The Rational Numbers

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The Rational Numbers
Fields
The system ™ of integers that we formally defined is an improvement algebraically on =
(we can subtract in ™). But ™ still has some serious deficiencies: for example, a simple
equation like $B  % œ # has no solution in ™. We want to build a larger number
system, the rational numbers, to improve the situation.
In Chapter 3, we introduced the idea of an algebraic structure called a field and we
proved, for example, that ™: is a field iff : is a prime number.
The fields axioms, as we stated them in Chapter 3, are repeated here for convenience.
Definition Suppose J is a set with two operations (called addition and multiplication)
defined inside J . J is called a field if the following “field axioms” are true.
1) There are elements ! − J and " − J (and ! Á "Ñ
2Ñ aB aC aD ÐB  CÑ  D œ B  ÐC  DÑ 2w Ñ aB aC aD ÐB † C ) † D œ B † ÐC † DÑ
(addition and multiplication are associative)
3Ñ aB aC B  C œ C  B
3w Ñ aB aC B † C œ C † B
(addition and multiplication are commutative)
4Ñ aB aC aD B † ÐC  DÑ œ B † C  B † D
(the distributive law connects addition and multiplication)
5Ñ aB B  ! œ B
5w ) aB ÐB Á ! Ê B † " œ BÑ
(0 and 1 are “neutral” elements for addition and multiplication. ! is called
the additive identity element and " is called the multiplicative identity element in
J)
6Ñ aB bC B  C œ !
(C is called an additive inverse of B)
6w Ñ aB ÐB Á ! Ê ÐbCÑ B † C œ "Ñ
Ðfor each B Á !, such a C is called a
multiplicative inverse of BÑ
In any field J , multiplication “B † C” is often written as just “BC”.
The (still informal) systems , ‘, and ‚ are other examples of fields. So is the
collection Ö+  ,È# À +ß , − ×, as you proved in a homework assignment.
Much of the material about fields in the next few pages is material you've seen before.
It's just collected in a systematic way here, partly for review.
Any particular field such as ™: ,  and ‘ is called a model for the field axioms. There
are significant differences between these models. For example,
ñ ™$ has only 3 elements, while ™& has 5 elements and  and ‘ are
infinite fields.
ñ In ™$ , "  "  " œ !à in ™& ß "  "  "  "  " œ !; in  and ‘, no
finite sum of "'s has ! for a sum. .
In other words, we cannot say that “all fields look alike.” In that respect, the field
axioms are quite different from the axioms P1-P5) for a Peano system: there, we were
able to argue that “all Peano systems look alike”  or, in more formal language, that any
two Peano systems are isomorphic.
An axiom system for which “all models are isomorphic” is called a categorical
axiom system. The field axioms are not categorical.
Some useful theorems can be proved just using the field axioms: these theorems therefore
apply in all fields. Proving them in the abstract is efficient; it saves the effort of proving
them over and over each time a new field comes up. For example, the statements in the
following theorem are true in every field  ™: Ð: a prime)ß ß ‘ß ‚ß ÞÞÞ .
Theorem 1 Suppose Bß ?ß @ are a members of a field J .
a) (Cancellation for addition) If B  ? œ B  @, then ? œ @.
b) (Cancellation for multiplication) If B Á ! and B? œ B@, then
? œ @Þ
c) The additive inverse of B is unique.
d) If B Á !, then the multiplicative inverse of B is unique.
e) The additive identity, !, is unique and the multiplicative identity, ", are
unique.
f) aB − J ß B † ! œ !
g) If ?@ œ !, then ? œ ! or @ œ !.
Proof
a) Suppose B  ? œ B  @Þ By Axiom 5), B has an additive inverse C . Then
C  ÐB  ?Ñ œ C  ÐB  @Ñ
ÐC  BÑ  ? œ ÐC  BÑ  @
ÐB  CÑ  ? œ ÐB  CÑ  @
!?œ!@
?!œ@!
? œ @Þ
(using Axiom 2)
(using Axiom 3)
(since C is an additive inverse for B)
(using Axiom 3)
(using Axiom 5)
b) If B Á !ß then B has a multiplicative inverse C (Axiom 6w ). So if B? œ B@,
then
CÐB?Ñ œ CÐB@Ñ
ÐCBÑ? œ ÐCB)@
ÐBCÑ? œ ÐBCÑ@
"†?œ"†@
?†"œ@†"
?œ@
(using Axiom 2w )
(using Axiom 3w )
(since C is a multiplicative inverse
for B)
(using Axiom 3w )
(using Axiom 5w )
c) Suppose Cß C w − J . If B  C œ ! and B  C w œ ! both are true,
then B  C œ B  C w Þ Adding C to both sides, we get
C  ÐB  CÑ œ C  ÐB  C w Ñ
ÐC  BÑ  C œ ÐC  BÑ  C w
ÐB  CÑ  C œ ÐB  CÑ  C w
!  C œ !  Cw
C  ! œ Cw  !
