Carl Friedrich GAUSS

(1777-1855), German mathematican, noted for his wide-ranging contributions to mathematics and physics, including number theory, analysis, differential geometry, geodesy, electromagnetism, astronomy and optics.

Gauss was born in Braunschweig (see map below) on April 1777.

 

1 : Braunschweig 2 : Göttingen


He studied ancient languages. At the age of 17 he became interested in mathematics and attempted a solution of the classical problem of constructing a regular heptagon, or seven-sided figure, with ruler and compass. He didn’t only succeed in proving that this construction was impossible, but went on to give methods of constructing figures with 17, 257 and 65, 537 sides.In so doing he proved that the construction, with compass and ruler, of a regular polygon with an odd number of sides was impossible only when the number of sides was a prime number of the series 3, 5, 17, 257, and 65, 537 or was a multiple of two or more of these numbers. With this discovery he gave up his intention to study languages and turned to mathematics.

He studied at the University of Göttingen from 1795 to 1798. For his doctoral thesis he submitted a proof that every algebraic equation has at least one root, or solution.
This theorem, which had challenged mathematicians for centuries, is still called
"THE FUNDAMENTAL THEOREM OF ALGEBRA"
"Every polynomial in x can be resoluted in factors from the first and second degree."
example:

x³ + 4x² + x -6 = (x-1).(x²+5x+6)

The theorem, in its most general form, is equivalent to the statement that every polynomial equation f(z) = 0, where z is a complex number, has at least one root, either real or complex. The coefficients of the polynomial may themselves be either real or complex. At the end of the 18th century (1798) Carl Friedrich Gauss proved this theorem.

By the time of Gauss, algebra had entered its modern phase. Attention shifted from solving polynomial equations to studying the structure of abstract mathematical systems whose axioms were based on the behavior of mathematical objects, such as complex numbers that mathematicians encountered when studying polynomial equations. Indeed at the end of the eighteenth century three men - Caspard Wessel (1745-1818), a self-taught Norwegian surveyor, Jean Robert Argand (1768-1822), an equally self-taught Swiss book-keeper, and Gauss himself - working independently, put forward geometrical interpretations of complex numbers. But in the words of G.H.Hardy, 'Gauss was the first mathematician to use complex numbers in a really confident and scientific way'. He also took the next step investigating the theory of complex functions. His volume on the theory of numbers, Disquisitiones Arithmeticae (Inquiries into Arithmetic, 1801), is a classic work in the field of mathematics.

Gauss next turned his attention to astronomy. A faint planetoid, Ceres, had been discovered in 1801; and because astronomers thought it was a planet, they observed it with great interest until losing sight of it. From the early observations Gauss calculated its exact position, so that it was easily rediscovered. He presented his results in a second great work , Theoria motus corporum coelestium, which was published in 1809


He also worked out a new method for calculating the orbits of heavenly bodies. In 1807 Gauss was appointed professor of mathematics and director of the observatory at Göttingen, holding both positions until his death there on February 23, 1855.
Although Gauss made valuable contributions to both theoritical and practical astronomy, his principal work was in mathematics and mathematical physics. On other example of Gauss' creativity in the number theory field can be found on the back page of one of his boyhood tables of logarithms : a shorthand statement of the prime number theorem. The proof for this theorem was found later independently by De la Vallée Poussin (Belgium) and Hademard (France).


Gauss' third major treatise is the Disquisitiones generales circa superficies curvas of 1827. In this work he not only provided the definitive treatment of the differential geometry of surfaces lying in three-dimensional space, but also advanced the totally new concept that a surface is a space in itself - an idea that was generalized further in the context of non-Euclidean geometry. In the first part of 19th century the German mathematician Carl Friedrich Gauss, the Russian mathematician Nikolay Ivanovich Lobachevsky, and Hungarian mathematician Jànos Bolyai independently demonstrated the possibility of constructing a consist system of geometry in which Euclid’s postulate of the unique parallel was replaced by a postulate stating that through any point not on a given straight line an infinite number of parallels to the given line could be drawn. But Gauss failed to publish these important findings because he wished to avoid publicity.

In probability theory, he developed the important method of least squares and the fundamental laws of probability distribution. The normal probability graph is still called the Gaussian curve.

 

The figure above represents the graph of the standardized normal Gauss curve with standarddeviation s =1, and the mean m = 0 and z = (X -  m )/s according to the formula:

Y = e-0.5z²

He made geodetic surveys, and applied mathematics to geodesy. With the German physicist Wilhelm Eduard Weber, Gauss did extensive research on magnetism and electricity. The Intensitas vis magneticae terrestris ad mensuram absolutam revocata (1832), the Allgemeine Theorie des Erdmagnetismus (1839) and the Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abstossungskräfte (1840) are among his most important works; the unit of intesity of magnetic fields is today called the gauss.He also carried out research in optics, particulary in systems of lenses. Scarcely a branch of mathematics or mathematical physics was untouched by Gauss.
His discoveries and insights pointed the way to much of the best mathematics of the later nineteenth century. He was indeed, as his contemporaries styled him, the 'PRINCE OF THE MATHEMATICIANS'.

 

Sources:


- "Gauss, Carl Friedrich,","Algebra","Theory of equations","Geometry"

Microsoft (R) Encarta Copyright (C)1994 Microsoft Corporation

- "Nieuwe Delta 4a", Wolters Leuven
- "Nieuwe Delta 6 analyse en stochastiek", Wolters Leuven

- "Makers of mathematics" Stuart Hollingdale

- http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Gauss.html

 

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