In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, there are several equivalent ways of defining the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s. One of them is that they form a
complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ordered field exists, and the existence proof consists of constructing a
mathematical structure
In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additiona ...
that satisfies the definition.
The article presents several such constructions. They are equivalent in the sense that, given the result of any two such constructions, there is a unique
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
of
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. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered fiel ...
between them. This results from the above definition and is independent of particular constructions. These isomorphisms allow identifying the results of the constructions, and, in practice, to forget which construction has been chosen.
Axiomatic definitions
An
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
atic definition of the real numbers is to define them as the elements of a complete ordered field. Precisely, this means the following. A ''model for the real number system'' consists of a set
, two distinct elements 0 and 1 of
, two
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
s + and × on
(called ''addition'' and ''multiplication'', respectively), and a
binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...
≤ on
, satisfying the following properties.
Axioms
#
forms a
field. In other words,
#* For all ''x'', ''y'', and ''z'' in
, ''x'' + (''y'' + ''z'') = (''x'' + ''y'') + ''z'' and ''x'' × (''y'' × ''z'') = (''x'' × ''y'') × ''z''. (
associativity
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
of addition and multiplication)
#* For all ''x'' and ''y'' in
, ''x'' + ''y'' = ''y'' + ''x'' and ''x'' × ''y'' = ''y'' × ''x''. (
commutativity
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
of addition and multiplication)
#* For all ''x'', ''y'', and ''z'' in
, ''x'' × (''y'' + ''z'') = (''x'' × ''y'') + (''x'' × ''z''). (
distributivity
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmeti ...
of multiplication over addition)
#* For all ''x'' in
, ''x'' + 0 = ''x''. (existence of additive
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), an ...
)
#* 0 is not equal to 1, and for all ''x'' in
, ''x'' × 1 = ''x''. (existence of multiplicative identity)
#* For every ''x'' in
, there exists an element −''x'' in
, such that ''x'' + (−''x'') = 0. (existence of additive
inverses)
#* For every ''x'' ≠ 0 in
, there exists an element ''x''
−1 in
, such that ''x'' × ''x''
−1 = 1. (existence of multiplicative inverses)
# (
, ≤) forms a
totally ordered set
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive) ...
. In other words,
#* For all ''x'' in
, ''x'' ≤ ''x''. (
reflexivity)
#* For all ''x'' and ''y'' in
, if ''x'' ≤ ''y'' and ''y'' ≤ ''x'', then ''x'' = ''y''. (
antisymmetry)
#* For all ''x'', ''y'', and ''z'' in
, if ''x'' ≤ ''y'' and ''y'' ≤ ''z'', then ''x'' ≤ ''z''. (
transitivity)
#* For all ''x'' and ''y'' in
, ''x'' ≤ ''y'' or ''y'' ≤ ''x''. (
totality)
# The field operations + and × on
are compatible with the order ≤. In other words,
#* For all ''x'', ''y'' and ''z'' in
, if ''x'' ≤ ''y'', then ''x'' + ''z'' ≤ ''y'' + ''z''. (preservation of order under addition)
#* For all ''x'' and ''y'' in
, if 0 ≤ ''x'' and 0 ≤ ''y'', then 0 ≤ ''x'' × ''y'' (preservation of order under multiplication)
# The order ≤ is ''complete'' in the following sense: every non-empty subset of
bounded above has a
least upper bound. In other words,
#* If ''A'' is a non-empty subset of
, and if ''A'' has an
upper bound, then ''A'' has a least upper bound ''u'', such that for every upper bound ''v'' of ''A'', ''u'' ≤ ''v''.
On the least upper bound property
Axiom 4, which requires the order to be
Dedekind-complete, implies the
Archimedean property
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields.
The property, typica ...
.
The axiom is crucial in the characterization of the reals. For example, the totally
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. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered fiel ...
of the rational numbers Q satisfy the first three axioms, but not the fourth. In other words, models of the rational numbers are also models of the first three axioms.
Note that the axiom is
not firstorderizable, as it expresses a statement about collections of reals and not just individual such numbers. As such, the reals are not given by a
first-order logic theory.
On models
Several models for axioms 1-4 are given down
below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
. Any two models for axioms 1-4 are isomorphic, and so up to isomorphism, there is only one complete ordered Archimedean field.
When we say that any two models of the above axioms are isomorphic, we mean that for any two models (
, 0
''R'', 1
''R'', +
''R'', ×
''R'', ≤
''R'') and (''S'', 0
''S'', 1
''S'', +
''S'', ×
''S'', ≤
''S''), there is a
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
''f'' :
→ ''S'' preserving both the field operations and the order. Explicitly,
*''f'' is both
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
and
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
.
*''f''(0
''R'') = 0
''S'' and ''f''(1
''R'') = 1
''S''.
*For all ''x'' and ''y'' in
, ''f''(''x'' +
''R'' ''y'') = ''f''(''x'') +
''S'' ''f''(''y'') and ''f''(''x'' ×
''R'' ''y'') = ''f''(''x'') ×
''S'' ''f''(''y'').
