transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
…
…
A real number is a number that may be approximated by rational numbers. Equipped with the operations of addition and multiplication induced from the rational numbers, real numbers form a field, commonly denoted $\mathbb{R}$. The underlying set is the completion of the ordered field $\mathbb{Q}$ of rational numbers: the result of adjoining to $\mathbb{Q}$ suprema for every inhabited bounded subset with respect to the natural ordering of rational numbers.
The set of real numbers also carries naturally the structure of a topological space and as such $\mathbb{R}$ is called the real line also known as the continuum. Equipped with both the topology and the field structure, $\mathbb{R}$ is a topological field and as such is the uniform completion of $\mathbb{Q}$ equipped with the absolute value metric.
Together with its cartesian products – the Cartesian spaces $\mathbb{R}^n$ for natural numbers $n \in \mathbb{N}$ – the real line $\mathbb{R}$ is a standard formalization of the idea of continuous space. The more general concept of (smooth) manifold is modeled on these Cartesian spaces. These, in turn are standard models for the notion of space in particular in physics (see spacetime), or at least in classical physics. See at geometry of physics for more on this.
The original idea of a real number came from geometry; one thinks of a real number as specifying a point on a line, with line understood as the abstract idea of the object that a pencil and a ruler draw on a piece of paper. (More precisely, given two distinct points on the line, called $0$ and $1$, you get a bijection between the points and the real numbers.)
Euclid (citing Eudoxus) dealt with ratios of geometric magnitudes, which give positive real numbers; an arbitrary real number is then a difference of ratios of magnitudes. However, the Greeks did not think of such ratios as numbers; that appears to have been an insight of the Arabs. See more at Eudoxus real number.
A big project of the 19th century (at least in hindsight) was the ‘arithmetisation of analysis’: showing how real numbers could be defined completely in terms of rational numbers (and the desired classes of functions on them could be defined in terms of the general point-set notion of function). Two successful approaches were developed in 1872, Richard Dedekind's definition of real numbers as certain sets of rational numbers (called Dedekind cuts) and Georg Cantor's definition as certain sequences of rational numbers (called Cauchy sequences).
A more modern approach is instead to characterise the properties that the set of real numbers must have and to prove that this is categorical (unique up to a unique bijection preserving those properties). Then the important result of the 19th-century programme is simply that this is consistent (that there exists at least one such set). One can even use Hilbert's or Tarski's axioms for geometry to do this characterisation, coming full circle back to geometry.
Exactly how to define or characterise real numbers is still important in constructive mathematics and topos theory with its internal logic. For more on this, see real numbers object and the examples below.
There are two basic approaches possible: to define what a real number is as a mathematical object, or to define the real line as a specific object in some previously known category.
There is a well-known algebraic (more or less) characterisation of the real line as the ‘terminal Archimedean ordered field’ (cf. Richman08) This can be interpreted as follows:
We speak of the such field because it is unique up to unique isomorphism by definition of terminal object.
In predicative mathematics, one can define $\mathbf{R}$ by fiat via the universal property of the real numbers as the terminal Archimedean ordered field. From this characterization, the real numbers are a terminal coalgebra of the identity endofunctor $X \mapsto X$ on the category of Archimedean ordered fields.
If quotient sets exist and sequence algebras of Archimedean ordered fields exist, then it is provable that $\mathbf{R}$ is Cauchy complete. From the definition of terminal object, $\mathbf{R}$ is an algebra of the endofunctor $X \mapsto \mathcal{C}(X)$ in the category of Archimedean ordered fields which takes Archimedean ordered fields $X$ to the Archimedean ordered field $\mathcal{C}(X)$ of equivalence classes of Cauchy sequences in $X$. Every algebra of the endofunctor $X \mapsto \mathcal{C}(X)$ in the category of Archimedean ordered fields is a Cauchy complete Archimedean ordered field.
Similarly, if power sets of Archimedean ordered fields exist, then it is provable that $\mathbf{R}$ is Dedekind complete. From the definition of terminal object, $\mathbf{R}$ is an algebra of the endofunctor $X \mapsto \mathcal{D}(X)$ in the category of Archimedean ordered fields which takes Archimedean ordered fields $X$ to the Archimedean ordered field $\mathcal{C}(X)$ of two-sided Dedekind cuts in $X$. Every algebra of the endofunctor $X \mapsto \mathcal{D}(X)$ in the category of Archimedean ordered fields is a Dedekind complete Archimedean ordered field.
