In class we have been discussing the work of Euler, and all his contributions to mathematics. While I had heard of Euler, and learned about topics that he discovered, I never knew anything about him or how much he contributed until now.

Leonhard Euler was not only a mathematician, but a physicist, astronomer, logician and an engineer whom made important strides in calculus, graph theory and number theory. He was born in Basel, Switzerland on April 15, 1707. Soon after, his family moved to a town called Riehen, where his father befriended the Bernoulli family, who ultimately had a large influence on Euler's work. At the age of only thirteen, Euler attended the University of Basel and graduated three years later with a masters in Philosophy. In 1727 Euler entered the Paris Academy Prize Problem where he had to find the best way to put a mast onto a ship. He ended up taking second place to Pierre Bouguer that year, however entered again and won twelve more times (https://en.wikipedia.org/wiki/Leonhard_Euler).

Euler worked in almost all areas of mathematics including geometry, physics, algebra, calculus, trigonometry and number theory. His works ended up occupying 60-80 quarto volumes, which is more than any other mathematician. He is also the only mathematician in history to have not one, but two numbers named after him. The first number, e, which is widely used in calculus and approximately equal to 2.71828. The other is called "Euler-Mascheroni constant Y (gamma) and is approximately equal to .57721, however it is not known if gamma is rational or irrational (https://en.wikipedia.org/wiki/Leonhard_Euler).

Mathematical notation is something that Euler made popular through his many textbooks as well. He was the first to denote the concept of a function, and write it as f(x) to denote the function

Another idea that Euler developed was the idea of number theory. Although he did not come up with number theory himself, he based his ideas of off the work of Pierre de Fermat and disproved some of his conjectures. Euler proved "the sum of the reciprocals of the primes diverges" and from here saw the connection between the Riemann zeta function and prime numbers. He also "proved Newton's Identities, Fermat's little theorem and Fermat's theorem on sums of two squares." He made contributions to Lagrange's four-square theorem and invented the toilet function. The toilet function is denoted as φ(

Graph theory was yet another mathematical concept that Euler helped develop throughout history. This began when he presented a problem known as the Seven Bridges of Königsberg and its solution. He realized that there is no way to cross all seven bridges only one time and end up in the same place one started, therefore there was no Eulerian circuit. He also discovered the formula V-E+F=2, relating to the number of vertices, faces, and edges there are in a convex polyhedron (https://en.wikipedia.org/wiki/Leonhard_Euler).

Along with all of the contributions mentioned above, he also contributed to applied mathematics, music, physics and astronomy. Euler was by far one of the greatest mathematicians of all time and without his work, mathematics and many other subjects would not be the same. I am happy to have learned so much about what he contributed to that has made my math experience so much better (https://en.wikipedia.org/wiki/Leonhard_Euler).

Leonhard Euler was not only a mathematician, but a physicist, astronomer, logician and an engineer whom made important strides in calculus, graph theory and number theory. He was born in Basel, Switzerland on April 15, 1707. Soon after, his family moved to a town called Riehen, where his father befriended the Bernoulli family, who ultimately had a large influence on Euler's work. At the age of only thirteen, Euler attended the University of Basel and graduated three years later with a masters in Philosophy. In 1727 Euler entered the Paris Academy Prize Problem where he had to find the best way to put a mast onto a ship. He ended up taking second place to Pierre Bouguer that year, however entered again and won twelve more times (https://en.wikipedia.org/wiki/Leonhard_Euler).

Euler worked in almost all areas of mathematics including geometry, physics, algebra, calculus, trigonometry and number theory. His works ended up occupying 60-80 quarto volumes, which is more than any other mathematician. He is also the only mathematician in history to have not one, but two numbers named after him. The first number, e, which is widely used in calculus and approximately equal to 2.71828. The other is called "Euler-Mascheroni constant Y (gamma) and is approximately equal to .57721, however it is not known if gamma is rational or irrational (https://en.wikipedia.org/wiki/Leonhard_Euler).

Mathematical notation is something that Euler made popular through his many textbooks as well. He was the first to denote the concept of a function, and write it as f(x) to denote the function

*f*to the argument*x*. He also introduced the Greek letter Σ for adding sums, the letter*i*to denote imaginary numbers and finally he popularized the Greek letter*π*which denotes "the ratio of a circle's circumference to its diameter." While many of Euler's proofs in calculus were not accepted, he made advancements in the development of the power series, and expressions of functions as the sum of infinitely many terms. He directly proved the power series for*e,*and the inverse tangent function which led him to solve the Basel problem in 1735. From here, he was able to use exponential functions and logarithms in analytic proofs and discover various logarithm functions using the power series. Euler was also able to define logarithms for negatives complex numbers, and the exponential function for complex numbers, which in turn he found related to trigonometric functions. He then came up with what is now known as Euler's Identity,**aka "the most remarkable formula in mathematics" according to Richard P. Feynman. He called it this because of all the single notations of addition, multiplication, exponentiation, equality and important constants (https://en.wikipedia.org/wiki/Leonhard_Euler).**Another idea that Euler developed was the idea of number theory. Although he did not come up with number theory himself, he based his ideas of off the work of Pierre de Fermat and disproved some of his conjectures. Euler proved "the sum of the reciprocals of the primes diverges" and from here saw the connection between the Riemann zeta function and prime numbers. He also "proved Newton's Identities, Fermat's little theorem and Fermat's theorem on sums of two squares." He made contributions to Lagrange's four-square theorem and invented the toilet function. The toilet function is denoted as φ(

*n*), and is "the number of positive integers less than or equal to the integer*n*that are coprime to*n."*From this function, he was able to generalize ideas from Fermat's little theorem and created what we now know as Euler's theorem. Finally, he further proved, from Euclid, the relationship between Mersenne primes and perfect numbers was a one to one function (https://en.wikipedia.org/wiki/Leonhard_Euler).Graph theory was yet another mathematical concept that Euler helped develop throughout history. This began when he presented a problem known as the Seven Bridges of Königsberg and its solution. He realized that there is no way to cross all seven bridges only one time and end up in the same place one started, therefore there was no Eulerian circuit. He also discovered the formula V-E+F=2, relating to the number of vertices, faces, and edges there are in a convex polyhedron (https://en.wikipedia.org/wiki/Leonhard_Euler).

Along with all of the contributions mentioned above, he also contributed to applied mathematics, music, physics and astronomy. Euler was by far one of the greatest mathematicians of all time and without his work, mathematics and many other subjects would not be the same. I am happy to have learned so much about what he contributed to that has made my math experience so much better (https://en.wikipedia.org/wiki/Leonhard_Euler).