In
mathematics, the
real
numbers include both
rational
numbers, such as 42 and −23/129, and
irrational numbers, such as
pi and the
square root of
two; or, a real number can be given by an infinite
decimal representation, such as
2.4871773339..., where the digits continue in some way; or, the
real numbers may be thought of as points on an infinitely long
number line.
These descriptions of the real numbers, while intuitively
accessible, are not sufficiently rigorous for the purposes of pure
mathematics. The discovery of a suitably rigorous definition of the
real numbers—indeed, the realization that a better definition was
needed—was one of the most important developments of 19th century
mathematics. Popular definitions in use today include
equivalence classes of
Cauchy sequences of rational numbers;
Dedekind cuts; a more sophisticated
version of "decimal representation"; and an axiomatic definition of
the real numbers as the unique
complete Archimedean ordered field. These definitions are all
described in detail below.
Basic properties
A real number may be either
rational
or
irrational; either
algebraic or
transcendental; and either
positive,
negative, or
zero.Real numbers are used to measure
continuous quantities. They may in
theory be expressed by
decimal
representations that have an infinite sequence of digits to the
right of the decimal point; these are often represented in the same
form as 324.823122147… The
ellipsis (three dots)
indicate that there would still be more digits to come.
More formally, real numbers have the two basic properties of being
an
ordered field, and having the
least upper bound property.
The first says that real numbers comprise a
field, with addition and multiplication
as well as division by nonzero numbers, which can be
totally ordered on a number line in a way
compatible with addition and multiplication. The second says that
if a nonempty set of real numbers has an
upper bound, then it has a
least upper bound. These two together define the
real numbers completely, and allow its other properties to be
deduced. For instance, we can prove from these properties that
every polynomial of odd degree with real coefficients has a real
root, and that if you add the square root of −1 to the real
numbers, obtaining the
complex
numbers, the resulting field is
algebraically closed.
Uses
In the physical sciences, most of the physical constants such as
the universal gravitational constant, and physical variables, such
as position, mass, speed, and electric charge, are modeled using
real numbers. Note importantly, however, that all actual
measurements of physical quantities yield
rational numbers because the precision of
such measurements can only be finite.
Computers cannot directly operate on real numbers, but only on a
finite subset of rational numbers, limited by the number of bits
used to store them. However,
computer algebra systems are able to
treat some
irrational numbers
exactly by storing their algebraic description (such as "sqrt(2)")
rather than their rational approximation.
A real number is said to be
computable if there exists an
algorithm that yields its digits. Because there are only
countably many algorithms, but an
uncountable number of reals,
almost all
real numbers are not computable. Some
constructivists accept the
existence of only those reals that are computable. The set of
definable numbers is broader, but
still only countable. If computers could use unlimited precision
real numbers (
real computation),
then one could solve
NPcomplete
problems, and even
#Pcomplete problems in
polynomial time, answering
affirmatively the
P = NP problem.
Unlimited precision real numbers in the physical universe are
prohibited by the
holographic
principle and the
Bekenstein
bound.
Mathematicians use the symbol
R (or alternatively,
\mathbb{R} , the letter "
R" in
blackboard bold, Unicode ℝ) to represent the
set of all real numbers. The
notation
R^{n} refers to an
n
dimensional space with real
coordinates; for example, a value from
R^{3} consists of three real numbers and
specifies a location in 3dimensional space.
In mathematics, real is used as an adjective, meaning that the
underlying field is the field of real numbers. For example
real
matrix,
real polynomial and
real Lie algebra. As a substantive, the term is
used almost strictly in reference to the real numbers themselves
(e.g., The "set of all reals").
History
Vulgar fractions had been used by
the
Egyptians around 1000 BC; the
Vedic "
Sulba Sutras" ("rule of chords" in, ca. 600 BC,
include what may be the first 'use' of
irrational numbers. The concept of
irrationality was implicitly accepted by early
Indian mathematicians since
Manava (c. 750–690 BC), who was aware that the
square roots of certain numbers such as
2 and 61 could not be exactly determined. Around 500 BC, the
Greek mathematicians led by
Pythagoras realized the need for
irrational numbers, in particular the irrationality of the
square root of 2.