C œ Cw
(using Axiom 2)
(using Axiom 3)
(because C is an additive inverse
for B)
(using Axiom 3)
(using Axiom 5).
Since B has a unique additive inverse, we can talk about the additive inverse of B and
give it a name: Ð  BÑÞ Then B  Ð  BÑ œ ! and (using Axiom 3) Ð  BÑ  B œ !Þ
The last equation says that B is the additive inverse for  B À that is  Ð  BÑ œ BÞ
This is not some profound fact; it's just a consequence of the notation we chose for the
additive inverse.
d) Suppose B Á !, that Cß C w − J . If BC œ " and BC w œ " both are true,
then BC œ BC w Þ Multiplying both sides by C , we get
CÐBCÑ œ CÐBC w Ñ
ÐCBÑC œ ÐCBÑC w
ÐBCÑC œ ÐBCÑC w
" † C œ " † Cw
C † " œ Cw † "
C œ Cw
(using Axiom 2w )
(using Axiom 23)
(since C is a multiplicative inverse
for B)
(using Axiom 3w )
(using Axiom 5w )
If B Á !, then B has a unique multiplicative inverse, so we can talk about the
multiplicative inverse of B and give it a name: B" Þ Then B † B" œ " and
(using Axiom 3w ) B" † B œ "Þ
The latter equation says that B is the multiplicative inverse for B" À that is ÐB" Ñ" œ BÞ
e) Suppose D − J and that D is an additive identity, that is, aB
B  D œ B œ B  !Þ Using part a) to cancel the B's, we get D œ !Þ
.
Suppose A − J and that A is a multiplicative identity, that is,
aB Á !ß BA œ B œ B † "Þ Using part b) to cancel the B's, we get A œ "Þ
f) Suppose B − J Þ 0 is the additive identity in J , so !  ! œ !Þ
Therefore
B † Ð!  !Ñ œ B † !
B†!B†!œB†!
(using Axiom 4)
Let C be the additive inverse of B † !, that is C œ  Ð! † BÑ
 Ð! † BÑ  ÐB † !  B † !Ñ œ  Ð! † BÑ  B † !
Ð  Ð! † BÑ  B † 0)  B † ! œ  Ð! † BÑ  B † !
!B†!œ !
B†!œ!
Part g) is left as an exercise.
ñ
In a field, we can also define subtraction and division.
Definition Suppose B and C are members of a field J .
a) We define the difference B  C œ B  Ð  CÑÞ
So subtraction is defined in terms of addition (adding the additive inverse).
b) If B Á !, we define the quotient C ƒ B œ CB" Þ
So division is defined in terms of multiplication (multiplying by the multiplicative
inverse).
Example Consider the field ™( œ Ö!ß "ß #ß $ß %ß &ß '× ( where "ß #ß ÞÞÞß ' are
abbreviations for the equivalence classes Ò"Óß Ò#Óß ÞÞÞß Ò'Ó)
a) The additive inverse of % is 3 because %  $ œ ! (and only $
has this property). Therefore we write  % œ $.
b) Subtraction: #  % œ #  Ð  %Ñ œ #  $ œ &Þ
c) The multiplicative inverse of $ is & because $ † & œ " (and only & has this
property). Therefore we write $" œ & (and &" œ $ÑÞ
d) Division:
# ƒ $ œ # † $" œ # † & œ $Þ
" ƒ $ œ " † $" œ " † & œ &Þ
Example If +ß ,ß - are members of any field J and + Á !, then the equation +B  , œ has a unique solution in J . We can simply use the “algebra of fields” contained in the
axioms and theorems to solve the equation (some detailed justifications  like references
to the repeated use of associativity and commutativity in ™  are omitted).
+B  , œ +B  ,  Ð  ,Ñ œ -  Ð  ,Ñ
+B  ! œ -  Ð  ,Ñ
+B œ -  ,
+" Ð+BÑ œ +" Ð-  ,Ñ
Using the additive
inverse of ,
+ Á ! so + has a
multiplicative inverse
" † B œ +" Ð-  ,Ñ
B œ +" Ð-  ,Ñ
For a more specific example, we solve the equation $B  % œ 2 in ™ ( . (See the
preceding example.)
$B  % œ #
$B  %  % œ #  %
$B  %  $ œ #  $
$B œ &
& † $B œ & † &
"†Bœ%
B œ %Þ
Constructing 
The preceding example shows that if we can enlarge the numbers system ™ to a field,
there will no longer be an issue about solving linear equations like $B  % œ #Þ We will
try to enlarge ™ in the most economical way we can. (Of course, what we are after is to
formally construct a field that acts just like the informal, familiar field of rational
numbers.)
In the work that follows, we can use all the properties of the integers that we have already
proven  for example, that addition and multiplication are commutative and associative,
the distributive law, that every integer has a unique additive inverse, that there are
cancellation rules for addition and multiplication (by a nonzero integer) in ™, etc. We
will often use these facts about integer arithmetic without actually citing the “chapter and
verse” reasons.
