axiomatic definition of the real numbers

Axiomatic definition of the real numbers

The real numbers consist of a set together with mappings +:× and :× and a relationMathworldPlanetmathPlanetmathPlanetmath <× satisfying the following conditions:

  1. 1.

    (,+) is an Abelian groupMathworldPlanetmath:

    1. (a)

      For a,b,c, we have

      a+b = b+a,
      (a+b)+c = a+(b+c),
    2. (b)

      there exists an element 0 such that a+0=a for all a,

    3. (c)

      every a has an inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (-a) such that a+(-a)=0.

  2. 2.

    ({0},) is an Abelian group:

    1. (a)

      For a,b,c, we have

      ab = ba,
      (ab)c = a(bc),
    2. (b)

      there exists an element 1{0} such that a1=a for all a,

    3. (c)

      every a{0} has an inverse a-1 such that a-1a=1.

  3. 3.

    The operationMathworldPlanetmath is distributive over +: If a,b,c, then

    a(b+c) =ab+ac,
    (b+c)a =ba+ca.
  4. 4.

    (,<) is a total orderMathworldPlanetmath:

    1. (a)

      (transitivity) if c, a<b, and b<c, then a<c,

    2. (b)

      (trichotomy) precisely one of the below alternatives hold:


    For convenience we make the following notational definitions: a>b means b<a, ab means either a<b or a=b, and ab means either b<a or a=b.

  5. 5.

    The operations + and are compatible with the order <:

    1. (a)

      If a, b, c and a<b, then a+c<b+c.

    2. (b)

      If a, b, c with a<b and 0<c, then ac<bc.

  6. 6.

    has the least upper bound property: If A, then an element M is an for A if

    a<M, for all aA.

    If A is non-empty, we then say that A is bounded from above. That has the least upper bound property means that if A is bounded from above, it has a least upper bound m. That is, A has an upper bound m such that if M is any upper bound from M, then mM.

Here it should be emphasized that from the above we can not deduce that a set with operations +,,< exists. To settle this question such a set has to be explicitly constructed. However, this can be done in various ways, as discussed on this page ( One can also show the above conditions uniquely determine the real numbers (up to an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath). The proof of this can be found on this page (

Basic properties

In condensed form, the above conditions state that is an ordered field with the least upper bound property. In particular (,+,) is a ring, and ({0},) is a group, and we have the following basic properties:

Lemma 1.

Suppose a,bR.

  1. 1.

    The additive inverse (-a) is unique (proof) (

  2. 2.

    The additive identity 0 is unique (proof) (

  3. 3.

    (-1)a=(-a) (proof) (

  4. 4.

    (-a)(-b)=ab (proof) (

  5. 5.

    0a=0 (proof) (

  6. 6.

    The multiplicative inverse a-1 is unique (proof) (

  7. 7.

    If a,b are non-zero, then (ab)-1=b-1a-1 (proof) (

In view of property 2, we can write simply -a instead of (-1)a and (-a).

Because of the additive inverse of a real number is unique (by property 1 above), and (-a)+a=a+(-a)=0, we see that the additive inverse of -a is a, or that -(-a)=a. Similarly, if a0, then a-10 (or we’ll end up with 1=aa-1=a0=0), and therefore by Property 6 above, a-1 has a unique multiplicative inverse. Since aa-1=a-1a=1, we see that a is the multiplicative inverse of a-1. In other words, (a-1)-1=a.

For a,b let us also define a-b=a+(-b), which is called the differencePlanetmathPlanetmath of a and b. By commutativity, a-b=-b+a.  It is also common to leave out the multiplication symbol and simply write ab=ab.  Suppose  a  and  b  is non-zero.  Then b divided ( by a is defined as


In consequence, if  a,b,c,d  and b,c,d are non-zero, then

  • abcd=bdac,

  • abb=a.

For example,

Title axiomatic definition of the real numbers
Canonical name AxiomaticDefinitionOfTheRealNumbers
Date of creation 2013-03-22 15:39:29
Last modified on 2013-03-22 15:39:29
Owner matte (1858)
Last modified by matte (1858)
Numerical id 17
Author matte (1858)
Entry type Definition
Classification msc 54C30
Classification msc 26-00
Classification msc 12D99
Related topic RealNumber