Formal derivation

Preliminaries

What is a number? It is just an abstract object that is only defined in relation to other numbers. We will approach this from a graph theory angle. All the possible number objects, in graph theory called vertices, are each connected to all other objects through directed edges shown as arrows in this diagram.

Out of this big mess we will select one single special element at random, shown in yellow here. We will also select sets of directed edges that follow certain rules.

We will call these directed edges unary open functions.

Traditionally functions have a defined domain and range. Since the names “mapping” and “transformation” also have special meanings in Mathematics and since the directed edges are most like unary functions we use the term open function for them, but it should be clear that they have an open domain; they can be applied to any other object, and that we use their range to define new sets, as explained below.

Definition

Here, open function f changes number object p into number object q or q = f(p).

Such an open function is conceptually similar a unary function, except that functions are defined as a mapping with a domain and a range. We use open functions to define new sets of numbers, and the domain is the set S of number objects already defined. This means we are able to apply any of our unary open functions to any number object in S. The result will either be a number object in S or not in S. In the latter case we can use this new number object to extend S.

We will define all the finite number sets that we need using one single initial number object 0 and two types of open functions, loops and chains.

Axiom 1.                  We select one number object 0.

We denote repeated applications of an open function to an object with a superscript. For open function f and all objects p:

$\begin{array}{l} f^{0}(p)=p \\ f^{n+1}(p)=f(f^{n}(p)) \\ \end{array}$

where n is a positive integer indicating the number of repeated applications. This is used as a shorthand notation; instead of saying f(f(f(p))) we can use f3(p).

From all the possible directed edges that connect all the number objects we select two types of open functions, chains and loops, which conform to certain rules:

Chain:

1. keeps generating new objects
2. is commutative

Loop of order n:

1. forms an n element cycle with any number object other than 0.
2. two objects define the loop between them
3. one loop per order
4. order 2 special case relating inverse chains

A formal definition of each of these rules follows.

An open function inc is called a chain if for any object p we have incn(p) ≠ incm(p) for all natural numbers n≠m and for all objects p.

Shown is a chain called inc that is repeatedly applied to a starting number object s. We can apply inc as often as we like and we will always get new number objects.

Furthermore we require that chains are commutative, so if inca and incb are different inc chains if a≠b and p is a number object and n and m are positive integers then incan(incbm(p))=incbm(incan(p))

An open function rotn is called a loop of order n (with respect to 0) if rotn(0)=0 and for all objects p≠0 we have rotnm(p) = p if and only if n divides m.

Chains and loops are both bijective. The inverse open function of loop or chain f(p) is f*(p) such that
f(f*(p)) = f*(f(p)) = p  for all p.

Axiom 2.                  For any number object p≠0 if rotn(p) = rotm(p) then n=m.

This states that two (non zero) objects define the loop  between them. Here is an example:

In this diagram there is a rot4 open function in red and the rot2 open function in blue. The black dotted arrow is not a valid alternative rot2 open function because it connects two objects that are already connected by a red arrow.

Axiom 3.                  For any  and n>0 there is only one unique set of n number objects generated by rotnm(p).

What these axioms state is best illustrated with an example. Here we have five objects a,b,c,d and e that are related with order 5 loops.

There are 4 different order 5 loops, shown as solid and dotted, red and blue arrows. They are interchangeable in the sense that there isn’t anything special about one particular one. If we pick one randomly to be called rot5, then the other three are rot52, rot53and rot54.

Axiom 3 states that there are no other order 5 loops containing either a,b,c,d or e.

Axiom 4.

Let x=incnm(0) and let y=inc*nm(0) for any positive integer n and m. Then x and y are related by the rot2 loop:
$\begin{array}{l} x=rot_{2} (y) \\ y=rot_{2} (x) \\ \end{array}$

This means that rot2 is a special loop open function that relates any inc chain to its inverse.

If we look at all the inc back boxes starting from our object zero we get a star shape like this.

For n>0 we relate the nth object on one chain with the nth object of another chain using rot open functions.

Axiom 5.                  For all n>0 there exist n distinct chains inc0, inc1,…,incn-1 and a loop rotn of order n such that roty(inc0x(0))=incyx(0) for all x>0 and all 0≤y≤n-1.

Some of these rot open functions are shown in the following diagram.

We have now defined the axes of our number system. All the inc chains are only connected at 0 but we can still get new number objects if we apply an inc chain to objects other than 0 on another inc chain.

For n >0 we define Mn to be set of all objects that can be obtained from 0 by applying the composition of a finite sequence of the open functions inc0, inc1,…,incn-1.

We use the Gaussian integers which is set M4 as an example set to illustrate the definition.

The grey objects are on the axes. The open functions shown as light and dark blue arrows can be viewed as the functions +1 and +i.

Permutability

In the diagram above we don’t need to define which direction is 1 and i, and -1 and -i. 1 and i are interchangeable, as well as 1 and -1, while 1 and -i are not.

From definitions 2 and 3 we know that for x≠0 and a prime order n, rotnm(x) for any m generates exactly n objects.

There are however n-1 loops of order n, namely rotn1, rotn2, … rotnn-1. These loop open functions are completely interchangeable.

We would like to allow a notation that is closer to conventional mathematical notation, but this requires that we randomly pick one chain to be in the 1 direction such that 1 = inc0(0).

