THE MATHEMATICS OF EULER. Introduction: The Master of Us All. (Dunham, Euler the Master xv). This quote by twentieth-century mathematician André Weil

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1 THE MATHEMATICS OF EULER Introduction: The Master of Us All All his life he seems to have carried in his head the whole of the mathematics of his day (Dunham, Euler the Master xv). This quote by twentieth-century mathematician André Weil describes a man whose life and work represent more mathematical discovery and innovation in more mathematical fields than perhaps any other mathematician to have ever lived. Most impressive about Leonhard Euler s contribution to mathematics is not necessarily his depth of study (although that is, in fact, impressive) but his breadth of study. As the careers of the Bernoullis, Newton, and Leibniz were starting to fade, and Gauss, Lagrange, and Laplace had not yet come onto the scene, Euler rose to prominence as the unparalleled mathematical genius of the eighteenth century (Sandifier, Early Mathematics xv). Over 550 books and papers were published during his lifetime, and between two hundred and three hundred papers were found posthumously (xvi). His accomplishments in the fields of number theory, complex analysis, algebra, geometry, combinatorics, and (perhaps most importantly) real and functional analysis were astoundingly important and vast, too numerous to even outline in such a brief space. As a result, we can only focus on his most significant contributions to various mathematical fields. Euler, however, was not completely infallible. His treatment of infinity left much to be desired, and can at best be considered hasty mathematics. Some examples of this in his writings are ( ) and ( ), both of which would be laughed at by modern mathematicians (Dunham, Euler the Master xvii). However, while some of his mathematics was sloppy by modern conventions, his confident treatment of topics such as the relativelyunknown infinite (later standardized by set theorists such as Georg Cantor) and his recognition of previously-unrecognized patterns led Pierre-Simon Laplace to exclaim, Read Euler, read Euler! He is the master of us all (xiii). While we cannot read all of Euler, a sample of his findings follows.

2 Primes and Perfect Numbers: A Contribution to Number Theory Euler s most significant contribution to number theory was in the area of perfect numbers. In Euler s time, the study of number theory was an afterthought compared to the new study of calculus. However, his correspondence with the mathematician Christian Goldbach in St. Petersburg led to his fascination with the subject and his work regarding prime numbers (Dunham, Euler the Master 6). To understand Euler s study of the perfect numbers, we start with a definition: Definition: A number is perfect if it is equal to the sum of its proper divisors. Therefore, we can say that 6 is a perfect number as and. Other perfect numbers are 28, 496, and Before Euler, these were the only known perfect numbers, found by the ancient Greek mathematicians (2). Euclid was the first mathematician to find a pattern for the perfect numbers, seen in the modernized theorem below: Theorem (Euclid): If is prime and if ( ), then is perfect (3). 1 This theorem was mathematically correct and provided a sufficient condition for a number to be perfect (4). Euler, however, desired to show that it was necessary that was an even perfect number, and he eventually proved the following theorem: Theorem (Euler): If is an even perfect number, then ( ), where is prime. 2 His proof related the sum of all of the whole divisors of a number to conditions for prime and perfect numbers, and thus was able to create an if-and-only-if statement regarding perfect numbers and their connection to primes that are powers of two less one (11). In his work, he proved that a power of two is never perfect, and that relative primes have similar properties to pairs of primes in terms of the sums of their whole number divisors (9). In all, his work in number theory led to many results, results that led to many more questions being asked, questions which still do not have resolution today. 1 For proof, see Dunham s Euler: The Master of Us All, page 3 2 For proof, see Dunham s Euler: The Master of Us All, page 10

3 The Great Theorem: A Contribution to Analysis Perhaps Euler s greatest contributions to the mathematical sciences were those regarding real and functional analysis. His ability to recognize patterns, manipulate results, and provide convincing and sound arguments provided a foundation for the new theory of calculus, mathematical analysis. Euler standardized many of the results used in analysis today, especially those regarding infinite series. While the Bernoullis had proven many results regarding the summation of the harmonic series, the problem of the series still posed an issue for mathematicians of the time (Dunham, Journey 212). Euler s first approach, guided by his teacher Johann Bernoulli, was to sum as many of the terms as possible and then attempt to recognize the sum as a well-known result. However, the result that Euler found, , was not recognizable as any famous number previously discovered. When Euler finally found the solution, he said that quite unexpectedly, [he] had found an elegant formula depending on. This was unexpected indeed, as the constant, related throughout history to circles and arcs, had been found to have a connection to the reciprocal of perfect squares. A sketch of the proof is below without it, the beauty of the result is lost on the reader. While the entirety of his proof is too long for this work 3, the most important aspects are shown below, as seen in Dunham s Journey Through Genius, pages : Theorem (Euler): Proof: To prove this, Euler used the infinite series ( ) [ ], which is the same as, which, by the Taylor series expansion of ( ), is equal to ( ). Using the zeroes of ( ) ( ), Euler was 3 See Dunham s Journey Through Genius, page 215, for full proof.

4 able to factor ( ) ( ) ( ) ( ) ( ) ( ), which can be rewritten as ( ) [( ) ( )] [( ) ( )]. Using the difference of squares, Euler found ( ) [ ] [ ] [ ]. This result equates a summation with a product, which is quite extraordinary. Multiplying out the infinite product gives a product the likes of ( ). It is easy to show that. Thus, looking back at our original function, ( ), we know that. Therefore, ( ). Therefore, crossmultiplying, we find that, which proves Euler s result. This also fit with Euler s approximation of the result, as. Euler did not stop with this result he continued to prove more results regarding infinite series (217). This example of clever use of mathematics to show a result that would otherwise be unexpected was not uncommon in Euler s work. These manipulations were what placed Euler among the great mathematicians in history. Euler and Science Euler s ideas did not only stay the pure mathematical realm but also applied to the natural sciences. Euler s work in the foundations of analysis led to a renewed interest and study in the field of mechanics, and from this study came a better understanding of elastic systems and ballistics. Euler also communicated with and assisted the great Russian scholars of astronomy, providing mathematical support and rigor to their field (Bogolyubov xiii). Perhaps most strikingly unique about Euler s findings is his work in music theory. While music was always seen to be mathematical, Euler worked with topics such as harmony, dissonance,

5 and the relationship between musical tones. These writings helped Euler to write what he called a new theory of music (335). While Euler s scientific works are often overlooked in the context of his work in pure mathematics, his applied works are no less important in fact, they give a certain aspect of credence to his mathematical genius. His applied works show that he was not only interested in mathematics for its own sake but also to apply that mathematics to the world around him using what he knew to form a better world, a better Creation in which to glorify God, which was his ultimate goal (McIntyre 3).

6 Bibliography Bogolyubov, N.N., G.K. Mikhailov and A.P Yushkevich, Euler and Modern Science. Washington, DC: Mathematical Association of America, Bradley, Robert E, Laurence A D'Antonio and C. Edward Sandifier, Euler at 300: An Appreciation. Washington, DC: Mathematical Association of America, Dunham, William. Euler: The Master of Us All. Washington, DC: Mathematical Association of America, Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley Science Editions, The Genius of Euler: Reflections on His Life and Work. Ed. William Dunham. Washington, DC: Mathematical Association of America, McIntyre, Dale L. "The God-Fearing Life of Leonhard Euler." Journal of the Association of Christians in the Mathematical Sciences (2006). Sandifier, C. Edward. How Euler Did It. Washington, DC: Mathematical Association of America, The Early Mathematics of Leonhard Euler. Washington, DC: Mathematical Association of America, 2007.