Real numbers are the set of numbers that can represent any point on a continuous one-dimensional line. They extend the rational numbers by including irrational numbers like \sqrt{2} and \pi. A real number x can be represented by a (possibly infinite) decimal expansion.
The field of real numbers, \mathbb R, can be constructed as the completion of the rational numbers \mathbb Q, using Dedekind cuts or Cauchy sequences. This process "fills the gaps" on the number line.
This defining property is known as the completeness axiom (or the least-upper-bound property). A consequence is the intermediate value theorem with proves that a continuous function on an interval must take on every value between its endpoints.
Geometrically, real numbers are represented on the number line, which establishes a total ordering where for any two distinct numbers, one must be greater than the other.
This structure is the basis for calculus and the measurement of continuous physical quantities, and for real analysis, the study of limits, continuity, and derivatives of functions on the real line.
KOLMOGOROV, A. N. and FOMIN, S. V., 1975. Introductory Real Analysis. 1st edition. New York: Dover Publications. ISBN 978-0-486-61226-3.
LEBL, Jiri, 2023. Basic Analysis I: Introduction to Real Analysis, Volume I. Las Vegas, NV: Independently published. ISBN 979-8-8519-4463-5.
LEBL, Jiri, 2023. Basic Analysis II: Introduction to Real Analysis, Volume II. Independently published. ISBN 979-8-8519-4597-7.