Monday, April 25, 2016

C.F. Gauss

            Carl Friedrich Gauss: Normal Distribution from Abnormal Genius

Mathematicians stand on each other's shoulders.” --C.F. Gauss
Johann Carl Friedrich Gauss was born in 1777 in Germany to a working class family. His father was not in support of his education, believing that he should go into the family trades. Luckily, his mother and uncle understood that his intelligence was genius, and insisted that he get a proper education. The Duke of Brunswick financially supported Gauss’s education at the Brunswick Collegium Carolinum. He entered at the age of 15, already with a deep understanding of arithmetical information and number theory. At this point he had already discovered Bode’s law of planetary distances, the binomial theorem for rational exponents, and the arithmetic-geometric mean. Before entering college at the University of Göttingen at the age of 18, Gauss had rediscovered the law of quadratic reciprocity, the prime number theorem, and related the arithmetic-geometric mean to infinite series of expansions.
Gauss’s original wonder towards the cosmos was driven by the need for a better understanding of navigation. In the early 18th century shipping was unanimous with economy, and there was a massive need to understand how the earth moved in relation to other celestial bodies.
Gauss’s developments in Astronomy instigated his most important contribution to statistics: The Gaussian (or Normal) Distribution. Astronomers at the time made calculations as to the position of stars and planets, but there was no standard to these calculations. Mathematicians and astronomers believed that there was a pattern in individual measurement errors and that this pattern could be harnessed to find a reasonable estimate of the true value. Laplace was the first to create an Error Curve in 1774, based on the assumption that errors were more likely to be small than large, and that there would be an even distribution of positive and negative values centered around the true mean.
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It was Gauss who improved Laplace’s theory with his Method of Least Squares. On January 1st, 1801, Giuseppe Piazzi claimed to have sighted a new planet and named it Ceres. Unfortunately, the celestial body vanished before proper observations were made to prove his claim. Using Laplace’s Error Curve theory, many astronomers began predicting where Ceres would reappear. Not surprisingly, it was Gauss, then just 24 years old, who predicted a completely different area of the sky and was correct. Gauss used his Method of Least Squares to predict the location.
Many mathematicians stood on the shoulders of Gauss after he published his Method of Least Squares. For example, the curve was later used during the development of the Central Limit Theorem. Gauss was more than just an astronomer or mathematician. His academic interests also included magnetism and non-Euclidean geometry (in fact, he was the first to discover how to draw a 17-sided polygon using only a compass and straightedge while in college). After publishing his Method of Least Squares, Gauss was appointed professor of mathematics and the director of the observatory at Gottingen University, a position he maintained until his death in 1855.
Gauss believed in spreading only the truth, a bias that prevented him from publishing theories that were incomplete or imperfect. Gauss was consistently ahead of his time. His advances in numerous fields have earned him the reputation today of being one of the greatest mathematicians of all time.  

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