Carl Friedrich Gauss - Студенческий научный форум

XI Международная студенческая научная конференция Студенческий научный форум - 2019

Carl Friedrich Gauss

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Carl Friedrich Gauss, original name Johann Friedrich Carl Gauss, (born April 30, 1777, Brunswick [Germany]—died February 23, 1855, Göttingen, Hanover), German mathematician, generally regarded as one of the greatest mathematicians of all time for his contributions to number theory, geometry, probability theory, geodesy, planetary astronomy, the theory of functions, and potential theory (including electromagnetism).


Gauss was a child prodigy. There are many anecdotes concerning his precocity as a child, and he made his first ground-breaking mathematical discoveries while still a teenager.

At just three years old, he corrected an error in his father payroll calculations, and he was looking after his father’s accounts on a regular basis by the age of 5. At the age of 7, he is reported to have amazed his teachers by summing the integers from 1 to 100 almost instantly (having quickly spotted that the sum was actually 50 pairs of numbers, with each pair summing to 101, total 5,050). By the age of 12, he was already attending gymnasium and criticizing Euclid’s geometry.

Gauss was the only child of poor parents. He was rare among mathematicians in that he was a calculating prodigy, and he retained the ability to do elaborate calculations in his head most of his life. Impressed by this ability and by his gift for languages, his teachers and his devoted mother recommended him to the duke of Brunswick in 1791, who granted him financial assistance to continue his education locally and then to study mathematics at the University of Göttingen from 1795 to 1798. Gauss’s pioneering work gradually established him as the era’s preeminent mathematician, first in the German-speaking world and then farther afield, although he remained a remote and aloof figure.

Scientific discoveries

At 15, Gauss was the first to find any kind of a pattern in the occurrence of prime numbers, a problem which had exercised the minds of the best mathematicians since ancient times. Although the occurrence of prime numbers appeared to be almost competely random, Gauss approached the problem from a different angle by graphing the incidence of primes as the numbers increased. He noticed a rough pattern or trend: as the numbers increased by 10, the probability of prime numbers occurring reduced by a factor of about 2 (e.g. there is a 1 in 4 chance of getting a prime in the number from 1 to 100, a 1 in 6 chance of a prime in the numbers from 1 to 1,000, a 1 in 8 chance from 1 to 10,000, 1 in 10 from 1 to 100,000, etc). However, he was quite aware that his method merely yielded an approximation and, as he could not definitively prove his findings, and kept them secret until much later in life.

Gauss’s first significant discovery, in 1792, was that a regular polygon of 17 sides can be constructed by ruler and compass alone. Its significance lies not in the result but in the proof, which rested on a profound analysis of the factorization of polynomial equations and opened the door to later ideas of Galois theory. His doctoral thesis of 1797 gave a proof of the fundamental theorem of algebra: every polynomial equation with real or complex coefficients has as many roots (solutions) as its degree (the highest power of the variable). Gauss’s proof, though not wholly convincing, was remarkable for its critique of earlier attempts. Gauss later gave three more proofs of this major result, the last on the 50th anniversary of the first, which shows the importance he attached to the topic.

Although he made contributions in almost all fields of mathematics, number theory was always Gauss’ favourite area, and he asserted that “mathematics is the queen of the sciences, and the theory of numbers is the queen of mathematics”. An example of how Gauss revolutionized number theory can be seen in his work with complex numbers (combinations of real and imaginary numbers).

Gauss gave the first clear exposition of complex numbers and of the investigation of functions of complex variables in the early 19th Century. Although imaginary numbers involving i(the imaginary unit, equal to the square root of -1) had been used since as early as the 16th Century to solve equations that could not be solved in any other way, and despite Euler’s ground-breaking work on imaginary and complex numbers in the 18th Century, there was still no clear picture of how imaginary numbers connected with real numbers until the early 19th Century. Gauss was not the first to intepret complex numbers graphically (Jean-Robert Argand produced his Argand diagrams in 1806, and the Dane Caspar Wessel had described similar ideas even before the turn of the century), but Gauss was certainly responsible for popularizing the practice and also formally introduced the standard notation a + bi for complex numbers. As a result, the theory of complex numbers received a notable expansion, and its full potential began to be unleashed.

At the age of just 22, he proved what is now known as the Fundamental Theorem of Algebra (although it was not really about algebra). The theorem states that every non-constant single-variable polynomial over the complex numbers has at least one root (although his initial proof was not rigorous, he improved on it later in life). What it also showed was that the field of complex numbers is algebraically "closed" (unlike real numbers, where the solution to a polynomial with real co-efficients can yield a solution in the complex number field).


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