Microworld:
Marden's
Theorem : (All in One)
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to visit the Microworld.
Author:
Dan
Kalman
Marden's
theorem concerns polynomials over the complex numbers.
As
a specific case, consider the polynomial p(z) = (z-a)(z-b)(z-c), where a,
b, and c are fixed complex numbers, and z is a complex variable. We know that
p has roots at a, b, and c, which we visualize as three points in the complex
plane. If those points are not colinear, they define a triangle within which
it is possible to inscribe an ellipse, tangent to the sides at their midpoints.
Marden's
theorem says that the foci of the ellipse are precisely the roots of the derivative,
p'(z).
In
this Microworld, you may experiment with this, and a more general version
of Marden's Theorem. And you may review properties of complex numbers and
the complex plane, both visually and algebraically. This Microworld will also
give you the opportunity to explore the geometry of ellipses inscribed in
triangles. This is a "hands on" opportunity to investigate some beautiful
correlations between algebra and geometry. At every step, the book invites
you to ask questions, and to see for yourself what the answers to those questions
are. You will enjoy this one!
Topics:complex numbers, complex arithmetic, cubic polynomials, geometry
Suggested Use: Experiment with the geometry of cubic polynomial equations.
Number of Pages: 10
Animation:Yes
Grade Level: Intermediate - Advanced College
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| author of this website, Mathwright 2000, MindScapes, | ||
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