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
, or more precisely, ,
By induction on one can prove that
The last claim follows since for (by induction), and . It follows that is an injection: If , then implies that , so .
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.
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 , it is shown that the rational numbers form a subfield of using explicit Dedekind cuts.
|Title||rational numbers are real numbers|
|Date of creation||2013-03-22 15:45:49|
|Last modified on||2013-03-22 15:45:49|
|Last modified by||matte (1858)|