Completeness of the real numbers
Encyclopedia
Intuitively, completeness implies that there are not any “gaps” (in Dedekind's terminology) or “missing points” in the real number line
. This contrasts with the rational number
s, whose corresponding number line has a “gap” at each irrational
value. In the decimal number system
, completeness is equivalent to the statement that any infinite string of decimal digits is actually the decimal representation for some real number.
Depending on the construction of the real numbers used may be completeness may take the form of an axiom
(the completeness axiom), or may be a theorem
proven from the construction. There are many equivalent
forms of completeness, the most prominent being Dedekind completeness (the least-upper-bound property
) and Cauchy completeness (completeness as a metric space).
s can be defined synthetically as an ordered field
satisfying some version of the completeness axiom
. Different versions of this axiom are all equivalent, in the sense that any ordered field that satisfies one form of completeness satisfies all of them. When the real numbers are instead constructed using a model, completeness becomes a theorem
or collection of theorems.
must have a least upper bound (or supremum). In a synthetic approach to the real numbers, this is the version of completeness that is most often stated as an axiom.
The rational number line
Q is not Dedekind complete. An example is the subset of rational numbers
The number 5 is certainly an upper bound for the set. However, this set has no least upper bound in Q: the least upper bound in this case is
which does not exist in Q, and for any upper bound x ∈ Q, there is another upper bound y ∈ Q with y < x.
Dedekind completeness is related to the construction of the real numbers using Dedekind cut
s. Essentially, this method defines a real number to be the least upper bound of some set of rational numbers.
In the order of the theory x Dedekind completeness can be generalized to any partially ordered set
. See completeness (order theory)
.
of real numbers converges.
The rational number line
Q is not Cauchy complete. An example is the following sequence of rational numbers:
Here the nth term in the sequence is the nth decimal approximation for pi
. Though this is a Cauchy sequence of rational numbers, it does not converge to any rational number. (In this real number line, this sequence converges to pi.)
Cauchy completeness is related to the construction of the real numbers using Cauchy sequences. Essentially, this method defines a real number to be the limit of a Cauchy sequence of rational numbers.
In mathematical analysis
, Cauchy completeness can be generalized to a notion of completeness for any metric space
. See complete metric space.
, and suppose that these intervals are nested in the sense that
The nested interval theorem states that the intersection
of all of the intervals is nonempty.
The rational number line
does not satisfy the nested interval theorem. For example, the sequence
is a nested sequence of closed intervals in the rational numbers whose intersection is empty. (In the real numbers, the intersection of these intervals contains the number pi
.)
states that every bounded sequence of real numbers has a convergent subsequence
. Again, this theorem is equivalent to the other forms of completeness given above.
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
. This contrasts with the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s, whose corresponding number line has a “gap” at each irrational
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
value. In the decimal number system
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
, completeness is equivalent to the statement that any infinite string of decimal digits is actually the decimal representation for some real number.
Depending on the construction of the real numbers used may be completeness may take the form of an axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
(the completeness axiom), or may be a theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
proven from the construction. There are many equivalent
Logical equivalence
In logic, statements p and q are logically equivalent if they have the same logical content.Syntactically, p and q are equivalent if each can be proved from the other...
forms of completeness, the most prominent being Dedekind completeness (the least-upper-bound property
Least-upper-bound property
In mathematics, the least-upper-bound property is a fundamental property of the real numbers and certain other ordered sets. The property states that any non-empty set of real numbers that has an upper bound necessarily has a least upper bound ....
) and Cauchy completeness (completeness as a metric space).
Forms of completeness
The real numberReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s can be defined synthetically as an ordered field
Ordered field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and...
satisfying some version of the completeness axiom
Completeness axiom
In mathematics the completeness axiom, also called Dedekind completeness of the real numbers, is a fundamental property of the set R of real numbers...
. Different versions of this axiom are all equivalent, in the sense that any ordered field that satisfies one form of completeness satisfies all of them. When the real numbers are instead constructed using a model, completeness becomes a theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
or collection of theorems.
Dedekind completeness
Dedekind completeness, also known as the least-upper-bound property, states that every nonempty set of real numbers having an upper boundUpper bound
In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is lesser than or equal to every element of S...
must have a least upper bound (or supremum). In a synthetic approach to the real numbers, this is the version of completeness that is most often stated as an axiom.
The rational number line
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
Q is not Dedekind complete. An example is the subset of rational numbers
The number 5 is certainly an upper bound for the set. However, this set has no least upper bound in Q: the least upper bound in this case is
which does not exist in Q, and for any upper bound x ∈ Q, there is another upper bound y ∈ Q with y < x.Dedekind completeness is related to the construction of the real numbers using Dedekind cut
Dedekind cut
In mathematics, a Dedekind cut, named after Richard Dedekind, is a partition of the rationals into two non-empty parts A and B, such that all elements of A are less than all elements of B, and A contains no greatest element....
s. Essentially, this method defines a real number to be the least upper bound of some set of rational numbers.
In the order of the theory x Dedekind completeness can be generalized to any partially ordered set
Partially ordered set
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
. See completeness (order theory)
Completeness (order theory)
In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set . A special use of the term refers to complete partial orders or complete lattices...
.
Cauchy completeness
Cauchy completeness is the statement that every Cauchy sequenceCauchy sequence
In mathematics, a Cauchy sequence , named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses...
of real numbers converges.
The rational number line
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
Q is not Cauchy complete. An example is the following sequence of rational numbers:
Here the nth term in the sequence is the nth decimal approximation for pi
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
. Though this is a Cauchy sequence of rational numbers, it does not converge to any rational number. (In this real number line, this sequence converges to pi.)
Cauchy completeness is related to the construction of the real numbers using Cauchy sequences. Essentially, this method defines a real number to be the limit of a Cauchy sequence of rational numbers.
In mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, Cauchy completeness can be generalized to a notion of completeness for any metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
. See complete metric space.
Nested intervals theorem
The nested interval theorem is another form of completeness. Let be a sequence of closed intervalsInterval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
, and suppose that these intervals are nested in the sense that
- .
The nested interval theorem states that the intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
of all of the intervals is nonempty.
The rational number line
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
does not satisfy the nested interval theorem. For example, the sequence
is a nested sequence of closed intervals in the rational numbers whose intersection is empty. (In the real numbers, the intersection of these intervals contains the number pi
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
.)
Monotone convergence theorem
The monotone convergence theorem (described as the fundamental axiom of analysis by ) states that every nondecreasing, bounded sequence of real numbers converges. This can be viewed as a special case of Dedekind completeness, but it can also be used fairly directly to prove the Cauchy completeness of the real numbers.Bolzano–Weierstrass theorem
The Bolzano–Weierstrass theoremBolzano–Weierstrass theorem
In real analysis, the Bolzano–Weierstrass theorem is a fundamental result about convergence in a finite-dimensional Euclidean space Rn. The theorem states thateach bounded sequence in Rn has a convergent subsequence...
states that every bounded sequence of real numbers has a convergent subsequence
Subsequence
In mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements...
. Again, this theorem is equivalent to the other forms of completeness given above.