Consider two inhabited subsets, $L$ and $U$, of a countable unbounded dense linear order, such as $\mathbb{Q}$ (the set of rational numbers) or $\mathbb{Z}[1/10]$ (the set of decimal fractions), such that:
We may define a Dedekind real number to be such a pair, which is also called a Dedekind cut.
If $x \coloneqq (L,U)$ is a Dedekind cut, then we write $a \lt x$ to mean that $a \in L$ and $x \lt b$ to mean that $b \in U$.
We may approximate a Dedekind cut $x$ as closely as we like by applying (*) as often as necessary. This will be only finitely often, for any fixed positive level of approximation, given initial upper and lower bounds (which exist since $L$ and $U$ are inhabited).
See Dedekind completion for more.
Classically, a real number can be given by an infinite Cauchy sequence of decimal fractions $\mathbb{Z}[1/10]$, each of which is a decimal fraction that approximates the real number to a given number of decimal places. However, many real numbers have several representations, i.e.
so we need to specify an equivalence relation on the Cauchy sequences. Thus, $\mathbb{R}$ is constructed as a subquotient of the function set $\mathbb{Z}[1/10]^{\mathbb{N}}$.
We can generalise this to any Cauchy sequence of rational numbers, and $\mathbb{R}$ is constructed as a subquotient of the function set $\mathbb{Q}^{\mathbb{N}}$.
This construction is equivalent to the construction by Dedekind cuts, at least assuming weak countable choice (which also follows from excluded middle). Thus it is popular in both classical mathematics and traditional constructive mathematics (which accepts countable choice). However, in stricter forms of constructive mathematics, including those used as internal languages in topos theory, the Cauchy reals and Dedekind reals are not equivalent. (On the other hand, by generalising to Cauchy nets, we recover the Dedekind reals again.)
See Cauchy real number and generalized Cauchy real number for more.
There is an algebraic (more or less) characterisation of the real line as the ‘initial sequentially modulated Cauchy complete archimedean field’. This can be interpreted as follows:
The initial sequentially modulated Cauchy complete ordered field results in the HoTT book real numbers.
There is a well-known algebraic (more or less) characterisation of the real line as the ‘Dedekind complete ordered field’, or sometimes the ‘Dedekind complete archimedean field’. This can be interpreted as follows:
In impredicative mathematics, we speak of the such field because it is unique up to unique isomorphism.
Assuming impredicative foundations, there is an archimedean field $\mathbb{R}$ which is Dedekind-complete, and into which every archimedean field embeds. Furthermore, every Dedekind-complete ordered field is isomorphic to $\mathbb{R}$, and uniquely so.
Construct $\mathbb{R}$ using, say, Dedekind cuts of rational numbers. Then it is well known how to prove these facts about $\mathbb{R}$, so we omit the proof for now.
However, we note that the proof is valid in weak foundations, in particular internal to any topos with a natural numbers object. One can actually work in even weaker foundations than that; see the constructions at real numbers object.
Even weaker foundations are possible if one allows the underlying set of $\mathbb{R}$ to be large.
However, if we are working in predicative mathematics with multiple universes in the foundation, the Dedekind real numbers would no longer be unique up to unique isomorphism, but rather there would be a set of Dedekind real numbers for each universe.
There is a well-known algebraic (more or less) characterisation of the real line as the ‘Cauchy complete ordered field’, or sometimes the ‘Cauchy complete archimedean field’. This can be interpreted as follows:
In impredicative mathematics, we speak of the such field because it is unique up to unique isomorphism.
Assuming impredicative foundations, there is an archimedean field $\mathbb{R}$ which is Cauchy-complete, and into which every archimedean field embeds. Furthermore, every Cauchy-complete ordered field is isomorphic to $\mathbb{R}$, and uniquely so.
Construct $\mathbb{R}$ using, say, Cauchy nets of rational numbers. Then it is well known how to prove these facts about $\mathbb{R}$, so we omit the proof for now.
However, we note that the proof is valid in weak foundations, in particular internal to any topos with a natural numbers object. One can actually work in even weaker foundations than that; see the constructions at real numbers object.
Even weaker foundations are possible if one allows the underlying set of $\mathbb{R}$ to be large.
However, if we are working in predicative mathematics with multiple universes in the foundation, the Cauchy real numbers would no longer be unique up to unique isomorphism, but rather there would be a set of Cauchy real numbers for each universe.
There is a characterisation of the real line as the ‘complete archimedean Tarski group’ due to Alfred Tarski. This can be interpreted as follows:
See Tarski's axiomatization of the real numbers for more information.
