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Introducing Mathwright Microworlds - Web Book: Cardano

James E. White

Our final example of an Interactive Web Book is Cardano, which develops and illustrates Cardano's Method for solving cubic equations. For that purpose, the book uses some tools that will offer a few experiments with complex algebra and then with abstract algebra. It finishes by demonstrating graphically an interesting property that cubic functions have when all of their roots are real. In that case, the three real roots are the projections of three equally spaced points on a certain circle (centered on the inflection point) to the real axis. This fact is non-intuitive and surprising, and it really benefits from the integration of two tools: One allows the reader to select cubics that satisfy the discriminant condition and to visualize how the shapes of cubic curves depend on the two parameters. The second tool demonstrates the above-mentioned fact, which is very helpful for a geometric understanding of Cardano's method.

The Cardano Interactive Web Book combines a variety of tools, including

  • graphing tools,
  • an abstract algebra "calculator" that allows the reader to explore some interesting properties of a certain complex algebra,
  • an expert system that solves cubic equations (among others) step-by-step, giving explanations for each step, and
  • many exercises that combine modern algebra with graphics.

It develops a novel and straightforward approach to Cardano's method, along with a simple means to understand it.

You may visit the Cardano Interactive Web Book by clicking the link or the picture above.

James E. White, "Introducing Mathwright Microworlds - Web Book: [i]Cardano[/i]," Convergence (December 2004)