Editors' Introductory Comments

G.T. Bagni (1958–2009) died on 10 June 2009 in a bicycle accident, while this paper was under review. Because of his untimely death, we do not know how he would have taken into account the reviewers' comments. However, we would have suggested a somewhat different structure of the paper and the clarification of several specialized terms, not expected to be understood by many readers to whom this book is addressed. Therefore, we provide below an outline of the rationale of the paper as it has been revised according to the reviewers' comments. We are indebted to P. Boero, B. D'Amore, and L. Radford for their help; they provided several explanatory comments and clarified unclear points. Their comments have been incorporated into the text and form the basis of most footnotes.

The rationale underlying this paper is as follows:

1. (i) The didactic phenomenon: Students at school are consciously or unconsciously hesitant, or even reluctant, to accept imaginary numbers (inasmuch as they have been taught for many years that square roots of negatives are strictly prohibited or nonsensical).

2. (ii) The historical context: Imaginaries entered mathematics, not as a theoretical algebraic construct (answering the pure algebraist's ‘a posteriori’ question: “how is it possible to extend real numbers, so that square roots of negatives make sense?”), but as an operational tool/trick to solve cubic equations (Section 5.4).

3. (iii) The empirical study: A pilot empirical study implemented this idea in the classroom (Section 5.5):

1. - To raise a meaningful mathematical problem: How to solve cubic equations?

2. - Using the “nonsensical” tool of square roots of negatives gives a “real” result.

3. […]