Geodesic
A Model of the Real Numbers
Canadian mathematical bulletin, Tome 6 (1963) no. 2, pp. 239-255

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The model of the real numbers described below was suggested by the fact that each irrational number ρ determines a linear ordering of J2, the additive group of ordered pairs of integers. To obtain the ordering, we define (m, n) ≤ (m', n') to mean that (m'- m)ρ ≤ n' - n. This order is invariant with group translations, and hence is called a "group linear ordering". It is completely determined by the set of its "positive" elements, in this case, by the set of integer pairs (m, n) such that (0, 0) ≤ (m, n), or, equivalently, mρ < n. The law of trichotomy for linear orderings dictates that only the zero of an ordered group can be both positive and negative. 
Trott, Stanton M. A Model of the Real Numbers. Canadian mathematical bulletin, Tome 6 (1963) no. 2, pp. 239-255. doi: 10.4153/CMB-1963-022-0
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     title = {A {Model} of the {Real} {Numbers}},
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