topic entry on real numbers


The real number system may be conceived as an attempt to fill in the gaps in the rational numberPlanetmathPlanetmathPlanetmath system. These gaps first became apparent in connection with the Pythagorean theoremMathworldPlanetmathPlanetmath, which requires one to extract a square root in order to find the third side of a right triangleMathworldPlanetmath two of whose sides are known. Hypossos, a student of Pythagoras, showed that there is no rational number whose square is exactly 2. In particular, this means that there is no rational number which may be used to describe the length of the diagonalMathworldPlanetmath of a square the length of whose sides is rational. This result ruined the philosophical program of Pythagoras, which was to describe everything in terms of whole numbers (or ratios of whole numbers) and, according to legend, resulted in Hypossos drowning himself. Eventually, geometers reconciled themselves to the existence of irrational magnitudes and Eudoxos devised his method of exhaustion which allowed one to prove results about irrational magnitudes by considerations of rational magnitudes which are smaller and larger than the the irrational magnitude in question.

Centuries later, Descartes showed how it is systematically possible to reduce questions of geometryMathworldPlanetmathPlanetmathPlanetmath to algebra. This brought up the issue of irrational numbers again — if one is going to reformulate everything in terms of algebra, then one cannot have recourse to defining magnitudes geometrically, but have to find some sort of number which can adequately represent things like the hypotenuseMathworldPlanetmath of a square with rational sides. At first, such problems of logical consistency were swept under the rug, but eventually mathematicians realized that their subject needed to be put on a firm logical foundation. In particular, Dedekind solved this difficulty by defining the real numbers as a certain type of partitionPlanetmathPlanetmathPlanetmath of the set of rational numbers which he termed a cut and defining operationsMathworldPlanetmath on these numbers, such as additionPlanetmathPlanetmath, subtractionPlanetmathPlanetmath, multiplication, and division in terms of operations on these partitions.

Index of entries on real numbers

The below list presents entries on real numbers in an order suitable for studying the subject.

  1. 1.

    Rational numbers

  2. 2.

    Axiomatic definition of the real numbers.

  3. 3.

    Constructions of real numbers (advanced):

    1. (a)
    2. (b)

      Cauchy sequencesMathworldPlanetmathPlanetmath (

    3. (c)

      CharacterizationMathworldPlanetmath of real numbers (

    4. (d)

      Reals not isomorphic to p-adic numbers (

  4. 4.
  5. 5.
  6. 6.

    Inequalities for real numbers (

  7. 7.
  8. 8.
  9. 9.
  10. 10.
  11. 11.

    Real numbers are uncountable (

  12. 12.
  13. 13.

    Operations for real numbers

    1. (a)
    2. (b)
    3. (c)
    4. (d)

      square root

    5. (e)
  14. 14.

    Topic entry on algebraicMathworldPlanetmath and transcendental numbersMathworldPlanetmath (

    1. (a)

      Irrational number (

    2. (b)

      Transcendental number

    3. (c)

      Algebraic numberMathworldPlanetmath (

  15. 15.

    Particular real numbers

    1. (a)

      natural log base

    2. (b)


    3. (c)

      Mascheroni constant

    4. (d)


There are many generalizationsPlanetmathPlanetmath of real numbers. These include the complex numbersMathworldPlanetmathPlanetmath, quaternions, extended real numbers, hyperreal numbers (, and surreal numbersMathworldPlanetmath.  Of course the field has many other field extensions, e.g. the field (x) of the rational functions.

Title topic entry on real numbers
Canonical name TopicEntryOnRealNumbers
Date of creation 2013-03-22 15:40:38
Last modified on 2013-03-22 15:40:38
Owner matte (1858)
Last modified by matte (1858)
Numerical id 22
Author matte (1858)
Entry type Topic
Classification msc 54C30
Classification msc 26-00
Classification msc 12D99
Related topic GelfandTornheimTheorem
Related topic PositivityInOrderedRing
Related topic TopicEntryOnRationalNumbers