Example research essay topic: Euclidean Geometry Number Theory - 704 words

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Carl Friedrich Gauss Carl Friedrich Gauss was a German mathematician and scientist who dominated the mathematical community during and after his lifetime. His outstanding work includes the discovery of the method of least squares, the discovery of non-Euclidean geometry, and important contributions to the theory of numbers. Born in Brunswick, Germany, on April 30, 1777, Johann Friedrich Carl Gauss showed early and unmistakable signs of being an extraordinary youth. As a child prodigy, he was self taught in the fields of reading and arithmetic. Recognizing his talent, his youthful studies were accelerated by the Duke of Brunswick in 1792 when he was provided with a stipend to allow him to pursue his education.

In 1795, he continued his mathematical studies at the University of G? tinge. In 1799, he obtained his doctorate in absentia from the University of Helmstedt, for providing the first reasonably complete proof of what is now called the fundamental theorem of algebra. He stated that: Any polynomial with real coefficients can be factored into the product of real linear and / or real quadratic factors. At the age of 24, he published Disquisitiones arithmetical, in which he formulated systematic and widely influential concepts and methods of number theory dealing with the relationships and properties of integers. This book set the pattern for many future research and won Gauss major recognition among mathematicians.

Using number theory, Gauss proposed an algebraic solution to the geometric problem of creating a polygon of n sides. Gauss proved the possibility by constructing a regular 17 sided polygon into a circle using only a straight edge and compass. Barely 30 years old, already having made landmark discoveries in geometry, algebra, and number theory Gauss was appointed director of the Observatory at G? tinge.

In 1801, Gauss turned his attention to astronomy and applied his computational skills to develop a technique for calculating orbital components for celestial bodies, including the asteroid Ceres. His methods, which he describes in his book Theory Motu's Corporum Coelestium, are still in use today. Although Gauss made valuable contributions to both theoretical and practical astronomy, his principle work was in mathematics, and mathematical physics. About 1820 Gauss turned his attention to geodesy the mathematical determination of the shape and size of the Earths surface to which he devoted much time in the theoretical studies and field work.

In his research, he developed the heliotrope to secure more accurate measurements, and introduced the Gaussian error curve, or bell curve. To fulfill his sense of civil responsibility, Gauss undertook a geodetic survey of his country and did much of the field work himself. In his theoretical work on surveying, Gauss developed results he needed from statistics and differential geometry. Most startling among the unpublished discoveries of Gauss is that of non-Euclidean geometry. With a fellow student at G? tinge, he discussed attempts to prove Euclid's parallel postulate Through a point outside of a line, one and only one line exists which is parallel to the first line.

As he got closer to solving the postulate, the closer he was to non-Euclidean geometry, and by 1824, he had concluded that it was possible to develop geometry based on the denial of the postulate. He did not publish this work, conceivably due to its controversial nature. Another striking discovery was that of noncommutative algebras, which has been known that Gauss had anticipated by many years but again failed to publish his results. In the 1820 s, in collaboration with Wilhelm Weber, he explored many areas of physics. He did extensive research on magnetism, and his applications of mathematics to both magnetism and electricity are among his most important works. He also carried out research in the field of optics, particularly in systems of lenses.

In addition, he worked with mechanics and acoustics which enabled him to construct the first telegraph in 1833. Scarcely a branch of mathematics or mathematical physics was untouched by this remarkable scientist, and in whatever field he labored, he made unprecedented discoveries. On the basis of his outstanding research in mathematics, astronomy, geodesy, and physics, he was elected as a fellow in many academies and learned societies. On February 23, 1855, Gauss died an honored and much celebrated man for his accomplishments.

Research essay sample on Euclidean Geometry Number Theory