## Extending the Real NumbersDavid R. Tribbledavid@tribble.com Revision 1.2, 2007-08-25 |

- Introduction
- More Numbers
- Even More Numbers
- Unreal Numbers
- How Many Numbers?
- Arithmetic
- Rings and Fields
- Integer Numbers
- Visualizing the Numbers
- Multiplicative Inverses
- Extending the Numbers
- Nested Hierarchies
- Complex Numbers
- n-Dimensional Numbers
- Still Bigger Sets
- An Uncountable Hierarchy
- Loose Ends
- References

The *real numbers* exist within standard arithmetic, and are conventionally
designated as the set **R**.
Our goal in this discussion is to create a set of numbers similar to the reals,
having essentially all the same properties as the reals and obeying most of the
same arithmetic rules.
(We're assuming the usual rules and arithmetic operations over the real numbers,
such as addition, multiplication, etc.)

We define a new type of number, which we'll call a *suprareal*,
that looks and acts (almost) identical to a real number,
but which is not a real itself.
Furthermore, we specify that these suprareals are members of set **H**
(the Greek letter *eta*), which is a set disjoint from **R**.

First we define how the suprareals are related to the reals.
Since we want the suprareals to exist outside the range of all reals
(i.e., sets **H** and **R** do not have any members in common),
we do this by establishing an *order relation*
between the suprareals and the reals:

Definition 1.

x<hfor allx∈Rand 0 <h∈H, and

x>hfor allx∈Rand 0 >h∈H.

This can be written in a slightly simpler form:

|x| < |h| for allx∈Randh∈H.

We can define set **H** more formally as:

Definition 2.

H^{+}= {h:h>xfor allx∈R}.

H^{−}= {h:h<xfor allx∈R}.

H=H^{−}∪H^{+}.

These definitions establish the *less than* (<) and
*greater than* (>)
ordering relations between the suprareals and the reals,
stating that the magnitude (absolute value) of every suprareal
is greater than the magnitude of every real number.
This guarantees that all of the suprareals lie outside the range of the reals.
We state this formally in our first theorem about suprareals:

Theorem 1.

For allh∈H,h∉R.

The proof is rather obvious, since by definition any member of **H** has a
magnitude greater than any member of **R**, therefore no member of **H**
can be a member of **R**.
In other words, no suprareal is a real.
The converse of this theorem is also true:

Theorem 2.

For allx∈R,x∉H.

In other words, no real is a suprareal. The proof is essentially the same as that for Theorem 1.

At this point in the discussion, astute readers will notice
that set **H** is empty, and that no suprareals actually exist.
This is a correct observation, because there is nothing in standard arithmetic
that in fact allows a suprareal to exist, i.e., there is no way to construct
a number with a magnitude greater than all real numbers.
So we must explicitly call suprareals into existence within our new system:

Axiom 1.

There exists η_{1}such thatx< η_{1}for allx∈R.

Or, equivalently:

There exists η_{1}∈H^{+}.

This establishes the existence of a single suprareal, η_{1}
(*eta*_{1}), which is a member of **H**.
It can be considered to be the "seed" or "primordial element"
of the set of suprareals, providing the foundation for them
in a way that is similar to the way **i** (√−1)
provides a foundation for the imaginary numbers.
For now we'll simply designate it the *fundamental constant*
of the suprareals.
(This will be discussed in more detail later on.)
We won't go into any details about what the actual *value* of
η_{1} is just yet, other than to note that it is positive.
Its existence alone is sufficient to continue the discussion.

Axiom 1 defines an extension to the system of standard real arithmetic.
This is a key point to be remembered, because anything we say about suprareals
applies only within our newly created framework, and is meaningless
within the framework of standard arithmetic.
In particular, we note that η_{1} is a number that exists
within our new system but is not a real,
and thus cannot exist within standard arithmetic.

At this stage of the discussion, set **H** is pretty sparse,
having the only existing suprareal to this point, η_{1},
as its only member.
So let's add a couple of rules to allow more suprareals to exist:

Axiom 2.

For allx∈R,x+η_{1}∈H.

Axiom 3.

For allx∈Rand x ≠ 0,xη_{1}∈H.

Now we have an entire universe of suprareals,
because for each real *x* there are corresponding suprareals
*x*+η_{1} and *x*η_{1}.
Note, however, that *x*η_{1} is specifically excluded from
being zero.
The reason for this is that allowing zero as a suprareal would introduce
an inconsistency with Definitions 1 and 2.
To see why this is so, we introduce the next theorem:

Theorem 3.

0 ∉H.

This is rather trivial to prove, since by Theorem 2, no member of **R**
is a member of **H**, i.e., no real is a suprareal,
and since 0 is a real, it cannot be a suprareal.

If we allowed 0 to be a suprareal, this theorem would be false, and
0 would be a member of **H**.
However, this would create an inconsistency with the ordering relations
of Definitions 1 and 2,
which state that all suprareals are greater than (or less than) all reals,
which would imply that 0 is greater than (or less than) itself.
Disallowing 0 as a suprareal eliminates this contradiction.

**H** is populated with a lot of suprareals at this point,
but the set is still not complete.
We provide a couple of more axioms which allow many more suprareals
to exist:

Axiom 4.

For allg∈H,g+η_{1}∈H.

Axiom 5.

For allg∈H,gη_{1}∈H.

These axioms lead us to two more theorems:

Theorem 4.(Partial closure under addition)

For allh,g∈H^{+},h+g∈H^{+}, and

for allh,g∈H^{−},h+g∈H^{−}.