Think of a rational number +, (, Á !). To name a rational number we need two integers
+ß , where , Á !Þ So, starting with ™, we could try to create  by defining a rational
number to be a pair of integers Ð+ß ,Ñ in which , Á !Þ
However, that attempt would give us “too many” different rationals. After all, we think
of "# ß #% ß $
' as being the same rational number. In the informal system of rationals,
+
, œ . iff +. œ ,- . So  if rational numbers are to be represented using pairs of
integers, we would want the pairs Ð+ß ,Ñ and Ð-ß .Ñ to represent the same rational number
iff +. œ ,- . We can accomplish this by using an equivalence relation.
Definition For Ð+ß ,Ñ and Ð-ß .Ñ − ™ ‚ Й  Ö!×Ñ, let
Ð+ß ,Ñ ¶ Ð-ß .Ñ
Theorem 2
iff +. œ ,-
¶ is an equivalence relation on ™ ‚ Й  Ö!×ÑÞ
Proof i) Ð+ß ,Ñ ¶ Ð+ß ,Ñ because +, œ ,+ in (the formal system) ™. Therefore ¶ is
reflexive.
ii) If Ð+ß ,Ñ ¶ Ð-ß .Ñ, then +. œ ,- , so -, œ .+ in ™Þ But that means
Ð-ß .Ñ ¶ Ð+ß ,Ñ, so ¶ is symmetric.
iii) Suppose œ
Ð+ß ,Ñ ¶ Ð-ß .Ñ
Ð-ß .Ñ ¶ Ð/ß 0 Ñ
so that
We want to show that Ð+ß ,Ñ ¶ Ð/ß 0 ÑÞ
+. œ ,œ -0 œ ./
(1)
(2)
Since Ð-ß .Ñ − ™ ‚ Й  Ö!×Ñ, we know . Á !
Case i) If - œ !, then +. œ ,- œ ,Ð!Ñ œ ! œ Ð!Ñ. , and canceling the
nonzero . gives + œ !Þ
Similarly, ./ œ -0 œ Ð!Ñ0 œ ! œ .Ð!Ñ, and canceling the
nonzero . gives / œ !.
Then Ð+ß ,Ñ œ Ð!ß ,Ñ ¶ Ð!ß 0 Ñ œ Ð/ß 0 ÑÞ
Case ii) If - Á !, then multiplying equations (1), (2) and using
commutativity and associativity gives Ð+0 ÑÐ-.Ñ œ Ð,/ÑÐ-.ÑÞ
Since -. Á ! (a theorem we proved in ™), we can cancel Ð-.Ñ from both
sides leaving +0 œ ,/Þ Therefore Ð+ß ,Ñ ¶ Ð/ß 0 ÑÞ
Therefore ¶ is transitive. ñ
Definition  œ ™ ‚ Й  Ö!×ÑÎ ¶ Þ A member of  is called a rational number.
According to the definition, a rational number is an equivalence class containing certain
pairs of integers. What do some of the equivalence classes look like?
ÞÞÞ œ ÒÐ  #  #ÑÓ œ ÒÐ  "ß  "ÑÓ œ ÒÐ"ß "ÑÓ œ ÒÐ#ß #ÑÓ œ ÞÞÞ
Going back to the intuitive motivation, we think of this /;?3@+6/8-/ -6+== as the
rational number 1.
Similarly, in the intuitive motivation, we think of the equivalence class
... œ ÒÐ  #ß %ÑÓ œ ÒÐ  "ß #ÑÓ œ ÒÐ  "ß #ÑÓ œ ÒÐ  #ß %ÑÓ œ ÞÞÞ œ ÒÐ  "(ß $%ÑÓ œ ÞÞÞ
... œ ÒÐ#ß  %ÑÓ œ ÒÐ"ß  #ÑÓ œ ÒÐ"ß  #ÑÓ œ ÒÐ#ß  %ÑÓ œ ÞÞÞ œ ÒÐ"(ß  $%ÑÓ œ ÞÞÞ
as the rational number  "# .
Arithmetic in 
We want to define addition and multiplication in : that is, we want to define addition
and multiplication of these equivalence classes. The definitions make use of
representatives of the equivalence classes  so, as always, we will have to check that the
addition and multiplication are well-defined.
Definition Suppose ÒÐ+ß ,ÑÓ and ÒÐ-ß .ÑÓ are in . Let
i) ÒÐ+ß ,ÑÓ  ÒÐ-ß .ÑÓ œ ÒÐ+.  ,-ß ,.ÑÓ
Of course, the definition of addition is motivated by the way, informally,
 ,-Ñ
that we think addition in the rationals should behave: +,  .- œ Ð+.,.
ii) ÒÐ+ß ,ÑÓ † ÒÐ-ß .ÑÓ œ ÒÐ+-ß ,.ÑÓ
Again, the definition is motivated by the fact that in the informal system of
ac
rationals, +, † .- œ ,.
