The real
numbers constitute a completion
of the rational
numbers, ratios of whole numbers, by filling in the gaps
so as to obtain
a continuum. There are several equivalent ways to do this; I chose a variant of
the Dedekind cut
approach, performed before the application of the standard completion of
addition by introduction of negative values.
Given the natural numbers, we must first establish the multiplication and
addition they support, then prove that both are Abelian, the addition is
cancellable and that the multiplication's restriction to positive naturals is
cancellable. It is then time to apply the standard completion trick to the
multiplication and obtain the positive ratios. We start with the collection of
pairs of positive naturals, {n←m: n, m in (|successor:{naturals})} and we
subject it to an equivalence relation (: (n←m) ← (p←q); n.q = m.p
:). The pair n←m, when understood modulo this equivalence, is denoted n/m
and the equivalence is read as n/m = p/q precisely if n.q = m.p
. Such a
pair, understood modulo this equivalence, is known as a positive
ratio
. There is a trivial embedding of the positive naturals in the
positive ratios via (: (n←1) ←n :{naturals}); I shall refer to the
left values of this as natural ratios
.
We induce a trivial multiplication on positive ratios by (n/m).(p/q) = (n.p)/(m.q) and an addition by (n/m)+(p/q) = (n.q+m.p)/(m.q). It is easy enough to infer that these are Abelian, from the corresponding properties of these two binary operators on naturals; and to infer that their action on natural ratios coincides with the result of doing the corresponding arithmetic in the naturals before embedding in the positive ratios. We must then establish that both are cancellable. The multiplication then forms a group by construction (we constructed it to be complete); the identity is then n/n, for arbitrary positive natural n, and the inverse of n/m is m/n for any naturals n, m. We extend the meaning of / to a binary operator on ratios by (n←m)/(p←q) = (n.q←m.p), noting that it coincides with / as already introduced for naturals when applied to natural ratios (m and q are then 1 and we get n/p).
The ordering of the naturals induces a natural ordering on the positive ratios: n/m compares to p/q exactly as n.q compares to m.p. Then n/m > p/q precisely when n.q > m.p in {naturals}, which is true precisely if there is some positive natural g for which n.q = g + m.p; this, in turn, implies n/m = g/(m.q) + p/q, so there is a positive ratio which, when added to p/q, yields n/m. Now, if there are positive naturals g, h for which n/m = p/q + g/h, we can infer n.q.h = m.p.h + m.q.g whence n.q.h > m.p.h whence, after cancelling h, n.q > m.p, so that n/m > p/q. Thus our ordering on the positive ratios can equally be stated as x > y precisely if there is some positive ratio z for which x = y+z.
Between any two distinct positive ratios, no matter how close,
there are as many positive ratios as there are natural numbers; all the same, it
turns out that there are not enough
of them. If we complete their
addition and try to use the result to define a two-dimensional plane, we can
define a squared distance
function (: x.x+y.y ←[x,y] :) and thus the
notion of an isometry, a linear map that preserves squared distance. We
discover that we only get a rather limited set of rotations (specifically, those
through angles seen in pythagorean
triangles or obtained from these by addition and subtraction. In
particular, we can't rotate through turn / n for any natural n other than 1, 2
and 4; we can construct plenty of pairs of lines that meet in angles through
which we can't rotate; and thus there are many radial lines, e.g. {[x,x]: x is a
positive ratio} which manage to pass from inside the unit circle, {[x,y]:
x.x+y.y=1}, to outside it without ever intersecting it. We thus need some way
to fill in the gaps
so as to avoid these deficiencies.
Now let a sub-set Q of {positive ratios} be termed initial
precisely
if x in Q implies Q subsumes {positive ratio y: x > y}; and understand two such
subsets as equivalent precisely if there is at most one ratio in one of them but
not in the other (although, of course, arbitrarily many of the infinitely many
equivalent pairs representing a single ratio may be in one but not the
other). Every non-empty initial has infinitely many members. I shall refer to
a non-empty proper initial sub-set of {positive ratios}, understood modulo this
equivalence, simply as a positive
; the collection of such is then
{positives}. The empty set is an initial sub-set of {positive ratios} and
serves as a zero; the set {positive ratios} itself is also initial in and serves
as an infinity; the former is non-empty and the latter is not proper, so neither
is included in {positives}.
When two positives are equivalent but distinct, with the single ratio in one
but not the other being some r, then the two positives are necessarily {x: x
< r} and {x: x ≤ r}, from the specification of their being initial; these
shall serve as the natural representation of the positive ratio r. When an
initial sub-set of the positive ratios is {x: x ≤ r} for some r, I shall
refer to it as a rational positive
and to r as its maximum; it should be
evident that an initial sub-set of the positives has a maximal element
precisely if it is a rational positive and this element is its maximum. I shall
refer to the rational positive that represents any natural ratio as a natural
positive
(not to be confused with the positive natural that it
represents).