This theorem states that the set of positive suprareals is *closed*
under addition.
The sum of any two members of **H**^{+} is always another member
of **H**^{+}.
The same is true for all of the negative suprareals as well,
i.e., **H**^{−} is closed under addition.

However, the entire set **H** is *not* closed under addition, because
the sum of two suprareals could be either another suprareal or it could be a
real.
For example, given
*h* = *x*+η_{1} for some real *x*,
and *g* = (−1)η_{1},
then *h*+*g*
= *x*+η_{1} + (−1)η_{1}
= *x*, which is a real.
If it were not for these kinds of sums, all of **H**
would be closed under addition.

Theorem 5.(Closure under multiplication)

For allh,g∈H,hg∈H.

This theorem states that the set of suprareals is closed under
multiplication.
This means that multiplying any two members of **H** results in another
member of **H**.
Multiplying suprareals always results in another suprareal,
and there are no special exceptions to worry about.
(We will see later that **H** is not a *ring*, but is part of
a larger set that is.)

It might be tempting to think that every suprareal is either
the sum of the fundamental constant η_{1} and some real number
(*x*+η_{1}), but this is not true.
We can create a suprareal sum that is greater than
any sum involving a real number:

η_{1}+η_{1}= 2η_{1}>x+η_{1}, for allx∈R.

Likewise, we can create a suprareal product that is greater than any product involving a real number:

η_{1}η_{1}= η_{1}^{2}>xη_{1}, for allx> 0 ∈R.

Similarly, for any given suprareal *g*+η_{1} or
*g*η_{1}, we can create a suprareal that is larger than it:

η_{1}^{3}>gη_{1}= (xη_{1})η_{1}=xη_{1}^{2}, for allg=xη_{1}andx> 0 ∈R.

In fact, it would appear that every suprareal can be represented as
a polynomial over powers of η_{1}.
This is a very important observation, so much so that
we can call this the "Fundamental Theorem of Suprareals" (FTSR):

Theorem FTSR.

For allh∈H,

h=x+x_{1}η_{1}^{1}+x_{2}η_{1}^{2}+x_{3}η_{1}^{3}+ ... +x_{n}η_{1}^{n},

wherex,x_{1},x_{2}, ...,x_{n}∈R, and at least onex_{j}≠ 0.

Some examples of polynomial suprareals:

1. η_{1}.

2. η_{1}^{2}.

3. η_{1}^{18}+ 1.

4. 2η_{1}^{2}− 3η_{1}^{1}+ 5.

5. −^{3}/_{4}η_{1}^{5}+^{7}/_{16}η_{1}^{3}+^{8}/_{3}η_{1}^{2}−^{9}/_{20}.

Since the magnitude of any given suprareal is greater than any
finite real number, we could say that the suprareals must be *non*finite,
or *in*finite numbers.
But we must take care when we talk about "infinite numbers".
There are various concepts of "infinity", "divergence", and "unboundedness"
within mathematics, and we want to be careful about how these concepts
apply to our suprareals.

Since the suprareals are designed to obey most of the rules of normal
arithmetic (which we will see later), they do not behave like the
*infinite ordinal numbers* nor the *transfinite cardinal numbers*
of standard set theory.
In particular, a given suprareal *h* obeys the inequality
*h* < 1+*h*, which is not how an infinite ordinal
or a transfinite cardinal behaves.

The term "infinite number" is an unfortunate one, and it probably makes more sense to consider the suprareals as "extremely large" numbers, or perhaps "unreal" numbers, to emphasize their relation to, and difference from, the reals. The only thing that really makes the suprareals different from the reals is the ordering relation of Definition 1. Thus it is more meaningful to think of the suprareals simply as a second set of real-like numbers that exist apart from the normal set of reals.

So we do not claim that the suprareals are infinite numbers at this point, and we want to preserve the meaning of ∞, the "usual" infinity, within our framework. We will go into more detail about this later on, but for now we simply mention this theorem in passing:

Theorem I.(Infinity)

For allh∈H, −∞ <h< +∞.

We might be tempted to assume that there are more suprareals than real numbers,
that **H** is a larger set than **R**.
However, this is not the case.

Theorem 6.

|H| = |R|.

We prove that the two sets are the same size
(or more precisely, that they have the same *cardinality*),
first by noting that every real *x* (other than 0)
can be mapped to a unique suprareal *x*η_{1}.
This shows that |**R**| ≤ |**H**|,
which we already suspected.
Next we note that since
each suprareal *g* can be represented
as a polynomial over powers of η_{1},
we can map each *g* to an n-tuple of reals
comprising the coefficients of the polynomial for *g*, which is
<*x*_{1}, *x*_{2}, ..., *x*_{n}>.
Furthermore, since every n-tuple of reals can be mapped one-to-one to a unique
real (one way to do this is to use Cantor's method of intertwining the digits of
each real in the tuple to create a single new real), it follows that every
suprareal can be uniquely mapped to a single real.
This shows that |**H**| ≤ |**R**|.
Thus |**R**| = |**H**|.
Or in other words, there are as many suprareals as there are real numbers.

Another way of looking at this is to realize that since every suprareal can be
written as an *n*-order polynomial, the *n* coefficients of each
suprareal are taken from **R**^{n} space.
And since the cardinality of every **R**^{n} space is the same as
the cardinality of **R** space, the cardinalities of **H** and **R**
are likewise the same.

Now we have to formalize the arithmetic operations we want to allow to apply to the suprareals. Our goal is that the suprareals behave pretty much the same under the usual arithmetic operations as the reals (possibly with a few caveats). We provide additional axioms to build upon:

Axiom 6.(Additive commutativity)

h+x=x+h, for allh∈Handx∈R.

