In mathematics, there are a number of ways of defining the real number system as an ordered field. The synthetic approach gives a list of axioms for the real numbers as a complete ordered field. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense that there is a model for the axioms, and any two such models are isomorphic. Any one of these models must be explicitly constructed, and most of these models are built using the basic properties of the rational number system as an ordered field. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, an ordered field is a field together with an ordering of its elements. ... This article is about a logical statement. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, axiomatic set theory is a rigorous reformulation of set theory in first-order logic created to address paradoxes in naive set theory. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...

Synthetic approach

The synthetic approach axiomatically defines the real number system as a complete ordered field. Precisely, this means the following. A model for the real number system consists of a set R, two distinct elements 0 and 1 of R, two binary operations + and * on R (called addition and multiplication, resp.), a binary relation ≤ on R, satisfying the following properties. In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... In mathematics, a binary relation (or a dyadic relation) is an arbitrary association of elements of one set with elements of another (perhaps the same) set. ...

1. (R, +, *) forms a field. In other words, In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

• For all x, y, and z in R, x + (y + z) = (x + y) + z and x * (y * z) = (x * y) * z. (associativity of addition and multiplication)
• For all x and y in R, x + y = y + x and x * y = y * x. (commutativity of addition and multiplication)
• For all x, y, and z in R, x * (y + z) = (x * y) + (x * z). (distributivity of multiplication over addition)
• For all x in R, x + 0 = x. (existence of additive identity)
• 0 is not equal to 1, and for all x in R, x * 1 = x. (existence of multiplicative identity)
• For every x in R, there exists an element −x in R, such that x + (−x) = 0. (existence of additive inverses)
• For every x ≠ 0 in R, there exists an element x⁻¹ in R, such that x * x⁻¹ = 1. (existence of multiplicative inverses)

2. (R, ≤) forms a totally ordered set. In other words, In mathematics, associativity is a property that a binary operation can have. ... Mathematical meaning A map or binary operation is said to be commutative when, for any x in A and any y in B . ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... For other uses, see identity (disambiguation). ... In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ... In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ...

• For all x in R, x ≤ x. (reflexivity)
• For all x and y in R, if x ≤ y and y ≤ x, then x = y. (antisymmetry)
• For all x, y, and z in R, if x ≤ y and y ≤ z, then x ≤ z. (transitivity)
• For all x and y in R, x ≤ y or y ≤ x. (totalness)

3. The field operations + and * on R are compatible with the order ≤. In other words, In set theory, a binary relation can have, among other properties, reflexivity or irreflexivity. ... In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X, if a is related to b and b is related to a, then a = b. ... In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ... In mathematics and set theory, a total order, linear order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ≤) on some set X. The relation is transitive, antisymmetric, and total. ...

• For all x, y and z in R, if x ≤ y, then x + z ≤ y + z. (preservation of order under addition)
• For all x and y in R, if 0 ≤ x and 0 ≤ y, then 0 ≤ x * y (preservation of order under multiplication)

4. The order ≤ is complete in the following sense: every non-empty subset of R bounded above has a least upper bound. In other words, In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually. ... In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound. ...

• If A is a non-empty subset of R, 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.

The final axiom, defining the order as Dedekind-complete, is most crucial. Without this axiom, we simply have the axioms which define a totally ordered field, and there are many non-isomorphic models which satisfy these axioms. This axiom implies that the Archimedean property applies for this field. Therefore, when the completeness axiom is added, it can be proved that any two models must be isomorphic, and so in this sense, there is only one complete ordered Archimedean field. In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually. ... In mathematics, a Dedekind cut in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for any element x in S, if a is in A and x ≤ a, then x is in A as well) and B is... In mathematics, an ordered field is a field together with an ordering of its elements. ... In mathematics (particularly abstract algebra), the Archimedean property is a property held by some ordered algebraic structures, and in particular by the ordered field of real numbers. ...