.
Notice that since ÒÐ+ß ,ÑÓ and ÒÐ-ß .ÑÓ − , we know that neither , nor . is !, and
therefore ,. Á !Þ Therefore ÒÐ+.  ,-ß ,.ÑÓ and ÒÐ+.  ,-ß ,.ÑÓ are also in . Adding
or multiplying two rational numbers gives another rational number.
Since ÒÐ"ß  #ÑÓ œ ÒÐ  #ß %ÑÓ and ÒÐ#ß $ÑÓ œ ÒÐ  %ß  'ÑÓß applying the
definitions to
ÒÐ"ß  #ÑÓ  ÒÐ#ß $ÑÓ
and
ÒÐ  #ß %ÑÓ  ÒÐ  %ß  'ÑÓ
should produce the same
answers as applying them to
and ÒÐ  #ß %ÑÓ † ÒÐ  %ß  'ÑÓ
ÒÐ"ß  #ÑÓ † ÒÐ#ß $ÑÓ
The next theorem guarantees that this is true.
Theorem 3 Addition and multiplication in  are well-defined.
Proof Suppose œ
Ð+ß ,Ñ ¶ Ð-ß .Ñ and
Ð/ß 0 Ñ ¶ Ð1ß 2Ñ
+. œ ,that is, suppose œ
/2 œ 0 1
1) Addition We need to show that
ÒÐ+ß ,ÑÓ  ÒÐ/ß 0 ÑÓ œ ÒÐ-ß .ÑÓ  ÒÐ1ß 2ÑÓ .
This is true iff
Ð+0  ,/ß ,0 Ñ ¶ Ð-2  .1ß .2Ñ
iff
.2Ð+0  ,/Ñ œ ,0 Ð-2  .1Ñ
iff
+.20  ,./2 œ ,-0 2  ,.0 1 (*)
But +. œ ,- and /2 œ 0 1, so the equation (*) just says
,-20  ,.0 1 œ ,-0 2  ,.0 1, which is true.
(1) and
(2)
2) Multiplication We need to show that
ÒÐ+ß ,ÑÓ † ÒÐ/ß 0 ÑÓ œ ÒÐ-ß .ÑÓ † ÒÐ1ß 2ÑÓ
ÒÐ+/ß ,0 ÑÓ œ ÒÐ-1ß .2ÑÓ
+/.2 œ ,0 -1 (**)
This is true
iff
iff
But equation (**) is true: it is just the result of multiplying together
the equations (1) and (2) in the hypothesis. ñ
Theorem 4 With addition and multiplication so defined,  is a field.
Proof We will prove that some of the field axioms are true for . The others (all just as
easy) are left as exercises. Suppose ÒÐ+ß ,ÑÓ and ÒÐ-ß .ÑÓ are in 
Axiom 3) Addition is commutative:
ÒÐ+ß ,ÑÓ  ÒÐ-ß .ÑÓ œ ÒÐ+.  ,-ß ,.ÑÓ
œ ÒÐ-,  .+ß .,ÑÓ (because addition and multiplication
in ™ are commutative)
œ ÒÐ-ß .ÑÓ  ÒÐ+ß ,ÑÓ
Axioms 5 and 5w ) Existence of identity elements
The equivalence classes ÒÐ!ß "ÑÓ and ÒÐ"ß "ÑÓ are the identity elements for
addition and multiplication in . For any ÒÐ+ß ,ÑÓ −  À
ÒÐ+ß ,ÑÓ  ÒÐ!ß "ÑÓ œ ÒÐ+ † "  , † !ß , † "ÑÓ œ ÒÐ+ß ,ÑÓ
and
ÒÐ+ß ,ÑÓ † ÒÐ"ß "ÑÓ œ ÒÐ+ † "ß , † "ÑÓ œ ÒÐ+ß ,ÑÓ
6 and 6w ) Existence of inverses For any ÒÐ+ß ,ÑÓ −  À
ÒÐ+ß ,ÑÓ  ÒÐ  +ß ,ÑÓ œ ÒÐ+ † ,  , † Ð  +Ñß , † ,ÑÓ
œ ÒÐ!ß , † ,ÑÓ œ ÒÐ!ß !ÑÓ
so ÒÐ+ß ,ÑÓ has an additive inverse in  À ÒÐ  +ß ,ÑÓ Þ
If ÒÐ+ß ,ÑÓ Á ÒÐ!ß !ÑÓ, then + Á ! so Ð,ß +Ñ − ™ ‚ (™  Ö!×) and
ÒÐ,ß +ÑÓ − . Then ÒÐ+ß ,ÑÓ † ÒÐ,ß +ÑÓ œ ÒÐ+,ß +,ÑÓ œ ÒÐ"ß "ÑÓÞ
Therefore ÒÐ+ß ,ÑÓ has a multiplicative inverse in  À ÒÐ,ß +ÑÓÞ
Just for convenience, we now assign some handy (and suggestive) names to the
equivalence classes.