For each order n we define a rotational constant of order n called rn such that r1=1 and rn=rotn(1). (“rotational” because they loop, not because they necessarily create a geometric rotation.)

This is just for convenience, so we can use familiar binary operations rather than the unary rot. Actually there is no privileged “1” object, the only object we define is “0” in Axiom 1.

Some lemmas

Lemma 1.              If rotnm is a loop of order n then rotnn-m is its inverse open function.

Proof: rotnm(rotnn-m(x))=rotnn(x)=x

Lemma 2.              Loops are commutative such that rotab(rotcd(x))=rotcd(rotab(x)).

Proof: rotab(rotcd(x))=rotacb+d(x)=rotcd(rotab(x))

Lemma 3.              Non-prime order loops contain sub-loops for dividing orders.

Proof: from Axiom 3 we get (rnm)n=rmand (rnm)m=rn

Lemma 4.              Even ordered loops contain negative rotational constants.

Proof: from Lemma 3 we get (r2n)n = -1

Objects a and b in set Mn can be generalised like this:

$\begin{array}{l} a,b\in M_{n} \\ a=\sum\limits_x {(r_{n} )^{x}a_{x} } \\ b=\sum\limits_x {(r_{n} )^{x}b_{x} } \\ \end{array}$

We call the numbers ax and bx the coordinates of a and b respectively.

We can define addition of a and b.

$a+b=\sum\limits_x {(r_{n} )^{x}(a_{x} } +b_{x})$

ax and bx are natural numbers.

Lemma 5.              Even ordered sets M2n are n dimensional, while the odd ordered sets M2n+1 are 2n+1 dimensional.

Proof:

M1=N, M2=Z

From Lemma 4: in an even order M2n there are only n dimensions because of the 2n powers of the rotational constant, half are inverses of each other.

Lemma 6.              (Mn,+) is a free abelian group of rank n if n is odd or of rank n/2 if n is even.

Proof: Let Ln be the set generated by repeated multiplication of rn. This set contains n elements including 1:

$L_{n} =\{1,r_{n} ,(r_{n} )^{2},...(r_{n} )^{n-1}\}$

Ln is closed under binary multiplication.

There is associativity since:

$(r_{n}^{a} r_{n}^{b} )r_{n}^{c} =r_{n}^{a+b+c} =r_{n}^{a} (r_{n}^{b} r_{n}^{c} )$

The identity element is 1.

Each element rna has an inverse rnn-a since

rna rnn-a = rnn = 1

Therefore Ln is a cyclic group of order n.

Set Mn is a group of Ln over N, where n>2.

Lemma 7.              The set Mn does not depend on the choice of object 0 in Axiom 1 in the sense that sets generated by different objects are isomorphic to each other.

Proof: We pick a new object Q as the origin and we define a mapping from each point p to p+Q in the new coordinate system. Inc open functions remain unchanged in the new coordinate system and we define a new open function qrot such that:

qrotn(x) = rotn(x-Q)+Q

Any number rotn(x) in the original system maps to number rotn(x)+Q in the new coordinate system.

rotn(x)+Q = qrotn(x+Q)

Neighbourhood and generation

A number p in Mn with all non-zero coordinates has n neighbours in order n, namely:

$\{p-r_{n}^{m} \vert 1\le m\le n\$

In an even ordered, n-dimensional, set M2n half of these are in the direction towards the origin, and half away from it. We call the ones towards the origin predecessors in order 2n and the ones away from the origin successors in order 2n. The predecessors of p are these:

$\{p-r_{n}^{m} \vert 1\le m\le n,r_{n}^{m} >0\}$

An odd ordered set M2n+1 is 2n+1 dimensional. Each point with all non-zero coordinates has 2n+1 neighbours. All the neighbours are predecessors order 2n+1.

If a number has coordinates that are zero then it has fewer neighbours. Let’s have a look at a diagram depicting the M4 set to illustrate this.

The grey numbers on the axes have only one predecessor order 4 each, while the white numbers have two predecessors order 4 each.

We can say that each number is defined by its predecessor(s), recursively until we reach zero. For example the number 3 is defined as 2+1. In effect “3” is shorthand for ((0+1)+1)+1.

The minimum number of inc chains to reach the number from zero is called the generation.

Using once again our complex integer grid to illustrate this:

The number inside each circle indicates the generation. Thus generation 1 is defined solely by the initial condition at the origin. Generation 2 is defined by generation 1 and so on. Generation is the distance from the origin and computed by the sum of the coordinates.

1.5.      General form

In an even order M2n there are only n dimensions because of the 2n powers of the rotational constant, half are inverses of each other.

For 0 < m < 2n-1 we only want the positive (r2n)m

The use absolute bars to denote this:

$for 0

As an example let’s look at order 4. The four powers of r4 are 1,i,-1,-i. We pick the two positive powers by specifying |r4|0 and |r4|1.

The general form for a number:

Odd orders 2n+1 have 2n+1 coordinates ak.

$p=\sum\limits_{k=0}^{2n} {a_{k} r_{2n+1}^{k}}$

Even orders 2n have n coordinates.

$p=\sum\limits_{k=0}^{n-1} {a_{k} \left| {r_{2n} } \right|^{k}}$

As a simplification we will use the following notation to stand for the general order n case.

$p=\sum\limits_k^ {a_{k} \left| {r_{n} } \right|^{k}}$