# 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.

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.

### Origin

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

### Ordering

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**.

## Also see

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

## Sources

- 1959: E.M. Patterson:
*Topology*(2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Topological Spaces: $\S 11$. Continuity on the Euclidean line - 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 1.1$. Number Systems - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 1.1$. Sets: Example $2$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): Notation and Terminology - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.2$: The set of real numbers - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Graphical Representation of Real Numbers - 1991: Felix Hausdorff:
*Set Theory*(4th ed.) ... (previous) ... (next): Preliminary Remarks - 2003: John H. Conway and Derek A. Smith:
*On Quaternions And Octonions*... (previous) ... (next): $\S 1$: The Complex Numbers: Introduction: $1.1$: The Algebra $\R$ of Real Numbers