Definition For any integers +ß , Ðwith , Á !Ñ, the rational number ÒÐ+ß ,ÑÓ is denoted
using fraction notation +, Þ In addition, we will also write an equivalence class
+
" œ ÒÐ+ß "ÑÓ simply as +Þ
Thus, a fractional notation like, say, $( is nothing but a convenient, handy definition of a
name for an equivalence class. It is a single, one-piece symbol standing for the
equivalence class ÒÐ$ß (ÑÓÞ The definition here of the symbol “ $( ” has nothing to do with
division.
However: As in any field, $ ƒ ( means $ † (" œ ÒÐ$ß "ÑÓ † ÒÐ(ß "ÑÓ"
œ ÒÐ$ß "ÑÓ † ÒÐ"ß (ÑÓ œ ÒÐ$ † "ß " † (ÑÓ œ ÒÐ$ß (ÑÓ œ $( Þ So it then turns out that the
symbol “ $( ” is the answer, in our formal system , to the problem “$ ƒ ( œ ?”.
This is good; if it didn't turn out that way, then the formal system  that we've
constructed wouldn't “act just like” the informal system of rationals after all.
Examples
ÒÐ"ß "ÑÓ œ "" . The equivalence class
"
"
is also just written as the rational number ".
The “"” 's in the pair Ð"ß "Ñ are the integer " À this integer, constructed earlier, is used
to construct a new object  the rational number ". Strictly speaking, the rational
number " is different from the integer "Þ
We saw a similar phenomenon earlier: when we constructed ™ß we saw that the integer "
was not the same as the whole number 1 (they are quite literally defined by different
sets). There are some additional comments about this at the conclusion of these notes.
ÒÐ!ß "ÑÓ œ !" . The equivalence class !" is also just written as the rational number !.
Since ÒÐ!ß "ÑÓ œ ÒÐ!ß ,ÑÓ for any nonzero integer , , we have !, œ !Þ
Since ÒÐ#ß "ÑÓ œ ÒÐ%ß #ÑÓ œ ÒÐ  "!ß  &ÑÓ, we have #" œ %# œ "!
& À because these
fractions are all just names agreed upon names for the same equivalence class. An
equivalence class like #" is also written more simply as the rational number #Þ
"
#
#
%
$
'
œ Ö ÞÞÞß Ð  $ß  'Ñß Ð  #ß  %Ñß Ð  "ß  #Ñß Ð"ß #Ñß Ð#ß %Ñß Ð$ß 'Ñß ÞÞÞ× and
œ Ö ÞÞÞß Ð  $ß  'Ñß Ð  #ß  %Ñß Ð  "ß  #Ñß Ð"ß #Ñß Ð#ß %Ñß Ð$ß 'Ñß ÞÞÞ× and
œ Ö ÞÞÞß Ð  $ß  'Ñß Ð  #ß  %Ñß Ð  "ß  #Ñß Ð"ß #Ñß Ð#ß %Ñß Ð$ß 'Ñß ÞÞÞ×
etc.
" #
$
"
#
$
# , % , and ' are names for the same equivalence class:
# œ % œ ' .
+
,
Notice that, in fraction notation,
œ
.
iff ÒÐ+ß ,ÑÓ œ ÒÐ-ß .ÑÓ
iff Ð+ß ,Ñ ¶ Ð-ß .Ñ iff +. œ ,-Þ
Examples Here once again are the definitions of addition and multiplication in   but
this time restated in terms of fractions. There is nothing to prove here. We are just
rewriting the definitions in a different notation. :
+
,
1)

.
œ
+.  ,+.
ÒÐ$ß %ÑÓ  ÒÐ#ß %ÑÓ œ ÒÐ#!ß "'ÑÓ œ ÒÐ&ß %ÑÓ
In fraction notation,
$
%

#
%
$†%#†%
"'
œ
œ
#!
"'
œ
&
%
ÒÐ$ß %ÑÓ  ÒÐ  "ß &ÑÓ œ ÒÐ"&  %ß #!ÑÓ œ ÒÐ""ß #!ÑÓ
In fraction notation,
$
%
+
,
2)
†
.
œ

"
&
œ
"&%
#!
œ
""
#!
+,.
ÒÐ  $ß %ÑÓ † ÒÐ"ß '!ÑÓ œ ÒÐ  $ß #%!ÑÓ œ ÒÐ  "ß )!ÑÓ
In fraction notation,
$
%
†
"
'!
œ
$
#%!
