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
(multiply, add)
that both are Abelian, the addition is
cancellable and that the
multiplication's restriction to positive naturals is cancellable. From this,
one can build a representation of ratios, the results of dividing a natural by a
positive natural, without needing the latter to be a factor of the former.
There are various ways to do that, so
I've picked one that happens to have some
convenient properties when I come to consider linear spaces. This gives
us natural ratios
of form n/m = m\n for m, n natural with m non-zero (so
positive); when n is also non-zero, n/m is a positive ratio
.
Orthodoxy generally formulates ratios as pairs of whole numbers subject to an equivalence, [n, p] ~ [m, q] precisely if n.q = p.m, which allows a definition of multiplication as [n, p].[m, q] = [n.m, p.q] and of addition as [n, p] +[m, q] = [n.q +p.m, p.q], yielding the familiar world of ratios if we interpret [m, q] as m/q. One can do essentially the same trick – with [n, p] ~ [m, q] precisely if n+q = m+p and suitably amended addition and multiplication – to complete an abelian cancellable addition, such as that on {naturals} or on {natural ratios}, so as to obtain an additive identity (if one was previously lacking) and additive inverses. I deliberately opt to not do this until later because working only with positives saves a whole lot of fiddly complications.
Between any two distinct natural ratios, no matter how close, there are as
many positive ratios as {naturals} has members; 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 natural ratio}
which manage to pass from inside the unit circle, {[x,y]: x.x+y.y=1}, to outside
it without ever intersecting it. The purpose of the real numbers is to provide
a way to fill in the gaps
so as to avoid these
deficiencies. (Technically, the algebraic numbers suffice to fill in the gaps
I've actually mentioned here; but there are other gaps, that it's harder to
explain, for which one needs the reals.)
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. (One could equally say only finitely many
in place
of at most one
, as it happens.) 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} is also initial in itself and serves as an infinity; the former is 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 represent the positive ratio r within {positives}. 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 (positive) natural as a positive ratio as
a natural positive
(not to be confused with the positive natural that it
represents).
We may define addition and multiplication 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 (as is only to be expected).
We obtain a natural order on initial subsets from: x ≥ y precisely if {r in y: r not in x} is finite; this arises precisely when x subsumes y or is equivalent to y. 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 convergent 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.
Within any ringlet we naturally obtain an image of the positive natural
numbers by repeated addition of the multiplicative identity; this forms a
sub-ringlet. In so far as the ringlet's addition supplies multiplicative
completions between the values of this positive natural sub-ringlet, we can
collect these together as positive ratios of the ringlet; these, likewise, form
a sub-ringlet. If this ringlet isn't additively complete (in particular, no sum
of its members delivers an additive identity) we get the usual ordering on it
induced by its addition (p +q > p for all positives p, q) and it forms an
ordered ringlet under this ordering. (That'll heppen precisely when the ringlet
has characteristic zero, I think.) This ordered ringlet of positive ratios is
necessarily a sub-ringlet of the centre of our given ringlet (that is, every
positive ratio commutes with all other values of the ringlet). If there is a
smallest positive ratio, then this ordered ringlet of positive ratios is
discrete and we'll get no limit points to extend it with; but if there is no
smallest positive ratio then there may be values of the ringlet that fit
in between
the ratios, in a similar manner to the real positives.
Performing the initial sub-set construction on the ordering of the positive ratios within a ringlet of character zero, when there is no smallest ratio, will get us a ringlet isomorphic to the real positives; but it's not a sub-ringlet of our original ringlet. So we must ask whether there is some way to identify members of the original ringlet that correspond to members of this real positive subringlet, derived from the original ringlet's positive ratios.
It's not immediately clear to me how to pin that concept down, but perhaps
we can take the union of the positives within each ordered subringlet of the
centre of a ringlet as a way to obtain the positives within
the given
ringlet, which would hopefully include any such limit points corresponding to
the real positives derived from the original ringlet's positive ratios.
Given our multiplication on {positives}, we can use repeated multiplication
to induce powers in the usual way. Initially this
gives us (: ({positives}: power(n) |{positives}) ←n |{naturals}) with
power(0, x) = 1 and power(1+n, x) = x.power(n, x) for each natural n and
positive x. Because multiplication on positives respects their ordering,
i.e. a > b
implies x.a > x.b
for every positive x, each
power aside from power(0) is monotonically
strictly increasing – for natural n > 0, with positives x > y,
we know power(n, x) > power(n, y). We can thus define, for each positive
natural n, ({positives}: power(1/n) |{positives}) by power(1/n, x) = {positive
ratio r: power(n, r) < x}; because power(n) is monotonic, this is an initial
sub-set of {positive ratios} which we can duly interpret as a positive
real.
