Leonard Euler

Leonhard Euler

of mathematics.   He hails from Switzerland, where he made numerous and diverse additions to his field.  In this paper, I will explain some of the innovations that changed the dynamics of the science, and math world.  His accomplishments are many, so I wont have a hard time picking just a few.

        One thing he developed that is relevant to any college math student, is the “e” (Euler's number, named after him), f(x) for functions, and “Pi”, as it relates to the ratio of a circles circumference to it's diameter.  He also was the first to dictate “i” for imaginary numbers.  In 0482, we used many of these same terms, so that class would look very different without Mr. Euler's contributions.

        In addition to mathematical notation, he also made additions to the field of complex analysis.  He introduced what we call scientific notation. He discovered what is now known as Euler's formula, that for any real number, “the complex exponential function satisfies: e^ipi = cos + isin“,  which has been called “the most remarkable formula in mathematics”.  Euler's Identity is e^ipi+1=0.  This is another remarkable formula, as it utilizes: e, pi, I, 1, and 0 which can be argued as the five most important constants in mathematics.

        Euler was also interested in the field of Number Theory.  One focus of Euler's work was to link the nature of prime distribution with ideas in analysis. He proved that the sum of the reciprocals of the primes diverges. In doing so, he discovered the connection between Riemann zeta function and prime numbers, known as the “Euler product formula for the Riemann zeta function”.  He also proved Newton's identities, Fermat's little theorem, Fermat's theorem on sums of two squares, and made distinct contributions to the Lagrange's four-square theorem. He also invented the “totient function”.