Definition:Real Number/Real Number Line

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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.


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.


The point representing the number $0$ (zero) is referred to as the origin.


The usual ordering on $\R$ is implemented on the real number line as:

$a > b$ if and only if $a$ is to the right of $b$
$a < b$ if and only if $a$ is to the left of $b$.

Also known as

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

Some just refer to it as the number line.

Also see

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