Furthermore, because each power(n) is continuous on {positives}, we can infer that power(1/n) is in fact reverse(power(n)). As composing powers corresponds to multiplying exponents, power(n)∘power(m) = power(n.m), we can duly construct power(n/m) = power(1/m)∘power(n) and extend power to (: ({positives}: power(r) |{positives}) ←r |{natural ratios}). This still has every power except power(0) strictly monotonically increasing. Furthermore, for each positive x, (: power(r, x) ←r |{positive ratios}) is also monotonically strictly increasing, which enables us to define, for positive p and x, power(p, x) = {positive ratio q: q < power(r, s) for some r in p, s in x} and, again, infer that this is an initial sub-set of {positive ratios} that we can interpret as a positive real. We are thus able to extend the definition of power to (: ({positives}: power(p) |{positives}) ←p |{positives}), every output of which is monotonically increasing, as is every output of its transpose.
As for the positive ratios we have naturals represented among the positives by the positive ratio corresponding to the rational representing each natural. For present purposes, write e = ({positives}: {positive ratio r: r ≤ n/1} ←n :{naturals} for the natural embedding of {naturals} in {positives}. We have an ordering on positives so we can again define:
for which, for any positive r, e(floor(r)) ≤ r ≤ e(ciel(r)) and
ciel(r) ≤ floor(r) +1, with equality in this last precisely when one of the
previous chain of inequalities is strict. This follows for essentially the same
reasons as made that true for positive ratios. We can thus express any positive
r as a sum of floor(r) and a fractional part
r −floor(r), albeit
this latter shall be zero (which isn't a positive) when r represents a natural.
This fractional part is necessarily < 1.
We can extend the numeral notation for naturals and positive ratios to infer one for positives. This notation potentially involves infinitely many digits in the fractional part of the number. As for the representation of positive ratios, the representation here depends on splitting the positive into its whole number part, provided by floor, and the fractional part by which it exceeds that. If the positive represents a whole number, i.e. is equivalent to {positive ratio r: r ≤ n/1} for some natural n, then its numeral notation is the usual numeral notation for that natural number, optionally followed by the fractional-part separator and arbitrarily many repeats of the digit representing zero. Likewise, for a positive that represents a ratio, any numeral denoting that ratio is a valid denotation for the positive.
It remains, then, to denote positives less than 1, that are not ratios. For a given number base b and fractional part r < 1, we have r.b < b and thus floor(r.b) in b has a representation as a digit. If we define
then our denotation for the fractional part of r is just, after an initial fractional-part separator, the sequence of digits denoting the naturals (all less than b) given by fractional(b, r). Note that, for a positive whose base b numeral terminates, fractional(b, r) is still an infinite sequence, with all entries zero after the terminating sequence ends.
Notice that this will never result in an endless sequence of the digit representing b −1, since the positive of which that might otherwise be a representation would necessarily be a rational with a terminating representation to base b. Allow, none the less, that a fractional part of form p&[n]&({b −1}: |{naturals}), with n +1 < b, can be transformed to the terminating fractional part p&[1+n]; and that a numeral whose fractional part is ({b −1}: |{naturals}) stands for the successor of the whole-number part of this numeral. That is, when writing a numeral to represent a positive, always use the form that terminates by preference but, when construction of a text having the form of a numeral results in one with an endless tail of b −1 digits, provide for it to be read as the positive whose terminating numeral is obtained from it by pruning the endless tail of b −1 digits and incrementing the last digit before it.
Suppose we have some mapping ({positives}: f |{naturals}). Let b be any number base > 2 and diag(f) = (: fractional(b, f(n), n) +1 mod b ←n :{naturals}). This digit sequence necessarily reads as a fractional part different from that of any output of f, since:
Consequently no mapping from the naturals to positives produces even
every positive less than 1, much less all positives. This implies that the
infinity of positives is unequivocally bigger
than that of naturals.
Note, however, that the collection of positives it's actually possible to represent is countable, since a representation of a positive is a finite text whose interpretation uniquely identifies that positive; this is a sub-set of the possible finite texts in whatever writing system, language and system of interpreting texts one uses; and that set of finite texts is countable.
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 natural m}; the ratio that represents 1 is repeat(1) = {relations}. 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; indeed, every positive is an infinite collection of positive ratios).
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).
Classically, one would define continuity of a function,
({reals}: f :{reals}), at a given input, x, by specifying that: for every
positive real e, there is some positive real h such that, for all real y
satisfying y+h > x and x+h > y, y is a right value of f with f(y)+e
> f(x) and f(x)+e > f(y)
. However, I have an intense dislike of the
challenge-response protocol involved here, due to there being no way to prove
any result of this kind without use of reductio ad absurdum.
So I'm trying to find an alternative.