The rational numbers are the set of all fractions:
They are an Abelian group under addition, and, if {0} is removed from
the set, they form an Abelian group under multiplication as well. Thus,
the rational numbers form a
field.
To make this concrete, we can construct a model for the rational
numbers. For its foundation, we will use the model for the integers,
, which we developed
here.
We'll
start by stating a set of axioms for the rational numbers, and then
build a model which we can prove satisfies the axioms.
(Incidentally, there is more than one way to state the axioms,
and I make no claim that the axiom set given here is in any way
"standard".)
Axioms for the Rational Numbers
We'll call the set of rational numbers
on this page.
(A.1) The number 0 is a rational number.
(A.2) The number 1 is a rational number, and 1 ≠ 0
(A.3) The rational numbers form an Abelian group under addition, denoted as "+", with 0 for the identity element.
Note that for the integers, we could simply define
multiplication as
repeated addition, and no new axioms were needed to describe it.
That's because the structure of the integers is determined
entirely by the behavior of addition; multiplication is in no way
fundamental to them. In
the case of the rational numbers it's not so simple, and the behavior
of multiplication helps to determine the basic structure of the set.
So we must build multiplication into the axioms.
(A.4) The set , which is the rational numbers without 0, forms an Abelian group under multiplication, denoted as "·" or "×", with 1 for the identity element.
We
now have addition and multiplication operations specified, but several
additional axioms will be needed to define how the two operations
interact. We'll also need a few more axioms to control the size
of the set, and give it a sensible overall "shape".
(A.5) 0 multiplied by any element is 0.
(A.6) [Distributive property]
Multiplication by a number, acting on addition, is a linear operator:
We've
now determined how addition and multiplication interact. But we
can find finite fields in which axioms (A.1) through (A.6) hold.
Such fields can be thought of as being "ring shaped" or having
"loops". Specification of a total ordering, and its interactions
with addition and multiplication, will force the rational numbers to
"stretch out" in a line.
(A.7) is totally ordered by the less-than relation, '<'.
The relation has the properties:
(R.1) [total order] Any pair of elements of may be compared
(R.2) [Transitivity]
(R.3) [Anti-symmetry]
For convenience, we will sometimes use "a>b" to mean "b<a".
(A.8) The '<' relation is compatible with
addition and multiplication. Specifically, it satisfies the
following additional relation properties:
(R.4) 0 < 1
(R.5) Addition preserves comparisons:
(R.6) Multiplication preserves comparisons:
(A.9) [Integer bounds]
Axioms (A.1)
through (A.8) could describe a structure which looked like many copies
of the rational numbers laid end to end. With the integer bounds
axiom, we'll restrict the rationals to things "no bigger than" normal
integers. Let's define the subsets of the rationals which can be
reached by repeatedly adding ±1 to 0:
And now we'll define the "integer subset" of the rationals:
We can now state the integer bounds axiom:
(A.10) [Rationality]
The
last thing we need to do is restrict the set to contain only those
numbers which are actually rational -- that is, values which can be
expressed as ratios of integers. We again use the set I which we defined in axiom (A.9):
A Model for the Rational Numbers
As a base to work from we will use the model of the integers we developed
here. For the rest of this page, let us define
Let the "<", "+", "×" operations on
be defined by restriction of the same operations we already defined on
.
Next, we form a set of
ordered pairs of integers:
We will refer to the first member of the pair as the "numerator" and the second member as the "denominator".
Define an equivalence relation on the set, using multiplication of integer quantities, which was previously defined:
Stepping outside the model for a moment, the relation will be true whenever
x/y =
z/w; equivalently, as long as neither
x nor
z is zero, the relation is true whenever
x/z = y/w. In
other words, this says that all ordered pairs which differ only by a
common factor in the numerator and denominator are equivalent.
We then form the set
In other words,
is the set of
equivalence classes in
Q_{0}, as determined by the ~ relation.
This is similar to discarding everything from the set except the
"reduced fractions" -- those for which the numerator and denominator
have no common divisor.
If we use [
q] to represent the equivalence class of
q, where
, then by using the relation "<" defined on the
integers, we can define "<" in
as the relation among
equivalence classes:
For this to be legitimate, it must be the case that it doesn't
matter what representatives of the equivalence classes we choose; we
must get the same result. That is, in fact, true, but we haven't
proved it, though the proof is straightforward (and we may add it to
this page eventually).
Multiplication is also easy to define:
And addition is nearly as simple:
Multiplication and addition are also well-defined by the above
statements only if it doesn't matter which members of the equivalence
classes we choose to evaluate them. Though that is in fact the
case, we have not proved it.
It is straightforward to ''prove'' the axioms which describe the
rational numbers within this model, thus showing that it is, indeed, a
model for the rationals.
Finally, as an aside, we exhibit a very small theorem:
Define
the set of all reduced fractions:
Then given any
element
, we will show that there exists a
reduced fraction,
, such that [
f] = [(
x,
y)]. Consider
the set of
all denominators for elements of
which are equivalent
to (
x,y):
Then
. But P is
well ordered
so
D must
have a smallest element. Call that element
W, and the
corresponding
numerator
Z. Then (
Z,W) must be the reduced fraction
which is equal to
(
x,y).
The rational numbers, in addition to being important in and of
themselves, are the one of the building blocks we use to construct the
real numbers.
Page not complete yet. We
plan to add the proof that the model satisfies the axioms, the proof
that the rationals are dense, and a few additional small theorems and
properties which can be derived easily from the axioms, which will show
that we really have constructed the familiar rational numbers here.
As of 11/2007 these have not yet been written up.This article was originally posted on Anarchopedia, in less complete form (and with rather more errors), in 2004.
Page
created on 11/18/2007