WebMath: Cubic equations

[James E. White]

Hi.  My name is James White, and I would like to invite you to visit the
Mathwright Library at:

http://www.mathwright.com  

	The Mathwright Library is a free collection of almost 150 interactive,
electronic mathematics and science "books." These books have been developed
by College and Secondary School mathematics and science teachers (and
sometimes by their students) since 1991. Initially funded by the National
Science Foundation and supported by the IBM Corporation, it has been in
place on the web since 1994. 
	Mathwright books are documents created with a mathematics authoring
program called Mathwright. The books are only available for the Microsoft
Windows ™ platform, and the Library is compatible with all versions of
Windows above 3.0, including Windows 98. The Library offers many services
to teachers who may be interested in offering web-based mathematics or
science courses, including: hosting their WorkBooks here, developing books
for them, and providing the Mathwright Author Kit so that they may create
their own materials.
	When you register in the Library on your first visit, you may download a
free program, called the Mathwright Library Player, which is a "reader" for
the books collected there. This reader is automatically installed with
icons on the Start Menu at that time. After registration, you may browse
the stacks, download any books or collections you please to your local
machine. The books you select are opened in your browser, and are also
installed with Start Menu icons, so that you may open and read them in the
future, whether or not you are connected to the web. 

	To give a sense of the sort of learning experience that is possible in a
Mathwright WorkBook, I would like to invite you to look at one of our new
books called "Cubic Equations". 
http://www.mathwright.com/book_pgs/book241.html 

	That WorkBook summarizes some original work by James White and Dan Kalman
and aims to present it in a form that will be accessible to an advanced
undergraduate.  It develops an approach to the study of Cardano's method
for solving cubic equations that discloses certain new symmetries and
points the way to generalization to higher degree equations.  Those
generalizations (to the quartic case) and remarks on difficulties with
higher-degree cases are developed in detail in a forthcoming paper by White
and Kalman.  

	The approach taken in this WorkBook is to develop the algebra of "cubic
numbers" which may be thought of either as 3x3 complex circulant matrices,
or as the complex group algebra on the cyclic group of order 3. 
Pedagogically, of course, we do not present them in this rarified light
here, but rather construct and explore them in our laboratories. 

	Another new feature of this WorkBook is that it gives the reader an
opportunity to experiment with certain rings of numbers, and to verify
empirically the key facts that we visit along the way to Cardano's
algorithm.  In fact, the principal aim of the WorkBook is to bring the
reader to such a clear understanding of the technique, its hidden structure
and symmetry, that she will be tempted to explore higher-degree cases on
her own thereafter.

	This WorkBook covers some elements of the theory of equations (but not
many), and is meant to guide the reader to her own questions by presenting
an old and venerable idea in a slightly new light.  There are many
exercises in the laboratories, some more challenging than others, but all
aimed at clarifying some point of the exposition.  

	If you would like to discuss any aspect of this WorkBook, or of the
Mathwright Library in general, you may write to me at 
mathwrig at gte.net

													Enjoy,
													Jim White



	
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