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Gauss, Karl Friedrich

mathematical arithmetic mathematics complex

[gows] (1777–1855) German mathematician: one of the greatest of all mathematicians.

Gauss was of the stature of and in range of interests he exceeded both. He contributed to all areas of mathematics and to number theory (higher arithmetic) in particular. His father was a gardener and merchant’s assistant; the boy showed early talent, teaching himself to count and read, correcting an error in his father’s arithmetic at age 3, and deducing the sum of an arithmetic series (a,a + b,a + 2b…) at the age of 10. Throughout his life he had an extraordinary ability to do mental calculations. His mother encouraged him to choose a profession rather than a trade, and fortunately friends of his schoolteacher presented him to the Duke of Brunswick when he was 14; the Duke thereafter paid for his education and later for a research grant. Gauss was grateful, and was deeply upset when the Duke was mortally wounded fighting Napoleon at Jena in 1806. Gauss attended the Collegium Carolinum in Brunswick and the University of Göttingen (1795–98). He devised much mathematical theory between the ages of 14 and 17; at 22 he was making substantial and frequent mathematical discoveries, usually without publishing them. After the Duke’s death he became director of the Observatory at Göttingen, and was able to do research with little teaching, as he preferred.

Up to the age of 20 Gauss had a keen interest in languages and nearly became a philologist; thereafter foreign literature and reading about politics were his hobbies (in both he had conservative tastes). When at 28 he was financially comfortable he married Johanne Osthof; unbelievably happy, Gauss wrote to his friend W Bolyai (1775–1856), ‘Life stands before me like an eternal spring with new and brilliant colours.’ Johanne died after the birth of their third child in 1809, leaving her young husband desolate and, although he married again and had three more children, his life was never the same and he turned towards reclusive mathematical research. This was done for his own curiosity and not published unless complete and perfect (his motto was ‘Few, but ripe’) and he often remained silent when others announced results that he had found decades before. The degree to which he anticipated a century of mathematics has become clear only since his death, although he won fame for his work in mathematical astronomy in his lifetime. Of the many items named after him, the Gaussian error curve is perhaps best known.

During his years at the Collegium Carolinum, Gauss discovered the method of least squares for obtaining the equation for the best curve through a group of points and the law of quadratic reciprocity. While studying at Göttingen he prepared his book Disquisitiones arithmeticae (Researches in Arithmetic), published in 1801, which developed number theory in a rigorous and unified manner; it is a book which, as Gauss put it, ‘has passed into history’ and virtually founded modern number theory as an independent discipline. Gauss gave the first genuine proof of the fundamental theorem of algebra: that every algebraic equation with complex coefficients has at least one root that is a complex number. He also proved that every natural number can be represented as the product of prime numbers in just one way (the fundamental theorem of arithmetic). The Disquisitiones discusses the binomial congruences x n = A (mod p ) for integer n , A and p prime; x is an unknown integer. The algebraic analogue of this problem is x n = A . The final section of the book discusses x n =1 and weaves together arithmetic, algebra and geometry into a perfect pattern; the result is a work of art.

Gauss kept a notebook of his discoveries, which includes such entries as

EYPHKA! num = ? + ? + ?

which means that any number can be written as a sum of three triangular numbers (ie ? =½ n ( n + 1) for n integral). Other entries such as ‘Vicimus GEGAN’ or ‘REV. GALEN’ inscribed in a rectangle have never been understood but may well describe important mathematical results, possibly still unknown.

The notebook and Gauss’s papers show that he anticipated non-Euclidean geometry as a boy, 30 years before J Bolyai (1802–60, son of Wolfgang) and that he found fundamental theorem of complex analysis 14 years earlier; that he discovered quaternions before and anticipated A-M Legendre (1752–1833), and in much of their important work. If he had published, Gauss would have set mathematics half a century further along its line of progress.

From 1801–20 Gauss advanced mathematical astronomy by determining the orbits of small planets such as Ceres (1801) from their observed positions; after it was first found and then lost by , it was rediscovered a year later in the position predicted by Gauss.

During 1820–30 the problems of geodesy, terrestrial mapping, the theories of surfaces and conformal mapping of one domain to another aroused his interest. Later, up to about 1840, he made discoveries in mathematical physics, electromagnetism, gravitation between ellipsoids and optics. He believed that physical units should be assembled from a few absolute units (mainly length, mass and time); an idea basic to the SI system. Gauss was a skilled experimentalist and invented the heliotrope, for trigonometric determination of the Earth’s shape, and, with W E Weber (1804–91), the electromagnetic telegraph (1833). From 1841 until his death Gauss worked on topology and the geometry associated with functions of a complex variable. He transformed virtually all areas of mathematics.

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