International Journal of Mathematics and Mathematical Sciences
Volume 12 (1989), Issue 3, Pages 603-613
doi:10.1155/S0161171289000736
Two constructions of the real numbers via alternating series
1Department of Applied Mathematics, University of the Witwatersrand, Johannesburg Wits 2050, South Africa
2Department of Mathematics, University of the Witwatersrand, Johannesburg Wits 2050, South Africa
Received 1 March 1988
Copyright © 1989 Arnold Knopfmacher and John Knopfmacher. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Two further new methods are put forward for constructing the complete
ordered field of real numbers out of the ordered field of rational numbers. The
methods are motivated by some little known results on the representation of real
numbers via alternating series of rational numbers. Amongst advantages of the
methods are the facts that they do not require an arbitrary choice of "base" or
equivalence classes or any similar constructs. The methods bear similarities to
a method of construction due to Rieger, which utilises continued fractions.