A Cauchy sequence of rational numbers is a sequence of rational numbers with the property that, for every rational number , there exists a natural number such that, for all natural numbers , the absolute value satisfies .
The set of real numbers is the set of equivalence classes of Cauchy sequences of rational numbers, under the equivalence relation if the interleave sequence of the two sequences is itself a Cauchy sequence. The real numbers form a ring, with addition and multiplication defined by
There is an ordering relation on , defined by if either or there exists a natural number such that for all . This definition is well-defined and does not depend on the choice of Cauchy sequences used to represent the equivalence classes.
One can prove that the real numbers form an ordered field and that they satisfy the Dedekind completeness property (also known as the least upper bound property): For every nonempty subset , if has an upper bound then has a lowest upper bound. It is also true that every ordered field with the least upper bound property is isomorphic to the real numbers.
Alternative definitions of the set of real numbers include:
Equivalence classes of decimal sequences (sequences consisting of natural numbers between 0 and 9, and a single decimal point), where two decimal sequences are equivalent if they are identical, or if one has an infinite tail of 9’s, the other has an infinite tail of 0’s, and the leading portion of the first sequence is one lower than the leading portion of the second.
Dedekind cuts of rational numbers (that is, subsets of with the property that, if and , then ).
The real numbers can also be defined as the unique (up to isomorphism) ordered field satisfying the least upper bound property, after one has proved that such a field exists and is unique up to isomorphism.
The main reason for introducing the reals is that the reals contain all limits. More technically, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section). This means the following:
A sequence of real numbers is called a Cauchy sequence if for any there exists an integer (possibly depending on ) such that the distance is less than provided that and are both greater than . In other words, a sequence is a Cauchy sequence if its elements eventually come and remain arbitrarily close to each other.
A sequence converges to the limit if for any there exists an integer (possibly depending on ) such that the distance is less than provided that is greater than . In other words, a sequence has limit if its elements eventually come and remain arbitrarily close to .
Every Cauchy sequence of real numbers is convergent.
That is, the reals are complete.
Note that the rationals are not complete. For example, the sequence , , , , , , is Cauchy but it does not converge to a rational number. (In the real numbers, in contrast, it converges to the square root of .)
The existence of limits of Cauchy sequences is what makes calculus work and is of great practical use. The standard numerical test to determine if a sequence has a limit is to test if it is a Cauchy sequence, as the limit is typically not known in advance.
For example the standard series of the exponential function
converges to a real number because for every the sums
can be made arbitrarily small by choosing sufficiently large. This proves that the sequence is Cauchy, so we know that the sequence converges even if we don’t know ahead of time what the limit is.
3 “The complete ordered field”
The real numbers are often described as “the complete ordered field,” a phrase that can be interpreted in several ways.
First, an order can be lattice complete. It’s easy to see that no ordered field can be lattice complete, because it can have no largest element (given any element , is larger), so this is not the sense that is meant.
Additionally, an order can be Dedekind-complete, as defined in the Definitions section. The uniqueness result at the end of that section justifies using the word “the” in the phrase “complete ordered field” when this is the sense of “complete” that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way.
These two notions of completeness ignore the field structure. However, an ordered group (and a field is a group under the operations of addition and subtraction) defines a uniform structure, and uniform structures have a notion of completeness (topology); the description in the Completeness section above is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion for metric spaces, since the definition of metric space relies on already having a characterisation of the real numbers.) It is not true that is the only uniformly complete ordered field, but it is the only uniformly complete Archimedean field, and indeed one often hears the phrase “complete Archimedean field” instead of “complete ordered field.” Since it can be proved that any uniformly complete Archimedean field must also be Dedekind complete (and vice versa, of course), this justifies using “the” in the phrase “the complete Archimedean field.” This sense of completeness is most closely related to the construction of the reals from Cauchy sequences (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals) and forms the uniform completion of it in a standard way.
But the original use of the phrase “complete Archimedean field” was by David Hilbert, who meant still something else by it. He meant that the real numbers form the largest Archimedean field in the sense that every other Archimedean field is a subfield of . Thus is “complete” in the sense that nothing further can be added to it without making it no longer an Archimedean field. This sense of completeness is most closely related to the construction of the reals from surreal numbers, since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield.
This article contains material from the http://en.wikipedia.org/wiki/Real_numbersWikipedia article on Real numbers which is incorporated herein under the terms of the http://en.wikipedia.org/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_LicenseGNU Free Documentation License.
|Date of creation||2013-03-22 11:52:22|
|Last modified on||2013-03-22 11:52:22|
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