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The reasonable Numbers

The rational numbers space the collection of every fractions:
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They room an Abelian team under addition, and, if 0 is eliminated fromthe set, they kind an Abelian team under multiplication together well. Thus,the reasonable numbers form afield.To make this concrete, we can construct a design for the rationalnumbers. Because that its foundation, us will use the version for the integers,
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, which we developed here.We"llstart through stating a collection of axioms for the rational numbers, and also thenbuild a version which we have the right to prove satisfies the axioms.(Incidentally, over there is more than one means to state the axioms,and ns make no claim that the axiom set given here is in any way"standard".)

Axioms because that the reasonable Numbers

We"ll call the collection of rational numbers ~ above this page.

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(A.1) The number0 is a rational number.

(A.2) The number 1 is a reasonable number, and 1 ≠ 0

(A.3) The rational numbers form an Abelian group under addition, denoted as "+", through 0 because that the identification element.Note that for the integers, we can simply definemultiplication asrepeated addition, and also no brand-new axioms were required to describe it.That"s due to the fact that the framework of the integers is determinedentirely through the actions of addition; multiplication is in no wayfundamental to them. Inthe situation of the rational number it"s no so simple, and also the behaviorof multiplication help to determine the basic structure that the set.So we must construct multiplication right into the axioms.

(A.4) The collection

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, which is the rational number without0, creates an Abelian group under multiplication, denoted together "�" or "�", with1 for the identification element.Wenow have addition and multiplication work specified, but severaladditional axioms will certainly be needed to specify how the two operationsinteract. We"ll also need a couple of more axioms to regulate the sizeof the set, and also give it a sensible as whole "shape".

(A.5) 0 multiplied by any element is 0.

(A.6) Multiplication by a number, acting on addition, is a direct operator:

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We"venow determined how enhancement and multiplication interact. However wecan find finite fields in i beg your pardon axioms (A.1) with (A.6) hold.Such fields can be assumed of as being "ring shaped" or having"loops". Specification that a complete ordering, and its interactionswith enhancement and multiplication, will pressure the rational number to"stretch out" in a line.

(A.7) is completely ordered by the less-than relation, "The relation has actually the properties:


(R.1) any kind of pair of elements of may be compared

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(R.2)

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(R.3)

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For convenience, we will sometimes use "a>b" to average "bA.8
) The "0 1

(R.5) enhancement preserves comparisons:

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(R.6) Multiplication conservation comparisons:

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(A.9) Axioms (A.1)through (A.8) could describe a structure which looked like countless copiesof the rational numbers laid end to end. V the integer boundsaxiom, we"ll border the rationals to things "no bigger than" normalintegers. Let"s specify the subsets that the rationals which can bereached by repeatedly including �1 to 0:

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And now we"ll define the "integer subset" that the rationals:
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We deserve to now state the integer limit axiom:
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(A.10) Thelast point we must do is limit the set to contain only thosenumbers which room actually rational -- that is, values which have the right to beexpressed as ratios that integers. Us again usage the set I which we characterized in axiom (A.9):

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A model for the rational Numbers

As a base to occupational from we will use the model of the integers we emerged here
. For the remainder of this page, let united state define
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Let the "Next, we kind a collection of ordered bag of integers:
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We will describe the first member that the pair together the "numerator" and also the 2nd member together the "denominator".Define one equivalence relationship on the set, utilizing multiplication of integer quantities, i m sorry was previously defined:
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Stepping external the design for a moment, the relation will be true whenever x/y = z/w; equivalently, as long as no x no one z is zero, the relation is true whenever x/z = y/w.Inother words, this claims that all ordered pairs which differ only by acommon variable in the numerator and denominator are equivalent. We then kind the collection
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In various other words, is the set of equivalence classes in Q
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This is similar to discarding whatever from the set except the"reduced fractions" -- those because that which the numerator and denominatorhave no usual divisor.If we use to represent the equivalence course of q, where
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, then by using the relation "integers, us can define "equivalence classes:
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For this to be legitimate, it have to be the instance that it doesn"tmatter what representatives of the equivalence classes we choose; wemust gain the very same result. That is, in fact, true, yet we haven"tproved it, despite the proof is simple (and us may add it tothis web page eventually).Multiplication is likewise easy to define:
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And enhancement is almost as simple:
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Multiplication and addition are likewise well-defined by the abovestatements just if the doesn"t issue which members the the equivalenceclasses we select to advice them. Though that is in reality thecase, we have actually not showed it.It is simple to ""prove"" the axioms which explain therational numbers in ~ this model, thus showing that it is, indeed, amodel because that the rationals.Finally, together an aside, we exhibit a very small theorem:
Definethe set of all reduced fractions:
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Then offered anyelement
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, we will display that over there exists areduced fraction,
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, such that = <(x,y)>. Considerthe set ofall platform for elements of
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which space equivalentto (x,y):
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then
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. Yet P is well orderedso D musthave a the smallest element. Call that facet W, and thecorrespondingnumerator Z. Then (Z,W) should be the lessened fractionwhich is equal to(x,y).