Small, Medium, and Large Hyperreal Numbers

Field Axioms

A “field” of numbers is any set of objects together with two operations, addition and multiplication that satisfy:

• The commutative laws of addition and multiplication,

&

• The associative laws of addition and multiplication,

&

• The distributive law of multiplication over addition,

• There is an additive identity, ,with for every number .

• There is an multiplicative identity, ,with for every number .

• Each number has an additive inverse, ,with .

• Each nonzero number has a multiplicative inverse, ,with.

The binomial expansion that follows is a consequence of the field axioms.

Hence this formula holds for any pair of numbers and in a field.

To compare sizes of numbers we need an ordering.

Ordered Field Axioms

A a number system is an ordered field if it satisfies the Field Axioms above and has a relation that satisfies:

Every pair of numbers a and b satisfies exactly one of the relations

, , or

If and ,then .

If , then .

If and ,then

In an ordered field the absolute value of a nonzero number is the larger of and .

We want to let be "small" write the differential approximation for ,

, with

Once we show that is small for limited , we have proved that .  Moreover, this is equivalent to "epsilon-delta" uniform approximation on compact sets.

Infinitesimal Numbers

A number in an ordered field is called infinitesimal if it satisfies

for any ordinary natural counting number,

Two hyperreal numbers and are said to be infinitely close, or differ by an infinitesimal, if is infinitesimal.  In this case we write .

This definition is intended to include as an infinitesimal.

NOTE: From now on the previously informal "approximately equal" notation "" is replaced by this precise definition.  An infinitesimal is a number that satisfies .  The point of this section is to show that the technical definition captures the intuitive ideas of Section 1.

Archimedes' Axiom is precisely the statement that the (Dedekind) "real" numbers have no positive infinitesimals.  Keisler's Algebra Axiom is the following:

Keisler's Algebra Axiom

The hyperreal numbers are an ordered field extension of the real numbers.  In particular, there is a positive hyperreal infinitesimal, , satisfying

for any ordinary natural counting number,

It follows from the laws of ordered algebra that there are many different infinitesimals.  For example, the law   ⇒ applied to and says .  Similarly, all the integer multiples of are distinct infinitesimals,

If is a natural number, , for any natural , because when is infinitesimal.

Magnifying the line by makes integer multiples of appear like the integers at unit scale.

Magnification at center with power is simply the transformation , so by laws of algebra, integer multiples of end up the same integers apart for magnification centered at zero.  Similar reasoning lets us place  , , ··· on a magnified line at one half the distance to , one third the distance, etc.

Where should we place the numbers , ···?  On a scale of , they are infinitely near zero, :

Magnification by reveals , but moves infinitely far to the right, for all natural

Laws of algebra dictate many "orders of infinitesimal" such as

The laws of algebra show that near every real number there are many hyperreals, say near

Medium and Large Numbers

A hyperreal number is called limited (or "finite in magnitude") if there is a natural number so that .  If there is no natural bound for a hyperreal number it is called unlimited (or "infinite").

Infinitesimal numbers are limited, being bounded by .

Theorem: Standard Parts of Limited Hyperreal Numbers

Every limited hyperreal number differs from some real number by an infinitesimal, that is, there is a real so that .  This number is called the "standard part" of , .

Proof

Define a Dedekind cut in the real numbers by and .   is the  real number defined by this cut.

A Curious "Paradox"

The real numbers are Dedekind complete.  Sometimes we think of this result as saying the real numbers are the points on a line with no gaps.  The Standard Part Theorem says all the limited hyperreals are clustered around real numbers.  When we take a line with no gaps and add lots of infinitesimals around each point, we create gaps!  The cut in the hyperreals consisting of all numbers that are either negative or infinitesimal on one hand or positive and non-infinitesimal on the other has no number at the cut.  There is no largest infinitesimal because twice that number would be infinitesimal and there is no smallest positive non-infinitesimal, because half of it would be infinitesimal, and then twice that also infinitesimal.

Our microscopic pictures of the hyperreal line do not reveal the gaps as long as we view the microscopic images as the image under similarity transformations   with hyperreal parameters.

Theorem: Computation Rules for Small,  Medium, and Large Numbers

(a) If and are limited, so are and

(b) If and δ are infinitesimal, so is ε + δ.

(c) If δ ≈ 0 and is finite, then . (finite x infsml = infsml)

(d) is still undefined and is unlimited only when .

Proof

These rules are easy to prove as we illustrate with (c).  If is limited, there is a natural number with .  The condition means

| δ | <

The uniform derivative of

Let's apply these rules to show that satisfies the differential approximation with when is limited.  We know by laws of algebra that

, with

If is limited and , (a) shows that is limited and that is also limited.  Condition (b) then shows that proving that for all limited

with whenever .  

Below we will see that this computation is logically equivalent to the statement that , uniformly on compact sets of the real line.  It is really no surprise that we can differentiate algebraic functions using algebraic properties of numbers.  This does not solve the problem of finding sound foundations for calculus using infinitesimals because we need to treat transcendental functions like sine, cosine, log.

Created by Mathematica  (September 22, 2004)