Advanced Microworlds
(Arranged alphabetically)
Cardano
Author(s): james white
Topics: cubic equations, equations, inflection points, graphing, factorization
of polynomials, maxima and minima, cubic polynomials, complex numbers (Level:
Advanced)
Description: This microworld 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 are to the quartic case.
Congruences
Author(s): ravinder kumar
Topics: congruence, systems of congruence equations, fermat theorem, chinese
remainder theorem, cayley tables, group suggested use: congruence equation and
systems of congruence equations solver (Level: Advanced)
Heron's
formula
Author(s): james white
Topics: geometry, heron's formula, triangles, optimization, extrema (Level:
Advanced)
Marden's
theorem
Author(s): dan kalman
Topics: complex numbers, complex arithmetic, cubic polynomials, geometry, marden's
theorem (Level: Advanced)
Description: Marden's theorem concerns polynomials over the complex numbers. Specifically, consider the polynomial p(z) = (z-a)(z-b)(z-c), where a, b, c are fixed complex numbers. The roots of p and the roots of the derivative p' have an interesting relationship.
Odds
and integrals
Author(s): james white
Topics: probability, integration, approximation, geometry, geometric construction,
vectors, permutations, barycentric, topology, simplex, simplicial complex, subdivision
(Level: Advanced)