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## Carl Friedrich Gauss / KARL FREE-drick GOUS /

###### Mathematics Ranking 4th of 46 1

A portrait of Gauss on a German stamp from 1955.

Carl Friedrich Gauss is one of the top three mathematicians, with Archimedes and Newton.

Bio–Ages 0–20

Gauss grew up in an unpainted shack and was the only son of poor, uneducated parents. He could add sums in his head from the time he was 3. At 15, Gauss made a thorough study of Greek and Latin language and literature. By 18, he had mastered the works of Lagrange and Newton.

At 19, Gauss discovered the first major new geometric construction since Greek times: how to construct a seventeen-sided polygon using only a compass and straightedge (see stamp below). At 20, he wrote his doctoral thesis in mathematics; his dissertation was the first complete proof of the fundamental theorem of algebra, only partially proven before him. Gauss with a 17-sided polygon, straightedge, and compass. Stamp from the German Democratic Republic in 1977.

It states that every integral rational equation in a single variable has at least one root. Stated more simply, the theorem means that any algebraic equation has at least one root. The corollary is even more important: Every equation will have as many roots as the highest power of the unknown. That is, x4 + 2x3 + 9 = 0 will have four roots; x3 + x2 + 2x + 4 = 0 will have three, and so on. By proving the fundamental theorem, Gauss took one more step toward systematizing algebra and generalizing its rules.(1)

As a consequence of proving the fundamental theorem of algebra Gauss solidified our understanding of imaginary numbers. Euler said, “...such expressions as √-1 and √-2 are impossible or imaginary numbers, since they represent roots of negative quantities; and of such numbers we may truly say that they are neither nothing, nor greater than nothing, nor less than nothing, which necessarily makes them imaginary or impossible.”(2) Gauss also showed that imaginary numbers, like real numbers, can be plotted on a graph. (See stamp to the right.) A graph showing the Gaussian Complex Number Plane with four complex numbers plotted as points in the plane. German stamp from 1977.

Mathematics–Number Theory

At 24, Gauss published his first major work, the Disquisitiones Arithmeticae (Arithmetic Researches), “one of the most brilliant achievements in the history of mathematics.”(3) In the Arithmetic Researches, he formulated systematic and widely influential concepts and methods of number theory–dealing with relationships and properties of integers (-2, -1, 0, +1, +2, etc.)–which, for him, was of paramount importance in mathematics.

Bio–Ages 24–30

Also at 24, he rediscovered the lost asteroid Ceres using advanced computational techniques. Gauss once wrote, “It is not knowledge, but the act of learning ... which grants the greatest enjoyment. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms again for another.”

At 30, in 1807, Gauss was named director of the University of Göttingen. The mathematician accomplished his rich insights in a cramped study, furnished only with a work table, desk, chair, and lamp. Although he set forth his theories in a clear, orderly fashion, many of them could not be understood by his colleagues. Gauss was puzzled by this fact. “If others would reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries.”

Mathematics–Geodesy, Probability (Normal Distribution Curve or Gaussian Error Curve)

By introducing what is now known as the Gaussian error curve, he showed how probability could be represented by a bell-shaped curve, commonly called the normal curve of variation or normal distribution curve. This curve is basic to descriptions of statistically distributed data. It is a theoretical curve that shows how often an experiment will produce a particular result. The curve is symmetrical and bell-shaped, showing that trials will usually give a result near the average, but will sometimes deviate by large amounts. The width of the “bell” indicates how much confidence one can have in the result of an experiment—the narrower the bell, the higher the confidence. The normal distribution curve is often used in connection with tests in schools. Generally, a teacher wants test results that conform to a normal distribution curve in which most of the test takers do moderately well (the middle of the bell); some do worse than average, and some do better (the sloping sides of the bell); and a very small number get very high or very low scores (the rim of the bell).

Mathematics–Non-Euclidean Geometry

Gauss also discovered non-Euclidean geometry, but because it ran counter to contemporary views, he feared publication. Gauss invented the term “Non-Euclidean Geometry.” The first step in the creation of non-Euclidean geometry was the realization that Euclid’s parallel postulate of his Elements could not be proved on the basis of the other nine axioms. Note that Euclid called his axioms either postulates (he included five) or common notions (also five). It was an independent assertion, and so it was possible to adopt a contradictory axiom and develop an entirely new geometry. Today, we recognize all four (Gauss, Bernhard Riemann, John Bolyai, and Nikolay Ivanovich Lobachevsky) as the originators of non-Euclidean geometry.(4) “The creation of non-Euclidean geometry was the most consequential and revolutionary step in mathematics since Greek times . . . (because) the axioms of geometry are our basic facts about physical space, and vast branches of mathematics and physical science use the properties of Euclidean geometry.”(5) (See also the separate section on Non-Euclidean Geometry in the Riemann biography)

Physical Sciences–Gauss’s Law

Gauss discovered two laws, called collectively Gauss’s theorem or Gauss’s law–one describing electric fluxes, the other magnetic fluxes. Gauss’s law for electricity states that the electric flux across any closed surface is proportional to the net electric charge enclosed by the surface. The law implies that like charges repel one another while unlike charges attract. Gauss’s law for magnetism states that the magnetic flux across any closed surface is zero. Mathematical formulations of these two laws are part of a group of four, called Maxwell’s equations, that provide the basis of the unified electromagnetic theory. Ampere’s law and Faraday’s law of induction complete the group of four.

About 1820, Gauss started focusing on geodesy–the mathematical determination of the shape and size of the Earth’s surface. He was involved in the first worldwide survey of the Earth’s magnetic field.

His intrinsic-surface theory inspired one of his students, Bernhard Riemann, to develop a general intrinsic geometry of spaces with three or more dimensions (called Riemannian geometry, a form of non-Euclidean geometry). About 60 years later, Riemann’s ideas formed the mathematical basis for Einstein’s general theory of relativity.

The magnetic unit, the gauss, is named for him.

Footnotes:

(1) Jane Muir, Of Men & Numbers The Story of the Great Mathematicians (New York, 1961), p. 162.

(2) Ibid., p. 163.

(3) Encyclopaedia Britannica, Macropaedia, Volume 19, 1993, 15th Edition, p. 697.

(4) William Dunham, Journey Through Genius The Great Theorems of Mathematics (John Wiley and Sons, Inc., New York, 1990, p. 57.

(5) Morris Kline, Mathematical Thought From Ancient to Modern Times (New York, 1972), pp. 879-880.

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