Rational Numbers From Projective Geometry

In this video we explain how rational numbers can be constructed starting with a projective plane where we can just draw points and lines. I define the important notion of the harmonic conjugate of a point, with respect to two other points on the line. I describe how a set of points on the line called a harmonic net (or net of rationality) can be obtained by repeated construction of harmonic conjugates. We also describe how this relates to another geometric construction of the rational points that uses the idea of parallel lines from affine geometry.

The uniqueness of the result of constructing the harmonic conjugate of A with respect to B and C is proved on pages 47 and 48 of Lawrence Edwards `Projective Geometry'. H.S.M. Coxeter discusses many details about harmonic nets from the point of view of projective geometry starting at page 30. Norman Wildberger gives an excellent description of how arithmetic can be thought of in terms of affine constructions in `The most fundamental and important problem in mathematics | Famous Math Problems 19a | NJ Wildberger` https://www.youtube.com/watch?v=Nu-YPJSNFpE and https://www.youtube.com/watch?v=t0aHtXud9r4

Видео Rational Numbers From Projective Geometry канала Richard Southwell