# Carl Friedrich Gauss

# Carl Friedrich Gauss

**1777-1855**

**German Mathematician, Astronomer and Physicist**

Carl Friedrich Gauss is considered to be one of the greatest mathematicians of all time, mentioned in the same breath as Archimedes (c. 287-212 b.c.) and Isaac Newton (1642-1727). He made revolutionary strides in pure mathematics, especially the theory of complex numbers, as well as important contributions to the fields of astronomy, geodesy, and electromagnetism. He showed how probability can be represented by a bell-shaped curve, a concept with major implications in understanding statistics. He was among the first to consider the possibility of a non-Euclidean geometry.

Gauss was born in Brunswick, Germany, the only son of an impoverished bricklayer and his wife. His gifts for mathematics and languages were apparent at an early age. Persistent efforts by impressed teachers and his devoted mother brought him to the attention of the Duke of Brunswick, who arranged financing for his continuation in secondary school and attendance at the University of Göttingen.

In his doctoral dissertation, obtained in absentia from the University of Helmstedt, Gauss completed the proof of the Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra states that any polynomial equation has solutions. It had been partially proven before Gauss, but he was able to demonstrate that every algebraic equation with complex coefficients (that is, coefficients given in terms of the imaginary square root of -1) has complex solutions. He managed to find a way to finish this proof without the actual manipulation of complex numbers, as their theory was yet to be well developed. In fact that task would soon be accomplished by Gauss himself.

At the age of 24, Gauss published his first great work, *Disquisitiones Arithmeticae*. This farranging treatise on number theory included the formulation of complex numbers as *a + b*√-1 where *a* and *b* are real numbers. Although it may seem that arithmetic involving imaginary numbers would not have many practical applications, in fact this mathematical construct has been instrumental in expressing and solving many physical problems. Later Gauss took a vital step in this direction by showing how complex numbers could be represented on an (x,y) plane.

Gauss also developed a way to use number theory to determine whether a regular polygon with a given number of sides could be geometrically constructed using a compass and ruler. Finding that a 17-sided polygon could be constructed, he developed a method to do so. His achievement was the first progress in this area since Euclid (c. 330-260 b.c.), who lived 2,000 years earlier.

In 1801, the asteroid Ceres was discovered, but astronomers were unable to calculate its orbit. Gauss developed a technique that predicted the orbital track with enough precision for the asteroid to be observed repeatedly as it continued on its path. He applied his technique again to the asteroid Pallas, refining it to account for the effects of the planets. Gauss's methods are still in use today, implemented by modern computers. In 1807 he became a professor at the University of Göttingen, and director of its observatory. He would remain there for the rest of his life.

Gauss started working in the field of geodesy, the measurement of the Earth's surface, in about 1820. He invented the *heliotrope*, a more accurate surveying instrument than was previously in use. It was in attempting to understand the distribution of repeated measurements that he introduced the famous bell curve, or *normal distribution of variation*.

Thinking about the curved surface of the Earth led Gauss back to pure mathematics, and he developed a way to determine characteristics of a surface by measuring the lengths of curves that lie upon it. This work would inspire his student, Bernhard Riemann (1826-1866), to embark upon the geometrical theories that would lay the foundation for general relativity. Gauss himself came to the conclusion that it was possible to have a self-consistent geometry in which Euclid's parallel axiom (that only one line can be drawn parallel to another line and through a point not on it) did not apply. But the conservative mathematician was reluctant to proclaim such an idea until it was independently published by Janos Bolyai (1802-1860) and Nicolai Lobachevsky (1792-1856) three decades later.

Gauss's work in mathematical physics contributed to potential theory and the development of the principle of conservation of energy. He calculated that the net electric flux through a surface, regardless of its shape, equals a constant times the net charge inside the surface, a result that has become known as Gauss' Law. He worked closely with the physicist Wilhelm Weber (1804-1891) on experiments in electromagnetism that would be instrumental in the development of telegraphy.

**SHERRI CHASIN CALVO**

## YOU, TOO, CAN BE A MATHEMATICAL GENIUS

The following story has often been told about Carl Gauss, though it may or may not actually be true. The math involved, however, is real. When Gauss was nine years old, his teacher gave his class the problem of adding the integers from 1 to 100. Gauss responded with the correct answer within seconds, astonishing his teacher. Gauss solved the problem by noticing a pattern: Whenthe greatest number in the series (100) is added to the least number (1), the result is 101; when the second highest number (99) is added to the second lowest number (2), the result is 101; when the third highest number (98) is added to the third lowest number (3), the result is 101; and so on. Because there are 50 pairs of integers between 1 and 100, the sum of these integers is 101 × 50 or 5,050. The pattern Gauss noticed can be used to find the sum of the integers from 1 to any even number—from 1 to 200, for example.

Step 1: Add 1 to the even number. 1 + 200 = 201

Step 2: Divide the even number by 2. 200 ÷ 2 = 100

Step 3: Multiply Step 1 by Step 2. 201 × 100 = 20,100

See if you can use these steps to find the sum of the integers from 1 to 1,000,000.

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**Carl Friedrich Gauss**