This section discusses a direct construction of the real numbers from
the integers (constructing the rationals at the same time) via the idea of
an almost linear map from to .

This approach is a particularly elegant and short one, and has the
rather pretty feature that addition in
corresponds to componentwise addition of integers, whereas
multiplication in corresponds to composition of maps.
However the approach is more sophisticated and requires
quite a lot of axiom-checking at the end. It also requires some
work with the notion of finiteness and with equivalence relations.

This web page is still under construction. The final version
will contain some (but not all) of the proofs omitted here at present.

Definition.

A function
is said
to be almost linear or almost a homomorphism if

(+)-()-()
,
has finitely many elements.

Note that the set here would be 0 if was a homomorphism
of respecting the addition stucture, i.e., if for all
,
we have (+)=()+()
. For such
the set
(+)-()-()
,
is 0 so has just one element, in particular has finitely many elements.

Definition.

Two functions
and
are almost equal
if

()-()
has finitely many elements. If ,
are almost equal then we
write
.

Proposition.

The relation of being almost equal
is an equivalence relation on almost linear maps
.

**Proof.**

Clearly
()-()
=0 so the reflexive axiom holds. If
then
()-()
=
-(()-())
so is finite, being the image under taking - of the finite set
showing that
. Therefore
and
is symmetric.

Finally, if
and
then both ~~=~~
()-()
and =
()-()
are finite, hence
()-()
~~+
~~
~~
~~~~
~~
is also finite. Hence transitivity.

Definition.

We write
for the equivalence class of .

If we pretend for a moment that is a real number that we know about,
we may define a function ()=
, where the square brackets
denote integer-part. This is almost linear as
+
(+)-2
and so (+)-()-()
0,1
.

Definition.

We define to be the set of
-equivalence classes of almost linear maps
.

Definition.

We define
+
=
where ()
is the function ()+()
.

Proposition.

This function + is well-defined.

Definition.

We define
=
where ()
is the function (())
.

Proposition.

This function is well-defined.

Definition.

For
,
we
define
if there is
such that for all
, ()-()0
.

Proposition.

The relation is well-defined and linearly orders .

Theorem.

The set with +,,
is an ordered
field. It also satisfies the Archimedean property
and is a complete ordered field in the sense that every
bounded monotonic sequence has a limit.

**Proof.**

Another long exercise (sigh).