Auteur principal			Co-auteurs(s)
Nom :	Dr. Jon Beal		Nom :	Dr. Michael McConnell
Courriel :	jbeal@clarion.edu		Courriel :	mmcconnell@clarion.edu
Institution ou compagnie:	Clarion University		Nom :
Département:	Mathematics		Courriel :
Ville :	Clarion		Nom :
État/Province :	PA		Courriel :
Pays :	USA		Nom :
Type de présentation:	Conférence :  50 minutes.		Courriel :
Conférence et numéro :	Derive & TI-CAS  ,    Numéro : D29		Horaire :   Local :	jeudi, 10h30   1752
Site Internet :
Titre de la communication :	Exploration of Alternative Addition and Multiplication using the TI-89
Résumé de la communication :
We define a new pair of operations on the real numbers by $a\oplus b=a+b-1$ and $a\odot b=a+b-ab$. These are not only binary operations on $\mathbb{R}$, but they give it a field structure that is isomorphic to the regular field operations. These new operations allow students to explore familiar ideas, specifically algebraic and analytic concepts, in an unfamiliar setting. This encourages them to look at the algebraic and analytic concepts beyond their intuitive understanding of the workings of the real numbers. One of the strengths of the TI-89 is that we can define functions in two variables to match these operations. First, we use the TI-89 to explore the algebraic properties of the operations, such as associativity, commutative, and order of operations. The solve command can be used to find identities for the operations. (This leads to surprising results.) We also explore the inverses of elements under the two operations, including which elements are not invertible. This leads to the question of solving polynomial equations involving these operations. This, in turn, leads to an expanded understanding of what polynomials are. We can now consider the formal derivative. One interesting question is how the formal derivative is related to the ``limit" definition of the derivative from analysis. An exploration of the formal derivative leads to an appropriate limit definition of the derivative from analysis. Other explorations will be discussed.