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.
Some just refer to it as the number line.
Some use the term number axis.
Also see
Hence from Metric Induces Topology, the real number line is also a topological space.
Sources
- 1919: Horace Lamb: An Elementary Course of Infinitesimal Calculus (3rd ed.) ... (previous) ... (next): Chapter $\text I$. Continuity: $1$. Continuous Variation
- 1957: Tom M. Apostol: Mathematical Analysis ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: $\text{1-4}$: Geometrical representation of real numbers
- 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 $2$
- 1971: Wilfred Kaplan and Donald J. Lewis: Calculus and Linear Algebra ... (previous) ... (next): Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-1}$: The Real Numbers
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): Notation and Terminology
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: $\S 1.1$: Real Numbers
- 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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): number line
- 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
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): number line
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): number line