Johann Carl Friedrich Gauss

Johann Carl Friedrich Gauss has been hailed as “The Prince of Mathematics”, and was one of the most history’s most influential men in numerous fields of mathematics, as well as many scientific fields. Born on April 30th, 1777, Gauss demonstrated prodigious intellect at a very young age. The first signs of his talent were revealed at the age of three, when he corrected an arithmetical error made by his father, a mason, while calculating a payroll. Into his schooling years in Germany, Gauss continued to demonstrate his unusual mathematical talent, most reputably when, when asked to add the numbers 1 to 100, he calculated the correct answer within seconds. He attended both the Collegium Carolinum and the University of Göttingen.

Gauss’ made his first significant breakthrough in 1796, during his second year at Göttingen. Two of his discoveries include his method for constructing the heptadecagon (a 17 sided figure) as well as his invention of modular arithmetic. His discovery of heptadecagonal constructability described the method by which all trigonometric functions of the shape (defined by 2π/17) could be expressed using non-trigonometric algebra. This includes square roots. His formula, which appeared five years later in his book Disquisitiones Arithmeticae, was as follows:

(1)
\begin{align} % MathType!MTEF!2!1!+- % feqaeaartrvr0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l % bbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0R % Yxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa % caGaaeqabaaaamaaaOqaaiaaigdacaaI2aGaci4yaiaac+gacaGGZb % WaaeWaaeaadaWcaaqaaiaaikdacqaHapaCaeaacaaIXaGaaG4naaaa % aiaawIcacaGLPaaacqGH9aqpcqGHsislcaaIXaGaey4kaSYaaOaaae % aacaaIXaGaaG4naaWcbeaakiabgUcaRmaakaaabaGaaG4maiaaisda % cqGHsislcaaIYaWaaOaaaeaacaaIXaGaaG4naaWcbeaaaeqaaOGaey % 4kaSIaaGOmamaakaaabaGaaGymaiaaiEdacqGHRaWkcaaIZaWaaOaa % aeaacaaIXaGaaG4naaWcbeaakiabgkHiTmaakaaabaGaaG4maiaais % dacqGHsislcaaIYaWaaOaaaeaacaaIXaGaaG4naaWcbeaaaeqaaOGa % eyOeI0IaaGOmamaakaaabaGaaG4maiaaisdacqGHRaWkcaaIYaWaaO % aaaeaacaaIXaGaaG4naaWcbeaaaeqaaaqabaaaaa!58FD! $16\cos \left( {\frac{{2\pi }}{{17}}} \right) = - 1 + \sqrt {17} + \sqrt {34 - 2\sqrt {17} } + 2\sqrt {17 + 3\sqrt {17} - \sqrt {34 - 2\sqrt {17} } - 2\sqrt {34 + 2\sqrt {17} } }$ \end{align}

Modular arithmetic describes the integer system in which numbers “wrap around” and repeat themselves after a given value, such as is used in clocks. The Disquisitiones Arithmeticae also proved the Fundamental Theory of Arithmetic, and dealt intensively with Number Theory and properties of integers.

Over the next ten years, after finishing his schooling, Gauss furthered his contributions to the world of mathematics, and through such discoveries as his study of asteroids and compass magnetism, broadened his impact on the scientific world. Gauss was a compulsive perfectionist, and refused to publish much of his work as a result. He insisted that that which he did publish be his very best work, and thus, many others took credit for discoveries which, as was not revealed until after his death, they had discovered years after him, but that he had not made public. Among these discoveries was the Cauchy Integral Theorem, a discovery of great magnitude in the field of analytical functions. This theorem, which was not revealed until years after Gauss’ death, was stated thus:

(2)
\begin{align} % MathType!MTEF!2!1!+- % feqaeaartrvr0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l % bbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0R % Yxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa % caGaaeqabaaaamaaaOqaamaapubabaGaamOzaaWcbaGaamyEaaqab0 % GaeSyeUhTaey4kIipakiaacIcacaWG6bGaaiykaiaadsgacaWG6bGa % eyypa0JaaGimaaaa!3E06! $\oint_y f (z)dz = 0$ \end{align}

Sources:

O'Connor, J. J., (06-12-96). Johann Carl Friedrich Gauss. A biography. Retrieved February 25, 2008, from
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Gauss.html

Weisstein, Eric W., (18-07-97). Gauss, Karl Friedrich (1777-1855). Retrieved February 25, 2008, from
http://scienceworld.wolfram.com/biography/Gauss.html

Rebertson, E. F., (20-06-04). Karl Friedrich Gauss. Retrieved February 25, 2008, from
http://www.shsu.edu/~icc_cmf/bio/gauss.html