Definition:Real Number/Real Number Line

Definition
From the Cantor-Dedekind Hypothesis, the set of real numbers is isomorphic to any infinite straight line.

The real number line is an arbitrary infinite straight line each of whose points is identified with a real number such that the distance between any two real numbers is consistent with the length of the line between those two points.


 * RealNumberLine.png

Thus we can identify any (either physically drawn or imagined) line with the set of real numbers and thereby illustrate truths about the real numbers by means of diagrams.

Also known as
Some texts refer to the real number line as  the Euclidean line.

Some just refer to it as the number line.

Some use the term number axis.

Also see

 * Real Number Line is Metric Space

Hence from Metric Induces Topology, the real number line is also a topological space.


 * Real Numbers form Vector Space