rational numbers are real numbers

Let us first show that the natural numbersMathworldPlanetmath 0,1,2, are contained in the real numbers as constructed above. Heuristically, this should be clear. We start with 0. By adding 1 repeatedly we obtain the natural numbers


To make this precise, let be the natural numbers. (We assume that these exist. For example, all the usual constructions of rely on the existence of the natural numbers.) Then we can define a map f: as

  1. 1.

    f(0)=0, or more precisely, f(0)=0,

  2. 2.

    f(a+1)=f(a)+1 for a.

By inductionMathworldPlanetmath on a one can prove that

f(a+b) = f(a)+f(b),
f(ab) = f(a)f(b),a,b


f(a) 0,awith equality only whena=0.

The last claim follows since f(a)>0 for a=1,2, (by induction), and f(0)=0. It follows that f is an injection: If ab, then f(a)=f(b) implies that f(a)=f(a)+f(b-a), so a=b.

To conclude, let us show that f() satisfies the Peano axiomsMathworldPlanetmath with zero element f(0) and sucessor operator

S:f() f()
k f(f-1(k)+1)

First, as f is a bijection, x=y if and only if S(x)=S(y) is clear. Second, if S(k)=0 for some k=f(a)f(), then a+1=0; a contradictionMathworldPlanetmathPlanetmath. Lastly, the axiom of induction follows since satisfies this axiom. We have shown that f() are a subset of the real numbers that behave as the natural numbers.

From the natural numbers, the integers and rationals can be defined as

= {-z:z},
= {ab:a,b{0}}.

Mathematically, and are subrings of that are ring isomorphicPlanetmathPlanetmathPlanetmath to the integers and rationals, respectively.

Other constructions

The above construction follows [1]. However, there are also other constructions. For example, in [2], natural numbers in are defined as follows. First, a set L is inductive if

  1. 1.


  2. 2.

    if aL, then a+1L.

Then the natural numbers are defined as real numbers that are contained in all inductive setsMathworldPlanetmath. A third approach is to explicitly exhibit the natural numbers when constructing the real numbers. For example, in [3], it is shown that the rational numbers form a subfieldMathworldPlanetmath of using explicit Dedekind cutsMathworldPlanetmath.


Title rational numbers are real numbers
Canonical name RationalNumbersAreRealNumbers
Date of creation 2013-03-22 15:45:49
Last modified on 2013-03-22 15:45:49
Owner matte (1858)
Last modified by matte (1858)
Numerical id 6
Author matte (1858)
Entry type Result
Classification msc 54C30
Classification msc 26-00
Classification msc 12D99
Related topic GroundField