The Extended Natural Number System
by
Huen Y.K.
CAHRC, P.O.Box 1003, Singapore 911101
http://web.singnet.com.sg/~huens/
email: huens@mbox3.singnet.com.sg
(A short communication - 1st released: 10/9/97.)
Abstract
It is proved that by increasing or decreasing the unit intervals of the natural number
system Nat(z) by multiplying these by a constant rational fraction which could be proper,
improper,
or integer, the resultant number sequences are always order invariant. To extend the number
family we can
start with the natural number sequence Nat(z)=z/(z-1) as the base which is redefined to
take
the general form GenNum(z) = z^(i*j)/(z^(i*j)-1) where i is integral but j can take any rational
number or integer values in the positive number range. A generalised equation of
divisibles given by
GenComp(z):= 1/(z^(i*j)*(z^(i*j)-1)) can be used to predict distributions of composites and
primes in these number systems. The whole family is called the extended natural number
system since this paper shows that both natural and rational number sequences are
govenered by the same generating functions. It is conjectured that irrational number
sequences do not belong to this family since such numbers do not have rational
fraction representations.
1. Introduction
The author proposes that the natural number system is not unique but that it is only a single
member, albeit a very unique and most studied one, within a general family which contains
an infinity of natural number systems and rational number systems. A proper number
system is one which always generate numbers defined within its own system when subjected
to arithmetic operations. Since divisions cannot be performed globally in number sequences,
such sequences are as order-invariant rather than as rings. This is called the extended
natural number system since rational numbers should be considered as forming a
continuum between integer numbers. In sequence algebra, both natural and rational
numbers are governed by the same generating functions which can also be used to predict
the distributions of primes and composites. A restriction is imposed that
the interval multipliers j's must not be irrational numbers since these numbers cannot be
expressed as rational fractions. In other words, there should be exact subdivisions
between successive integers and this is guaranteed so long as interval multipliers
are rational numbers.
2. The Extended Natural Number System
The standard sequence algebraic generating function defined for Nat(z) is z/(z-1).
From it, one could derive Even(z),
Odd(z),
Comp(z), Prime(z) and and other interesting sequences pertaining to Nat(z) [17,18,19]. In
this paper, an extended form of Nat(z) called GenNum(z) is defined in equation (1) and an
extended equation of divisibles called GenComp(z) is defined in equation (4).
GenNum(z):=z^(i*j)/(z^(i*j)-1);
.............................................(i*j)
...........................................z
..........ExtendNum(z):= ------------ ........................................(1).
..........................................(i*j)
........................................z......- 1
The original Nat(z) sequence is simply a special case where both i and j equal unity and this
is confirmed by finite expansions as shown in equation (2). Since j is the interval
multiplier, it could take any proper or improper rational fraction including integers values.
Nat(z):=series(sum(sum(z^(i*j)/(z^(i*j)-1),i=1..1),j=1..1),z=infinity,5);
...........................................................1.........1........1...........1
...........................Nat(z) := 1 + 1/z + ---- + ---- + ---- + O(----)
..............................................................2........3........4............5
...........................................................z..........z........z............z
.............................................................= {0,1,2,3,4,........}. ..................(2).
The first member of the rational number system family is obtained by putting i = 1 and j = 1/2
as shown in equation (3). Other members are obtained by assigning j to decreasing reciprocal integer
fractions of 1/3,1/4,1/5, and so on. Number sequences with arbitrary rational fractional
intervals are more complicated but these are still governed by the same generating
functions and are thus integral in the extended natural number system.
GenNum(z):=series(sum(sum(z^(i*j)/(z^(i*j)-1),i=1..1),j=1/2..1/2),z=infinity,5);
Ratnum(z) :=
...................1/2...................3/2....1............5/2......1.............7/2......1............9/2..........1
......1 + (1/z).....+ 1/z + (1/z)....+ ---- + (1/z).....+ ---- + (1/z)....+ ---- + (1/z).....+ O(----)
......................................................2.......................3.........................4..........................5
....................................................z.......................z..........................z..........................z
..................= {0,1/2,1,3/2,2,5/2,3,7/2,4,9/2,..................} ....................(3).
