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The Rational Numbers

The rational numbers are the set of all fractions:

    file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_HKFHIM.png

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, file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_KInVd3.png, 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 file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_fJkEpr.png 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 file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_fjzYTA.png, 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:

file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_TJJCXj.png

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) file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_fJkEpr.png is totally ordered by the less-than relation, '<'.

The relation has the properties:

(R.1) [total order] Any pair of elements of file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_fJkEpr.png may be compared

file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_JqDAzu.png

(R.2) [Transitivity]

file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_sG9uWq.png

(R.3) [Anti-symmetry]

file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_RI997l.png

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:

file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_7hLv7U.png

(R.6)    Multiplication preserves comparisons:

file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_pjjJmT.png

(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:

    file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_r77OVl.png

And now we'll define the "integer subset" of the rationals:

    file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_wY5J1T.png

We can now state the integer bounds axiom:

    file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_M3xyUA.png

(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):

    file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_gxt7oo.png

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

    file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_G7j6Qx.png

Let the "<", "+", "" operations on file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_d7b1k0.png be defined by restriction of the same operations we already defined on file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_KInVd3.png.

Next, we form a set of ordered pairs of integers:

    file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_U5WBLa.png

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:

    file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_5i2qVw.png

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

    file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_TcaqBZ.png

In other words,  file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_fJkEpr.png is the set of equivalence classes in Q0, 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 file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_tMNiJz.png,  then by using the relation "<" defined on the integers, we can define "<" in file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_fJkEpr.png as the relation among equivalence classes:

    file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_gXg4Oh.png

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:

    file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_VIy0WL.png

And addition is nearly as simple:

    file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_BbCsyi.png

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:

file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_HQB287.png

Then given any element file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_pqukcx.png, we will show that there exists a reduced fraction, file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_ZoVqKq.png, such that [f] = [(x,y)]. Consider the set of all denominators for elements of file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_eR8mgR.png which are equivalent to (x,y):

file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_TYNMnJ.png

Then file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/eqe_temp_image_JqBZyz.png. 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