œ
"
)! Þ
Examples
1)
+,
+-
2)
+
,
,
-
because ÒÐ+,ß +-ÑÓ œ ÒÐ,ß -ÑÓ because Ð+,ß +-Ñ ¶ Ð,ß -Ñ
because +,- œ +-,
So, in fraction notation, a common factor can be “canceled” from numerator and
denominator
œ

,
œ
+,  ,,†,
œ
,Ð+  -Ñ
,†,
œ
+,
(canceling a common factor)
3) We proved that an object (in any field) has a unique additive inverse. As in any field,
we write the additive inverse of +, as  +, ß so +,  Ð  +, Ñ œ !Þ
Since +,  Ð +, Ñ œ +, ,Ð+Ñ,
œ ,Ð+ ,Ð+ÑÑ
œ ,!# œ !, we see that
#
#
+
,
is an additive inverse for +, Þ But  +, is the one and only element in  which added
to +, produces !. Therefore it must be that  +, œ +
, in .
+
+
Moreover, , œ , (because we proved, in ™, that +, œ Ð  +ÑÐ  ,ÑÑ.
Therefore 
4)
+
,
œ
+
,
+
,
+
,
œ
+
,
œ
for any
+
,
−
since Ð+ß ,Ñ ¶ Ð  +ß  ,ÑÞ
+
,
5) Subtraction:

.
œ
+
,
 Ð  .- Ñ œ
+
,

.
œ
+.  ,,.
Å
subtraction is defined in any field by
“add the additive inverse”
6) We proved that a nonzero object (in any field) has a unique multiplication inverse. If
! Á +, − ß then the multiplicative inverse of +, is denoted (as in any field) by Ð +, Ñ" .
Since ,+ † +, œ ", +, is a multiplicative inverse for +, in . Since Ð +, Ñ" is the one and
only rational which multiplied times +, produces ", it must be that +, œ Ð +, Ñ" in .
For example: Ð #$ Ñ" œ
$
#
, #" œ Ð #" Ñ" œ
If , Á !, ," œ Ð ", Ñ" œ
"
#
and
"
,
7) Division: If .- Á !, then +, ƒ .- œ +, † Ð .- Ñ" œ +, † .- œ +.
,(the old “invert and multiply" rule”).
Since we already observed that a rational number, in fractional form, can be
thought of as a division, we can also write this computation as
+
,
.
œ
+
,
ƒ
.
œ
+.
,-
As illustrated earlier: $ ƒ ( œ
#
$
%
(
œ
#
$
ƒ
%
(
$
"
ƒ
(
"
œ
œ
#
$
†
(
%
$
"
œ
†
"%
"#
"
(
œ $( Þ
œ
(
'
All the familiar manipulations involving addition, subtraction, multiplication and division
from the informal system of rationals can be shown to be true in the formal system .
At this point, we will assume that all such rules have been proven and we will use them
freely, without further comment. However, rules for manipulating inequalities in  still
need to be explored and justified.
Inequalities in 
Of course, we have an idea in the informal system of rationals what “positive” and
“negative” mean, and how relations like  and Ÿ work. But in the formal system 
that we have defined, we need to give definitions for positive, negative,  ß and Ÿ
show see what properties they have (hopefully, they will “act just like” our informal
notions).
Definition A nonzero rational number +, is called positive if it is possible to write +, œ .where - and . are both in .
Equivalently, we can say that +, is positive if it is possible to write +, œ .where the integers -ß . are both not in   because, in that case, we have +, œ .
where
 -ß  . are both in ,
Definition A nonzero rational number +, is called negative if it is possible to write
where exactly one of the two integers -ß . is in .
+
,
œ
.
The rational ! is by definition, neither positive nor negative.
Theorem 5 For every
+
,
+
,
+
,
+
,
− , exactly one of the following statements is true:
œ!
is positive
is negative
Proof If +, Á ! and +, is not positive, then it must be that exactly one of + or , is in  so
+
, is negative. Therefore one of the three statements must be true.
By definition, if +, œ !, then +, is neither positive nor negative. So we only need to
consider whether it is possible for +, to be both positive and negative.
Suppose +, is positive, where +ß , − Þ Suppose +, œ .- . It cannot be that exactly
one of -ß . is in  because then the equation +. œ ,- would have member of 
on
one side but not the other.
Therefore +, is not also negative.
-0 œ ./ holds in ™. This is impossible since one side of the equation is in  but
the other isn't. ñ
Example
#œ
"
#
#
"
œ
œ
#
" ,
"
# ,

"
$
are all positive, and
# ß and % ß
$
$
$
#
( œ ( œ ( , and " œ  # are
all negative.
Theorem 6 Suppose B œ
+
,
− Þ Then B is positive iff  B is negative.
Proof If B is positive, then we can write B œ +, œ .- where both -ß . − Þ Then
 B œ  .- œ . where . −  and  -  . Therefore
 B is negative.
The proof of the converse is left as an exercise. ñ
Theorem 7 If B and C are positive rationals, then B  C and BC are positive.
Proof We can write B œ
+
,
and C œ
B  C œ +,  .- œ
+BC œ +, † .- œ ,.
Þ
.
where all of +ß ,ß -ß . − Þ Then
+.  ,,.