When we say that any two models of the above axioms are isomorphic, we mean that for any two models (R, 0_R, 1_R, +_R, *_R, ≤_R) and (S, 0_S, 1_S, +_S, *_S, ≤_S), there is a bijection f : R → S preserving both the field operations and the order. Explicitly, A bijective function. ...

• f is both 1-1 and onto.
• f(0_R) = 0_S and f(1_R) = 1_S.
• For all x and y in R, 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 R, x ≤_R y if and only if f(x) ≤_S f(y).

In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... ↔ ⇔ ≡ logical symbols representing iff. ...

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.

Construction from Cauchy sequences

If we have a space where Cauchy sequences are meaningful (such as a metric space, i.e., a space where distance is defined, or more generally a uniform space), a standard procedure to force all Cauchy sequences to converge is adding new points to the space (a process called completion). By starting with rational numbers and the metric d(x,y) = |x − y|, we can construct the real numbers, as will be detailed below. (A different metric on the rationals could result in the p-adic numbers instead.) The plot of a Cauchy sequence shown in blue, as versus If the space containing the sequence is complete, the ultimate destination of this sequence, that is, the limit, exists. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In the mathematical field of topology, a uniform space is a set with a uniform structure. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... The title given to this article is incorrect due to technical limitations. ...

Let R be the set of Cauchy sequences of rational numbers. Cauchy sequences (x) and (y) can be added, multiplied and compared as follows: In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...

(x_n) + (y_n) = (x_n + y_n)

(x_n) × (y_n) = (x_n × y_n)

(x_n) ≥ (y_n) if and only if for every rational ε > 0, there exists an integer N such that x_n ≥ y_n - ε for all n > N.

Two Cauchy sequences are called equivalent if the sequence (x_n - y_n) has limit 0. This does indeed define an equivalence relation, it is compatible with the operations defined above, and the set R of all equivalence classes can be shown to satisfy all the usual axioms of the real numbers. We can embed the rational numbers into the reals by identifying the rational number r with the equivalence class of the sequence (r,r,r, …). In mathematics, an equivalence relation is a binary relation between two elements of a set which groups them together as being equivalent in some way. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out... In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ...

The only real number axiom that does not follow easily from the definitions is the completeness of ≤. 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, there is a rational number L such that L < s for some s in S. Now define sequences of rationals (u_n) and (l_n) as follows:

Set u₀ = U and l₀ = 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 obviously defines two Cauchy sequences of rationals, and so we have real numbers l=(l_n) and u=(u_n). 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 b < u = l. Since (l_n) is monotonic increasing it is easy to see that b < l_n 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.

A practical and concrete representative for an equivalence class representing a real number is provided by the representation to base b -- in practice, b is usually 2 (binary), 8 (octal), 10 (decimal) or 16 (hexadecimal). For example, the number π = 3.14159... corresponds to the Cauchy sequence (3,3.1,3.14,3.141,3.1415,...). Notice that the sequence (0,0.9,0.99,0.999,0.9999,...) is equivalent to the sequence (1,1.0,1.00,1.000,1.0000,...); this shows that 0.999... = 1. Binary is quite hard The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ... The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. ... For other uses, see Decimal (disambiguation). ... In mathematics and computer science, hexadecimal, base-16, or simply hex, is a numeral system with a radix, or base, of 16, usually written using the symbols 0–9 and A–F, or a–f. ... In mathematics, the recurring decimal 0. ...

Construction by Dedekind cuts

A Dedekind cut 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. In detail, one can make the following definitions. (These are of value in extending some definitions to combinatorial game theory.) In mathematics, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards... Mathematicians playing Konane at a Combinatorial game theory workshop (for technical content, see external link) This article is on the theory of combinatorial games. ...

When using Dedekind cuts to construct the real numbers, we can embed the rational numbers into the reals by identifying the rational number r with the Dedekind cut This embedding preserves the meaning of the comparison operators and the arithmetic operations as defined below on Dedekind cuts. In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ...