For classication purepose, equation (3) is also called a rational number system
because integer numbers integer numbers can be considered as special rationa numbers
of the form 1/1,2/1,3/1 and so on.
Table 1 gives a summary of finite expansions of the first five members of each family.
Table 1 - The first five members of the two families, i.e.,
natural number systems and rational number systems
------------------------------------------------------------------------------------------
Natural number systems:
i=1; j=1: (The first member is Nat(z)).
{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,...............................}
i=1; j=2; (This is in fact Even(z)).
{0,2,4,6,8,10,12,14,16,18,20,22,24,26,...............................}
i=1; j=3:
{0,3,6,9,12,15,18,21,24,27,30,33,36,39,..............................}
i=1; j=4:
{0,4,8,12,16,20,24,28,32,36,40,44,48,52,.............................}
i=1; j=5:
{0,5,10,15,20,25,30,35,40,45,50,55,60,65,70,......................}
Rational number systems:
i=1; j=1/1: (this is identical to the first member in the natural number family, i.e. Nat(z)).
i=1; j=1/2:
{0,1/2,1,3/2,2,5/2,3,7/2,4,9/2,5,11/2,6,13/2,7,15/2,8,17/2,9,19/2,10................}
i=1; j=1/3:
{0,1/3,2/3,1,4/3,5/3,2,7/3,8/3,3,10/3,11/3,4,13/3,14/3,5,16/3,17/3,6,19/3,20/3,7,22/3,23/3,8
,25/3,26/3,9,28/3,29/3,10,.......................}
i=1; j=1/4:
{0,1/4,1/2,3/4,1,5/4,3/2,7/4,2,9/4,10/4,11/4,3,13/4,14/4,15/4,4,17/4,18/4,19/4,5,21/4,22/4,23/4
,6,25/4,26/4,27/4,7,29/4,30/4,31/4,8,33/4,34/4,35/4,9,37/4,38/4,39/4 ,10,........}
i=1; j=1/5:
{0,1/5,2/5,3/5,4/5, 1,6/5,7/5,8/5,9/5,2,11/5,12/5,13/5,14/5,3,16/5,17/5,18/5,19/5,4,21/5,22/5,
23/5,24/5,5,26/5,27/5,28/5,29/5,6,31/5,32/5,33/5,34/5,7,............................}
-------------------------------------------------------------------------------------------------------------
Distributions of composites can be predicted for all members of the family using the
extended form of the equation of divisibles defined in equation (4) [17,18,19]:
Comp(z):=sum(1/(z^(i*j)*(z^(i*j)-1)),i=2..infinity);
.....................................infinity
..................................... -----
.......................................\ ..................1
....................Comp(z):=....)........------------------- ..........................................(4).
......................................./..............(i j)......(i j)
......................................-----.......z.......(z..... - 1)
.....................................i = 2
The choice of intervals between successive integers does not affect the index value of the
lowerbound i = 2. This is because divisors are only available for testing compositeness
starting with the third number which in the case of the Nat(z) is the integer 2. If the interval is
1/2, then the third member will take the integer value of 1 but this does not affect the
lowerbound value which still remains at i = 2 since this third integer is given by i*j = 2*(1/2) =
1.
Equation (5) shows that the composites numbers are predicted correctly including the
number of ways of factoring.
Comp(z):=sort(series(sum(sum(1/(z^(i*j)*(z^(i*j)-1)),i=2..20),j=1..1),z=infinity,20));
.................................1.......2.........2........1........2.......4.......2......2......3.......4
.............Comp(z):= ---- + ---- + ---- + ---- + --- + --- + --- + --- + --- + --- .....(5).