, and
Since +.  ,-ß ,.ß and +- − , the sum and product are positive. ñ
In the discussion that follows, the notation is cleaner if we simply refer to rationals as
Bß Cß Dß ÞÞÞ − . We don't need the “fractional forms” B œ +, ß ÞÞÞ Þ
We can use the term “positive” to define order relations  and Ÿ in .
Definition For Bß C − , we say
BC
BŸC
iff C  B is positive
iff C  B is positive or B œ C
(We also write B  C and B Ÿ C as C  B and C BÞÑ
In particular, this means (fortunately for our intuition) that
!  B iff
and
B!
iff
B  ! œ B is positive
!  B is positive
iff  B is positive
iff Ð  BÑ is negative
iff B is negative
Example Using  , we can rewrite some of our previous observations (which ones?) in
a new way:
") Since ! is not positive, !  ! is false.
2) aB − ß exactly one of the following is true:
B  !ß B œ !ß B  !
3) aB −  B  ! iff  B  !Þ
Note, just for emphasis: this statement implies that
 B  ! iff  Ð  BÑ œ B  !ÞÑ
4) a B a C − ß if B  ! and C  !, then B  C  ! and BC  !.
Some of the most important facts about  are listed in the following theorem. You can
also formulate an analogous theorem for Ÿ .
Theorem 8 For all Bß Cß Dß A − ß
a)
b)
c)
d)
e)
f)
g)
h)
If B  C and C  D , then B  D (the relation “  ” is transitive)
If B  C , then B  D  C  D
If B  C and D  !, then BD  CD
If B  C and D  !, then BD  CD
If B Á !, then B#  !
If B  C , then  B   C .
If BC  !, then both Bß C are positive or both are negative
If B  C and D  A, then B  D  C  AÞ
Proof We will prove a few of the statements to indicate how the arguments go.
a) Suppose B  C and C  D .
Then C  B and D  C are both positive.
So ÐC  BÑ  ÐD  CÑ œ D  B is positive.
Therefore B  DÞ
ÐDefinition of “  ” Ñ
Ð Theorem 7 Ñ
ÐDefinition of “  ” Ñ
d) If B  C, then C  B is positive
If D  !, then !  D œ  D is positive
Therefore  DÐC  BÑ is positive
DÐB  CÑ is positive
BD  BC is positive
BD  CD
Ð Definition of “  ” Ñ
ÐDefinition of “  ” Ñ
ÐDefinition of “  ” Ñ
ÐTheorem 7Ñ
e) If B is positive, then B † B œ B# is positive,
that is, B#  !Þ
ÐTheorem (Ñ
If B is negative, then  B is positive
ÐTheorem 'Ñ
so Ð  BÑ † Ð  BÑ œ B# is positive.
ÐTheorem (Ñ
g) The proof is by contraposition.
If B is positive and C is negative, then B and  C are both
positiveß
ÐTheorem 'Ñ
so  BC is positive
ÐTheorem (Ñ
Then  Ð  BCÑ œ BC is negative
ÐTheorem 'Ñ
so BC 
Î !Þ
ÐDefinition of “  ”Ñ
In a similar way, you can also show that it is impossible to have B negative and C
positive.
The remaining parts of the theorem are left as exercises.
ñ
Definition A relation V on a set \ is called antisymmetric iff
aB aC − \ (BVC • CVBÑ Ê B œ C
“Antisymmetry” for a relation V means more than “not symmetric.” It is the
extreme opposite of “symmetry” in the following sense: À
“symmetric”
“not symmetric”
which is equivalent to
“antisymmetric”
aB aC − \ ÐBVC Ê CVBÑ
µ ÐaB aC − \ ÐBVC Ê CVBÑÑ
Î
bB bC − \ ÐBVC • CVBÑ
Î
aB aC − \ ÐBVC Ê CVBÑ
Definition A relation V on \ is called a linear ordering (or total ordering) iff V is
i) reflexive, transitive, and antisymmetric, and
ii) aB aC − \ ÐBVC ” CVB Ñ
The pair Ð\ß VÑ is then called a linearly ordered set.
It is easy to show that Ÿ is a linear ordering on . (What about  ?)
Theorem * Ÿ is a linear ordering on .
Proof Exercise.
Concluding Comments
1) We constructed the rationals from the integers, the integers from the whole numbers,
and whole numbers from sets. If everything is “unpacked,” each rational number is a
(complicated) set. For example, what is the rational number “2” ?
The rational number # œ ÒÐ#ß "ÑÓ
œ Ö..., Ð  %ß  #Ñß Ð  #ß  "Ñß Ð#ß "Ñß Ð%ß #Ñß ÞÞÞ×
(the #'s in the ordered pairs are integer #'s )
Each member of this equivalence class is an ordered pair of integers, and each
ordered pair Ð+ß ,Ñ is a set ÖÖ+×ß Ö+ß ,××. We will “unpack” just one of those ordered
pairs, say Ð#ß "Ñ À
Ð#ß "Ñ œ ÖÖ#×ß Ö#ß "××, where # and " here represent the integers # and 1.
