Does Computer Algebra help at all learning about real numbers?

The motivation for this paper arises from the work of Monaghan et al. [13] that considers the possibility of using Computer Algebra tools to improve students' comprehension about real numbers. These authors claim that Computer Algebra helps balance the conceptual/procedural aspects involved in teaching this. We think the idea of using symbolic computation packages to learn about real numbers is interesting, but potentially dangerous since most real numbers are not computable. In our paper, we offer a different way of exploiting such a possibility. We specify that only a certain class of real numbers should be didactically approached via Computer Algebra, namely, real algebraic numbers. We think that most commercially available packages have a poor performance for manipulating (i.e. defining, performing arithmetic operations, comparing, etc.) these numbers. Thus we introduce, succinctly, the theoretical foundations (Thom's Lemma) of an algorithmic package that achieves these tasks. Moreover, we will consider some alternative approach to irrational, real algebraic numbers, that conceptually links the closed-form (or “key stroke”) presentation style while highlighting the approximation process. We finally show how the two approaches relate to each other, and we discuss didactical and scientific benefits and disadvantages for each.