Pafnuty Chebyshev
Encyclopedia
Pafnuty Lvovich Chebyshev ( – ) was a Russian
mathematician
. His name can be alternatively transliterated
as Chebychev, Chebysheff, Chebyshov, Tschebyshev, Tchebycheff, or Tschebyscheff (the latter two are French and German transcriptions).
, province of Kaluga. His father, Lev Pavlovich, was a Russian nobleman and wealthy landowner. Pafnuty Lvovich was first educated at home by his mother Agrafena Ivanovna (in reading and writing) and by his cousin Avdotya Kvintillianovna Sukhareva (in French and arithmetic). Chebyshev mentioned that his music teacher also played an important role in his education, for she "raised his mind to exactness and analysis."
A physical handicap (of unknown cause) affected Chebyshev's adolescence and development. From childhood, he limped and walked with a stick and so his parents abandoned the idea of his becoming an officer in the family tradition. His disability prevented his playing many children's games and he devoted himself instead to mathematics.
In 1832, the family moved to Moscow
, mainly to attend to the education of their eldest sons (Pafnuty and Pavel, who would become lawyers). Education continued at home and his parents engaged teachers of excellent reputation, including (for mathematics and physics) P.N. Pogorelski, held to be one of the best teachers in Moscow and who had taught (for example) the writer Ivan Sergeevich Turgenev
.
, N.E. Zernov and D.M. Perevoshchikov of whom it seems clear that Brashman had the greatest influence on Chebyshev. Brashman instructed him in practical mechanics and probably showed him the work of French engineer J.V. Poncelet
.
In 1841 Chebyshev was awarded the silver medal for his work "calculation of the roots of equations" which he had finished in 1838. In this, Chebyshev derived an approximating algorithm for the solution of algebraic equations of nth degree based on Newton's algorithm
. In the same year he finished his studies as "most outstanding candidate".
In 1841, Chebyshev's financial situation changed drastically. There was famine in Russia and his parents were forced to leave Moscow. Although they could no longer support their son, he decided to continue his mathematical studies and prepared for the master examinations, which lasted six months. Chebyshev passed the final examination in October 1843 and, in 1846, defended his master thesis "An Essay on the Elementary Analysis of the Theory of Probability." His biographer Prudnikov suggests that Chebyshev was directed to this subject after learning of recently-published books on probability theory or on the revenue of the Russian insurance industry.
's works were rediscovered by P. N. Fuss and were being edited by V. Ya. Bunyakovsky
, who encouraged Chebyshev to study them. This would come to influence Chebyshev's work. In 1848, he submitted his work The Theory of Congruences for a doctorate, which he defended in May 1849. He was elected an extraordinary professor
at St Petersburg University in 1850, ordinary professor in 1860 and, after 25 years of lectureship, he became merited professor in 1872. In 1882 he left the university and devoted his life to research.
During his lectureship at the university (1852–1858), Chebyshev also taught practical mechanics at the Alexander Lyceum
in Tsarskoe Selo (now Pushkin), a southern suburb of St Petersburg.
His scientific achievements were the reason for his election as junior academician
(adjunkt) in 1856. Later, he became an extraordinary (1856) and in 1858 an ordinary member of the Imperial Academy of Sciences
. In the same year he became an honorary member of Moscow University. He accepted other honorary appointments and was decorated several times. In 1856, Chebyshev became a member of the scientific committee of the ministry of national education. In 1859, he became an ordinary member of the ordnance department of the academy with the adoption of the headship of the commission for mathematical questions according to ordnance and experiments related to ballistics. The Paris academy
elected him corresponding member in 1860 and full foreign member in 1874. In 1893, he was elected honorable member of the St. Petersburg Mathematical Society
, which had been founded three years earlier.
Chebyshev died in St Petersburg on 26 November 1894.
, statistics
and number theory
. Chebyshev's inequality
says that if
is a random variable
with standard deviation
σ, the probability that the outcome of
is no less than 
away from its mean is no more than 
:
Chebyshev's inequality is used to prove the weak law of large numbers.
The Bertrand–Chebyshev theorem (1845|1850) states that for any
, there exists a prime number 
such that 
. It is a consequence
of Chebyshev inequalities for the number
of prime numbers less than 
, which state that
is of the order of 
. A more precise form is given by the celebrated prime number theorem
: the
quotient of the two expressions approaches 1 as
tends to infinity.
n mathematics. Among his well-known students were the prolific mathematicians Dmitry Grave
, Aleksandr Korkin
, Aleksandr Lyapunov
and Andrey Markov
. According to the Mathematics Genealogy Project
, Chebyshev has 7,483 mathematical "descendants" as of 2010.