Axiom 7.(Multiplicative commutativity)

hx=xh, for allh∈Handx∈R.

These axioms lead naturally to a few more useful theorems (which are provided without proof):

Theorem 7a.(Additive commutativity)

h+g=g+h, for allh,g∈H.

Theorem 7b.(Multiplicative commutativity)

hg=gh, for allh,g∈H.

Theorem 8a.(Additive associativity)

(h+x)+y=h+(x+y), for allh∈Handx,y∈R.

Theorem 8b.(Additive associativity)

(h+g)+x=h+(g+x), for allh,g∈Handx∈R.

Theorem 8c.(Additive associativity)

(h+g)+u=h+(g+u), for allh,g,u∈H.

Theorem 9a.(Multiplicative associativity)

(hx)y=h(xy), for allh∈Handx,y∈R.

Theorem 9b.(Multiplicative associativity)

(hg)x=h(gx), for allh,g∈Handx∈R.

Theorem 9c.(Multiplicative associativity)

(hg)u=h(gu), for allh,g,u∈H.

Theorem 10.(Additive identity)

0+h=h, for allh∈H.

Theorem 11.(Multiplicative identity)

1•h=h, for allh∈H.

Note that 0 and 1 are the additive and multiplicative identities
for the suprareals, but they are not members of **H** themselves
(because they are reals).

Theorem 12a.(Distributive law)

h(x+y) =hx+hy, for allh∈Handx,y∈R.

Theorem 12b.(Distributive law)

h(g+x) =hg+hx, for allh,g∈Handx∈R.

Theorem 12c.(Distributive law)

h(g+u) =hg+hu, for allh,g,u∈H.

We provide a proof of the last theorem to give some idea of how most of these are proved:

Proof 12c.

1.h=x+x_{1}η_{1}^{1}+x_{2}η_{1}^{2}+ ... +x_{n}η_{1}^{n}. (FTSR)

2.g=y+y_{1}η_{1}^{1}+y_{2}η_{1}^{2}+ ... +y_{n}η_{1}^{n}. (FTSR)

3.u=v+v_{1}η_{1}^{1}+v_{2}η_{1}^{2}+ ... +v_{n}η_{1}^{n}. (FTSR)

4. Thenh(g+u) =

(x+x_{1}η_{1}^{1}+x_{2}η_{1}^{2}+ ... +x_{n}η_{1}^{n}) ((y+y_{1}η_{1}^{1}+y_{2}η_{1}^{2}+ ... +y_{n}η_{1}^{n}) + (v+v_{1}η_{1}^{1}+v_{2}η_{1}^{2}+ ... +v_{n}η_{1}^{n})).

5. Andhg+hu=

(x+x_{1}η_{1}^{1}+x_{2}η_{1}^{2}+ ... +x_{n}η_{1}^{n}) • (y+y_{1}η_{1}^{1}+y_{2}η_{1}^{2}+ ... +y_{n}η_{1}^{n}) +

(x+x_{1}η_{1}^{1}+x_{2}η_{1}^{2}+ ... +x_{n}η_{1}^{n}) • (v+v_{1}η_{1}^{1}+v_{2}η_{1}^{2}+ ... +v_{n}η_{1}^{n}).

6. Expanding (4) yields

[x(y+v) +x(y_{1}+v_{1})η_{1}^{1}+x(y_{2}+v_{2})η_{1}^{2}+ ... +x(y_{n}+v_{n})η_{1}^{n}] +

[x_{1}(y+v)η_{1}^{1}+x_{1}η_{1}^{1}(y_{1}+v_{1})η_{1}^{1}+x_{1}η_{1}^{2}(y_{2}+v_{2})η_{1}^{1}+ ... +x_{1}η_{1}^{n}(y_{n}+v_{n})η_{1}^{1}] +

[x_{2}(y+v)η_{1}^{2}+x_{2}η_{1}^{1}(y_{1}+v_{1})η_{1}^{2}+x_{2}η_{1}^{2}(y_{2}+v_{2})η_{1}^{2}+ ... +x_{2}η_{1}^{n}(y_{n}+v_{n})η_{1}^{2}] +

... +

[x_{n}(y+v)η_{1}^{n}+x_{n}η_{1}^{1}(y_{1}+v_{1})η_{1}^{n}+x_{n}η_{1}^{2}(y_{2}+v_{2})η_{1}^{n}+ ... +x_{n}η_{1}^{n}(y_{n}+v_{n})η_{1}^{n}].

7. Expanding (5) yields the same as (6).

8. Thush(g+u) =hg+hu.

We note that *additive inverses* and
subtraction are also defined over the suprareals:

Theorem 13.(Additive inverse)

For allh∈H, −hexists such that −h+h= 0.

Theorem 14.(Subtraction)

h−g=h+ −g, for allh,g∈H.

Some example equations involving suprareals:

1. (4η_{1}^{2}+ η_{1}^{1}− 5) + (−2η_{1}^{3}− 3η_{1}^{1}+ 6) = (−2η_{1}^{3}+ 4η_{1}^{2}− 2η_{1}^{1}+ 1).

2. (η_{1}^{2}+^{15}/_{8}η_{1}^{1}−^{3}/_{2}) + (−^{1}/_{2}η_{1}^{2}+^{3}/_{4}η_{1}^{1}+^{3}/_{5}) = (−^{1}/_{2}η_{1}^{2}+^{21}/_{8}η_{1}^{1}−^{9}/_{10}).

