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 HyperRealsAxioms for the HyperRealsSome 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 numbersContains infinitely large numbersHas 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 AxiomTransfer 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: aR bR 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) problemExtend 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 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: +is infinitesimal.b+ is finite but not infinitesimal.b+c is finite (possibly infinitesimal).H+ and H+b are infinite.

(Theorem cont)

Products:andb are infinitesimal b*c is finite but not infinitesimal.H *b and H*K are infinite.

Roots:If >0, is infinitesimal.If b>0, is finite but not infinitesimal.If H>0, is infinite.

n n b

n H

(Theorem cont)

Quotients:b, H, andbH are infinitesimal b/c is finite but not infinitesimal.b/ , H/andH/b are infinite provided0.

Indeterminate Forms

Indeterminate Form

infinitesimal

finite(equal to 1)

infinite

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

Examples

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 a2. If a b, then b a3. If a b and b c then a b.

(i.e., is an equivalence relation.)

Theorem 4.

Assume a b, Then1. 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 number2. 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. st(-a) = -st(a).2. st(a+b) = st(a) + st(b).3. st(a-b) = st(a) - st(b).4. st(ab) = st(a) * st(b).5. If st(b) , then st(a/b) = st(a)/st(b).0

(theorem 5 cont.)

6. st(a)n = st(an).

7. .8. .

0, ( ) ( )n nIf a then st a st a

, ( ) ( )If a b then st a st b

Example 1: st(a)

Assume c 4 and c 4.

2

2

2 24 ( 6)( 4)( ) ( )

16 ( 4)( 4)

6 ( 6)( )

4 ( 4)

( ) (6) 4 6 10 5

( ) (4) 4 4 8 4

c c c cst st

c c c

c st cstc st c

st c st

st c st

Example 2: st(a)

Assume H is a positive infinite hyperreal number.

3 2 3 3 2

3 2 3 3 2

1 2 1 2

1 2 1 2

1 2

1 2

2 5 3 (2 5 3 )( ) ( )7 2 4 (7 2 4 )

2 5 3 (2 5 3 )( )7 2 4 (7 2 4 )

(2) (5 ) (3 ) 2 0 0 2

(7) (2 ) (4 ) 7 0 0 7

H H H H H H Hst st

H H H H H H H

H H st H Hst

H H st H H

st st H st H

st st H st H

Example 3: st(a)

Assume e is a nonzero infinitesimal.

(5 25 )( ) ( )5 25 (5 25 )(5 25 )

(5 25 ) (5 25 )( ) ( )25 (25 )

(5 25 ) (5) ( 25 )

(5) (25 ) 5 5 10

st st

st st

st st st

st st