The crater Chebyshev
on the Moon
and the asteroid
2010 Chebyshev
are named in his honour.
Russians
The Russian people are an East Slavic ethnic group native to Russia, speaking the Russian language and primarily living in Russia and neighboring countries....
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. His name can be alternatively transliterated
Romanization of Russian
Romanization of the Russian alphabet is the process of transliterating the Russian language from the Cyrillic alphabet into the Latin alphabet...
as Chebychev, Chebysheff, Chebyshov, Tschebyshev, Tchebycheff, or Tschebyscheff (the latter two are French and German transcriptions).
Early years
One of nine children,Biography in MacTutor Archive Chebyshev was born in the village of Okatovo in the district of BorovskBorovsk
Borovsk is a town and the administrative center of Borovsky District of Kaluga Oblast, Russia, located just south from the oblast's border with Moscow Oblast. It is situated on the Protva River, about southwest of Moscow.Population: 12,000 ....
, province of Kaluga. His father, Lev Pavlovich, was a Russian nobleman and wealthy landowner. Pafnuty Lvovich was first educated at home by his mother Agrafena Ivanovna (in reading and writing) and by his cousin Avdotya Kvintillianovna Sukhareva (in French and arithmetic). Chebyshev mentioned that his music teacher also played an important role in his education, for she "raised his mind to exactness and analysis."
A physical handicap (of unknown cause) affected Chebyshev's adolescence and development. From childhood, he limped and walked with a stick and so his parents abandoned the idea of his becoming an officer in the family tradition. His disability prevented his playing many children's games and he devoted himself instead to mathematics.
In 1832, the family moved to Moscow
Moscow
Moscow is the capital, the most populous city, and the most populous federal subject of Russia. The city is a major political, economic, cultural, scientific, religious, financial, educational, and transportation centre of Russia and the continent...
, mainly to attend to the education of their eldest sons (Pafnuty and Pavel, who would become lawyers). Education continued at home and his parents engaged teachers of excellent reputation, including (for mathematics and physics) P.N. Pogorelski, held to be one of the best teachers in Moscow and who had taught (for example) the writer Ivan Sergeevich Turgenev
Ivan Turgenev
Ivan Sergeyevich Turgenev was a Russian novelist, short story writer, and playwright. His first major publication, a short story collection entitled A Sportsman's Sketches, is a milestone of Russian Realism, and his novel Fathers and Sons is regarded as one of the major works of 19th-century...
.
University studies
In summer 1837, Chebyshev passed the registration examinations and, in September of that year, began his mathematical studies at the second philosophical department of Moscow University. His teachers included N.D. BrashmanNikolai Brashman
Nikolai Dmitrievich Brashman was a Russian mathematician of Austrian origin. He was a student of Joseph Johann Littrow, and the advisor of Pafnuty Chebyshev....
, N.E. Zernov and D.M. Perevoshchikov of whom it seems clear that Brashman had the greatest influence on Chebyshev. Brashman instructed him in practical mechanics and probably showed him the work of French engineer J.V. Poncelet
Jean-Victor Poncelet
Jean-Victor Poncelet was a French engineer and mathematician who served most notably as the commandant general of the École Polytechnique...
.
In 1841 Chebyshev was awarded the silver medal for his work "calculation of the roots of equations" which he had finished in 1838. In this, Chebyshev derived an approximating algorithm for the solution of algebraic equations of nth degree based on Newton's algorithm
Newton's method
In numerical analysis, Newton's method , named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. The algorithm is first in the class of Householder's methods, succeeded by Halley's method...
. In the same year he finished his studies as "most outstanding candidate".
In 1841, Chebyshev's financial situation changed drastically. There was famine in Russia and his parents were forced to leave Moscow. Although they could no longer support their son, he decided to continue his mathematical studies and prepared for the master examinations, which lasted six months. Chebyshev passed the final examination in October 1843 and, in 1846, defended his master thesis "An Essay on the Elementary Analysis of the Theory of Probability." His biographer Prudnikov suggests that Chebyshev was directed to this subject after learning of recently-published books on probability theory or on the revenue of the Russian insurance industry.
Adult years
In 1847, Chebyshev promoted his thesis pro venia legendi "On integration with the help of logarithms" at St Petersburg University and thus obtained the right to teach there as a lecturer. At that time some of Leonhard EulerLeonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
's works were rediscovered by P. N. Fuss and were being edited by V. Ya. Bunyakovsky
Viktor Bunyakovsky
Viktor Yakovlevich Bunyakovsky was a Russian mathematician, member and later vice president of the Petersburg Academy of Sciences.He worked in theoretical mechanics and number theory , and is credited with an early discovery of the Cauchy-Schwarz inequality, proving it for the infinite dimensional...
