Plane Geometry - Bruce E. Shapiro

SECTION 14. BETWEENNESS 69By the Ruler postulate (axiom 14.11) there exists a coordinate functiong : l ↦→ R. Definec = −g(P )and define h : l ↦→ R byh(x) = g(x) + cThen h is a coordinate function by Lemma 14.23.Since h(P ) = 0, it must be the case that h(Q) ≠ 0 because h is 1-1. Wehave two cases to consider: h(Q) > 0 or h(Q) < 0.If h(Q) > 0 then h has the desired properties and the theorem is proven.If h(Q) < 0, define a new function j : l → R by j(x) = −h(x). Then bylemma 14.24, j is a coordinate function. Since it has the required properties,the theorem is proven.Figure 14.3: Circles that intersect in the real plane do not necessarily intersectin the rational plane.Distances Must Be Real. The following examples illustrate why rulers(hence distances) require real numbers and not rational numbers.Example 14.4 The distance between the points (1, 0) and (0, 1) in P is√2.Example 14.5 Find the intersection of the line y = x and the unit circleusing whatever knowledge you may already have of circles and triangles.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.