The
Middle Ages saw the acceptance of
zero,
negative,
integral and
fractional numbers, first by Indian
and
Chinese mathematicians, and
then by
Arabic
mathematicians, who were also the first to treat irrational
numbers as algebraic objects, which was made possible by the
development of
algebra. Arabic
mathematicians merged the concepts of "
number" and "
magnitude" into a more general idea
of real numbers.
The Egyptian
mathematician Abū Kāmil Shujā ibn
Aslam (c. 850–930) was the first to accept irrational
numbers as solutions to
quadratic
equations or as
coefficients in an
equation, often in the form of square
roots,
cube roots and
fourth roots.
In the 18th and 19th centuries there was much work on irrational
and
transcendental numbers.
Lambert (1761) gave the
first flawed proof that π cannot be rational;
Legendre (1794) completed the proof, and showed
that π is not the square root of a rational number.
Ruffini (1799) and
Abel (1842) both constructed proofs of
Abel–Ruffini theorem:
that the general
quintic or higher
equations cannot be solved by a general formula involving only
arithmetical operations and roots.
Évariste Galois (1832)
developed techniques for determining whether a given equation could
be solved by radicals which gave rise to the field of
Galois theory.
Joseph Liouville (1840) showed that neither
e nor
e^{2} can be a root of an integer
quadratic equation, and then
established existence of transcendental numbers, the proof being
subsequently displaced by Georg Cantor (1873).
Charles Hermite (1873) first proved that
e is
transcendental, and
Ferdinand
von Lindemann (1882), showed that π is transcendental.
Lindemann's proof was much simplified by Weierstrass (1885), still
further by
David Hilbert (1893), and
has finally been made elementary by
Hurwitz and
Paul
Albert Gordan.
The development of
calculus in the 1700s
used the entire set of real numbers without having defined them
cleanly. The first rigorous definition was given by
Georg Cantor in 1871. In 1874 he showed that
the set of all real numbers is
uncountably
infinite but the set of all
algebraic numbers is
countably infinite. Contrary to widely held
beliefs, his first method was not his famous
diagonal argument, which he
published in 1891. See
Cantor's first
uncountability proof.
Definition
Construction from the rational numbers
The real numbers can be constructed as a completion of the rational
numbers in such a way that a sequence defined by a decimal or
binary expansion like {3, 3.1, 3.14, 3.141, 3.1415,...}
converges to a unique real number. For
details and other constructions of real numbers, see
construction of the real
numbers.
Axiomatic approach
Let
R denote the
set of all real numbers. Then:
The last property is what differentiates the reals from the
rationals. For example, the set of
rationals with square less than 2 has a rational upper bound (e.g.,
1.5) but no rational least upper bound, because the
square root of 2 is not rational.
The real numbers are uniquely specified by the above properties.
More precisely, given any two Dedekindcomplete ordered fields
R_{1} and
R_{2},
there exists a unique field
isomorphism
from
R_{1} to
R_{2}, allowing us to think of them as
essentially the same mathematical object.
For another axiomatization of
R, see
Tarski's axiomatization of
the reals.
Properties
Completeness
The main reason for introducing the reals is that the reals contain
all
limits. More technically,
the reals are
complete (in
the sense of
metric spaces or
uniform spaces, which is a different sense
than the Dedekind completeness of the order in the previous
section). This means the following:
A
sequence
(
x_{n}) of real numbers is called a
Cauchy sequence if for any
ε > 0 there exists an integer
N (possibly
depending on ε) such that the
distance

x_{n} −
x_{m}
is less than ε for all
n and
m that are both
greater than
N. In other words, a sequence is a
Cauchy sequence if its elements
x_{n} eventually come and remain
arbitrarily close to each other.
A sequence (
x_{n})
converges to the
limit x if for any ε > 0 there exists an
integer
N (possibly depending on ε) such that the distance

x_{n} −
x is less
than ε provided that
n is greater than
N. In
other words, a sequence has limit
x if its elements
eventually come and remain arbitrarily close to
x.