, who encouraged Chebyshev to study them. This would come to influence Chebyshev's work. In 1848, he submitted his work The Theory of Congruences for a doctorate, which he defended in May 1849. He was elected an extraordinary professor
Professor
A professor is a scholarly teacher; the precise meaning of the term varies by country. Literally, professor derives from Latin as a "person who professes" being usually an expert in arts or sciences; a teacher of high rank...
at St Petersburg University in 1850, ordinary professor in 1860 and, after 25 years of lectureship, he became merited professor in 1872. In 1882 he left the university and devoted his life to research.
During his lectureship at the university (1852–1858), Chebyshev also taught practical mechanics at the Alexander Lyceum
Tsarskoye Selo Lyceum
The Imperial Lyceum in Tsarskoye Selo near Saint Petersburg also known historically as the Imperial Alexander Lyceum after its founder the Emperor Alexander I with the object of educating youths of the best families, who should afterwards occupy important posts in the Imperial service.Its...
in Tsarskoe Selo (now Pushkin), a southern suburb of St Petersburg.
His scientific achievements were the reason for his election as junior academician
Academician
The title Academician denotes a Full Member of an art, literary, or scientific academy.In many countries, it is an honorary title. There also exists a lower-rank title, variously translated Corresponding Member or Associate Member, .-Eastern Europe and China:"Academician" may also be a functional...
(adjunkt) in 1856. Later, he became an extraordinary (1856) and in 1858 an ordinary member of the Imperial Academy of Sciences
Russian Academy of Sciences
The Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....
. In the same year he became an honorary member of Moscow University. He accepted other honorary appointments and was decorated several times. In 1856, Chebyshev became a member of the scientific committee of the ministry of national education. In 1859, he became an ordinary member of the ordnance department of the academy with the adoption of the headship of the commission for mathematical questions according to ordnance and experiments related to ballistics. The Paris academy
French Academy of Sciences
The French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research...
elected him corresponding member in 1860 and full foreign member in 1874. In 1893, he was elected honorable member of the St. Petersburg Mathematical Society
St. Petersburg Mathematical Society
The St. Petersburg Mathematical Society is a society of St. Petersburg mathematicians. It was dissolved and subsequently revived twice, each time changing its name:* St. Petersburg Mathematical Society...
, which had been founded three years earlier.
Chebyshev died in St Petersburg on 26 November 1894.
Mathematical contributions
Chebyshev is known for his work in the field of probabilityProbability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
, statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
. Chebyshev's inequality
Chebyshev's inequality
In probability theory, Chebyshev’s inequality guarantees that in any data sample or probability distribution,"nearly all" values are close to the mean — the precise statement being that no more than 1/k2 of the distribution’s values can be more than k standard deviations away from the mean...
says that if
is a random variableRandom variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
with standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
σ, the probability that the outcome of
is no less than away from its mean is no more than :
Chebyshev's inequality is used to prove the weak law of large numbers.
The Bertrand–Chebyshev theorem (1845|1850) states that for any
, there exists a prime number such that . It is a consequenceof Chebyshev inequalities for the number
of prime numbers less than , which state that is of the order of . A more precise form is given by the celebrated prime number theoremPrime number theorem
In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....
: the
quotient of the two expressions approaches 1 as
tends to infinity.Legacy
Chebyshev is considered a founding father of RussiaRussia
Russia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...
n mathematics. Among his well-known students were the prolific mathematicians Dmitry Grave
Dmitry Grave
Dmitry Aleksandrovich Grave was a Russian and Soviet mathematician.Naum Akhiezer, Nikolai Chebotaryov, Mikhail Kravchuk, and Boris Delaunay were among hisstudents.-Brief history:...
, Aleksandr Korkin
Aleksandr Korkin
Aleksandr Nikolayevich Korkin was a Russian mathematician. He made contribution to the development of partial differential equations. After Chebyshev, Korkin was the most important initiator of the formation of the Saint Petersburg Mathematical School....
, Aleksandr Lyapunov
Aleksandr Lyapunov
Aleksandr Mikhailovich Lyapunov was a Russian mathematician, mechanician and physicist. His surname is sometimes romanized as Ljapunov, Liapunov or Ljapunow....
and Andrey Markov
Andrey Markov
Andrey Andreyevich Markov was a Russian mathematician. He is best known for his work on theory of stochastic processes...
. According to the Mathematics Genealogy Project
Mathematics Genealogy Project
The Mathematics Genealogy Project is a web-based database for the academic genealogy of mathematicians. As of September, 2010, it contained information on approximately 145,000 mathematical scientists who contribute to "research-level mathematics"...
, Chebyshev has 7,483 mathematical "descendants" as of 2010.
