nLab
real number

Contents

Idea

A real number is something that may be approximated by rational numbers. Equipped with the operations of addition and multiplication induced from the rational numbers, real numbers form a number field, 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 n\mathbb{R}^n for natural numbers nn \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.

History

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 00 and 11, 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.

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.

Definitions and characterizations

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.

Dedekind cuts

Consider two inhabited subsets, LL and UU, of \mathbb{Q} (the set of rational numbers) such that:

  • If aLa \in L, then bLb \in L for some b>ab \gt a.
  • If bUb \in U, then aUa \in U for some a<ba \lt b.
  • If a<ba \lt b are rational numbers, then aLa \in L or bUb \in U. (*)
  • If aLa \in L and bUb \in U, then a<ba \lt b.

We may define a Dedekind real number to be such a pair, which is also called a Dedekind cut.

If x(L,U)x \coloneqq (L,U) is a Dedekind cut, then we write a<xa \lt x to mean that aLa \in L and x<bx \lt b to mean that bUb \in U.

We may approximate a Dedekind cut xx 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 LL and UU are inhabited).

See Dedekind cut for more.

Cauchy sequences

Classically, a real number can be given by an infinite sequence of rational numbers, each of which is a decimal fraction that approximates the real number to a given number of decimal places. We can generalise this to any Cauchy sequence of rational numbers. However, now each real number has several representations, 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{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 for more.

The complete ordered field

There is a well-known algebraic (more or less) characterisation of the real line as the ‘complete ordered field’, or sometimes the ‘complete archimedean field’. This can be interpreted as follows:

  • A field is well known in algebra; if it matters, we mean a Heyting field.
  • An ordered field means a linearly ordered field.
  • An archimedean field is an ordered field in which every element is bounded above by a natural number, so it has no infinite? elements (and thus no non-zero infinitesimal elements).
  • An ordered field is complete if it is Dedekind-complete.
  • Alternatively, an archimedean field is complete if it is terminal in the category of archimedean fields.

We speak of the? such field because it is unique up to unique isomorphism.

Theorem

There is an archimedean field \mathbb{R} which is both Dedekind-complete and terminal among archimedean fields. Furthermore, every Dedekind-complete ordered field is isomorphic to \mathbb{R}. (By abstract nonsense, we already know that every terminal archimedean field is isomorphic to \mathbb{R} and that \mathbb{R} has only the identity automorphism, so isomorphisms to it are unique.)

Proof

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.

The locale of real numbers

Consider a binary relation \sim on rational numbers satisfying these four properties:

  • If aba \geq b, then aba \sim b.
  • If abcda \geq b \sim c \geq d, then ada \sim d.
  • If ab>cda \sim b \gt c \sim d, then ada \sim d.
  • If bcb \sim c whenever a<ba \lt b and c<dc \lt d, then ada \sim d.

These relations form a frame, which we may interpret (by definition) as the locale of real 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.

\mathbb{R} as a terminal coalgebra

The real line \mathbb{R}, or at least the positive real line +\mathbb{R}^+, may be characterized as the terminal coalgebra for an endofunctor

Let Pos be the category of posets. Consider the endofunctor

F 1:PosPos F_1\colon Pos \to Pos

that acts by ordinal product? with ω\omega

F 1:XXω. F_1\colon X \mapsto X \cdot \omega \,.
Proposition

The terminal coalgebra of F 1F_1 is order isomorphic to the non-negative real line +\mathbb{R}^+, with its standard order.

Proof

This is theorem 5.1 in

There are more and similar characterizations along these lines. One is an example at final coalgebra.

Topologies

There are alternative topologies on \mathbb{R} sometimes considered:

  • the discrete topology (not very interesting),
  • the order topology? or absolute-value topology? (the usual topology),
  • the upper-interval topology? or the lower-interval topology?,
  • the K-topology.

Another variant of \mathbb{R} as a topological space is the

Generalisations

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 sets \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 pp-adic numbers for various prime numbers pp are variations on the theme of real numbers; real numbers may be thought of as 00-adic numbers. Similarly, the real numbers are characteristic-00 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, 00-adic, characteristic-00, commutative numbers.

Lorentzian spacetime dimensionspin groupnormed division algebrabrane scan entry
3=2+13 = 2+1Spin(2,1)SL(2,)Spin(2,1) \simeq SL(2,\mathbb{R})\mathbb{R} the real numbers
4=3+14 = 3+1Spin(3,1)SL(2,)Spin(3,1) \simeq SL(2, \mathbb{C})\mathbb{C} the complex numbers
6=5+16 = 5+1Spin(5,1)SL(2,)Spin(5,1) \simeq SL(2, \mathbb{H})\mathbb{H} the quaternionslittle string
10=9+110 = 9+1Spin(9,1) somesenseSL(2,𝕆)Spin(9,1) \simeq_{some\,sense} SL(2,\mathbb{O})𝕆\mathbb{O} the octonionsheterotic/type II string

References

A formalization of the real numbers in homotopy type theory is in

For more see the references at analysis.

Revised on March 4, 2014 00:02:44 by Urs Schreiber (89.204.139.164)