Citations:rational numbers

Set of all rational numbers

1. • 1978, Felix Kaufmann, Paul Foulkes (translator), Brian McGuinness (editor), The Infinite in Mathematics: Logico-mathematical writings, page 105,

     Just as inadmissable as the definition of irrational numbers by means of a cut in the domain of rational numbers is the definition of irrational number identifying it with a bounded sequence.

   • 1996 [1941], Richard Courant, Herbet Robbins, Ian Stewart (revision), What is Mathematics?, 2nd Edition, page 81,

     Since the rational numbers have been shown to be denumerable, one might suspect that any infinite set is denumerable, and that this is the ultimate result of the analysis of the infinite.

   • 1992, S. C. Malik, Savita Arora, Mathematical Analysis, 2nd Edition, 2005 Reprint, page 20,

     We shall, however, show that the property of completeness does not hold good for the ordered field of rational numbers, i.e., the ordered field ℚ of rationals is not order complete.

   • 2006, Graham Oppy, Philosophical Perspectives on Infinity, page 26,

     First, we review his method for showing that the cardinality of the rational numbers is the same as the cardinality of the natural numbers.

   • 2005, Piotr Błaszczyk, Mode of Existence of Real Numbers, Anna-Teresa Tymieniecka (editor), Analecta Husserliana:The Yearbook of Phenomenological Research, Volume LXXXVIII, page 152

     And even if the process of extending the field of the rational numbers is aimed at setting the fundaments of calculus...

   • 2007, Thorsten Riedel, On the Construction of Class Fields by Picard Modular Forms, Rolf-Peter Holzapfel, A. Muhammed Uludağ, Musaaki Yoshida (editors), Arithmetic and Geometry Around Hypergeometric Functions, page 272

     Every abelian extension of the rational numbers is contained in a cyclotomic field.