3. (3η_{1}^{2}+ 2) • (η_{1}^{1}−^{1}/_{2}) = (3η_{1}^{3}−^{3}/_{2}η_{1}^{2}+ 2η_{1}^{1}− 1).

As mentioned earlier,
Theorems 4 and 5 state that the suprareals are almost closed under
addition and multiplication.
Axioms 6 and 7 state that addition and multiplication are also
*commutative* operations.
Theorem 12c further states that these operations are *distributive*
as well.
Theorem 13 defines *additive inverses* for the suprareals.
With all of these properties, the positive suprareals and
the negative suprareals almost form *commutative rings*,
(**H**^{+}, +, •) and
(**H**^{−}, +, •).
However, because the additive and multiplicative identities (0 and 1) are not
members of **H**^{+} or **H**^{−},
this prevents them from forming rings.

Also recall that there are exceptional cases where adding suprareals results in
reals,
which prevents the entire set of suprareals **H** from forming a
commutative ring.
However, since suprareal sums are either suprareals or reals,
and the additive and multiplicative identities 0 and 1 are reals,
we can combine the suprareals and the reals into a set that is
completely closed under addition and multiplication:

Theorem 15a.(Closure under addition)

For allh,g∈H∪R,h+g∈H∪R.

Theorem 15b.(Closure under multiplication)

For allh,g∈H∪R,h•g∈H∪R.

Theorem 15c.(Additive identity)

For allh∈H∪R, 0 ∈H∪Rand 0+h=h.

Theorem 15d.(Multiplicative identity)

For allh∈H∪R, 1 ∈H∪Rand 1•h=h.

Theorem 15e.(Additive inverse)

For allh∈H∪R, −h∈H∪Rand −h+h= 0.

This means that the combined set **H** ∪ **R** is a ring:

Theorem 16.

(H∪R, +, •) is a commutative ring.

The reals also form a *field*, which means that they form a ring
that is closed under the division operation, or more precisely,
that every real except zero has a *multiplicative inverse*.
We have not defined division or multiplicative inverses for the suprareals,
however, so (**H** ∪ **R**, +, •)
is *not* a field.
(Later we will see that it is part of a larger set that is, though.)

If we restrict the coefficients of each suprareal polynomial to the integers,
we get a subset of the suprareals that act something like real integers.
That is, each suprareal that is a polynomial with integer coefficients
over powers of η_{1} is something we can call a *suprainteger*:

Definition 3.

supraintegerp=k+k_{1}η_{1}^{1}+k_{2}η_{1}^{2}+ ... +k_{n}η_{1}^{n},

wherek,k_{1},k_{2}, ...,k_{n}∈Z, and at least onek_{j}≠ 0.

For convenience, we place these numbers into their own set **H**_{I},
which is a subset of **H**:

Definition 4.

H_{I}= {p:pis a suprainteger }.

Like the suprareals, the supraintegers are almost closed under addition and are completely closed under multiplication:

Theorem I-1.(Partial closure under addition)

For allp,q∈H_{I}^{+},p+q∈H_{I}^{+}, and

for allp,q∈H_{I}^{−},p+q∈H_{I}^{−}.

Theorem I-2.(Closure under multiplication)

For allp,q∈H_{I},pq∈H_{I}.

While **H**_{I}^{+} and
**H**_{I}^{−} are closed under addition and
multiplication, the additive and multiplicative identities 0 and 1 are
not members of either set.
In addition, **H**_{I} is not completely closed under addition,
because some supraintegers sums are reals.
However, combining set **H**_{I} with the integers produces a set
that is completely closed under addition and multiplication:

Theorem I-3.(Additive closure)

For allp,q∈H_{I}∪Z,p+q∈H_{I}∪Z.

Theorem I-4.(Additive identity)

For allp∈H_{I}∪Z, 0 ∈H_{I}∪Zand 0+p=p.

Theorem I-5.(Multiplicative identity)

For allp∈H_{I}∪Z, 1 ∈H_{I}∪Zand 1•p=p.

The combined set is therefore a ring:

Theorem I-6.

(H_{I}∪Z, +, •) is a commutative ring.

A few examples of supraintegers arithmetic:

1. η_{1}+ η_{1}= 2η_{1}.

2. 2η_{1}+ 3η_{1}= 5η_{1}.

3. (−7η_{1}^{3}− 4η_{1}^{2}+ 1) + (2η_{1}^{2}+ 11η_{1}^{1}− 9) = −7η_{1}^{3}− 2η_{1}^{2}+ 11η_{1}^{1}− 8.

4. 8η_{1}^{2}+ −8η_{1}^{2}= 0.

5. 2η_{1}• 3η_{1}= 6η_{1}^{2}.

6. (−7η_{1}^{3}− 4η_{1}^{2}+ 1) (2η_{1}^{2}+ 11η_{1}^{1}− 9) = −14η_{1}^{5}− 85η_{1}^{4}− 107η_{1}^{3}+ 38η_{1}^{2}+ 11η_{1}^{1}− 9.

The simplest subset of supraintegers are the "counting" suprareals:

Definition 5.

p=k+η_{1}, for allk∈Z.

This defines the sequence:

..., −3+η_{1}, −2+η_{1}, −1+η_{1}, η_{1}, 1+η_{1}, 2+η_{1}, 3+η_{1}, ...

An interesting thing to notice about this sequence is that there is no least positive suprainteger:

Theorem I-7.

k+η_{1}> 0, for allk∈Z.

