The axiom posits a fundamental relationship between real numbers and points on a line. It states that the points on a line can be put into a one-to-one correspondence with real numbers, allowing for the measurement of distances between any two points. This correspondence effectively creates a coordinate system on the line. The distance between two points is then defined as the absolute value of the difference of their corresponding coordinates. For instance, if point A corresponds to the number 2 and point B corresponds to the number 5, then the distance between A and B is |5 – 2| = 3.
This foundational concept underpins numerous geometric proofs and calculations. By assigning numerical values to points, geometric problems can be translated into algebraic ones, thereby broadening the range of problem-solving techniques available. Its incorporation into a system of axioms provides a rigorous basis for establishing geometric theorems. Historically, the formalization of this connection between numbers and geometry contributed significantly to the development of analytic approaches in this branch of mathematics.
