Transcript Introduction to the HyperReals An extension of the Reals with infinitely small and infinitely large numbers.

Introduction to
the HyperReals
An extension of the Reals
with infinitely small and
infinitely large numbers.
Introduction to the
HyperReals
Descriptive introduction
“Pictures” of the HyperReals
Axioms for the HyperReals
Some Properties (Theorems) of
HyperReals
Descriptive introduction
A complete ordered field extension of the
Reals
(in a similar way that the Reals is a complete
ordered field extension of the Rationals)
Contains infinitely small numbers
Contains infinitely large numbers
Has the same logical properties as the
Reals
The HyperReals
Near the Reals
The HyperReals
Far from the Reals
Axioms for the HyperReals
Axioms common to the Reals
Algebraic Axioms
Order Axioms
Completeness Axiom
Axioms unique to the HyperReals
Extension Axiom
Transfer Axiom
Algebraic Axioms
 Closure laws: 0 and 1 are numbers. If a and b are
numbers then so are a+b and ab.
 Commutative laws: a + b = b + a and ab = ba
 Associative laws: a + (b + c) = (a + b) +c and
a(bc)=(ab)c
 Identity laws: 0 + a = a and 1a = a
 Inverse laws:
For all a, there exists number –a such that a + (-a) = 0 and
if a  0, then there exist number a-1 such that a(a-1) = 1
 Distributive law: a( b+c ) = ab + ac
Order Axioms
The is a set P of positive numbers which satisfies:
 If x, y are elements of P then x + y is an element of P.
 If x, y are elements of P then xy is an element of P.
 If x is a number then exactly one of the following must
hold:
x = 0,
x is an element of P or
-x is an element of P.
Definition of < and >
a < b
if and only if
(b - a) is an element of P
i.e. (b - a) is positive
a > b if and only if b < a
Properties of <
0 < 1
 Transitive law:
If a<b and b<c then a<c.
 Trichotomy law:
Exactly one of the relations a<b, a = b, b<a, holds.
 Sum law:
If a<b, then a+c<b+c.
 Product law:
If a<b and 0<c, then ac<bc.
 Root law:
For a>0 and positive integer n,
there is a number b>0 such that bn = a.
Completeness Axiom
A number b is said to be an upper bound of a
set of numbers A if b  x for all x in A.
A number c is said to be an least upper bound
of the set of numbers A if c is an upper bound
of A and b  c for all upper bounds b of A.
Completeness Axiom:
Every non-empty set of numbers which is
bounded above has a least upper bound.
Axioms unique to the
HyperReals
Extension Axiom
Transfer Axiom
Note: Actually these axioms are all that are
needed as for the HyperReals as the previous
axioms can be derived from these two axioms.
Extension Axiom
 The set R of real numbers is a subset of the set R* of
hyperreal numbers.
 The order relation <* on R* is an extension of the
order < on R.
 There is a hyperreal number  such that 0 <* 
and
 <* r for each positive real number r.
 For each real function f there is a given hyperreal
function f* which has the following properties
domain(f) = R  domain(f*)
 f(x) = f*(x) for all x in domain(f)
range(f) = R  range(f*)
(Extension actually applies to any standard set built from the Reals.)
Transfer Axiom
 (Function version)
Every real statement that holds for one or more particular
real functions holds for the hyperreal extensions of
these functions
 (Full version)
Every standard statement (about sets built from the Reals)
is true if and only if the corresponding non-standard
statement (about sets built from HyperReals - formed by
adding the * operator) is true.
Example: Deriving the Commutative Laws
from the Extension and Transfer Axioms
 Commutative laws for the Reals:
S: aR bR a+b=b+a and a•b=b•a.
The Extension axiom gives us R*, +*, •*, and * and
the Transfer axiom tells us that S*, the commutative
laws for the HyperReals is true.
 Commutative laws for the HyperReals:
S*: a*R* b*R* a+*b=b+*a and a•*b=b•*a.
Definition: Infinitesimal
A HyperReal number b is said to be:
positive infinitesimal if b is positive but
less than every positive real number.
negative infinitesimal if b is negative but
greater than every negative real number.
infinitesimal if b is either positive
infinitesimal, negative infinitesimal, or 0.
Definitions:
Finite and Infinite
A HyperReal number b is said to be:
finite if b is between two real numbers.
positive infinite if b is greater than every
real number.
negative infinite if b is smaller than every
real number.
infinite if b is positive infinite or negative
infinite.
Theorem:
The only real infinitesimal number is 0.
 Proof:
Suppose s is real and infinitesimal.
Then exactly one of the following is true:
s is negative, s = 0, or s is positive.
If s is negative then it is a negative infinitesimal and hence
r < s for all negative real numbers r. Since s is negative
real then s < s which is nonsense.
Thus s is not negative.
Likewise if s is positive we get s < s. So s is not positive.
Hence s = 0.
The Standard Part Principle
Theorem:
For every finite HyperReal number b, there is
exactly one real number r infinitely close to b.
Definition:
If b is finite then the real number r, with r b, is
called the standard part of b.
We write r = std( b ).
Proof of the Standard Part Principle
Uniqueness:
Suppose r, s  R and r  b and s  b.
Hence r  s.
We have r-s is infinitesimal and real.
The only real infinitesimal number is 0.
Thus r-s = 0 which implies r = s.
Existence:
Since b is finite there are real numbers s and t with s < b < t.
Let A = { x | x is real and x < b }. A is non-empty since it contains s
and is bounded above by t. Thus there is a real number r which is
the least upper bound of A.
We claim r  b.
Suppose not. Thus r  b and Hence r-b is positive or negative.
Case r-b is positive. Since r-b is not a positive infinitesimal there is a
positive real s, s < r-b which implies b < r-s so that r-s is an upper
bound of A. Thus r-s  r but r-s < r. Thus r-b is not positive.
Case r-b is negative. Since r-b is not a negative infinitesimal there is a
negative real s, r-b<s which implies r-s < b so r-s is in A and
hence r  r-s but r < r-s, since s<0.
Thus r-b is infinitesimal. So r  b.
Infinite Numbers Exist
Let  be a positive infinitesimal.
Thus 0 <  < r for all positive real number r.
Let r be a positive real number. Then so is 1/r.
Therefore 0 <  < 1/r and so 1/ > r.
Let H = 1/ . Thus H > r for all positive real
number r.
Therefore H is an infinite number.
General Approach to Using
the HyperReals
Start with standard (Real) problem
Extend to non-standard (HyperReal) - adding *
Find solution of non-standard problem
Take standard part of solution to yield standard
solution - removing *
Note: In practice we normally switch between
Real and HyperReal without comment.
Theorem 1: Rules for Infinitesimal,
Finite, and Infinite Numbers
Th. Assume that  and d are infinitesimals; b,c
are hyperreal numbers which are finite but
not infinitesimal; and H, K are infinite
hyperreal numbers; and n an integer . Then
Negatives:
- is infinitesimal.
-b is finite but not infinitesimal.
-H is infinite.
(Theorem cont)
Reciprocals:
1/ is infinite.
1/b is finite but not infinitesimal.
1/H is infinitesimal.
Sums:
 +d is infinitesimal.
b+ is finite but not infinitesimal.
b+c is finite (possibly infinitesimal).
H+ and H+b are infinite.
(Theorem cont)
Products:
*d and *b are infinitesimal
b*c is finite but not infinitesimal.
H *b and H*K are infinite.
Roots:
If  >0,
n
 is infinitesimal.
If b>0, n b is finite but not infinitesimal.
If H>0, n H is infinite.
(Theorem cont)
Quotients:
/ b, / H, and b/ H are infinitesimal
b/c is finite but not infinitesimal.
b/ , H/, and H/b are infinite provided   0.
Indeterminate Forms
Examples
Indetermina
te Form
infinitesimal
finite
(equal to 1)
infinite
/d
2/
/
/2
H/K
H/H2
H/H
H2/H
H*
H*(1/H2)
H*(1/H)
H2*(1/H)
H+K
H+(-H)
(H+1)+(-H)
H+H
Theorem 2
1. Every hyperreal number which is between two
infinitesimals is infinitesimal.
2. Every hyperreal number which is between two
finite hyperreal numbers is finite.
3. Every hyperreal number which is greater than
some positive infinite number is positive
infinite.
4. Every hyperreal number which is less than
some negative infinite number is negative
infinite.
Definitions:
Infinitely Close
Two numbers x and y are said to be
infinitely close ( written x  y) if and only
(x-y) is infinitesimal.
Theorem 3.
Let a, b, and c be hyperreal numbers. Then
1. a  a
2. If a  b, then b  a
3. If a  b and b  c then a  b.
(i.e.,  is an equivalence relation.)
Theorem 4.
Assume a  b, Then
1. If a is infinitesimal, so is b.
2. If a is finite, so is b.
3. If a is infinite, so is b.
Definition: Standard Part
Let b be a finite hyperreal number.
The standard part of b, denoted by
st(b), is the real number which is
infinitely close to b.
Note this means:
1. st(b) is a real number
2. b = st(b) +  for some infinitesimal .
3. If b is real then st(b) =b.
Theorem 5.
Let a and b be finite hyperreal numbers.
Then
1.
2.
3.
4.
st(-a) = -st(a).
st(a+b) = st(a) + st(b).
st(a-b) = st(a) - st(b).
st(ab) = st(a) * st(b).
5. If st(b)  0 , then st(a/b) = st(a)/st(b).
(theorem 5 cont.)
6. st(a)n = st(an).
n
n st (a)
If
a

