On the other hand, there are still uncountably many q (including all natural numbers greater than 1) for which there is only one base-q expansion of 1, other than the trivial 1.000.... This result was first obtained by Paul Erdős, Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the Komornik–Loreti constant q = 1.787231650.... In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the Thue–Morse sequence, which does not repeat. A more far-reaching generalization addresses the most general positional numeral systems. They too have multiple representations, and in some sense the difficulties are even worse. For example: In the balanced ternary system, = 0.111... = 1.111.... In the reverse factorial number system (using bases 2!,3!,4!,... for positions after the decimal point), 1 = 1.000... = 0.1234.... Impossibility of unique representation That all these different number systems suffer from multiple representations for some real numbers can be attributed to a fundamental difference between the real numbers as an ordered set and collections of infinite strings of symbols, ordered lexicographically. Indeed, the following two properties account for the difficulty: If an interval of the real numbers is partitioned into two non-empty parts L, R, such that every element of L is (strictly) less than every element of R, then either L contains a largest element or R contains a smallest element, but not both. The collection of infinite strings of symbols taken from any finite "alphabet", lexicographically ordered, can be partitioned into two non-empty parts L, R, such that every element of L is less than every element of R, while L contains a largest element and R contains a smallest element. Indeed, it suffices to take two finite prefixes (initial substrings) p1, p2 of elements from the collection such that they differ only in their final symbol, for which symbol they have successive values, and take for L the set of all strings in the collection whose corresponding prefix is at most p1, and for R the remainder, the strings in the collection whose corresponding prefix is at least p2. Then L has a largest element, starting with p1 and choosing the largest available symbol in all following positions, while R has a smallest element obtained by following p2 by the smallest symbol in all positions. The first point follows from basic properties of the real numbers: L