Transreal numbers: Mathematics or Reverie?

Abstract

Mathematics, known as an exact science, has the concept of number as one of its fundamental ideas. Nevertheless history shows that the understanding we have of numbers is not a dogma, but a perception which changes with time and with the needs which arise in the development of society. Some chapters of this story have already been written: we went from fractional to irrational numbers, from positives to negatives and from reals to imaginaries. At the present time we live among the emergence of a new set of numbers. James A. D. W. Anderson, a computer scientist, has proposed the transreal numbers, a set where division by zero is allowed. As happens with a theory in its initial state, the transreals are now undergoing a moment of affirmation in the academy. We bring to the light this discussion. There is no doubt that the treansreals are a novel concept in mathematics. However, although, according to Anderson, they are applied in computing, the transreals have not yet caused greater interest in the mathematical community. Do they have a fragile and inconsistent structure? Are they consistent, but irrelevant? Are they destined to supply interesting results in the future or is their usefulness restricted to the philosophical and poetic reveries they provide? Introduction The set of transreal numbers, denoted by R , is an extension of the set of real numbers. James A. D. W. Anderson, the proponent of this new set of numbers, postulates the existence, beyond the real numbers, of three new elements, namely −∞, ∞ and Φ, called, respectively, negative infinity, infinity and nullity. Thus, R = R ∪ {−∞, ∞, Φ}. Transreal arithmetic is total, that is, the result of any one addition, subtraction, multiplication or division, among transreal numbers, is a transreal number. In particular division by zero is allowed. Anderson define = −∞, = ∞ e = Φ (ANDERSON, 2005). A pictorial image of the transreal line is shown below. * Tiago is a doctoral student in the Graduate Program in History of Sciences, Techniques and Epistemology Federal University of Rio de Janeiro, professor at the Federal Institute of Education, Science and Technology of Rio de Janeiro campus Volta Redonda and master in mathematics from the Institute of Mathematics Federal University of Rio de Janeiro. 1 James A. D. W. Anderson is currently a teacher and research at the School of Systems Engineering, University of Reading, England.