Volume 12 No 4
Volume 12 No 4
- Classical Versus Computer Algebra Methods in Elementary Geometry
- Relations Between Some Characteristic Lengths in A Triangle
- The Q-way of Doing Analysis
- On One Unusual Method of Computation of Limits of Rational Functions in the Program Mathematica®
- Some Reflections on Cas Assisted Proofs of Theorems
Classical Versus Computer Algebra Methods in Elementary Geometry
Pavel Pech
University of South Bohemia, Czech Republic pech@pf.jcu.cz
Computer algebra methods based on results of commutative algebra like Groebner bases of ideals and elimination of variables make it possible to solve complex, elementary and non elementary problems of geometry, which are difficult to solve using a classical approach. Computer algebra methods permit the proof of geometric theorems, automatic derivation and discovery of formulas, construction of objects which have given properties and which cannot be easily done with a ruler and compass, etc. On the other hand classical methods offer better insight into geometric situations, show the beauty of geometry and enable better understanding the problem.
Using a few examples from the geometry of polygons in a plane the strengths and weaknesses of both methods are demonstrated. Obstacles students encounter in automatic proving are also discussed.
Relations Between Some Characteristic Lengths in A Triangle
Wolfram Koepf and Markus Brede
University of Kassel, P.O. Box 10 13 80, D-34109 Kassel, Germany
koepf@mathematik.uni-kassel.de mbrede@mathematik.uni-kassel.de
The paper’s aim is to note a remarkable (and apparently unknown) relation for right triangles, its generalisation to arbitrary triangles and the possibility to derive these and some related relations by elimination using Groebner basis computations with a modern computer algebra system.
Reinhard Oldenburg
Albrechtstr. 5, 37085 Göttingen, Germany roldenburg@gmx.de
The subject of q-calculus is a rich source of learning activities that address concept formation, exploration, variation, rich training and proof. This paper is a survey of the subject that shows how high school students using a computer algebra system can explore the field and prove some results.
On One Unusual Method of Computation of Limits of Rational Functions in the Program Mathematica®
Jaroslav Hora and Pavel Pech2
horajar@kmt.zcu.cz Dept. of Mathematics, Faculty of Education, University of West Bohemia, Klatovska 51, 320 13 Plzen, Czech Republic.
pech@pf.jcu.czFaculty of Education, University of South Bohemia, Jeronymova 10, 371 15 Ceske Budejovice, Czech Republic.
Computing limits of functions is a traditional part of mathematical analysis which is very difficult for students. Now an algorithm for the elimination of quantifiers in the field of real numbers is implemented in the program Mathematica. This offers a non-traditional view on this classical theme.
Some Reflections on CAS Assisted Proofs of Theorems
Thierry Dana-Picard
Department of Applied Mathematics, Jerusalem College of Technology, Havaad Haleumi Str. 21, POB 16031, Jerusalem 91160 Israel dana@jct.ac.il
A mathematician's work consists of proving theorems, calculating, and making mathematics understandable. An assistant for all three components is a Computer Algebra System. We describe and discuss various CAS-assisted processes for proving theorems, and discuss the constraints which can appear regarding efficiency, confidence in the result and mathematical insight during the computerised work. Two examples are described with some details and general structures are displayed.
Overview