...................................4.........6........8.......9.......10.....12.....14.....15....16....18
.................................z.........z........z.........z.........z.......z........z......z.......z.......z
An alternative formulation which enables one to trace all the factoring divisors used for each
integer number is show in equation (6):
Comp(z):=sort(series(sum(sum(x^i*y^j/(z^(i*j)*(z^(i*j)-1)),i=2..20),j=1..1),z=infinity,20));
.....................2........3.........3........2.......4........2........5........2..........6.......4........3........2
...................x..y....x..y......x..y + x..y....x..y + x..y.....x..y + x..y......x..y + x..y + x..y + x
Comp(z):= ---- + ---- + ----------- + ----------- + ----------- + -------------------------
......................4.........9...............6..................8...............10........................12
....................z..........z...............z..................z...............z...........................z
................7........2.........5........3.........8........4........2.........9........6........3.......2
..............x..y + x..y......x..y + x..y......x..y + x..y + x..y......x..y + x..y + x..y + x..y
...........+ ----------- + ----------- + ------------------ + ------------------------- ..............(6).
....................14 ................15.....................16..................................18
...................z....................z......................z....................................z
Table 2 - Factoring Of the first 10 composites in Nat(z).
The divisors 1 and the integer number itself must be excluded
for use in factoring. The number of ways of factoring is given
by the numerator coefficients of Comp(z) in equation (5).
-----------------------------------------------------------------------------
4={2*2}
6={2*3, 3*2)
8={2*4, 4*2}
9={3*3}
10={5*2,2*5}
12={6*2,4*3,3*4,2*6}
14={7*2,2*7}
15={5*3,3*5}
16={8*2,4*4,2*8}
18={9*2,6*3,3*6,2*9}
-----------------------------------------------------------------------------
In equation (7) and Table 3, the accuracy of Comp(z) is also tested for the first rational
number sequence with j = 1/2, i.e. half a unit interval.
Comp(z):=sort(series(sum(sum(1/(z^(i*j)*(z^(i*j)-1)),i=2..20),j=1/2..1/2),z=infinity,20));
....................1........2........2........2.......4.........2.........3........4.......4........2......6.......2.......4......6
Comp(z):= ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + --- + --- + --- + --- + --- + ---
......................2........3.........4........5.......6.........7.........8........9.....10......11....12.....13....14......15
....................z.........z........z.........z........z.........z.........z.........z......z.........z......z........z.......z........z
..........4.......2.......7.......2...........25/2..........9/2.............15/2.............21/2..............27/2
......+ --- + --- + --- + --- + (1/z).......+ (1/z).....+ 2 (1/z)......+ 2 (1/z)......+ 2 (1/z)
............16.....17.....18......19
...........z.......z........z.......z
...................33/2............35/2.............39/2
......+ 2 (1/z).......+ 2 (1/z).......+ 2 (1/z) ............................................(7).
The factoring can be traced using an alternative formulation as shown in equation (8):
Comp(z):=sort(series(sum(sum(x^i*y^j/(z^(i*j)*(z^(i*j)-1)),i=2..20),j=1/2..1/2),
z=infinity,20));
......................................................................2..1/2........3..1/2....2..1/2.........4..1/2....2..1/2
........25/2..5..1/2..........9/2..3..1/2........1.......x..y...........x..y....+ x..y............x..y....+ x..y
(1/z)........x..y......+ (1/z).....x..y....+ O(---).+ -------..+ ----------------- + -----------------
............................................................20..........2..................3..............................4
............................................................z...........z..................z..............................z
.........5..1/2.....2..1/2.............6..1/2....4...1/2....3..1/2....2..1/2.........7..1/2....2..1/2
........x..y....+ x..y................x..y......+ x..y.....+ x..y....+ x..y............x..y....+ x..y
......+ ----------------- + ------------------------------------- + -----------------
..................5.............................................6............................................7
................z.............................................z............................................z
