rational numbers are real numbers
Let us first show that the natural numbers are
contained in the real numbers as constructed above.
Heuristically, this should be clear. We start with .
By adding 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 as
-
1.
, or more precisely, ,
-
2.
for .
By induction on one can prove that
and
The last claim follows since for (by induction), and . It follows that is an injection: If , then implies that , so .
To conclude, let us show that
satisfies the Peano axioms with zero element and
sucessor operator
First, as is a bijection, if and only if
is clear.
Second, if for some , then ; a contradiction.
Lastly, the axiom of induction follows since satisfies this axiom.
We have shown that 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
Mathematically, and are subrings of that are
ring isomorphic 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 is inductive if
-
1.
,
-
2.
if , then .
Then the natural numbers are defined as real numbers that are contained in all
inductive sets.
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 subfield
of
using explicit Dedekind cuts
.
References
- 1 H.L. Royden, Real analysis, Prentice Hall, 1988.
- 2 M. Spivak, Calculus, Publish or Perish.
- 3 W. Rudin, Principles of mathematical analysis, McGraw-Hill, 1976.
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 |