Contributions of Leonhard Euler to mathematics
His seminal work had a profound impact in numerous areas of mathematics and he is widely credited for introducing and popularizing modern notation and terminology.
, the complex exponential function satisfies Richard Feynman called this "the most remarkable formula in mathematics".
, i, 1, and 0, arguably the five most important constants in mathematics, as well as the four fundamental arithmetic operators: addition, multiplication, exponentiation, and equality.
The development of calculus was at the forefront of 18th-century mathematical research, and the Bernoullis—family friends of Euler—were responsible for much of the early progress in the field.
While some of Euler's proofs may not have been acceptable under modern standards of rigor, his ideas were responsible for many great advances.
In doing so, he united two disparate branches of mathematics and introduced a new field of study, analytic number theory.
[6] Euler's great interest in number theory can be traced to the influence of his friend in the St. Peterburg Academy, Christian Goldbach.
Euler made progress toward the prime number theorem and conjectured the law of quadratic reciprocity.
The two concepts are regarded as the fundamental theorems of number theory, and his ideas paved the way for Carl Friedrich Gauss.
[8] The city of Königsberg, Kingdom of Prussia (now Kaliningrad, Russia) is set on the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges.
In addition, his recognition that the key information was the number of bridges and the list of their endpoints (rather than their exact positions) presaged the development of topology.
He introduced a formula governing the relationship between the number of edges, vertices, and faces of a convex polyhedron.
Euler characteristic, which may be generalized to any topological space as the alternating sum of the Betti numbers, naturally arises from homology.
Most of Euler's greatest successes were in applying analytic methods to real world problems, describing numerous applications of Bernoulli's numbers, Fourier series, Venn diagrams, Euler numbers, e and π constants, continued fractions and integrals.
He also facilitated the use of differential equations, in particular introducing the Euler–Mascheroni constant: 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 integrate music theory as part of mathematics.
This part of his work, however did not receive wide attention and was once described as too mathematical for musicians and too musical for mathematicians.