..............8..1/2....4..1/2.....2..1/2...........9..1/2.....6..1/2.....3..1/2.....2..1/2
............x..y....+..x..y....+..x..y..............x...y....+..x..y....+..x..y....+..x..y
......+ --------------------------- + -------------------------------------
............................8..................................................9
..........................z..................................................z
............10..1/2.....5..1/2.....4..1/2.....2..1/2.........11..1/2....2...1/2
...........x...y....+..x..y....+..x..y....+..x...y.............x...y....+..x....y
......+ -------------------------------------- + ------------------
.................................10..............................................11
................................z................................................z
.............12..1/2.....8..1/2....6..1/2......4..1/2.....3..1/2.....2..1/2.........13..1/2.....2..1/2
............x...y....+..x..y....+..x..y.....+..x..y....+..x..y....+..x..y.............x...y.....+..x..y
......+ ---------------------------------------------------------- + ------------------
........................................12....................................................................13
.......................................z.....................................................................z
...........14..1/2........7..1/2......4..1/2.......2...1/2
.........x....y.......+..x..y......+..x..y......+..x..y
......+ --------------------------------------
....................................14
...................................z
...........15..1/2.....10..1/2.....6..1/2.....5..1/2.....3..1/2.....2..1/2
.........x....y.....+..x...y....+...x..y.....+..x..y....+..x..y....+..x..y
......+ -----------------------------------------------------------
.........................................................15
........................................................z
............16..1/2......8..1/2.....4..1/2.....2..1/2........17.1/2......2..1/2
..........x....y.....+..x..y....+..x..y....+..x..y.............x...y.....+..x..y
......+ -------------------------------------- + ------------------
........................................16.....................................17
.......................................z.......................................z
......18..1/2.....12..1/2......9..1/2.....6..1/2.....4..1/2.....3..1/2....2...1/2.........19..1/2.....2..1/2
....x....y.....+..x...y.....+..x..y....+..x..y....+..x..y....+..x..y....+..x..y..............x...y....+..x..y
+ --------------------------------------------------------------------- + ------------------
...................................................18....................................................................19
.................................................z.......................................................................z
......7...1/2.....3..1/2.........21/2......9..1/2.....3..1/2........27/2.....5..1/2.....3..1/2.......15/2
+ (x...y.....+..x..y....) (1/z).......+ (x..y....+..x..y....) (1/z)......+ (x..y....+..x..y....) (1/z)
............7..1/2......5..1/2........35/2.....11..1/2......3...1/2.........33/2
......+ (x..y.....+..x..y.....) (1/z)......+ (x...y......+..x..y.....) (1/z)
............13..1/2.....3..1/2...........39/2
......+ (x....y.....+ x...y......) (1/z) ............................................................(8).
Table 3 - Factoring of composites in the rational number sequence
with interval of j = 1/2. The divisor of 1/2 and the rational number itself must
be excluded in factoring. The number of ways of factoring is given by
the numerator coefficients of equation (8).
---------------------------------------------------------------------------------------------
2= {1}
3= {1,3/2}
4= {1,4/2}
9/2= {3}
5= {1,5/2}
6= {1,6/4,6/3,6/2}
7= {1,7/2}
15/2= {5,3 }
8= {1,8/4,8/2}
9= {1,9/6,9/3,9/2}
10= {1,10/5,10/4,10/2}
21/2= { 7,3}
11= {1,11/2}
12= {1,12/8,12/6,12/4,12/3,12/2}
13={1,13/2}
27/2= {9,3 }
14={1,14/7,14/4,14/2}
15={1,15/10,15/6,15/5,15/3,15/2}
16= {1,16/8,16/4,16/2}
33/2= {11,3}
17= {1,17/2}
35/2={7,5}
18={1,18/12,18/9,18/6,18/4,18/3,18/2}
19={1,19/2}
39/2=(13,3}
--------------------------------------------------------------------------------------------------------------
Table 4 - Summary Of Composites For the First 5 members of
The natural number subfamily and the rational number subfamily
-----------------------------------------------------------------------------------------------------------------
Natural composite number sequences with multiples of unit intervals j.