This is a direct consequence of the more general fact that there is no least positive suprareal. Likewise, there is no largest negative suprainteger:

Theorem I-8.

k−η_{1}< 0, for allk∈Z.

These observations are of importance to the next section.

It is easy to visualize the set of real numbers **R** as a
continuous line (the "real number line"), centered at zero and
extending infinitely far to the right through the positive reals
and infinitely far to the left through the negative reals.

Visualizing the set of suprareals **H** is a little trickier, however,
because even though the suprareals act like real numbers in almost every
respect, they do not form a single contiguous set.
More precisely, the positive suprareals (**H**^{+}) are separated
from the negative suprareals (**H**^{−}) by a kind of "gap".
This can be seen by starting on the positive suprareal line at
η_{1} and moving progressively to the left by
repeatedly subtracting 1, and noting that there is no least positive suprareal.
By the same token, start on the negative suprareal line
at −η_{1} and moving progressively to the right,
repeatedly adding 1, and note that there is no greatest negative suprareal.
This is formalized in these simple theorems
(which are generalizations of Theorems I-7 and I-8,
and are direct consequences of Axiom 2):

Theorem 17a.

For allh∈H^{+},h−1 ∈H^{+}.

Theorem 17b.

For allh∈H^{−},h+1 ∈H^{−}.

In other words, there is no point where the negative suprareals "meet" the positive suprareals. In fact, not only do the two half-sets not meet, there is a large gap separating them, as large as the entire real number line. Remember that all positive suprareals are greater than all the reals, and likewise all negative suprareals are less than all the reals (by Definitions 1 and 2).

So we're left with a split number line, with the right half containing
the positive suprareals and the left half containing the negative suprareals.
To form a visualization that makes sense, we mark the fundamental
constant +η_{1} on the positive half
and −η_{1} on the negative half.
These are the "center points" for the two suprareal half-sets.

Combining the two sets **R** and **H** gives us an
*extended number line* with zero in the center.
In other words, we can fit the real number line between the halves of the
suprareal line, capturing the essence of the ordering relationship
between the negative suprareals, the reals, and the positive suprareals.

As useful as this diagram is, it's important to realize that visualizing
the union of the reals and the suprareals (**R** ∪ **H**)
as a single contiguous extended real number line is misleading,
because there is no continuity or "connecting points" between the negative reals
and the negative suprareals,
and likewise none between the positive reals and the positive suprareals.
No matter how far to the right we proceed along the positive reals,
we never reach a "greatest positive" real that is adjacent to a
"least positive" suprareal.
There is no point where the reals "meet" the suprareals, on either the positive
or the negative half of the lines.
Thus the union of the two sets contains two "gaps", or discontinuities,
between them.

Earlier it was implied that defining multiplicative inverses
(a reciprocal or division operator) over the suprareals
could possibly make them a field.
This is not quite correct; what happens is that once multiplicative inverses
are introduced into our system, a new kind of number is brought into existence.
Given any real *x* greater than 1,
its reciprocal ^{1}/_{x} is a positive real number
less than 1 but greater than 0.
Similarly, given any positive suprareal *h*, which by definition is
greater than any real, its reciprocal ^{1}/_{h}
is a new positive number that is less than any real but still
greater than 0.
We formalize this with an axiom that defines into existence a specific
multiplicative inverse of the suprareals:

Addendum 1 |
---|

While Axiom 8 is correct as shown, the equivalent statement for it
is incorrect.
We can define a multiplicative inverse for
In order to allow Specifically, how do we determine whether such a series converges or
not?
Until we can answer these questions, we can simply choose the simpler
approach for now and restrict the suprareals and i-suprareals to being only
polynomials over
This means, of course, that suprareals in general do |

Or, equivalently:Axiom 8.

There existsε_{1}=^{1}/_{η1}, such that 0 <ε_{1}<xfor allx∈R.

For allh∈Hwhereh> 0, there existsε=^{1}/_{h},

such that 0 <ε<xfor allx∈R.

This new fundamental constant *ε*_{1} and all the other
numbers derivable from it are *infinitesimal* numbers, because they are
smaller than any real but still greater than zero.
For now, however,
we'll simply call them the i-suprareals ("inverse suprareals").
For convenience, we place them into their own set **L**:

Definition 6.

L= {^{1}/_{h}:h∈H}.

Like the suprareals, it appears that every i-suprareal can be represented as
a polynomial over powers of *ε*_{1}:

Theorem FTI.

For allv∈L,

v=x_{1}ε_{1}^{1}+x_{2}ε_{1}^{2}+x_{3}ε_{1}^{3}+ ... +x_{n}ε_{1}^{n},

wherex_{1},x_{2}, ...,x_{n}∈R, and at least onex_{j}≠ 0.

As was noted earlier, the suprareals by themselves are not a field (because dividing a suprareal by some other number does not always produce another suprareal), but the union of the suprareals, the reals, and the i-suprareals is indeed a field:

Theorem 18a.(Closure over addition)

For allu,v∈H∪R∪L,u+v∈H∪R∪L.

Theorem 18b.(Closure over multiplication)

For allu,v∈H∪R∪L,u•v∈H∪R∪L.

Theorem 18c.(Closure over multiplicative inversion)

For allu≠ 0 ∈H∪R∪L,^{1}/_{u}∈H∪R∪L.

Theorem 18d.(Additive identity)

For allu∈H∪R∪L, 0+u=u.

Theorem 18e.(Additive inverse)

For allu∈H∪R∪L, −u+u= 0.

Theorem 18f.(Multiplicative identity)

For allu∈H∪R∪L, 1•u=u.

