论直线的定义及相关基本性质

In this paper, the shortcomings of both Euclidean definition of straight line and the definition of straight line merely based on the length metrology and taking \'the shortest distance between two points\' as its essential attribute are pointed out; the concepts of the point divergence, the point\'s duality of isolation and divergence and the geometrically-full point are proposed; the concepts of the straightness of the points alignment of a line, the length invariance during the reshaping of a piece of line and the point content in the length metrology of a piece of line are also proposed; the novel interetations of the concepts of real numbers, extremities, infinities, infinitesimals and duality-number pairs in number theory are made; and then, based on these concepts and concept-interpretations, the definition of the true straight line applicable to geometrical free space and the elemental Euclidean plan is worked out and, furthermore, being within both these two specified geometrical spaces, the validity of the propositions including the certainty of the existence of a straight line through any two points and that\'the straight-line distance is the shortest distance between two specified points and the line is the unique one through the two points\' are proved; Meanwhile, the general definition of straight line applicable to arbirary geometrical surfaces as well as to geometrical free space is given more strictly. This work has revealed the essential relationship between lines and points, improves relevant items of the fundamental theory of geometry and several fundamental concepts in number theory and provides a new thinking perspective for more thoroughly understanding the meanings of differential and integral, for getting more intensive knowledge of the relationship between continuity and discontinuity and for making better responses to the fourth one among Hilbert\'s 23 important mathematical problems and so on.