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,

a_1 + a_2 = a_2 + a_1 & a_1  a_2 = a_2  a_1

• The associative laws of addition and multiplication,

a_1 + (a_2 + a_3) = (a_1 + a_2) + a_3 & a_1  (a_2  a_3) = (a_1  a_2)  a_3

• The distributive law of multiplication over addition,

a_1  (a_2 + a_3) = a_1  a_2 + a_1  a_3

• There is an additive identity, 0,with 0 + a = a for every number a.

• There is an multiplicative identity, 1,with 1  a = a for every number a ≠0.

• Each number a has an additive inverse, −a,with a + (−a) = 0.

• Each nonzero number a≠0 has a multiplicative inverse, 1/a ,with a  1/a = 1.

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

(x + Δx)^3 = x^3 + 3x^2Δx + ((3x + Δx)  Δx)  Δx

Hence this formula holds for any pair of numbers x and Δx 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

a = b, a<b, or b<a

• If a<b and b<c,then a<c.

• If a<b, then a + c<b + c.

• If 0<a and 0<b,then 0<a  b .

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

We want to let Δx = δx be "small" write the differential approximation for y = f[x] = x^3,

f[x + δx] - f[x] = f ' [x]  δx + ε  δx

(x + δx)^3 - x^3 = 3x^2δx + ε  δx , with ε = ((3x + δx)  δx)

Once we show that ε is small for limited x, we have proved that f^′[x] = 3 x^2.  Moreover, this is equivalent to "epsilon-delta" uniform approximation on compact sets.

Infinitesimal Numbers

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

| δ | <1/m for any ordinary natural counting number, m = 1, 2, 3, ⋯

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

This definition is intended to include 0 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 δ≈0.  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, δ, satisfying

0<δ<1/m for any ordinary natural counting number, m = 1, 2, 3, ⋯

It follows from the laws of ordered algebra that there are many different infinitesimals.  For example, the law  a<ba + c<b + c applied to a = 0 and b = c = δ says δ<2δ.  Similarly, all the integer multiples of δ are distinct infinitesimals,

⋯< -3δ< -2δ< -δ<0<δ<2δ<3δ<⋯

If k is a natural number, k δ<1/m, for any natural m, because δ<1/(k  m) when δ is infinitesimal.

Magnifying the line by 1/δ makes integer multiples of δ appear like the integers at unit scale.


Magnification at center c with power 1/δ is simply the transformation x 1/δ (x - c), so by laws of algebra, integer multiples of δ end up the same integers apart for magnification centered at zero.  Similar reasoning lets us place  δ/2, δ/3, ··· on a magnified line at one half the distance to δ, one third the distance, etc.


Where should we place the numbers δ^2, δ^3···?  On a scale of δ, they are infinitely near zero, 1/δ (δ^2 - 0) = δ≈0:

Magnification by 1/δ^2 reveals δ^2, but moves δ infinitely far to the right, 1/δ^2 (δ - 0) = 1/δ>m for all natural m = 1, 2, 3, ⋯


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 π = 3.14159⋯

⋯<π - 3δ<π - 2δ<π - δ<π<π + δ<π + 2δ<π + 3δ<⋯

Medium and Large Numbers

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

Infinitesimal numbers are limited, being bounded by 1.

Theorem: Standard Parts of Limited Hyperreal Numbers

Every limited hyperreal number x differs from some real number by an infinitesimal, that is, there is a real r so that x≈r.  This number is called the "standard part" of x, r = st[x].


Define a Dedekind cut in the real numbers by A = { s : s≤x } and B = {s : x<s }.  st[x] 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 (x, y) 1/δ (x - a, y - b)  with hyperreal parameters.

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

(a) If p and q are limited, so are p + q and p  q

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

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

(d) 1/0 is still undefined and 1/x is unlimited only when x≈0.


These rules are easy to prove as we illustrate with (c).  If q is limited, there is a natural number with | q | <k.  The condition δ≈0 means

| δ | < 1/(k  m)
, so | q  δ | <1/m proving that q  δ≈0.

The uniform derivative of x^3

Let's apply these rules to show that f[x] = x^3 satisfies the differential approximation with f^′[x] = 3 x^2 when x is limited.  We know by laws of algebra that

(x + δx)^3 - x^3 = 3x^2δx + ε  δx , with ε = ((3x + δx)  δx)

If x is limited and δx≈0, (a) shows that 3x is limited and that 3x + δx is also limited.  Condition (b) then shows that ε = ((3x + δx)  δx) ≈0 proving that for all limited x

f[x + δx] - f[x] = f^′[x]  δx + ε  δx

with ε≈0 whenever δx≈0.  

Below we will see that this computation is logically equivalent to the statement that Underscript[Lim, Δx0] ((x + Δx)^3 - x^3)/(Δx) = 3x^2, 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)