Cardano: An Adventure in Algebra in 8 Parts

by James White
In collaboration with Dan Kalman


This Interactive Web Book is a demonstration of a possible shape for mathematics books of the future. It presents its story in the familiar way that a static text might present it, but with the exception that the pages of the story "come to life" and offer the reader the opportunity to make and test hypotheses, to experiment and explore in a visual and interactive way many of its main constructions and concepts. For many readers, this active participation in the story can add a dynamic dimension that will help them visualize certain of its ideas for the first time. Of course, you may download and extract the Word 2000 version here in order to print and read it in the traditional way as static text off-line, but these interactions are not ancillary; they are from the beginning an essential part of the narrative.

Cardano is not actually intended to teach the mathematical topics it develops to any particular student target audience. It covers a range of topics from high school to graduate-level mathematics. It moves at warp speed across several centuries of mathematics, beginning perhaps in the 16th Century, and ending with some results obtained by the authors, and published in the November 2001 issue of The American Mathematical Monthly. But it was written so that almost anyone who has studied a little algebra can "jump in" and play with it. Some topics that it covers are accessible to high school students, others to university students of Modern Algebra and Theory of Equations, and others may be of interest to graduate students, teachers, and professional mathematicians. So the aim is not to teach the mathematics but to demonstrate the range and the efficacy of a new style of web pedagogy and of authorship.

It may appear that the interactions that one finds on the eight exploration pages are simply Java applets. While they behave like applets, they are different from them for several reasons. Perhaps the most important from the viewpoint of authorship and web design is that they were not written in Java, but were created in a high level object-oriented mathematics scripting language called MathScript. Further the visual design was graphical "point-and-click" or "What you see is what you get." This combination, using our new Mathwright32 Author™ program, produces efficient Java code, but does not require any knowledge of Java itself. It is much simpler to write books with than Java.

One might also point to the range of resources available to this book, and to every interactive web book that uses the MathwrightWeb Control. It uses a symbolic Expert System, computer algebra, sprite animation and graphics, command-line tools, and a special-purpose command language to represent and manipulate ring-theoretic objects. All of this is immediately available to the book. A Java applet would, in principle, have to download these resources each time this book (or a similar book) was read.

The mathematical documentation for this story was created fairly easily using Design Science MathPage™ technology with Microsoft Word 2000™. And the mathematical interactions were created with Mathwright32 Author™. The Cardano Book may be read in Microsoft Windows™ (95, 98, Me, 2000, or XP) using Internet Explorer Browser 5.0 or later. In order to read it, you must download and install the free Personal MathwrightWeb Control. Once you install the control, please check that the browser is prepared to read the books. Be sure to download Version 1.004 (after Feb. 22, 2002 or later).

Your browser must be able to use ActiveX controls. Microsoft Internet Explorer 5.0 Browser (or later) is so equipped. You should check that the Security Settings under Tools, Internet Options, Security for the Internet, Custom Level has:

The Interactive Web Book: Cardano


Table of Contents

Graphing Polynomial Functions
Graphing and Factoring Polynomials
Inflection Points and Roots
When do Cubics have real roots?
Complex Arithmetic
Algebra of Cubic Numbers
Cardano's Method
Real Roots of Cubic Equations