*For all ''x'' and ''y'' in
, ''x'' ≤
''R'' ''y''
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
''f''(''x'') ≤
''S'' ''f''(''y'').
Tarski's axiomatization of the reals
An alternative synthetic
axiomatization
In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contai ...
of the real numbers and their arithmetic was given by
Alfred Tarski
Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...
, consisting of only the 8
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
s shown below and a mere four
primitive notions: a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
called ''the real numbers'', denoted
, a
binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...
over
called ''order'', denoted by
infix
An infix is an affix inserted inside a word stem (an existing word or the core of a family of words). It contrasts with '' adfix,'' a rare term for an affix attached to the outside of a stem, such as a prefix or suffix.
When marking text for i ...
<, a
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
over
called ''addition'', denoted by infix +, and the constant 1.
''Axioms of order'' (primitives:
, <):
Axiom 1. If ''x'' < ''y'', then not ''y'' < ''x''. That is, "<" is an
asymmetric relation.
Axiom 2. If ''x'' < ''z'', there exists a ''y'' such that ''x'' < ''y'' and ''y'' < ''z''. In other words, "<" is
dense in
.
Axiom 3. "<" is
Dedekind-complete. More formally, for all ''X'', ''Y'' ⊆
, if for all ''x'' ∈ ''X'' and ''y'' ∈ ''Y'', ''x'' < ''y'', then there exists a ''z'' such that for all ''x'' ∈ ''X'' and ''y'' ∈ ''Y'', if ''z'' ≠ ''x'' and ''z'' ≠ ''y'', then ''x'' < ''z'' and ''z'' < ''y''.
To clarify the above statement somewhat, let ''X'' ⊆
and ''Y'' ⊆
. We now define two common English verbs in a particular way that suits our purpose:
:''X precedes Y'' if and only if for every ''x'' ∈ ''X'' and every ''y'' ∈ ''Y'', ''x'' < ''y''.
:The real number ''z separates'' ''X'' and ''Y'' if and only if for every ''x'' ∈ ''X'' with ''x'' ≠ ''z'' and every ''y'' ∈ ''Y'' with ''y'' ≠ ''z'', ''x'' < ''z'' and ''z'' < ''y''.
Axiom 3 can then be stated as:
:"If a set of reals precedes another set of reals, then there exists at least one real number separating the two sets."
''Axioms of addition'' (primitives:
, <, +):
Axiom 4. ''x'' + (''y'' + ''z'') = (''x'' + ''z'') + ''y''.
Axiom 5. For all ''x'', ''y'', there exists a ''z'' such that ''x'' + ''z'' = ''y''.
Axiom 6. If ''x'' + ''y'' < ''z'' + ''w'', then ''x'' < ''z'' or ''y'' < ''w''.
''Axioms for one'' (primitives:
, <, +, 1):
Axiom 7. 1 ∈
.
Axiom 8. 1 < 1 + 1.
These axioms imply that
is a
linearly ordered abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
under addition with distinguished element 1.
is also
Dedekind-complete and
divisible.
Explicit constructions of models
We shall not prove that any models of the axioms are isomorphic. Such a proof can be found in any number of modern analysis or set theory textbooks. We will sketch the basic definitions and properties of a number of constructions, however, because each of these is important for both mathematical and historical reasons. The first three, due to
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
/
Charles Méray,
Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and
the axiomatic foundations of arithmetic. His ...
/
Joseph Bertrand and
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
all occurred within a few years of each other. Each has advantages and disadvantages. A major motivation in all three cases was the instruction of mathematics students.
Construction from Cauchy sequences
A standard procedure to force all
Cauchy 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. More precisely, given any small positive distance, all but a finite numbe ...
s in a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
to converge is adding new points to the metric space in a process called
completion.
is defined as the completion of Q with respect to the metric , ''x''-''y'', , as will be detailed below (for completions of Q with respect to other metrics, see
''p''-adic numbers.)
Let ''R'' be the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of Cauchy sequences of rational numbers. That is, sequences
: ''x''
''1'', ''x''
''2'', ''x''
''3'',...
of rational numbers such that for every rational , there exists an integer ''N'' such that for all natural numbers , . Here the vertical bars denote the absolute value.
Cauchy sequences (''x''
''n'') and (''y''
''n'') can be added and multiplied as follows:
: (''x''
''n'') + (''y''
''n'') = (''x''
''n'' + ''y''
''n'')
: (''x''
''n'') × (''y''
''n'') = (''x''
''n'' × ''y''
''n'').
Two Cauchy sequences are called ''equivalent'' if and only if the difference between them tends to zero.
This defines an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
that is compatible with the operations defined above, and the set R of all
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es can be shown to satisfy
all axioms of the real numbers. We can
embed
Embedded or embedding (alternatively imbedded or imbedding) may refer to:
Science
* Embedding, in mathematics, one instance of some mathematical object contained within another instance
** Graph embedding
* Embedded generation, a distributed ...
