MODULE 7 - LEONHARD PAUL EULER (1707 – 1783)

LEONHARD PAUL EULER (1707 – 1783)

FATHER OF TOPOLOGY

Born	15 April 1707, Basel, Switzerland (1707-04-15)
Died	18 September 1783 (aged 76), St. Petersburg, Russia
Residence	Prussia, Russia, Switzerland
Nationality	Swiss
Fields	Mathematician and Physicist
Institutions	Imperial Russian Academy of Sciences ,Berlin Academy
Alma mater	University of Basel
Doctoral advisor	Johann Bernoulli

Leonhard Paul Euler (15 April, 1707 – 18 September, 1783) was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany. He wrote 886 books in mathematics. Euler is the father of topology. 

Euler made important discoveries in fields as diverse as calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also renowned for his work in mechanics, optics, and astronomy

EULER CONTRIBUTIONS:

Euler worked in almost all areas of mathematics: geometry, calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and other areas of physics. He is a seminal figure in the history of mathematics; if printed, his works, many of which are of fundamental interest, would occupy between 60 and 80 quarto volumes. Euler's name is associated with a large number of topics.

MATHEMATICAL NOTATION

Ø   f(x): he introduced the concept of a function and was the first to write f(x) to denote the function f applied to the argument x.

Ø   Trigonometric Functions: He also introduced the modern notation for the trigonometric functions,

Ø   e,i,Σ,π : The letter “e”  for the base of the natural logarithm (now also known as Euler's number), the Greek letter “Σ” for summations and the letter “i” to denote the imaginary unit. The use of the Greek letter “π” to denote the ratio of a circle's circumference to its diameter was also popularized by Euler, although it did not originate with him.

ANALYSIS

Ø  Exponential  Expansion: Euler is well-known in analysis for his frequent use and development of power series, the expression of functions as sums of infinitely many terms,

Ø  Power Series Expansions: Euler discovered the power series expansions for e and the inverse tangent function.

Ø  Complex Exponential Function: He also defined the exponential function for complex numbers, and discovered its relation to the trigonometric functions. For any real number 

Ø   Euler created the theory of hyper geometric series, q-series, hyperbolic trigonometric functions and the analytic theory of continued fractions.

NUMBER THEORY

Ø  Prime Number: He found a formula for prime number. Ie. X²-X-41=one prime number. If we substitute the value X=1 to 40 then we get the prime number.

Ø  Mersenne Prime: By 1772 Euler had proved that 2³¹ − 1 = 2,147,483,647 is a Mersenne prime. It may have remained the largest known prime until 1867.

Ø  Totient Function: He also invented the totient function φ(n) which is the number of positive integers less than the integer n that are coprime to n. Using properties of this function, he generalized Fermat's little theorem to what is now known as Euler's theorem.

Ø  Euler's interest in number theory can be traced to the influence of Christian Goldbach, his friend in the St. Petersburg Academy.  He proved that the sum of the reciprocals of the primes diverges. Euler proved Newton's identities, Fermat's little theorem, Fermat's theorem on sums of two squares,

GEOMETRY

 Ü   Euler line , Euler's circle

GRAPH THEORY

             Seven Bridges of Königsberg: In 1736, Euler solved the problem known as the Seven Bridges of Königsberg. The city of Königsberg, Prussia was set on the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges. The problem is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point. It is not: there is no Eulerian circuit. This solution is considered to be the first theorem of graph theory, specifically of planar graph theory.

V − E + F = 2 : Euler also discovered the formula V − E + F = 2 relating the number of vertices, edges, and faces of a convex polyhedron and hence of a planer graph.

PHYSICS AND ASTRONOMY

• Euler-Bernoulli Beam Equation: Euler helped develop the Euler-Bernoulli beam equation, which became a cornerstone of engineering.
• His work in astronomy was recognized by a number of Paris Academy Prizes over the course of his career.  His calculations also contributed to the development of accurate longitude tables.  In addition, Euler made important contributions in optics.

APPLIED MATHEMATICS:

 ·            F₅:  He found the symbol of F₅   ie.  F₅   =4,294,967,297 = 641 X 6 700 417

·            Music: One of Euler's more unusual interests was the application of mathematical ideas in music. In 1739 he wrote the Tentamen novae theoriae musicae, hoping to eventually incorporate musical theory as part of mathematics.