Brain Teaser Solution:

Connecting Points in the Euclidean Plane
While Avoiding the Rational Lattice

RETURN

Partition the real plane into three subsets:

R, rational points - the set of all points, (Xr,Yr), both coordinates rational (this constitutes the Rational Lattice);

I, irrational points - the set of all points, (Xi,Yi), both coordinates irrational;

Q, quasi-rational points - the set of all points, (Xq,Yq), one and only one coordinate rational.

Problem:
Show also that any two points in the complete plane, (I unionQ unionR), may be connected by a path crossing no other rational points.
Preface to Solutions: note that

a. (Lemma) for any line on the plane with irrational slope, at most one rational point can lie on it. (Two rational points on the same line would define its slope to be rational.)

b. (Definitions) Any irrational coordinate identifies a path through the rational lattice, i.e. no line y=ri or x=ri, ri an irrational number, goes through a rational point.

c. (Definition & Lemma) the set of all lines such that y=ri or x=ri, ri an irrational number, constitutes the Non-rational Grid. The horizontal lines, all y such that y=ri, intersect with all vertical lines, all x such that x= rj.

Solutions and Sketch of Proof

1. All Irrational and Quasi-Irrational Points lie on the Non-rational Grid. Thus a path exists between any two such points which avoids all Rational Points.

2. For any rational point, A, a line with an irrational slope can be constructed which passes through the rational point. This line intersects the Non-rational Grid. Thus any other rational point, B, can be connected to A, without involving points other than A and B on the Rational Lattice.

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