j=1: {4,6,8,9,10,12,14,15,18,20,21,22,24,25,26,27,28,30,32,33,34,35,38,.....}
j=2: {8,12,16,18,20,24,28,30,32,36,40,42,44,48,50,52,54,56,60,64,66,......}
j=3: {12,18,24,27,30,36,42,45,48,54,60,63,66,72,75,78,81,84,90,96,99,102,105,108,114,..}
j=4:
{16,24,32,36,40,48,56,60,64,72,80,84,88,96,100,104,108,112,120,128,132,136,140,144,....}
j=5:
{20,30,40,45,50,60,70,75,80,90,100,105,110,120,125,130,135,140,150,160,165,170,....}
----------------------------------------------------------------------------------------------------------------
Rational composite number sequences with fractional unit intervals j.
j=1/2:
{2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
25/2 9/2 15/2 21/2 27/2 33/2 35/2 39/2}
j=1/3:
{2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
34/3 38/3 44/3 46/3 49/3 40/3 52/3 55/3 56/3
50/3 4/3 8/3 10/3 14/3 16/3 20/3 22/3 25/3
26/3 28/3 32/3 35/3 58/3}
j=1/4:
{1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
19 25/2 9/2 15/2 21/2 27/2 33/2 35/2 39/2 3/2 9/4 5/2 7/2
15/4 21/4 11/2 25/4 13/2 27/4 33/4 17/2 35/4 19/2 39/4 45/4
23/2 49/4 51/4 55/4 57/4 29/2 31/2 63/4 65/4 69/4 37/2 75/4 77/4}
j=1/5:
{2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
6/5 8/5 9/5 12/5 14/5 16/5 18/5 4/5
21/5 22/5 24/5 26/5 27/5 28/5 32/5 33/5
34/5 36/5 38/5 39/5 42/5 44/5 46/5 48/5
49/5 51/5 52/5 54/5 56/5 57/5 58/5 62/5
63/5 64/5 66/5 68/5 69/5 72/5 74/5 76/5
77/5 78/5 81/5 82/5 84/5 86/5 87/5 88/5
91/5 92/5 93/5 94/5 96/5 98/5 99/5}
j=1/6:
{1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
34/3 38/3 44/3 46/3 49/3 40/3 52/3 55/3
56/3 50/3 4/3 8/3 10/3 14/3 16/3 20/3
22/3 25/3 26/3 28/3 32/3 35/3 58/3 25/2
9/2 23/3 49/6 55/6 29/3 31/3 65/6 2/3
5/3 7/3 41/3 85/6 43/3 91/6 47/3 95/6
11/3 25/6 13/3 17/3 35/6 19/3 53/3 115/6
59/3 119/6 37/3 77/6 15/2 21/2 27/2 33/2
35/2 39/2 3/2 5/2 7/2 11/2 13/2 17/2
19/2 23/2 29/2 31/2 37/2}
-----------------------------------------------------------------------------------------------------------------------
Rational number systems are related to Nat(z) as these can be generated by Nat(z)*j where j
takes reciprocal integer values.
Table 5 - Summary Of Primes For the First 5 members of
The natural number family and the rational number family
-----------------------------------------------------------------------------------------------------------------
Primes in natural number sequences with multiples of unit intervals j:
j=1: {2,3,5,7,11,13,17,19,23,29,31,37,41,.....}
j=2: {6,10,14,22,26,34,38,46,58,62.......................}
j=3: {9,15,21,33,39,51,53,57,69,87,93,111,.................}
j=4: {12,20,28,44,52,68,76,92,116,124,136,...............}
j=5: {15,25,35,45,55,65,85,95,115,145,155,.....}
----------------------------------------------------------------------------------------------------------------
Primes in rational number sequences with submultiples of unit intervals j:>
j=1/2;
{5/2,7/2,11/2,13/2,17/2,19/2,23/2,29/2,31/2,37/2,.......}
j=1/3;
{7/3,11/3,13/3,17/3,19/3,23/3,29/3,31/3,37/3,41/3,43/3,47/3,53/3,59/3,........}
j=1/4:
{2/4,3/4,5/4,7/4,11/4,13/4,17/4,19/4,23/4,29/4,31/4,37/4,41/4,43/4,47/4,53/4,59/4,61/4,67/4,7
1/4,73/4,.................}
j=1/5
{3/5,7/5,11/5,13/5,17/5,19/5,23/5,29/5,31/5,37/5,41/5,43/5,47/5,53/5,59/5,61/5,67/5,71/5,73/5
,79/5,83/5,89/5,97/5,.............}
j=1/6:
{3/6,11/6,13/6,17/6,19/6,23/6,29/6,31/6,37/6,43/6,47/6,53/6,59/6,61/6,67/6,71/6,73/6,79/6,83/
6,89/6,97/6,101/6,103/6,107/6,109/6,113/6,......................}
-----------------------------------------------------------------------------------------------------------------------
3. GenNum(z) and RatNum(z)
Any number sequence with uniform interval is order-invariant. This is because
an order-invariant number sequence always has 1/z^0 = 1 as its first member. Thus
when such a number sequence is exponentiated, this first member when multiplied
into the other sequence will not affect its order. Other members of the second sequence
will only cause shifting of the resultant sequence but they will not affect the order either.
