Section 3.2 Ordered Fields... We have two operations, + and , called addition and multiplication, such thatthe following properties apply: Commutative Property Associative Property Com
Trang 1Chapter 3
The Real Numbers
Trang 2Section 3.2 Ordered Fields
Trang 3We have two operations, + and , called addition and multiplication, such that
the following properties apply:
Commutative Property
Associative Property
Commutative Property
Associative Property
These 11 axioms are called the field axioms
The real numbers are an example of an ordered field
A1 For all x, y , x + y and if x = w and y = z, then x + y = w + z
A2 For all x, y , x + y = y + x.
A3 For all x, y, z , x + ( y + z) = (x + y) + z.
A4 There is a unique real number 0 such that x + 0 = x for all x
A5 For each x there is a unique real number x such that x + ( x) = 0.
M1 For all x, y , x y and if x = w and y = z, then x y = w z
M2 For all x, y , x y = y x.
M3 For all x, y, z , x ( y z) = (x y) z.
M4 There is a unique real number 1 such that 1 0 and x 1 = x for all x
M5 For each x with x 0, there is a unique real number 1/ x such that x (1/ x) = 1.
DL For all x, y, z , x ( y + z) = (x y) + (x z)
Mult Identity Add Inverse
Mult Inverse Distributive Law
Additive Identity
Addition
Multiplication
Trang 4In addition to the field axioms, the real numbers also satisfy four order axioms.
These axioms indentify the properties of the relation “<”
O1 For all x, y , exactly one of the relations x = y, x > y, or x < y holds.
trichotomy law
Note: a > b means b < a.
O2 For all x, y, z , if x < y and y < z, then x < z. transitive property
O3 For all x, y, z , if x < y then x + z < y + z.
O4 For all x, y, z , if x < y and z > 0, then x z < y z.
Let x, y, and z be real numbers.
(a) If x + z = y + z, then x = y (b) x 0 = 0.
(c) – 0 = 0 (d) (–1) x = – x.
(e) x y = 0 iff x = 0 or y = 0 (f) x < y iff – y < – x.
(g) If x < y and z < 0, then x z > y z.
Theorem 3.2.1
Our first theorem shows how the axioms may be used to derive some familiar algebraic properties
We illustrate the proofs by doing parts (a) and (d)
Trang 5Let x, y and z be real numbers
(a) If x + z = y + z, then x = y.
If x + z = y + z, then (x + z) + (– z) = ( y + z) + (– z)
x + [z + (– z)] = y + [z + (– z)]
x + 0 = y + 0
by A5 ( add inverse )
and A1 ( addition )
by A3 ( assoc property )
by A5 ( add inverse )
x = y by A4 ( add identity )
Theorem 3.2.1
Proof:
Trang 6Let x, y and z be real numbers
Theorem 3.2.1
(d) For any real x, (–1)x = – x.
Question: What is –1? Answer: –1 is the number which when added to 1 gives 0
Question: What is –x? Answer: – x is the number which when added to x gives 0.
Question: So how do we show that (–1)x = – x?
Answer: We show that (–1)x satisfies definition of – x
Namely, when (–1)x is added to x, the result is 0.
Proof:
We must show that x + (–1)x =
0
We have x + (–1)x = x + x (–
1)
by M2 ( commutative )
= x (1) + x (–
1)
by M4 ( mult identity )
by DL ( distributive law )
by A5 ( add inverse )
= x [1 + (–1)]
= x 0
by part (b)
= 0 Thus (–1)x = – x by the uniqueness of – x in A5
Trang 7For a more unusual example of an ordered field, let be the set of all rational functions.
Any mathematical system that satisfies these fifteen axioms is called an ordered field
But there are other examples as well The rational numbers are another example
The real numbers are an example of an ordered field
Example 3.2.6
That is, is the set of all quotients of polynomials A typical element of looks like
where the coefficients are real numbers and b k 0
Using the usual rules for adding, subtracting, multiplying, and dividing polynomials,
it is not difficult to verify that is a field
We can define an order on by saying that a quotient such as above is positive iff a n and b k have the same sign; that is, a nb k > 0
For example, since (3)(7) > 0
But since (4)(–7) < 0
,
n n k k
2 5
0,
x
5
0,
Trang 8If p/q and f/g are rational functions, then we say that
That is,
Practice 3.2.7*
Which is larger, We have
The verification that “>” satisfies the order axioms is Exercise 11
It turns out that the ordered field has a number of interesting properties,
as we shall see later
There is one more algebraic property of the real numbers to which we give special
attention because of its frequent use in proofs in analysis, and because it may not familiar
iff 0
q g q g
iff 0
2 2
3
or ?
7
x x
4 2
21
0
7
x x
2 2
3
7
x
x
Trang 9Thus we suppose
that x > y and we must show that there exists an > 0 such that x > y +
Theorem 3.2.8
Let x, y such that x y + for every > 0 Then x y.
Proof: We shall establish the contrapositive
By axiom O1 (the trichotomy law), the negation of x y is x > y.
Question: If x > y, what positive can we add on to y so that x > y + ?
We could take equal to half the distance from x to y.
Let = (x – y) /2 Since x > y, > 0 Furthermore,
as required
,
y y x
Trang 10Recall the definition of absolute value from Section 1.4.
Definition 3.2.9
Let x, y and let a > 0 Then
The basic properties of absolute value are summarized in the following theorem
Theorem 3.2.10
(a) |x| 0, (b) |x | a iff a x a,
(c) |x y| = |x| |y|, (d) |x + y| |x| + |y|
If x , then the absolute value of x, denoted by | x|, is defined by
We will prove parts (b) and (d)
, if 0,
| |
, if 0
x
Trang 11Let x, y and let a > 0 Then
Proof:
Since x = | x | or x = |x|, it follows that |x | x | x|.If |x| a, then we have
Conversely, suppose that a x a.If x 0, then | x | = x a.
And if x < 0, then |x| = x a In both cases, |x| a.
Let x, y and let a > 0 Then
Proof:
As in part (b), we have
Adding the inequalities together, we obtain
which implies that |x + y| |x| + |y| by part (b)
| | | |
| |x x | | and | |x y y | | y
(| | | |)x y x y | | | |,x y
Trang 12Part (d) of Theorem 3.2.10 is referred to as the triangle inequality:
| x + y | | x | + | y |.
Its name comes from its being used with vectors in the plane, where |x| represents
the length of vector x.
x
y
x + y
It says that the length of one side of a triangle is less than or equal to the sum of the
lengths of the other two sides