The crater Chebyshev
Chebyshev (crater)
Chebyshev is a large lunar crater that lies in the southern hemisphere on the far side of the Moon. The somewhat smaller crater Langmuir is intruding into the east-southeastern rim of Chebyshev, forming a chain of large craters with Brouwer on Langmuir's eastern rim.The outer rim of this walled...
on the Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
and the asteroid
Asteroid
Asteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...
2010 Chebyshev
2010 Chebyshev
2010 Chebyshev is an asteroid from the asteroid belt discovered on October 13, 1969 by Burnasheva, B. at Nauchnyj.-References:...
are named in his honour.
See also
- Chebyshev cube root
- Chebyshev distanceChebyshev distanceIn mathematics, Chebyshev distance , Maximum metric, or L∞ metric is a metric defined on a vector space where the distance between two vectors is the greatest of their differences along any coordinate dimension...
- Chebyshev filterChebyshev filterChebyshev filters are analog or digital filters having a steeper roll-off and more passband ripple or stopband ripple than Butterworth filters...
, a family of filterFilter (signal processing)In signal processing, a filter is a device or process that removes from a signal some unwanted component or feature. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal...
s in electronicsElectronicsElectronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...
and signal processingSignal processingSignal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time... - Chebyshev functionChebyshev function[Image:ChebyshevPsi.png|thumb|right|The Chebyshev function ψ, with x [Image:ChebyshevPsi.png|thumb|right|The Chebyshev function ψ, with x ...
, in number theory - Chebyshev polynomialsChebyshev polynomialsIn mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and...
/Chebyshev form - Chebyshev inequality, in probability theory
- Chebyshev's sum inequalityChebyshev's sum inequalityIn mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that ifa_1 \geq a_2 \geq \cdots \geq a_nandb_1 \geq b_2 \geq \cdots \geq b_n,then...
- Chebyshev's equation
- Chebyshev linkageChebyshev linkageThe Hoekens linkage is a four-bar mechanism that converts rotational motion to approximate straight-line motion. The Hoekens linkage is a cognate linkage of the Chebyshev linkage.-See also:*Straight line mechanism*Peaucellier–Lipkin linkage...
, a straight line generating linkageLinkage (mechanical)A mechanical linkage is an assembly of bodies connected together to manage forces and movement. The movement of a body, or link, is studied using geometry so the link is considered to be rigid. The connections between links are modeled as providing ideal movement, pure rotation or sliding for... - Roberts–Chebyshev theorem, on the generation of cognate coupler-curves
- Chebyshev–Markov–Stieltjes inequalities
- Chebychev–Grübler–Kutzbach criterionChebychev–Grübler–Kutzbach criterionThe Chebychev–Grübler–Kutzbach criterion determines the degree of freedom of a kinematic chain, that is, a coupling of rigid bodies by means of mechanical constraints...
for the mobility analysis of linkagesLinkage (mechanical)A mechanical linkage is an assembly of bodies connected together to manage forces and movement. The movement of a body, or link, is studied using geometry so the link is considered to be rigid. The connections between links are modeled as providing ideal movement, pure rotation or sliding for... - Chebyshev nodesChebyshev nodesIn numerical analysis, Chebyshev nodes are the roots of the Chebyshev polynomial of the first kind. They are often used as nodes in polynomial interpolation because the resulting interpolation polynomial minimizes the Runge's phenomenon.-Definition:...
- Chebyshev rational functionsChebyshev rational functionsIn mathematics, the Chebyshev rational functions are a sequence of functions which are both rational and orthogonal. They are named after Pafnuty Chebyshev...
- Chebyshev's biasChebyshev's biasIn prime number theory, Chebyshev's bias is the phenomenon that most of the time, there are more primes of the form 4k + 3 than of the form 4k + 1, up to the same limit. This phenomenon was first observed by Chebyshev in 1853....
- Discrete Chebyshev polynomials
External links
- Mechanisms by Chebyshev - Short 3d films - embodiment of Tchebishev's inventions
- Biography, another one, and yet another (all in RussianRussian languageRussian is a Slavic language used primarily in Russia, Belarus, Uzbekistan, Kazakhstan, Tajikistan and Kyrgyzstan. It is an unofficial but widely spoken language in Ukraine, Moldova, Latvia, Turkmenistan and Estonia and, to a lesser extent, the other countries that were once constituent republics...
). - Œuvres de P.L. Tchebychef (in FrenchFrench languageFrench is a Romance language spoken as a first language in France, the Romandy region in Switzerland, Wallonia and Brussels in Belgium, Monaco, the regions of Quebec and Acadia in Canada, and by various communities elsewhere. Second-language speakers of French are distributed throughout many parts...
)