Solving equations, an elegant legacy

摘要

Solving equations-one of the primary themes in mathematics. In discussing this my secondary theme is that classical and modern mathematics are tightly intertwined, that contemporary mathematics contributes real insight and techniques to understand traditional problems. In the long second section I discuss some procedures that help to solve equations. There the discussion of symmetry is extensive because it is treated so inadequately as a fundamental thread throughout mathematics courses. The third section gives two different techniques to prove that equations have solutions. Most of the sections are independent; thus you can skip to examples that are more appealing. One ingredient in solving equations that I have not emphasized adequately is the basic role of inequalities. They are lurking here and there: the Euclidean algorithm and the application of the Brouwer fixed point theorem, to name two less obvious instances. To give inequalities their due would have changed the character of this article, which is drawn from the longer version [10]. [References: 15]