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Construction of the real numbers
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In mathematics,
there are several 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.
[edit] 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.
1. (R,
+, *) forms a field. In other words,
·
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)
·
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,
·
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)
·
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.
When we say
that any two models of the above axioms are isomorphic, we mean that for any
two models (R, 0R, 1R, +R,
*R, ≤R) and (S, 0S,
1S, +S, *S, ≤S),
there is a bijection
f : R → S preserving both the field operations and the
order. Explicitly,
- f is both injective
and surjective.
- f(0R)
= 0S and f(1R) = 1S.
- 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).
[edit] 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/Charles
Méray, Richard Dedekind and Karl
Weierstrass/Otto Stolz 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.
[edit] Construction
from Cauchy sequences
If we have a
space where Cauchy sequences are meaningful (such as a
`rational' metric space, i.e., a space in which distance is
defined and takes rational values, 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.)
Let R be
the set of Cauchy sequences of rational numbers. That
is, sequences
x1,x2,x3,...
of rational numbers such that for every rational ε > 0, there exists an
integer N such that for all natural numbers m,n > N,
|xm-xn|<ε. Here the vertical bars denote
the absolute value.
Cauchy
sequences (x) and (y) can be added and multiplied as follows:
(xn) + (yn)
= (xn + yn)
(xn) × (yn)
= (xn × yn).
Two Cauchy
sequences are called equivalent if and only if the difference between
them tends to zero.
Comparison
between two cauchy sequences is possible as such : (xn) ≥ (yn)
if and only if x is equivalent to y or there exists an integer N
such that xn ≥ yn for all n > N.
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.
This is remarkable because not all of these axioms necessarily apply to the
rational numbers, which are being used to construct the sequences themselves.
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, …).
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, there is a
rational number L such that L < s for some s in S.
Now define sequences of rationals (un) and (ln)
as follows:
Set u0 = U and
l0 = L.
For each n
consider the number:
mn = (un + ln)/2
If mn
is an upper bound for S set:
un+1 = mn
and ln+1 = ln
Otherwise set:
ln+1 = mn
and un+1 = un
This obviously
defines two Cauchy sequences of rationals, and so we have real numbers l
= (ln) and u = (un). It is easy to
prove, by induction on n that:
un is an upper bound for S for all
n
and:
ln 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 (un − ln) is 0, and so l
= u. Now suppose b < u = l is a smaller upper
bound for S. Since (ln) is monotonic increasing it is
easy to see that b < ln for some n. But ln
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.
An advantage of
this approach is that it does not use the linear order of the rationals, only
the metric. Hence it generalizes to other metric spaces.
[edit] 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.
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:[1]
-
is not empty
-
- r is closed
downwards. In other words, for all
such that
, if
then
- r 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
the rational numbers into the reals by identifying the rational number
with the set of all smaller
rational numbers
.[1]
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.
- Negation
is a special case of subtraction:
- Defining multiplication
is less straightforward.[1]
- 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.
If a nonempty set
of real numbers has any upper
bound in
, then it
has a least upper bound in
that is
equal to
.[1]
As an example
of a Dedekind cut representing an irrational
number, we may take the positive
square root of 2. This can be defined by the set
.[2]
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 for any
positive rational
with
, there is a
rational
with
and
The choice
works. Then
but to show
equality requires showing that if
is any rational number less than 2,
then there is positive
in
with
.
An advantage of
this construction is that each real number corresponds to a unique cut.
[edit] Stevin's construction
It has been
known since Simon Stevin[3]
that real numbers can be represented by decimals. We can take the infinite
decimal expansion to be the definition of a real number, defining
expansions like 0.9999... and 1.0000... to be equivalent, and define the
arithmetical operations formally. This is equivalent to the constructions by
Cauchy sequences or Dedekind cuts and incorporates an explicit modulus of convergence. Similarly, another radix can be used.
Weierstrass attempted to construct the reals but did not entirely succeed. He
pointed out that they need only be thought of as complete aggregates (sets) of
units and unit fractions.[4]
This
construction has the advantage that it is close to the way we are used to
thinking of real numbers and suggests series expansions for functions. A
standard approach to show that all models of a complete ordered field are
isomorphic is to show that any model is isomorphic to this one because we can
systematically build a decimal expansion for each element.
[edit] Construction
using hyperreal numbers
As in the hyperreal
numbers, one constructs the hyperrationals *Q from the
rational numbers by means of an ultrafilter.
Here a hyperrational is by definition a ratio of two hyperintegers.
Consider the ring B of all limited (i.e. finite)
elements in *Q. Then B has a unique maximal
ideal I, the infinitesimal numbers. The quotient ring B/I
gives the field R of real numbers. Note that B
is not an internal set in *Q. Note that this
construction uses a non-principal ultrafilter over the set of natural numbers,
the existence of which is guaranteed by the axiom
of choice.
It turns out
that the maximal ideal respects the order on *Q. Hence the
resulting field is an ordered field. Completeness can be proved in a similar
way to the construction from the Cauchy sequences.
[edit] Construction
from surreal numbers
Every ordered
field can be embedded in the surreal
numbers. The real numbers form a maximal subfield that is Archimedean
(meaning that no real number is infinitely large). This embedding is not
unique, though it can be chosen in a canonical way.
[edit] Construction
from Z (Eudoxus reals)
A relatively
less known construction allows to define real numbers using only the additive
group of integers
with different versions.[5][6][7]
The construction has been formally verified by the IsarMathLib
project.[8]
Shenitzer[9]
and Arthan refer to this construction as the Eudoxus reals.
Let an almost
homomorphism be a map
such that the set
is finite. We
say that two almost homomorphisms
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
denotes the real number represented by
an almost homomorphism
we say that
if
is bounded or
takes an infinite number of positive
values on
. This defines the linear
order relation on the set of real numbers constructed this way.
[edit] Other constructions
|
“
|
Few mathematical structures have
undergone as many revisions or have been presented in as many guises as the
real numbers. Every generation reexamines the reals in the light of its
values and mathematical objectives.[10]
|
”
|
As a reviewer
of one noted: "The details are all included, but as usual they are tedious
and not too instructive."[16]
4.
^ V. Dantscher, "Vorlesungen über
die Weierstrass'sche Theorie der irrationalen Zahlen" , Teubner (1908)
9.
^ Shenitzer, A. (1987) A topics course
in mathematics. The Mathematical Intelligencer 9, no. 3, 44--52.
11.
^ N.G. de Bruijn, N. G. Construction of
the system of real numbers. (Dutch) Nederl. Akad. Wetensch. Verslag Afd.
Natuurk. 86 (1977), no. 9, 121–125.
12.
^ N. G. de Bruijn,Defining reals without
the use of rationals. Nederl. Akad.Wetensch. Proc. Ser. A 79 = Indag. Math. 38
(1976), no. 2, 100–108
13.
^ Rieger, G. J. A new approach to the
real numbers (motivated by continued fractions). Abh. Braunschweig.Wiss. Ges.
33 (1982), 205–217
14.
^ Knopfmacher, Arnold; Knopfmacher, John
Two concrete new constructions of the real numbers. Rocky Mountain J. Math. 18
(1988), no. 4, 813–824.
15.
^ Knopfmacher, Arnold; Knopfmacher, John
A new construction of the real numbers (via infinite products). Nieuw Arch.
Wisk. (4) 5 (1987), no. 1, 19–31.
16.
^ MR693180 (84j:26002) review of A new
approach to the real numbers (motivated by continued fractions) by Rieger, G.
J.