Theorem 18g.(Multiplicative inverse)

For allu≠ 0 ∈H∪R∪L,u•^{1}/_{u}= 1.

Theorem 19.

(H∪R∪L, +, •) is a field.

To visualize adding the i-suprareals to the extended real number line, note that all of the positive i-suprareals fall between zero and all of the positive reals, and likewise for the negative i-suprareals, which fall between all of the negative reals and zero. So if we visualize the normal real number line as being centered at zero, we can split it into two half-lines and a point, i.e., the negative reals, zero, and the positive reals.

The negative i-suprareals are then inserted between the negative half-line
and zero, and the positive i-suprareals are inserted between zero and the
positive half-line.
Positive and negative *ε*_{1} are used as the
"center points" of the i-suprareal half-lines.

We can now show the hierarchy of i-suprareals, reals, and suprareals by listing their half-lines according to the ordering of the points contained within them:

For all positiveε∈L,x∈R,h∈H,

−∞ < −h< −x< −ε< 0 < +ε< +x< +h< +∞

H^{−}<R^{−}<L^{−}<0<L^{+}<R^{+}<H^{+}

The visualization shown earlier of the reals nested within the suprareals
suggests that we can continue to extend the suprareals even further, adding
another layer of numbers surrounding the suprareals.
If we call the suprareals as they have been defined earlier as
η_{1}-numbers, we can likewise define new η_{2}-numbers
into existence.
To make things a little clearer, we'll rename
set **H** as **H**_{1}, since it is composed of all the
η_{1}-numbers derived from η_{1}.

Just as we defined a fundamental element η_{1}
which is greater than any real number,
likewise we now define another fundamental element η_{2}
which is greater than any η_{1}-number
(and which is obviously greater than any real number as well):

Axiom 9.

There exists η_{2}∈H_{2}such thatg< η_{2}for allg∈H_{1}.

All the new suprareals derived from the fundamental constant η_{2}
are contained in the new set **H**_{2}.

We can continue defining more layers of suprareals indefinitely, defining into existence an entire hierarchy of suprareals:

Each ηAxiom 9.(Generalized)

There exists η_{i+1}∈H_{i+1}such thatg< η_{i+1}for allg∈H_{i}.

It appears that each higher level of suprareals can be represented as
polynomials over powers of the fundamental η_{i} for that level:

Theorem G-FTSR.

For allg∈H_{i},g=u+u_{1}η_{i}^{1}+u_{2}η_{i}^{2}+u_{3}η_{i}^{3}+ ... +u_{n}η_{i}^{n},

whereu,u_{1},u_{2}, ...,u_{n}∈H_{i−1}, and at least oneu_{j}≠ 0.

All of the **H**_{j} sets together form a nested hierarchy of
suprareals, where the positive and negative halves of each
set **H**_{j} encloses set **H**_{j−1},
forming a nested grouping all the way down to **H**_{1},
which encloses **R**.

Earlier we defined a fundamental i-suprareal constant
*ε*_{1}
corresponding to the fundamental suprareal η_{1}, and likewise
we can define an entire hierarchy of sub-infinitesimals
*ε*_{i} corresponding to each η_{i}.

Or, equivalently:Axiom 10.

There existsε_{i+1}=^{1}/_{ηi+1}, such that 0 <ε_{i+1}<ε_{i}.

For allg_{i}∈H_{i}^{+}there existsε_{i}=^{1}/_{gi},

such that 0 <ε_{i}<ε_{i−1}<x, for allg_{i−1}∈H_{i−1}^{+}for allx∈R.

All inverse suprareals can be expressed as polynomials over powers of
*ε*_{i}:

Theorem G-FTI.

For allg∈L_{i},g=v+v_{1}ε_{i}^{1}+v_{2}ε_{i}^{2}+v_{3}ε_{i}^{3}+ ... +v_{n}ε_{i}^{n},

wherev,v_{1},v_{2}, ...,v_{n}∈L_{i−1}, and at least onev_{j}≠ 0.

All of the suprareal sets and i-suprareal sets can be collected together into a single set. This single mega-set, combined with the real numbers, contains all of the reals, all of the suprareals, and all of the i-suprareals:

Definition 7.

R_{E}=R∪H_{1}∪L_{1}∪H_{2}∪L_{2}∪ ...

**R**_{E} is the set of "extended reals",
and contains all of the numbers, real and non-real, defined in our system.

Up to this point, we've defined suprareal fundamental constants
η_{i} and their corresponding inverse suprareal constants
*ε*_{i} = ^{1}/_{ηi}.
But we have not defined an η_{0}.
Since the suprareals in **H**_{i} form a hierarchy greater than the
reals, and the inverse suprareals **L**_{i} form a hierarchy
less than the reals, it seems logical to conclude that we can define
**H**_{0} to be the set of reals, **R**.
Thus:

Theorem 20.(H_{0})

H_{0}=L_{0}=R.

Since **H**_{−0} is simply **H**_{0},
**L**_{0} is just another name for **H**_{0},
which is just another name for **R**.
Furthermore, since η_{0} is the fundamental constant of
**H**_{0}, and since its reciprocal
^{1}/_{η0} is the fundamental contant of
**L**_{0}, and both of these sets are actually the same set
**R**, it follows that η_{0} is its own inverse in **R**.
That is to say:

Theorem 21.(η_{0})

η_{0}=^{1}/_{η0}= 1.