Descending deeper (see the similar comments in the notes “Constructing the
Integers”): the integer= # and " are certain equivalence classes of pairs of whole
numbers:
The integer 2 is the set (equivalence class) ÒÐ#ß !ÑÓ œ ÖÐ#ß !Ñß Ð$ß "Ñß Ð%ß #Ñß ÞÞÞ ×
The integer " is the set (equivalence class) ÒÐ"ß !ÑÓ œ ÖÐ"ß !Ñß Ð#ß "Ñß Ð$ß #Ñß ÞÞÞ ×
So the ordered pair of integers
Ð#ß "Ñ œ ÖÖ#×ß Ö#ß "×× œ ÖÖÒÐ#ß !ÑÓ×ß ÖÒÐ#ß !ÑÓß ÒÐ"ß !ÑÓ××
œ ÖÖÖÐ#ß !Ñß Ð$ß "Ñß Ð%ß #Ñß ÞÞÞ ××ß ÖÖÐ#ß !Ñß Ð$ß "Ñß Ð%ß #Ñß ÞÞÞ ×ß ÖÐ"ß !Ñß Ð#ß "Ñß Ð$ß #Ñß ÞÞÞ ××× (*)
(where on the preceding line, the numbers in the ordered pairs are whole numbers.
So each one of the ordered pairs of integers in the rational number 2 is really a set like (*)
But to go further, each one of the ordered pairs in each set like (*) can be further
unpacked: for example, consider the single ordered pair Ð#ß "Ñ of whole numbers that
occurs in (*). Each whole number was defined earlier as a set, so
whole number pair Ð#ß "Ñ œ ÖÖ#×ß Ö#ß "×× œ ÖÖÖgß Ög×××ß ÖÖgß Ög××ß Ög×××
“What is #?” can rapidly become mind-boggling. The thing to remember in years to
come is not the details of how each rational number is constructed but that
i) each rational can be built up from sets  so sets, as far as we have gone, are
proving to be an adequate set of building blocks for mathematics, and
ii) beyond that, all we need to know about “#” to do mathematics is “how does 2
behave?”
2) As we constructed them, the integers are not members of . However, it is not hard
to see that
the system of rational numbers
ÖÞÞÞß  #ß  "ß !ß "ß #ß ÞÞÞ×
“acts just like”
the system of integers
ÖÞÞÞß  #ß  "ß !ß "ß #ß ÞÞÞ×
(the more technical language is that the two systems are isomorphic).
So we can think of the system of rationals ÖÞÞÞß  #ß  "ß !ß "ß #ß ÞÞÞ× inside  as being
an “exact photocopy” of the system of integers ÖÞÞÞß  #ß  "ß !ß "ß #ß ÞÞÞ×. If we agree
to ignore the difference between the photocopy and the real thing, then we can write
™ © .
3) Of course the number system  is for many purposes a big improvement over ™. For
example, it's a field and ™ isn't, and linear equations in , but not in ™, can always be
solved. But  still has some major algebraic shortcomings. For example, such a simple
equation as B# œ # cannot be solved in . (Why not?)
To deal with this difficulty requires enlarging the number system again to include new
number(s) with square #. The informal number system ‘ has such numbers: „ È # Þ
Unfortunately won't have time in this course to formally construct ‘Þ It turns out,
in that construction, that a real number is an ordered pair ÐEß FÑ, where E and
F are two special subsets of  that form a “cut” in . For example,
È# œ ÐEß FÑ where E œ Ö; −  À ; #  #× and F œ Ö; −  À ; #  #×Þ
Informally, ‘ is a field  just as  is  but, in ‘, there is a solution for B# œ #. ‘ must
have some additional property(s) making it a “more special” field than : some
property that guarantees that there are also “irrational” numbers.
The number system ‘ is a very powerful system: it's all you needed to be able to do
calculus. But even ‘ is not completely satisfactory algebraically. Such as simple
equation as B#  " œ ! has no solution in ‘.
For a very substantial part of mathematics, one more enlargement finally does the trick:
the field ‘ is enlarged to the set of complex numbers ‚. Each element of ‚ is an ordered
pair of real numbers Ð+ß ,Ñ, and addition and multiplication are defined in ‚ in such a
way as to make ‚ into a field.
In high school, you probably learned to write complex number in a form like
+  ,3 and -  .3ß and you probably we told to compute
Ð+  ,3ÑÐ-  .3Ñ œ +-  ,-3  +.3  ,.3#
œ Ð+-  ,.Ñ  Ð,-  +.Ñ3 because 3# œ  "
This is the informal motivation for the definition of multiplication in a formal
construction of ‚ À
Ð+ß ,Ñ † Ð-ß .Ñ œ Ð+-  ,.ß ,-  +.Ñ
In ‚, it turns out (and it's not easy to prove) that
every polynomial equation
+8 D 8  ÞÞÞ  +" D  +! œ !
has a solution.
This result is called the Fundamental Theorem of Algebra.
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