It is easy to see that every convergent sequence is a Cauchy
sequence. An important fact about the real numbers is that the
converse is also true :
 Every Cauchy sequence of real numbers is
convergent.
That is, the reals are complete.
Note that the rationals are not complete. For example, the sequence
(1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ...) is Cauchy but it does
not converge to a rational number. (In the real numbers, in
contrast, it converges to the positive
square root of 2.)
The existence of limits of Cauchy sequences is what makes
calculus work and is of great practical use. The
standard numerical test to determine if a sequence has a limit is
to test if it is a Cauchy sequence, as the limit is typically not
known in advance.
For example, the standard series of the
exponential function
 \mathrm{e}^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}
converges to a real number because for every
x the
sums
 \sum_{n=N}^{M} \frac{x^n}{n!}
can be made arbitrarily small by choosing
N sufficiently
large. This proves that the sequence is Cauchy, so we know that the
sequence converges even if the limit is not known in advance.
"The complete ordered field"
The real numbers are often described as "the complete ordered
field", a phrase that can be interpreted in several ways.
First, an order can be
latticecomplete. It is easy to see that no
ordered field can be latticecomplete, because it can have no
largest element (given any element
z,
z + 1 is larger), so this is not the sense that
is meant.
Additionally, an order can be
Dedekindcomplete, as defined in the
section
Axioms. The uniqueness result at the end
of that section justifies using the word "the" in the phrase
"complete ordered field" when this is the sense of "complete" that
is meant. This sense of completeness is most closely related to the
construction of the reals from Dedekind cuts, since that
construction starts from an ordered field (the rationals) and then
forms the Dedekindcompletion of it in a standard way.
These two notions of completeness ignore the field structure.
However, an
ordered group (in this
case, the additive group of the field) defines a
uniform structure, and uniform structures have
a notion of
completeness ;
the description in the section
Completeness above
is a special case. (We refer to the notion of completeness in
uniform spaces rather than the related and better known notion for
metric spaces, since the definition of
metric space relies on already having a characterisation of the
real numbers.) It is not true that
R is the
only uniformly complete ordered field, but it is the only
uniformly complete
Archimedean
field, and indeed one often hears the phrase "complete
Archimedean field" instead of "complete ordered field". Since it
can be proved that any uniformly complete Archimedean field must
also be Dedekindcomplete (and vice versa, of course), this
justifies using "the" in the phrase "the complete Archimedean
field". This sense of completeness is most closely related to the
construction of the reals from Cauchy sequences (the construction
carried out in full in this article), since it starts with an
Archimedean field (the rationals) and forms the uniform completion
of it in a standard way.
But the original use of the phrase "complete Archimedean field" was
by
David Hilbert, who meant still
something else by it. He meant that the real numbers form the
largest Archimedean field in the sense that every other
Archimedean field is a subfield of
R. Thus
R is "complete" in the sense that nothing further
can be added to it without making it no longer an Archimedean
field. This sense of completeness is most closely related to the
construction of the reals from
surreal
numbers, since that construction starts with a proper class
that contains every ordered field (the surreals) and then selects
from it the largest Archimedean subfield.
Advanced properties
The reals are
uncountable; that is,
there are strictly more real numbers than
natural numbers, even though both sets are
infinite. In fact, the
cardinality of the reals equals
that of the set of subsets (i.e., the power set) of the natural
numbers, and
Cantor's
diagonal argument states that the latter set's cardinality is
strictly bigger than the cardinality of
N. Since
only a countable set of real numbers can be
algebraic,
almost
all real numbers are
transcendental. The nonexistence of a
subset of the reals with cardinality strictly between that of the
integers and the reals is known as the
continuum hypothesis. The continuum
hypothesis can neither be proved nor be disproved; it is
independent from the
axioms of set theory.
