered on the fact that he was interested in mathematics and took the initiative to pursue his dreams and goals when other resources weren’t there.