Consider binary relations $\sim$ on a countable inhabited dense linear order without endpoints, such as the rational numbers, satisfying these four properties:
The collection of all such relations form a frame, which we may interpret (by definition) as the locale of real numbers. It can also be defined as the localic completion of the rational numbers.
We may then define a localic real number to be a point of this locale.
This agrees with the notion of Dedekind real number, even in very weak (predicative and constructive) foundations.
See locale of real numbers for more.
The unit interval of the real numbers $[0,1]$ could be constructed as a terminal coalgebra of an endofunctor in the category of intervals. Let $(\mathbb{R},0,+,-,1,\cdot,\lt)$ be an ordered field where $0 \lt 1$, with a monotone $f:[0,1]\to \mathbb{R}$ such that $f(0) = 0$ and $f(1) = 1$. The set $\mathbb{R}$ of real numbers is the initial such ordered field.
See also: dyadic interval coalgebra, decimal interval coalgebra, rational interval coalgebra.
The positive real line $\mathbb{R}^+$ may be characterized as the terminal coalgebra for an endofunctor
Let Pos be the category of posets with a forgetful functor
Consider the endofunctor
defined as the ordinal product?
for $\omega \in Pos$, where $\omega \cdot X$ is the cartesian product $U(\omega) \times U(X)$ with the lexicographic order.
The terminal coalgebra of $F_1$ is order isomorphic to the non-negative real line $\mathbb{R}^+$, with its standard order.
This is theorem 5.1 in (Pavlovic–Pratt 1999).
There are many ways of setting up this description of $\mathbb{R}^+$, depending on the coalgebra structure $\mathbb{R}^+ \to \omega \cdot \mathbb{R}^+$ chosen. Here is one: there are evident poset isomorphisms $\mathbb{R}^+ \cong [1, \infty)$ and $\omega \cong \mathbb{N}_{\geq 2} = \{n \in \mathbb{N}: n \geq 2\}$. Define a map
where $\alpha(x)$ is the smallest integer strictly greater than $x$, and $\beta(x) = 1/(\alpha(x) - x)$. The stream of integers $a_n = \alpha(\beta^n(x))$ gives a continued fraction representation of $x$ in the form
and the resulting bijection $[1, \infty) \to \mathbb{N}_{\geq 2} \times \mathbb{N}_{\geq 2} \times \ldots$, sending $x$ to $(a_0, a_1, \ldots)$, is in fact a poset isomorphism if we endow the right-hand side with the lexicographic order.
Another way, which circumvents the use of isomorphisms $\mathbb{R}^+ \cong [1, \infty)$ and $\omega \cong \mathbb{N}_{\geq 2}$, is to define
where $\alpha(x)$ is the floor of $x$, and $\beta(x) = 1/(1 - x + \alpha(x)) - 1$. Then $a_n = \alpha(\beta^n(x))$ gives a continued fraction representation of $x$ in the form
and the resulting bijection $\mathbb{R}_+ \to \omega \times \omega \times \ldots$, sending $x$ to $(a_0, a_1, \ldots)$, is again a poset isomorphism if we endow the right-hand side with the lexicographic order.
There are more and similar characterizations along these lines. An interesting application of the coalgebraic views is that they allow defining morphisms between the reals. A coalgebaic version Dedekind cuts or Conway numbers makes $\mathbb{R}$ into a compact category where the addition arises as the biproduct. The real vector spaces are the $\mathbb{R}$-enriched bicompletions and linear operators arise as the $\mathbb{R}$-enriched Kan extensions. (Pavlovic 2020)
The smooth real numbers (say in a smooth topos) are only an ordered local ring rather than an ordered field, because there might be non-zero non-invertible nilpotent infinitesimals in the smooth real numbers. Nevertheless, the quotient of the smooth real numbers by the ideal of non-invertible elements is one of the ordered field of real numbers as defined above.
There are alternative topologies on $\mathbb{R}$ sometimes considered:
Another variant of $\mathbb{R}$ as a topological space is the
The term ‘real number’ was originally introduced to indicate that one is not considering the generalistion to complex numbers or other kinds of hypercomplex numbers. Accordingly, that term ‘real’ may sometimes be used for another generalisation of real numbers to indicate again that one is not considering a complexification.
