Construction of the transreal numbers

The authors are Tiago Soares dos Reis and Walter Gomide. James Anderson appears on the archive in an editorial capacity.

The transreal numbers, proposed by James Anderson, are an extension of the real numbers. This new set is closed under the four arithmetical operations: addition, subtraction, multiplication and division. In particular, division by zero is allowed. Anderson introduced the transreals intuitively and axiomatically. In this paper we propose a construction of the transreals from the reals. Thus the transreal numbers and their arithmetic arise as consequences of real numbers. We define the set of transreal numbers as a certain class of subsets of ordered pairs of real numbers and we show that, in an appropriate sense, there is a copy of the real numbers in this new set.