Elementary Functions

Abstract

The main theme of this book has been the tension between arithmetic and geometry and its creative role in the development of mathematics. The story of $\sqrt 2$ is an excellent example of such tension and its beneficial effects: geometry confronted arithmetic with the diagonal of the unit square, arithmetic expanded its concept of number in response, and the new number $\sqrt 2$ proved its worth by giving new insight into the old numbers, for example, by generating integer solutions of the equation x² - 2y² = 1 (Section 8.5). In other cases, geometry was not so much a source of conflict with arithmetic as a source of immediate insight; for example, in generating Pythagorean triples by the chord construction (Section 4.3) or in guaranteeing unique prime factorization in the Gaussian integers by the triangle inequality (Section 7.5).