Given all of this, we can simplify our notation somewhat by renaming each set
**L**_{i} as its equivalent set **H**_{−i}.
So where each set **H**_{i} has a fundamental contant
η_{i}, likewise each set **H**_{−j} has a
fundamental contant
η_{−j}
= *ε*_{j}
= ^{1}/_{ηj}.
This makes the set names **L**_{j} redundant, as we now can consider
every suprareal, inverse suprareal, or real as a member of some suprareal set
**H**_{i}.

[This may require some adjustments to the notion that 0 and 1 are not "proper" suprareals, since they are not members of any setH_{i}where i ≠ 0, but they are members ofH_{0}. Taken as a whole, the union of allH_{i}sets is therefore a field and a "complete" superset ofR.]

We can go another step further and extend the suprareals over the complex field. Just as a complex real number is a combination of real and imaginary parts, a complex suprareal is likewise a combination of suprareal and imaginary suprareal parts:

Definition 8.

z=u+ iv, for someu,v∈H_{1}.

Applying this to the entire hierarchy of suprareals,
for every suprareal and i-suprareal set **H**_{i}
there is a corresponding set **CH**_{i}
that extends those suprareals into the complex plane:

Definition 9.

CH_{i}= {z:z=u+ivfor allu,v∈H_{i}} for alli∈Z.

Note that as a special case **CH**_{0} is just **C**,
the standard set of complex numbers.

The final step is to combine all of the complex suprareals, complex i-suprareals, and complex reals into a single set, the extension of the complex numbers:

Definition 10.

C_{E}=C∪CH_{1}∪CH_{−1}∪CH_{2}∪CH_{−2}∪ ...

The diagram below illustrates one way of visualizing set **C**_{E}.
The dashed lines represent the discontinuities between the subsets comprising
the extended complex plane.
The diagram is simplified, showing only two levels of suprareals and
i-suprareals instead of the entire (infinite) hierarchy.

We can also extend the suprareals into quaternion space, creating numbers
of the form i*h*+j*g*+k*u*.

Our final step is to extend the suprareals into n-dimensional number spaces.
The approach is to define n-tuples
<*v*_{1}, *v*_{2}, *v*_{3}, ...>,
where each *v*_{i} is a member of **R**_{E}
or **C**_{E}.

If we consider that zero is surrounded by the entire set of i-suprareals,
the negative i-suprareals on the left and the positive i-suprareals on the
right, we could say that the i-suprareals form a "cloud" or "halo"
(borrowing a term from non-standard analysis
^{[2]})
of non-real numbers around zero.

Taking this concept one step further, we can combine the i-suprareals with all of the reals in such a way that every real is surrounded by its own halo of i-suprareals:

Definition 11.

R_{L}=R∪ {x+ε:x∈Randε∈L}.

Or, equivalently:

R_{L}= {x+ε:x∈Randε∈ {0} ∪L}.

This defines set **R**_{L} having all the reals as members,
and for each real *x*, all of the i-suprareals *x*+*ε*
surrounding *x* as well.

We can likewise define a set of the suprareals that includes the i-suprareal halos around each suprareal:

Definition 12.

H_{L}=H∪ {h+ε:h∈Handε∈L}.

Since the i-suprareals are defined in terms of the
fundamental constant *ε*_{1},
which in turn is defined in terms
of the fundamental constant η_{1}, we can rewrite the i-suprareals
as polynomials over negative powers of η_{1}
(where *ε*_{1}^{n}
= ^{1}/_{η1n}
= η_{1}^{−n}):

For allv∈L,

v=x_{1}η_{1}^{−1}+x_{2}η_{1}^{−2}+ ... +x_{n}η_{1}^{−n},

wherex_{1},x_{2}, ...,x_{n}∈R, and at least onex_{j}≠ 0.

Furthermore, the members of **H**_{L} are
sums of suprareals and i-suprareals.
It is therefore apparent that the members of this new set can be
described as polynomials over *all* of the integral powers of
η_{1}
(positive and negative powers included):

Theorem FTSR-L.

For allh∈H_{L},

h=y_{m}η_{1}^{−m}+ ... +y_{2}η_{1}^{−2}+y_{1}η_{1}^{−1}+x+x_{1}η_{1}^{1}+x_{2}η_{1}^{2}+ ... +x_{n}η_{1}^{n},

wherey_{m}, ...,y_{2},y_{1},x,x_{1},x_{2}, ...,x_{n}∈R, and at least onex_{j}≠ 0.

And finally, we can combine all of these numbers with the infinitesimals into a single set:

Definition 13.

R_{U}=L∪R_{L}∪H_{L}.

Every element in this set is either a real or a suprareals, an i-suprareal, a real plus an i-suprareal, or a suprareal plus an i-suprareal.

Up to this point, we have defined a countable hierarchy of suprareals and
inverse suprareals, i.e., the sets **H**_{n} for every
integer *n*.
(Remember that **H**_{0} is a special case and is the same as
**R**.)

But we can create an uncountable hierarchy of suprareal sets by defining an suprareal constant for every real:

Axiom 9-U.(Generalized uncountable)

For allx∈Randx> 0,

there exists η_{x}∈H_{x}such thatr< η_{x}for allr∈R,

and η_{x}< η_{y}for allx,y∈Rwherex<y.

Axiom 10-U.(Generalized uncountable)

For all η_{x}∈H_{x},

there exists^{1}/_{ηx}=ε_{x}∈H_{−x}such that 0 <ε_{x}<rfor allr∈R.

In other words, for every real *x* there is a suprareal η_{x},
which is the fundamental constant for an entire set of suprareals
**H**_{x}.
Every suprareal *h* in **H**_{x} can be written as a
polynomial over integer powers of η_{x},
thus *h* = f(η_{x}).
Furthermore, for every suprareal *h*_{x} in **H**_{x}
and every suprareal *h*_{y} in **H**_{y},
*h*_{x} < *h*_{y}
when *x* < *y*.