It can be seen that the number sequences listed in Table 1 can be
generated by GenNum(z) and Ratnum(z) and these all have uniform intervals and
are therefore order-invariant.
(i) GenNum(z) sequences:
GenNum(z):=sum(z^i/(z^i-1),i=1..5);
...................................................2...........3...........4...........5
.................................................z...........z...........z...........z..............z
........................GenNum(z) := ----- + ------ + ------ + ------ + ------
................................................2............3............4............5........z - 1
..............................................z - 1......z - 1......z - 1......z - 1
GenNum1:=series((z/(z-1))^2,z=infinity,10);
......................................................3........4.........5........6........7........8............1
................GenNum1 := 1 + 2/z + ---- + ---- + ---- + ---- + ---- + ---- + O(----)
........................................................2........3........4.........5........6........7.............8
......................................................z.........z........z.........z.........z.........z.............z
GenNum2:=series((z^2/(z^2-1))^2,z=infinity,20);
..............................................2........3.........4.........5........6......7.......8..........1
..................GenNum2 := 1 + ---- + ---- + ---- + ---- + --- + --- + --- + O(---)
................................................2........4.........6.........8.......10.....12.....14........16
..............................................z.........z.........z.........z........z........z........z..........z
GenNum3:=series((z^3/(z^3-1))^2,z=infinity,30);
..............................................2.........3.........4........5......6......7......8............1
..................GenNum3 := 1 + ---- + ---- + ---- + --- + --- + --- + --- + O(---)
................................................3..........6........9......12.....15....18.....21.........24
..............................................z..........z.........z........z.......z.......z.......z............z
GenNum4:=series((z^4/(z^4-1))^2,z=infinity,40);
...............................................2........3........4........5.......6.......7.......8.........1
...................GenNum4 := 1 + ---- + ---- + --- + --- + --- + --- + --- + O(---)
.................................................4........8.......12......16.....20.....24.....28......32
...............................................z.........z........z........z.......z........z.......z........z
GenNum5:=series((z^5/(z^5-1))^2,z=infinity,50);
...............................................2........3......4.......5.......6.......7.......8...........1
...................GenNum5 := 1 + ---- + --- + --- + --- + --- + --- + --- + O(---)
.................................................5.......10....15.....20.....25.....30.....35..........40
...............................................z........z.......z.......z.......z........z........z............z
(ii) RatNum(z) sequences:
RatNum1:=series((z^1/(z^1-1))^2,z=infinity,10);
.............................................3........4.........5........6........7........8........9........10...........1
.........RatNum1 := 1 + 2/z + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + O(---)
...............................................2........3.........4........5.........6........7........8........9...........10
.............................................z........z.........z.........z.........z.........z.........z..........z...........z
RatNum2:=sort(series((z^(1/2)/(z^(1/2)-1))^2,z=infinity,10));
.........................1..................5........7.........9.......11.......13......15......17..............7/2................9/2
RatNum2 := O(----) + 3/z + ---- + ---- + ---- + ---- + ---- + ---- + ---- + 8 (1/z)....+ 10 (1/z)
...........................9..................2........3.........4........5.........6........7........8
.........................z..................z.........z..........z........z..........z.......z.........z
.