We may define addition and multiplicatio on {positives} by the simple expedient of applying to every member: that is, x.y = {r.s: r in x, s in y} and x+y = {r+s: r in x, s in y}. We must, again, show that each is cancellable and that the multiplication is complete (hence forms a group). The latter is achieved by identifying the inverse of a positive, x, as {positive ratio r: for all s in x, r/s < 1}. We can extend the addition via {}+x = x to make {} serve as additive identity, 0; extending multiplication by using the same rule as for positives, we obtain {}.x = {} for every initial x, so 0 breaks multiplicative cancellability.
We obtain a natural order on initial subsets from: x ≥ y precisely if x subsumes y, extended via our equivalence to also allow x ≥ y when x and y are equivalent (so x may lack one member of y, its maximum, if it has one). This leads conveniently to any non-empty set of initial subsets having a least upper bound (its union) and a greatest lower bound (its intersection), although the latter may be empty and the former might be infinity. When a set of positives thus combined is finite, these shall be maximal and minimal elements of the set (hence neither zero nor infinity). If we have an infinite sequence of initial subsets, we can derive from it two (non-strictly) monotonic sequences, one increasing, the other decreasing, by taking the intersection and union (respectively) of all entries in the sequence past each given point in it. By taking the union of the former and the intersection of the latter we obtain lower and upper bounds (respectively) on what values sub-sequences of the original sequence can converge to. If the original sequence does converge, these two bounds shall coincide and give us the limiting value. In particular, whenever a sequence is convergent, we are assured that it does have a limit; note, however, that even the limit of a sequence of positives may be {}, i.e. zero.
I generally prefer to work with {positives} rather than {reals} wherever I can; it remains that the reals do have a proper place in mathematics and so it is desirable to actually show how they may be obtained from the positives.
Finally, we are ready to obtain the real numbers by completing the addition;
a real number
, or simply real
, is a pair x←y of positives,
understood modulo the equivalence (: (x←y)←(u←v); x+v = y+u
:). The pair x←y, understood modulo this equivalence, is denoted
x−y. We can induce addition, multiplication and an ordering on {reals},
from those on {positives}, as
(x←y) < (u←v)precisely if
x+v < y+u.
Then (: (x+a←a) ←x, a in positives :) embeds the initial sub-sets of {positive ratios} in {reals} while preserving arithmetic structure and ordering. We can extend − to apply to reals via (x←y)−(u←v) = (x+v)←(y+u), which naturally interacts faithfully with this embedding.
Each natural number is the set of earlier naturals; thus the natural 0 is {} and the natural 1 is {0} = {{}} = (: {}←{} :). Among subsets of {positive ratios}, {} does indeed again serve as the additive identity, although the multiplicative identity, the rational positive that represents the natural 1, is {n←m: 0 in n in m, m is natural}; the ratio that represents 1 is n←n = {n} for arbitrary positive natural n. We could define a real to be a pair of initial sub-sets of {positive ratios} subject to the same equivalence, but this would then have left us with the natural 1 = {}←{} as one of the pairs we can interpret as a real serving to represent 0. Rather than overly encumber contexts with the need to be clear about how to read {}←{}, I chose to have reals be pairs of positives; then no real happens to be equal, as a relation, to any natural or ratio (since no positive is a natural).
The real a−a, for arbitrary positive a (all such reals are trivially
equivalent), serves as additive identity and (: (x←y)←(y←x) :)
equips every real with an additive inverse. A real that is > the additive
identity is described as positive (and is the real which represents some
positive); a real that is < the additive identity is described
as negative
(and is the additive inverse of some positive real).
The embedding of initial subsets of {positive ratios} in {reals} embeds {},
the additive identity, as a←a for arbitrary positive a; for any positive
ratio r, using our prior embedding in {positives} as {x: x≤r} (or,
equivalently, {x: x<r}) we thus obtain a real ({x: x≤r}+a)←a for
arbitrary positive a that represents the ratio r. I refer to the additive
identity of the reals, the images of rational positives under the embedding of
positives in {reals} and the additive inverses of these as rational
;
a positive rational
is then always the real that represents some rational
positive. I refer to the images of natural positives, as embedded in {reals},
along with their additive inverses and the additive identity, as whole
(the whole reals are the integers, in effect).
In practice it is seldom interesting to distinguish a whole positive real, a natural positive or a natural ratio from the (positive) natural they represent; nor to distinguish a positive rational or a rational positive from the positive ratio they represent. None the less, it is technically necessary to draw these distinctions: they are distinct relations, so it is important to be specific about which of them any given relation accepts as a left or right value; and it is at least desirable that an entity of one type should not also be understandable as an entity of another type unless it, via relevant embedding, represents the same value either way (as {} does when understood as the additive identity both among initial sub-sets of {positive ratios} and among naturals).
Written by Eddy.