0,
then
st
(
a
)

7.
.
8. If a  b, then st (a)  st (b) .
Example 1: st(a)
Assume c  4 and c  4.
c  2c  24
(c  6)(c  4)
st (
)  st (
)
2
c  16
(c  4)(c  4)
c6
st (c  6)
 st (
)
c4
st (c  4)
st (c)  st (6) 4  6 10 5


 
st (c)  st (4) 4  4 8 4
2
Example 2: st(a)
Assume H is a positive infinite hyperreal number.
2 H 3  5 H 2  3H
H 3 (2 H 3  5H 2  3H )
st (
)  st ( 3
)
3
2
3
2
7 H  2H  4H
H (7 H  2 H  4 H )
1
2
1
2
2  5 H  3H
st (2  5 H  3H )
 st (
)
1
2
1
2
7  2H  4H
st (7  2 H  4 H )
1
2
st (2)  st (5 H )  st (3H ) 2  0  0 2



1
2
st (7)  st (2 H )  st (4 H ) 7  0  0 7
Example 3: st(a)
Assume e is a nonzero infinitesimal.

 (5  25   )
st (
)  st (
)
5  25  
(5  25   )(5  25   )
 (5  25   )
 (5  25   )
 st (
)  st (
)
25  (25   )

  st (5  25   )   st (5)  st ( 25   )
  st (5)  st (25   )  5  5  10