William Dunham – Euler The Master of Us All

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Leonhard Euler was one of the most prolific mathematicians that have ever lived. This book examines the huge scope of mathematical areas explored and developed by Euler, which includes number theory, combinatorics, geometry, complex variables and many more. The information known to Euler over 300 years ago is discussed, and many of his advances are reconstructed. Readers will be left in no doubt about the brilliance and pervasive influence of Euler’s work.

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Euler and Complex Variables In his 1637 masterpiece Geometrie, Rene Descartes addressed the perplexing matter of square roots of negative numbers. "Neither the true (i.e., positive) nor the false (i.e., negative) roots are always real," he wrote, "sometimes they are imaginary." The term "imaginary" is hardly one to inspire confidence. It sounds slightly delusional, as though a discussion of imaginaries ought to begin with the phrase "Once upon a time." Imaginary numbers seem unlikely to have any real significance. Nothing could be farther from the truth. Once mathematicians overcame their squeamishness about square roots of negatives, they discovered that such entities played a critical role in mathematics. Complex numbers (as they now prefer to be called) were anything but a worthless sidebar. On the contrary, the complex realm not only offered exciting new challenges but also provided unexpected information about the real numbers embedded within. Mathematicians thereby saw the familiar through an unfamiliar-but indisputably useful-lens. Not surprisingly, a full understanding and acceptance of complex variables did not happen at once. In this chapter we shall consider the origins of the subject before examining the discoveries of one of its great pioneers, Leonhard Euler, whose mathematical imagination was perfectly suited to the mathematical imaginary.

William Dunham is the Truman Koehler Professor of Mathematics at Muhlenberg College in Allentown, Pennsylvania. He has won two awards for excellence in expository writing from The Mathematical Association of America: the 1993 George Plya Award, and in 1997, the Trevor Evans Award. His books Journey Through Genius: The Great Theorems of Mathematics, and The Mathematical Universe have both been selections of the Book-of-the-Month Club and the History Book Club. He has also been the recipient of several grants from the National Endowment for the Humanities to fund seminars on the great theorems of mathematics in historical context.