Two concrete new constructions of the real numbers

Abstract

Two 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 known theorems on so-called Engel and Sylvester series. Amongst advantages of the methods are the facts that they do not require an arbitrary choice of base, or any equivalence classes or similar constructs. Introduction. By old theorems of Lambert (1770) and Engel (1913) (see Perron [2]), every real number A has unique representation as the sum of series 1 1 1 , A = o0 H 1 1 1 1 = ( o , i , 2 , . . . ) » l CL\Q>2 CL\Q a; > 2 for i > 1. Further, A is rational if and only if a?;+i = a?; for all sufficiently large i. An analogous representation (see [2]) of Lambert (1770) and Sylvester (1880) states that every real 1 1 1 A = a0- 1 1 h 1 = ( ( 0 , i , 2 , . . . ) ) » a U2 an say, where the a?; are integers defined uniquely by A, such that a > 2 and â _|_i > ai{ai — 1) + 1 for i > 1. Further, A is rational if and only if a^+i = ui(ui — 1) + 1 for all sufficiently large i. In certain ways, these representations may be compared with that by simple continued fractions, and are even simpler than the latter. The main purpose of this note is to justify this remark by deriving some elementary further properties of the Engel-Lambert and SylvesterLambert representations, and (with these and the above-mentioned Received by the editors on April 14, 1986. AMS (1980) Classification Number. 10A30. Copyright (©1988 Rocky Mountain Mathematics Consortium