Q into R by identifying the rational number ''r'' with the equivalence class of the sequence .
Comparison between real numbers is obtained by defining the following comparison between Cauchy sequences: if and only if
''x'' is equivalent to ''y'' or there exists an integer ''N'' such that for all
.
By construction, every real number ''x'' is represented by a Cauchy sequence of rational numbers. This representation is far from unique; every rational sequence that converges to ''x'' is a representation of ''x''. This reflects the observation that one can often use different sequences to approximate the same real number.
The only real number axiom that does not follow easily from the definitions is the completeness of ≤, i.e. the
least upper bound property. It can be proved as follows: Let ''S'' be a non-empty subset of R and ''U'' be an upper bound for ''S''. Substituting a larger value if necessary, we may assume ''U'' is rational. Since ''S'' is non-empty, we can choose a rational number ''L'' such that for some ''s'' in ''S''. Now define sequences of rationals (''u''
''n'') and (''l''
''n'') as follows:
:Set ''u''
0 = ''U'' and ''l''
0 = ''L''.
For each ''n'' consider the number:
:''m''
''n'' = (''u''
''n'' + ''l''
''n'')/2
If ''m''
''n'' is an upper bound for ''S'' set:
: ''u''
''n''+1 = ''m''
''n'' and ''l''
''n''+1 = ''l''
''n''
Otherwise set:
: ''l''
''n''+1 = ''m''
''n'' and ''u''
''n''+1 = ''u''
''n''
This defines two Cauchy sequences of rationals, and so we have real numbers and . It is easy to prove, by induction on ''n'' that:
: ''u''
''n'' is an upper bound for ''S'' for all ''n''
and:
: ''l''
''n'' is never an upper bound for ''S'' for any ''n''
Thus ''u'' is an upper bound for ''S''. To see that it is a least upper bound, notice that the limit of (''u''
''n'' − ''l''
''n'') is 0, and so ''l'' = ''u''. Now suppose is a smaller upper bound for ''S''. Since (''l''
''n'') is monotonic increasing it is easy to see that for some ''n''. But ''l''
''n'' is not an upper bound for S and so neither is ''b''. Hence ''u'' is a least upper bound for ''S'' and ≤ is complete.
The usual
decimal notation
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numer ...
can be translated to Cauchy sequences in a natural way. For example, the notation π = 3.1415... means that π is the equivalence class of the Cauchy sequence (3, 3.1, 3.14, 3.141, 3.1415, ...). The equation
0.999... = 1 states that the sequences (0, 0.9, 0.99, 0.999,...) and (1, 1, 1, 1,...) are equivalent, i.e., their difference converges to 0.
An advantage of constructing R as the completion of Q is that this construction is not specific to one example; it is used for other metric spaces as well.
Construction by Dedekind cuts
A
Dedekind cut
In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the r ...
in an ordered field is a partition of it, (''A'', ''B''), such that ''A'' is nonempty and closed downwards, ''B'' is nonempty and closed upwards, and ''A'' contains no
greatest element. Real numbers can be constructed as Dedekind cuts of rational numbers.
For convenience we may take the lower set
as the representative of any given Dedekind cut
, since
completely determines
. By doing this we may think intuitively of a real number as being represented by the set of all smaller rational numbers. In more detail, a real number
is any subset of the set
of rational numbers that fulfills the following conditions:
#
is not empty
#
#
is closed downwards. In other words, for all
such that
, if
then
#
contains no greatest element. In other words, there is no
such that for all
,
* We form the set
of real numbers as the set of all Dedekind cuts
of
, and define a
total ordering on the real numbers as follows:
* We
embed
Embedded or embedding (alternatively imbedded or imbedding) may refer to:
Science
* Embedding, in mathematics, one instance of some mathematical object contained within another instance
** Graph embedding
* Embedded generation, a distributed ...
the rational numbers into the reals by identifying the rational number
with the set of all smaller rational numbers
. Since the rational numbers are
dense, such a set can have no greatest element and thus fulfills the conditions for being a real number laid out above.
*
Addition
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
.
*
Subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
.
where
denotes the
relative complement
In set theory, the complement of a set , often denoted by (or ), is the set of elements not in .
When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is ...
of
in
,
*
Negation
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
is a special case of subtraction:
* Defining
multiplication
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
is less straightforward.
** if
then
** if either
or
is negative, we use the identities
to convert
and/or
to positive numbers and then apply the definition above.
* We define
division in a similar manner:
** if
then
** if either
or
is negative, we use the identities
to convert
to a non-negative number and/or
to a positive number and then apply the definition above.
*
Supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
. If a nonempty set
of real numbers has any upper bound in
, then it has a least upper bound in
that is equal to
.
As an example of a Dedekind cut representing an
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
, we may take the
positive square root of 2. This can be defined by the set
. It can be seen from the definitions above that
is a real number, and that
. However, neither claim is immediate. Showing that
is real requires showing that
has no greatest element, i.e. that for any positive rational
with
, there is a rational
with