Introduction 3 appendices. Later, when ready to go deeper or to check an omitted proof, a reading of the relevant appendices should satisfy the reader’s hunger for more detail. Finally we would like to discuss “visual aids”—that is, the various kinds of diagrams and pictures that make it easier for a student to internalize a complicated mathematical concept upon meeting it for the first time. Both of the authors have been very actively involved with the development of software tools for creating such mathemati- cal visualizations and with investigating techniques for using them to enhance the teaching and learning of mathematics, and paradoxically that has made it difficult for us to choose appropriate figures for our text. Indeed, recent advances in technology, in particular the explo- sive development of the Internet and in particular of the World Wide Web, have not only made it easy to provide visual material online, but moreover the expressiveness possible using the interactive and an- imated multimedia tools available in the virtual world of the Internet far surpasses that of the classic static diagrams that have traditionally been used in printed texts. As a result we at first considered omit- ting diagrams entirely from this text, but in the end we decided on a dual approach. We have used traditional diagrams in the text where we felt that they would be useful, and in addition we have placed a much richer assortment of visual material online to accompany the text. Our publisher, the American Mathematical Society, has agreed to set aside a permanent area on its own website to be devoted to this book, and throughout the text you will find references to this area that we will refer to as the “Web Companion”.1 Here, organized by chapter and section, you will find visualizations that go far be- yond anything we could hope to put in the pages of a book—static diagrams, certainly, but in addition Flash animations, Java applets, QuickTime movies, Mathematica, Matlab, Maple Notebooks, other interactive learning aids, and also links to other websites that contain material we believe will help and speed your understanding. And not only does this approach allow us to make much more sophisticated visualizations available, but it also will permit us to add new and improved material as it becomes available. 1 Its URL is
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