Pafunty Lvovich Chebyshev Biography

World of Mathematics on Pafunty Lvovich Chebyshev

Chebyshev has given his name to results in probabilityand analysis, one of the first Russian mathematicians by birth to be so recognized. His work reflected a great deal of mathematical sophistication, making connections between different areas and generalizing techniques. He played a primary role in establishing a viable mathematical curriculum at St. Petersburg University, which laid the foundations for subsequent achievements in Russian mathematics.

Pafnuty Lvovich Chebyshev was born on May 16, 1821 in Okatovo in the Kaluga region of Russia. His father, Lev Pavlovich Chebyshev, was a former army officer. In 1832, the family moved to Moscow, where Chebyshev was educated at home under the instruction of an author of popular arithmetics. As a result, he was well prepared when he enrolled in Moscow University in 1837. There, Chebyshev studied physics and mathematics.

In 1841, Chebyshev graduated with a bachelor's degree in mathematics and within two years he had passed his master's examination. Chebyshev's thesis, entitled "An Essay on an Elementary Analysis of the Theory of Probability," dealt with the derivation of a law of large numbers (one of a whole group of results that indicated the increasing reliability of experimental results the larger the number of trials) using the methods of analysisof which he had already shown himself a master. In general, Chebyshev was looking for derivations of the leading results of probability by methods that could not be faulted for rigor, but which were not dependent on mathematical ideas that seemed out of proportion to the depth of the subject.

Chebyshev could not find employment in Moscow. As in the German university system, it was necessary to produce a thesis to earn the privilege of teaching and Chebyshev's thesis examined integration by means of logarithms, a topic straight out of analysis. After its acceptance, Chebyshev joined the faculty at St. Petersburg University in 1847, lecturing on algebra, number theory, and probability. In addition to his teaching, he helped prepare a new edition of Leonhard Euler's papers on number theory. Not far removed from this task was the subject of the theory of congruences, which Chebyshev dealt with in his doctoral thesis. He defended the thesis in 1849 and received a prize from the Russian Academy of Sciences.

Between the years 1850 and 1860 Chebyshev spent much of his time working on questions of mechanical engineering. During this period he moved up the academic ladder and by 1859 he was a senior academician in the St. Petersburg Academy. The subject of hinges led Chebyshev to consider problems of best approximation to a function, one of the results of which was later known as the Chebyshev polynomials. In addition to his own polynomials, he studied other systems of what are called orthogonal polynomials as well (the orthogonality refers to an independence they have in being needed to represent a given function).

Establishes Russian Tradition in Probability

There had been work done in Russia on probability before Chebyshev, but the number and quality of the students picked up considerably as a result of his efforts. One contributing factor was his generosity with his time, which explains the impressive list of students who chose to study with him. He kept an open house for students and continued to do so even after his retirement. In his own work, Chebyshev preferred to use elementary methods, and that may have given his work an appearance of comprehensibility.

Among the subjects to which Chebyshev contributed was the distribution of prime numbers. There had been much work devoted to the question of whether the apparent irregularities in the distribution of prime numbers (any number only divisible by one and itself) disappeared as one looked at larger initial segments of the positive whole numbers. Chebyshev was able to get a decent approximation for the number of prime numbers less than a fixed number compared to known functions of that fixed number, but he did not prove that there was a limiting value. He did, however, demonstrate a conjecture that if n is greater than 3, there is always at least one prime between n and 2 n-2.

In addition to his own teaching, Chebyshev was also active in improving the quality of mathematics and physics teaching on a national basis. He built a calculating machine in the late1870s, although this was more as a demonstration of the potential usefulness of mechanical devices than as a genuine aid to calculation. He was perhaps the creator of the tradition in Russia of making probability a part of the general mathematical curriculum. In light of the profound contributions made to the subject by Andrei Kolmogorov, this was no slight step in the development of probability.

Chebyshev's virtuosity as an analyst enabled him to derive results in probability that had previously been used without being well understood. He was able to take advantage of material ndeveloped by the French probabilist I.J. Bienaymé to produce a convincing demonstration of the law of large numbers. Chebyshev was recognized at home and abroad, being the first Russian to be elected a foreign member of the Paris Académie des Sciences. At the time of his death on December 8, 1894, the Russian mathematical tradition was stronger than ever before, thanks to his work and that of his students.