...................11/2....................13/2............1/2.............15/2............3/2..............17/2
......+ 12 (1/z)......+ 1 + 14 (1/z).....+ 2 (1/z).....+ 16 (1/z).....+ 4 (1/z)....+ 18 (1/z)
...................5/2
......+ 6 (1/z)
RatNum3:=sort(series((z^(1/3)/(z^(1/3)-1))^2,z=infinity,5));
.....................................13/3..................7......10.....13................1/3.............2/3............7/3
......RatNum3 := O((1/z)........) + 4/z + ---- + ---- + ---- + 2 (1/z)....+ 3 (1/z)....+ 8 (1/z)
................................................................2.......3.......4
..............................................................z........z.......z
........................4/3.............5/3..............10/3..............11/3.............8/3
...........+ 5 (1/z).....+ 6 (1/z).....+ 11 (1/z)......+ 12 (1/z)......+ 9 (1/z).....+..1
RatNum4:=sort(series((z^(1/4)/(z^(1/4)-1))^2,z=infinity,3));
...............................5/2................9...............1/4.............3/4.............5/4.............1/2.............7/4
RatNum4 := O((1/z).....) + 5/z + ---- + 2 (1/z).....+ 4 (1/z)....+ 6 (1/z)....+ 3 (1/z)....+ 8 (1/z)
......................................................2
....................................................z
...........................9/4............3/2
......+ 1 + 10 (1/z).....+ 7 (1/z)
RatNum5:=sort(series((z^(1/5)/(z^(1/5)-1))^2,z=infinity,3));
...............................13/5................11.............3/5.............4/5............1/5............2/5.............7/5
RatNum5 := O((1/z).......) + 6/z + ---- + 4 (1/z)....+ 5 (1/z)....+ 2 (1/z)....+ 3 (1/z).....+ 8 (1/z)
.........................................................2
.......................................................z
..................8/5.................9/5..............6/5...............11/5...............12/5
.....+ 9 (1/z).....+..10 (1/z).....+..7 (1/z).....+..12 (1/z)......+..13 (1/z)......+..1
RatNum6:=sort(series((z^(1/6)/(z^(1/6)-1))^2,z=infinity,3));
....................................8/3............13.............1/6..................5/6............1/3.............7/6
.....RatNum6 := O((1/z) ) + 7/z + ---- + 2 (1/z)....+ 1 + 6 (1/z)....+ 3 (1/z)....+ 8 (1/z)
........................................................2
......................................................z
.......................2/3...............3/2..............5/3...............11/6............4/3...............13/6
..........+ 5 (1/z).....+ 10 (1/z).....+ 11 (1/z).....+ 12 (1/z)......+ 9 (1/z).....+ 14 (1/z)
..........................7/3...............5/2.............1/2
...........+ 15 (1/z).....+ 16 (1/z).....+ 4 (1/z)
4. Conclusions
Natural number systems with uniform subdivision of intervals between successive
integers are order-invariant and are described by
GenNum(z)=z^(i*j)/(z^(i*j)-1) and the distrbutions of composites by
GenComp(z)=1/(z^(i*j)*(z^(i*j)-1)).
Also for GenComp(z), the lowerbound of i = 2 applies to the whole family. This shows that
natural numbers and rational numbers are closely related. However irrational numbers
cannot be included in this family because these cannot find exact rational fraction forms.
It is noted from Table 4 that higher members of the family of natural number systems have
distributions of primes quite different from that of Nat(z). In the case of the family of rational
number systems, a general pattern can be observed that the primes all have prime
numerators irrespective of the values of reciprocal integer in j. If j is a prime, then that prime
will be missing in a specific rational number system.
It is also observed that irrational numbers are not included in the family because such
numbers cannot be expressed as rational fractions. Some rational numbers amongst the
systems are recurrent but these are not irrational since they have exact rational fraction
forms.