The extended real numbers include $\pm\infty$ as well as the real numbers; one may speak of finite numbers or bounded numbers to indicate that one is not considering this extension. Lower reals, upper reals, and MacNeille reals are related generalisations studied in constructive mathematics, although with excluded middle they are (at least if bounded) the same as ordinary real numbers; one may speak of located numbers to indicate that one is not considering such extensions.
Surreal numbers and the hyperreal numbers of nonstandard analysis are two ways to include infinite and infinitesimal versions of real numbers (besides the trivial case of $\pm\infty$); one may speak of standard numbers to indicate that one is not considering such extensions (although the precise meaning of ‘standard’ depends on the universe that one is working in).
In descriptive set theory, one often says ‘real number’ for an element of Baire space $\mathbb{N}^{\mathbb{N}}$. This is not really a generalisation; by the Schroeder-Bernstein theorem, the underlying sets of $\mathbb{R}$ and $\mathbb{N}^{\mathbb{N}}$ are isomorphic. Constructively, $\mathbb{N}^{\mathbb{N}}$ can still be thought of as the set of irrational numbers, so this use of the term may actually be a restriction.
Floating-point numbers? are often used in computer programming to represent real numbers, but they do not behave very well; one may speak of infinite-precision numbers to indicate that one's programming environment models ‘real real numbers’.
As mentioned above, the $p$-adic numbers for various prime numbers $p$ are variations on the theme of real numbers; real numbers may be thought of as $0$-adic numbers. Similarly, the real numbers are characteristic-$0$ numbers since they are based on the prime field $\mathbb{Q}$; one could also start the construction with a different characteristic (although it makes more sense to get analogues of complex numbers than of real numbers).
Finally, one can consider points on a noncommutative line instead of the usual commutative numbers.
So in summary, this page is about the real, finite, located, standard, analytic, infinite-precision, $0$-adic, characteristic-$0$, commutative numbers.
in constructive analysis one may use the completion monad for dealing with real numbers
exceptional spinors and real normed division algebras
Lorentzian spacetime dimension | $\phantom{AA}$spin group | normed division algebra | $\,\,$ brane scan entry |
---|---|---|---|
$3 = 2+1$ | $Spin(2,1) \simeq SL(2,\mathbb{R})$ | $\phantom{A}$ $\mathbb{R}$ the real numbers | super 1-brane in 3d |
$4 = 3+1$ | $Spin(3,1) \simeq SL(2, \mathbb{C})$ | $\phantom{A}$ $\mathbb{C}$ the complex numbers | super 2-brane in 4d |
$6 = 5+1$ | $Spin(5,1) \simeq$ SL(2,H) | $\phantom{A}$ $\mathbb{H}$ the quaternions | little string |
$10 = 9+1$ | Spin(9,1) ${\simeq}$ “SL(2,O)” | $\phantom{A}$ $\mathbb{O}$ the octonions | heterotic/type II string |
For more see the references at analysis.
Review and history:
As the coalgebra of the real interval:
A definition of the real numbers as the terminal Archimedean ordered field and as a complete Archimedean ordered field:
A compact category $\mathbb{R}$ making the $\mathbb{R}$-linear operators into $\mathbb{R}$-enriched Kan extensions:
and monoidal computers_, Logic and Structure in Computer Science 1000-1070, doi:10.1007/978-3-031-24117-8, arXiv:2007.10057
The definition of the real numbers in constructive analysis as Cauchy real numbers, namely as regular Cauchy sequences of rational numbers is due to:
Errett Bishop: Foundations of Constructive Analysis, McGraw-Hill (1967)
Errett Bishop, Douglas Bridges: Constructive Analysis, Grundlehren der mathematischen Wissenschaften 279, Springer (1985) [doi:10.1007/978-3-642-61667-9]
Direct formalization of the definition of Cauchy real numbers from Bishop (1967) in Agda:
Martin Lundfall, Formalizing real numbers in Agda (2015) [pdf, pdf, github]
Zachary Murray, Constructive Analysis in the Agda Proof Assistant [arXiv:2205.08354, github]
review:
and closely related constructions in Coq have been implemented in
following a monadic re-formulation (via the completion monad) due to
Overview of implementation of real numbers in proof assistants (cf. constructive analysis):
A novel construction principle of the type of real numbers, as a higher inductive-inductive type in univalent homotopy type theory, not reliant on representatives by sequences of rational numbers, and with provable Cauchy completeness even without extra axiom of countable choice is laid out (cf. HoTT book real numbers) in
Something like this has been implemented in Coq:
Last revised on January 28, 2024 at 04:12:30. See the history of this page for a list of all contributions to it.