Certain arithmetic operations and set-theoretic notions which apply to the real numbers can be defined correspondingly for Dedekind cuts as follows:

1. Comparison. Two Dedekind cuts, (A_x, B_x) and (A_y, B_y) are equal: A comparison is an evaluation of similarities and differences - described by Gregory Bateson in his book Mind and Nature as the two quanta of experience. ... In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ...

and (A_x, B_x) is less than, or equal to, (A_y, B_y):

2. Addition. The sum of two Dedekind cuts: 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ... Addition is one of the basic operations of arithmetic. ...

3. Subtraction is defined analogously to addition. 5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is the inverse of addition. ...

4. Multiplication. The product of two Dedekind cuts, in case In mathematics, multiplication is an elementary arithmetic operation. ...

5. Division. The quotient of two Dedekind cuts, in case In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... In mathematics, a quotient is the end result of a division problem. ...

6. Completeness. The supremum of a set of Dedekind cuts which is bounded above: Look up completeness in Wiktionary, the free dictionary. ... In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ...

and the infimum of a set of Dedekind cuts which is bounded below: In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is less than or equal to all other elements of the subset. ...

Construction by decimal expansions

We can take the infinite decimal expansion to be the definition of a real number, considering expansions like 0.9999... and 1.0000... to be equivalent, and define the arithmetical operations formally. Similarly another radix can be used. This is a special case of the construction by Cauchy sequences. The radix (Latin for root), also called base, is the number of various unique symbols (or digits or numerals) a positional numeral system uses to represent numbers. ...

Construction from ultrafilters

As in the hyperreal numbers, construct ^*Q from the rational numbers using an ultrafilter. Take the ring of all elements in ^*Q whose absolute value is less than some nonzero natural number (it doesn't matter which). This ring has a unique maximal ideal, the infinitesimal numbers. Factoring a ring by a maximal ideal gives a field, in this case the field of reals. It turns out that the maximal ideal respects the order on ^*Q, so the field we get is an ordered field. Completeness can be proved in a similar way to the construction from the Cauchy sequences. The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of calculus by Newton and Leibniz. ... In mathematics, especially in order theory, an ultrafilter is a subset of a partially ordered set (a poset) which is maximal among all proper filters. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, more specifically in ring theory a maximal ideal is an ideal which is maximal (with respect to set inclusion) amongst all proper ideals, i. ... Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

Construction from surreal numbers

Every ordered field can be embedded in the surreal numbers. The real numbers form the largest subfield that is Archimedean (meaning that no real number is infinitely large). In mathematics, the surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similar to superreal numbers and hyperreal numbers. ... In mathematics, the Archimedean property of an ordered algebraic structure, such as a linearly ordered group, and in particular of the real numbers, is the property of having no (non-zero) infinitesimals. ...

Construction from the group of integers

A relatively less known construction allows to define real numbers using only the additive group of integers. Different versions of this construction are described in [1], [2] and [3]. The construction has been formally verified by the IsarMathLib project [4]. Automated theorem proving (ATP) or automated deduction, currently the most well-developed subfield of automated reasoning (AR), is the proving of mathematical theorems by a computer program. ...

Let an almost homomorphism be a map such that the set is finite. We say that two almost homomorphisms f,g are almost equal if the set is finite. This defines an equivalence relation on the set of almost homomorphisms. Real numbers are defined as the equivalence classes of this relation. To add real numbers defined this way we add the almost homomorphisms that represent them. Multiplication of real numbers corresponds to composition of almost homomorphisms. If [f] denotes the real number represented by an almost homomorphism f we say that if f is bounded or f takes an infinite number of positive values on . This defines the linear order relation on the set of real numbers constructed this way. In mathematics and set theory, a total order, linear order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ≤) on some set X. The relation is transitive, antisymmetric, and total. ...

See also

• Constructivism (mathematics)#Example from real analysis