Abstract
This chapter derives basic properties of least upper bounds and explores their relationship with the Archimedean property. On an arbitrary line in a Euclidean/LUB plane (which has been built into an ordered field) real multiples of points are defined and their algebraic properties derived. These properties are used to show the existence of an order-preserving isomorphism between the set of all real numbers and the whole line. The chapter ends with coordinatization of a Euclidean/LUB plane.
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Notes
- 1.
The existence of \(\varphi\) is guaranteed by Property R.5 of Definition NEUT.2 and Axiom REF.
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© 2015 Springer International Publishing Switzerland
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Specht, E.J., Jones, H.T., Calkins, K.G., Rhoads, D.H. (2015). A Line as Real Numbers (REAL); Coordinatization of a Plane (RR). In: Euclidean Geometry and its Subgeometries. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-23775-6_18
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DOI: https://doi.org/10.1007/978-3-319-23775-6_18
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-23774-9
Online ISBN: 978-3-319-23775-6
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