Download Math 308: Defining the rationals and the reals

Math 308: Defining the rationals and the reals.
From the integers to the rationals…….
N = {1,2,3,4, , , ,} (natural numbers)
Z = {0,1,-1,2,-2,3,-3, , , ,} (integers)
Z* = Z – {0} = { 1,-1,2,-2,3,-3, , , ,} (non-zero integers)
Addition is a function from Z x Z to Z
It could be written as S (for Sum): Z  Z  Z
For example S ( (3,4) ) = 7 given by ( (3,4), 7) in (Z xZ) x Z
Instead, standard notation is different: we write 3 + 4 = 7.
Similarly, multiplication is really a function P (for product):
P: Z  Z  Z and we write 3  4  P((3,4))  12 . It is also
customary to write ab as an abbreviation for the product a  b
From the set theoretic definition of natural numbers N given last
time (and a mild generalization to a definition of Z, which we
omit) it is possible (not easy) to prove the “standard algebra rules”
for integers, and to define the relation a  b as we usually
understand it. We then define the relation a  b on Z to mean
a  b or a  b
In this note, we show how addition, multiplication, and the 
relation extend from the integers Z to the rational numbers Q.
Let S  Z  Z *
be the set of ordered pairs (n, d ) of integers with d  0
We will define an equivalence relation on S based on crossmultiplication of fractions:
3 6
 if and only if 3  8  6  4 . In more detail
4 8
Definition of rational numbers Q: The rational numbers Q is the
set of equivalence classes for the relation  on S  Z  Z * given
by ( N , D)  (n, d ) if and only if Nd  nD .
(We need to Prove it’s an equivalence relation)
The equivalence class [ (N,D) ] of ( N , D)  S is usually written
N
 [( N , D)]  {( n, d )  Z  Z * | nD  Nd }
as a fraction symbol
D
3
For example  {(3,4), (3,4), (6,8), (6,8), (9,12), (9,12), , , }
4
Note that the integers Z can be viewed as a subset of Q, namely
n
as the image of the injective map i : Z  Q defined by i (n) 
1
Theorem 1: There are well defined functions
p r ps  rq
Addition:  : Q  Q  Q and given by  
and
q s
qs
p r pr
Multiplication  : Q  Q  Q given by  
q s qs
Proof that multiplication on Q is well defined.
We need to show: if a)
r r'
p p'
p r p' r '

and b)  , then c)   
s s'
q q'
q s q' s'
By definition of the equivalence relation,
a) says pq'  p ' q and b) says rs '  r ' s
pr p' r '
c) says qs  q ' s '
( pr)(q' s' )  ( p' r ' )(qs)
From a) and b) (“multiply equals by equals”) pq' rs '  p ' qr ' s .
Rearrange using commutative and associative laws for multiplying
integers to obtain c) QED
Exercise: Prove that addition on Q is well defined.
Finally, we can define a relation < on Q as follows: Assume
p r
q  0 and s  0 . Then  if and only if ps  qr
q s
Note: the assumption on q and s is justified by the fact that every
fraction equivalence class has a representative with positive
3
5
?
denominator. For example, to see if
, rewrite the
4
6
3
5
?
inequality as
, i.e. is  3  6  4  5 ?
4
6
5
3

No, so the correct relation is
6 4
Exercise: the < relation on Q as given above is well defined.
Hint: you need to show the following:
r r'
p p'
p r
p' r '