For example, we can compare various suprareals:

h= f_{h}(η_{1}) =x_{0}+x_{1}η_{1}^{1}+x_{2}η_{1}^{2}+ ... +x_{k}η_{1}^{k}

g= f_{g}(η_{0.5}) =y_{0}+y_{1}η_{0.5}^{1}+y_{2}η_{0.5}^{2}+ ... +y_{m}η_{0.5}^{m}

u= f_{u}(η_{π}) =z_{0}+z_{1}η_{π}^{1}+z_{2}η_{π}^{2}+ ... +z_{n}η_{π}^{n}

Since 0.5 < 1 < π, we know that
η_{0.5}
< η_{1}
< η_{π}, which tells us that
f_{g}(η_{0.5})
< f_{h}(η_{1})
< f_{u}(η_{π}), and therefore
*g* < *h* < *u*.

We can combine all of the **H**_{x} sets into a single set:

Definition 14.

H_{R}= {h:h∈H_{x}} for allx∈R.

Or, equivalently:

H_{R}= ∪H_{x}for allx∈R.

Since |**R**| = **c**, and since |**H**_{x}| = **c**
for every for each real *x* in **R**,
there are **c**^{2} suprareals in the union of all the
**H**_{x} sets.
By cardinal exponentiation, **c**^{2}
= **c**,
so |∪**H**_{x}| = **c**.
In other words, **H**_{R} is the same size as **R** and **H**.

The definition of the suprareals could be simpler.
In particular, there seem to be too many axioms, which could probably be
replaced by a more general rule, something like the *transfer principle*
^{[4]}
of *non-standard analysis*
^{[2]},
which would allow most of the
arithmetic operators (order, addition, and multiplication) to be defined
over the suprareals with a single rule.
In particular, Axiom 6 (additive commutativity) and
Axiom 7 (multiplicative commutativity) should be theorems instead of
axioms.

There is also the question of how operations more complex than arithmetic
addition and multiplication may be defined over the suprareals.
Analytic functions require the notion of *limit*, and it is not yet clear
how **lim**_{h→∞} f(*h*)
is to be defined for *h* in **H**.

Are the suprareals the same as the *hyperreals*
^{[1]}
of non-standard analysis?
They certainly share some of the same properties, notably their ordering
relation to the reals.
It could be that a given member of **R**_{L}, being the sum of
a real and an i-suprareal *x*+*ε*, is equivalent to a
non-standard real *r* and one of the infinitesimals in its halo,
with *x* being the standard part of *r*,
thus *x* = *st*(*r*)
and *ε* = *r* − *st*(*r*).

In other ways, though, the suprareals do not appear to behave like the
hyperreals.
Specifically, the infinite set of standard real numbers is not definable within
*internal set theory*
^{[5]}
(specifically, all infinite sets of IST contain a non-standard element,
and all sets of IST containing only standard elements are always finite),
yet **R** is clearly definable as a proper subset of
**R**_{E} and **R**_{L}.
Also, the hyperreals ***R** do not form a metric space, but
it appears that the suprareal sets **H**, **R**_{E}, and
**L** ∪ **R**_{L} ∪ **H**_{L}
do.

Furthermore, reciprocals of non-standard infinitesimals are called
*infinite numbers* (because they are larger than any reals),
whereas the suprareals, while also being reciprocals of infinitesimals,
are not considered to be numbers with infinite magnitudes,
but simply comprise a different set of numbers similar to, but separate from,
the reals.
This could be just a difference in nomeclature, though,
if it turns out that the suprareals and the hyperreals are equivalent.
The hyperreals are also called *illimited* numbers, which is probably the
more "correct" term.

Another issue to discuss in more detail is the fact that
each region (or level) of suprareals is *unconnected* to any other region.
For example, each *h* = *x*η_{i} is greater than
every *g* = *y*+η_{i} (for positive reals *x,y*).
Furthermore, each *h* in **H**_{i} is greater than
every *g* in **H**_{i−1}.
All of these suprareals exist within different, unconnected regions of
suprareal space.
The suprareals form a highly unconnected set, in fact.

The author does not posess the mathematical expertise to answer these questions definitively, so perhaps others with a deeper understanding of these things can provide better answers.

- [1]
**Hyperreal numbers** -
Wikipedia:
http://en.wikipedia.org/wiki/Hyperreal_number,

MathWorld: http://mathworld.wolfram.com/HyperrealNumber.html. - [2]
**Nonstandard Analysis** -
Wikipedia:
http://en.wikipedia.org/wiki/Nonstandard_analysis,

MathWorld: http://mathworld.wolfram.com/NonstandardAnalysis.html. - [3]
**Nonstandard Analysis** - Abraham Robinson, 1974, ISBN 0-691-04490-2.
- [4]
**Transfer Principle** -
Wikipedia:
http://en.wikipedia.org/wiki/Transfer_principle,

MathWorld: http://mathworld.wolfram.com/TransferPrinciple.html. - [5]
**Internal Set Theory** -
Wikipedia:
http://en.wikipedia.org/wiki/Internal_set_theory,

MathWorld: http://mathworld.wolfram.com/SaturatedEnlargement.html.

This document: http://david.tribble.com/text/hnumbers.html.

*
Copyright ©2006 by David R. Tribble, all rights reserved.
Permission is granted to reference, quote, and link to this document provided
that appropriate credit is given to the orginal author.
*