It is conjectured that the integer number system and the reciprocal integer number system
form a continuum of natural numbers. In other words, reciprocal integer numbers should be
considered as natural numbers as well. The generality goes further than this as even
improper ration fractions form part of the continuum.
Although the natural number system with unity number interval is the most well
known and studied, an infinity of number systems with uniform intervals which are
either multiples or submultiples of unity intervals can be formed. These are legitimate
numer systems. The author conjectures that all such number systems are order-invariant.
5. References
(Not all references are directly relevent. References in sequence algebra are all
included as these are not generally available outside this URL site.)
1. Huen Y.K.: A Simple Introduction To Sequence
Algebra,
URL site: http://web.singnet.com.sg/~huens/
2. Huen Y.K.: The Canonical Generating Function or
CGF(z) - a Swiss-knife function. URL site: http://web.singnet.com.sg/~huens/ .
3. Huen Y.K.: Information Contents Of Number
Theoretic
Functions. URL site: http://web.singnet.com.sg/~huens/ .
4. Huen Y.K.: In Search Of Exotic Arithmetic Operators,
URL site: http://web.singnet.com.sg /~huens/ .
5. Huen Y.K.: Visual Solutions Of Number Theoretic
Functions in Multidimensional Sequence Space, URL site: http://web.singnet.com.sg
/~huens/ .
6. Huen Y.K.: Final Value Theorems Applied To Number
Sequences -- its strengths and weaknesses, URL site: http://web.singnet.com.sg /~huens/ .
7. Huen Y.K.: Unsolved Problems In Sequence Algebra,
URL site: http://web.singnet.com.sg /~huens/ .
8. Huen Y.K.: Explicit Formulation For Modular
Arithmetic
In Sequence Algebra, URL site: http://web.singnet.com.sg /~huens/ .
9. Huen Y.K.: Cyclic Generating Functions In Sequence
Algebra, URL site: http://web.singnet.com.sg /~huens/ .
10. Huen Y.K. : Methods Of Developing Sequence
Algebraic Formulations For Comp(z) and Prime(z). URL site: http://web.singnet.com.sg
/~huens/ .
11. Huen Y.K. : Information Contents Of Hypothetical
DNA Sequences. URL site: http://web.singnet.com.sg/~huens/ .
12. Huen Y.K. : Composite Number Sequence
Challenge 1/97. URL site: http://web.singnet.com.sg/~huens/ .
13. Huen Y.K. : Lemmata, Corollaries, And Theorems In
Sequence Order Analysis. URL site: http://web.singnet.com.sg/~huens/ .
14. Huen Y.K. : Is it a number line or a number
generator? URL site: http://web.singnet.com.sg/~huens/ .
15. Huen Y.K. : Some Interesting Contiguity Properties
Of Odd(z)^2. URL site: http://web.singnet.com.sg/~huens/ .
16. Huen Y.K. : Sequence Algebraic Modelling Of Pixel
Colours. URL site: http://web.singnet.com.sg/~huens/ .
17. Huen Y.K.: A Matrix Map for Prime and Non-prime Numbers, INT. J. Math. Educ. Sci.
Technol., 1994, VOL. 25, NO.6, pp 913-920.
18. Huen Y.K.: Some Interesing Properties Of The Natural Number System, Int. J. Math.
Educ.
Sci. Technol., 1996, VOL.27, NO. 5, 685-691.
19. Huen Y.K.: Visual algebra and its applications, INT. J. Math. Educ. Sci. Technol.,1997,
VOL.28 NO.3, 333-344.
20. Huen Y.K.: Twin primes revisited: INT. J. Math. Educ. Sci. Technol., 1997,
VOL.??,NO.?, ???-???. (In the press as proof paper mes 100488).
21. Huen Y.K.: Is Pie Periodic? : INT. J. Math. Educ. Sci. Technol., 1997, VOL??,NO.?,???-
???,(In the press as proof paper mes100495).
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