Suppose 
and  . Then  if and only if
s
s
'
q q'
q s
q' s'
Finally, define relation a  b on Q to mean a  b or a  b
Theorem 2 : With the definitions of addition, multiplication, and
a  b given above, Q is an ordered field. That is, field axioms hold
and so do the order axioms:
For all a, b, c  Q
O1: a  b or b  a
O2: If a  b and b  a , then a  b
O3: If a  b and b  c , then a  c
O4: If a  b then a  c  b  c
O5: If a  b and 0  c then ac  bc
Proof. (Tedious, optional) Exercise. In each case you will need to
use the corresponding result for integers.
From the rationals to the reals………
The ancient Greeks knew that some numbers that occur as lengths
cannot be expressed as rational numbers. The simplest example:
Let x be the length of the diagonal of a square with side 1. Then a
geometry argument shows that the product x 2  xx  2 . However,
p
it is easy to show: for any rational number x  , the number
q
2
 p
p2
x     2 is not equal to 2. (We did it earlier this semester.)
q
q
2
It follows that not all lengths are rational numbers. Similarly, Ross
1.2 shows that many numbers that are roots of equations with
integer coefficients cannot be rational.
So there is a tricky question: just what numbers are there on the
(real) number line besides rational numbers? It was not answered
until the end of the nineteenth century. One answer was given by
Richard Dedekind and is described in Ross, Section 1.6.
The following theorem tells you everything you need to know,
even without a prior definition of the real numbers.
Theorem 3:
 The real numbers R are a set that (strictly) contains the
rational numbers Q.
 Addition and multiplication of real numbers extends (agrees
with the previous definition of) addition and multiplication
of rational numbers (still satisfying field axioms)
 The  relation extends from Q to R (still satisfying order
axioms)
 The real numbers satisfy the least upper bound (LUB)
property described below, but the rational numbers do not.
Suppose S  R . In other words, let S be a subset of the real
numbers.
z
---------y---------yyyyyy-------yyyyy-----------yy--------x---x--Definition:
1) x  R is an upper bound for S means: y  x for all y  S .
2) z  R is a least upper bound for S means
 z is an upper bound for S and
 if x is any upper bound for S, then z  x
Example 1: Let S = {r  R | r  2} . Then 2 and 4.099 are upper
bounds for S. Furthermore 2 is a least upper bound for S.
Proof that 2 is a LUB for S.
a) it’s an upper bound because r in S implies r < 2
b) suppose 2 is not a least upper bound for S. Then there exists a
smaller upper bound q with q < 2. But this can’t be true
because q < 2 implies q/2 + 1 is in S and q < q/2 +1, hence q
is not an upper bound for S, contradiction.
Example 2: Let S  {r  R | r 2  2} . Then 1.415 is an upper bound
for S. It is not obvious whether S has a least upper bound. The fact
that S does have a least upper bound z that satisfies z 2  2 is a
consequence of
Least Upper Bound Property for Real Numbers: Any subset of
R that has an upper bound has a least upper bound.
The fact that the LUB property holds for R is described in various
ways: R is complete; R has no gaps, etc.
The LUB Property allows us to understand the meaning of infinite
decimal expansions.
Two examples: what is meant by
x = 0.345345345 ……. and
y = 0.101001000100001000001..... ?
1. Let X be the set of rational numbers
{0.345, 0.345345, 0.34534345, , , , , }
Then 1 is an upper bound for X .
By the LUB property, X has a least upper bound.
That least upper bound is x = 0.345345345345345……… =
345/999
2. Similarly, let Y be the set
{0.1, 0.101, 0.101001, 0.1010010001, , , ,}
Then 0.2 is an upper bound for Y.
By the LUB property, Y has a least upper bound.
That least upper bound is y = 0.101001000100001000001…….
Exercise Prove that there is a real number z with z 2  2 as follows.
Let S = {x  R | x  0and x 2  2} and T = {x  T | x  0and x 2  2}
Prove each of the following:
a) S has an upper bound . Let z = LUB of S.
b) If x  S then there exists x ' S with x'  x
c) z  S
d) If If y  T then there exists y ' T with y '  y
e) z  T
f) z 2  2
Homework: Ross Read 1.3 and 1.4
In Section 1.3, Exercises (p 18) 3.3, 3.4 ,3.5
Section 1.4 Exercises (p25) 4.2 (first column) 4.4, 4.6, 4.8