The real numbers form a
metric space:
the distance between
x and
y is defined to be the
absolute value

x −
y. By virtue of being a
totally ordered set, they also carry an
order topology; the
topology arising from the metric and the one
arising from the order are identical. The reals are a
contractible (hence
connected and
simply connected),
separable metric space of
dimension 1, and are
everywhere dense. The real numbers are
locally compact but not
compact. There are various properties that
uniquely specify them; for instance, all unbounded, connected, and
separable
order topologies are
necessarily
homeomorphic to the
reals.
Every nonnegative real number has a
square
root in
R, and no negative number does. This
shows that the order on
R is determined by its
algebraic structure. Also, every polynomial of odd degree admits at
least one real root: these two properties make
R
the premier example of a
real closed
field. Proving this is the first half of one proof of the
fundamental theorem of
algebra.
The reals carry a canonical
measure, the
Lebesgue measure, which is the
Haar measure on their structure as a
topological group normalised such that the
unit interval [0,1] has measure
1.
The supremum axiom of the reals refers to subsets of the reals and
is therefore a secondorder logical statement. It is not possible
to characterize the reals with
firstorder logic alone: the
LöwenheimSkolem theorem
implies that there exists a countable dense subset of the real
numbers satisfying exactly the same sentences in first order logic
as the real numbers themselves. The set of
hyperreal numbers satisfies the same first
order sentences as
R. Ordered fields that satisfy
the same firstorder sentences as
R are called
nonstandard models of
R. This is what makes
nonstandard analysis work; by proving
a firstorder statement in some nonstandard model (which may be
easier than proving it in
R), we know that the
same statement must also be true of
R.
Generalizations and extensions
The real numbers can be generalized and extended in several
different directions:
 The complex numbers contain
solutions to all polynomial equations and
hence are an algebraically
closed field unlike the real numbers. However, the complex
numbers are not an ordered field.
 The affinely
extended real number system adds two elements +∞ and −∞. It is
a compact space. It is no longer a
field, not even an additive group; it still has a total order; moreover, it is a complete lattice.
 The real projective line
adds only one value ∞. It is also a compact space. Again, it is no
longer a field, not even an additive group. However, it allows
division of a nonzero element by zero. It is not ordered
anymore.
 The long real line pastes
together ℵ_{1}* + ℵ_{1} copies of the real line
plus a single point (here ℵ_{1}* denotes the reversed
ordering of ℵ_{1}) to create an ordered set that is
"locally" identical to the real numbers, but somehow longer; for
instance, there is an orderpreserving embedding of ℵ_{1}
in the long real line but not in the real numbers. The long real
line is the largest ordered set that is complete and locally
Archimedean. As with the previous two examples, this set is no
longer a field or additive group.
 Ordered fields extending the reals are the hyperreal numbers and the surreal numbers; both of them contain
infinitesimal and infinitely large
numbers and thus are not Archimedean.
 Selfadjoint operators on a Hilbert space (for example, selfadjoint
square complex matrices) generalize
the reals in many respects: they can be ordered (though not totally
ordered), they are complete, all their eigenvalues are real and they form a real
associative algebra. Positivedefinite operators correspond to
the positive reals and normal
operators correspond to the complex numbers.
"Reals" in set theory
In
set theory, specifically
descriptive set theory the
Baire space is used as a surrogate
for the real numbers since the latter have some topological
properties (connectedness) that are a technical inconvenience.
Elements of Baire space are referred to as "reals".
See also
Notes
 Scott
Aaronson, NPcomplete Problems and Physical Reality,
ACM SIGACT News, Vol. 36,
No. 1. (March 2005), pp. 3052.
 T. K. Puttaswamy, "The Accomplishments of Ancient Indian
Mathematicians", pp. 410–1, in
 Jacques Sesiano, "Islamic mathematics", p. 148, in
References
 Georg Cantor, 1874, "Über eine
Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen",
Journal für die Reine und Angewandte Mathematik, volume
77, pages 258262.
 Robert Katz, 1964, Axiomatic Analysis, D. C. Heath and
Company.
 Edmund Landau, 2001, ISBN
082182693X, Foundations of Analysis, American Mathematical
Society.
 Howie, John M., Real Analysis, Springer, 2005, ISBN
1852333146
External links