Thursday, 27 January 2011

Carl Friedrich Gauss (30 April 1777 – 23 February 1855)

Carl Friedrich Gauss
Oil painting by G. Biermann
Two hundred years ago, one of the greatest mathematical thinkers ever to have lived was in his prime. In 1807, at the age of 30, Carl Friedrich Gauss was appointed Professor of Astronomy at Göttingen. But this did not inhibit his prolific work in pure mathematics. In 1811, he published a paper on the meaning of integrals with complex limits and examined the dependence of such integrals on the chosen path of integration.

Remarkable as it is, it is not so much this particular piece of work we wish to celebrate in this article. In mathematics Gauss is remembered for so many things, but perhaps most often for the Gaussian or "Normal" statistical distribution.

Gauss was a child prodigy. There is a story, which may or may not be true, about his remarkable mathematical ability at primary school. His teacher, realising Gauss had completed a task well ahead of the rest the class, asked Gauss to calculate the sum of the numbers from 1 to 100. The child thought for a few seconds, then told her the answer: 5050. He had added 1 and 99, 2 and 98, etc, to turn the sum into 49 x 100, after which he added the only unpaired numbers, 50 and 100.

When Gauss was only 14, Charles William Ferdinand, the Duke of Brunswick, who was always on the lookout for bright students, paid for the boy to go to the university in Brunswick.

Subsequently, he studied further in Göttingen. 1796 was Gauss’s most prolific year, in which he documented at least 5 major discoveries and proofs in number theory and geometry. He demonstrated the construction of a 17-sided polygon using only straight-edge and compass, something that had eluded all mathematicians since the ancient Greeks.

Gauss sorely wanted to be accepted by the elite group of Parisian mathematicians. But he presented his ideas in cryptic ways and didn’t believe in showing much working and this ultimately went against him. He was to spend all his life in what is now Germany.

In 1801, there was great excitement in the world of astronomy as an 8th planet was discovered between Mars and Jupiter. It was named Ceres, and today we know it as the largest asteroid. This discovery was considered a great omen for a new dawn of science at the beginning of the 19th century.

Shortly afterwards, however, the astronomers lost sight of the newly discovered object. Gauss announced he knew where to find it, using mathematical techniques now used to analyse data of many different types. To solve the problem of measurements of Ceres, Gauss had invented the Gaussian (or Normal) Distribution, now used widely in statistics, enabling patterns to be seen in seemingly random data.

The breakthrough was his realisation that the measurements, accurate and inaccurate, would, when plotted on a graph, be distributed around the true value in the shape of a bell-shaped curve. With this and other tools, statistical analysis has become a powerful weapon to analyse data, to test hypotheses, to separate statistical fact from fiction.

Gauss had an aversion to teaching and said that a professor “loses his precious time” lecturing students. This led to his taking the non-teaching post as Professor of Astronomy in the University of Göttingen in 1807 and he spent the rest of his career tracking the paths of planets.

The Gaussian distribution crops up everywhere; it is one of the statistician’s tools for understanding the real world, in the fields of chemistry, medicine, engineering, finance and many others.
Although arguably the most famous of Gauss’s mathematical innovations, the Gaussian distribution is probably not the greatest of his many mathematical achievements.

Despite his enormous influence in the fields of statistics and astronomy, the properties of numbers was Gauss’s true mathematical penchant. He once said “Maths is the Queen of the Sciences and Number Theory the Queen of Maths.”
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