Download THE INFINITE CALCULUS

preLiminarY inFiniTe caLculuS
By
Zuhair Abdulgafoor Al-Johar
Iraq,Baghdad
2004
Preface
This account is only a simple trial to endeavour into the infinite numerical world,
seeking its best representation and possible calculus, a quest that proved over
centuries to be delusive! I just hope that the current study would provide a good
beginning towards solving such an extremely difficult puzzle.
Zuhair Abdulgafoor Al-Johar
II
Contents
Introduction:
IV
-The numerical systems
Material:
V
-The Gamma numericals
Translation from decimal system
Derivation and Counting
Algebra
Numerical alignments
Vertical and Horizontal operations
Arithmetic and Geometric operations
Mixed operations
Comparisons and the role of zero
Results:
VI
VIII
XI
XIV
-The infinite calculus
Representation of infinity of continuum by the Gamma numerical system
Summations and Products
Discriminative & Indiscriminative summations
Subtraction and Ratios
XV,XVI
Roots and Complex numericals
XVIII
Mixed algebra
Comparative issues and the role of zero
XIX
-Additional Results : Exponents
XX
Roots of finite numbers
00
XXI
The complementary ratios
-Questionable Results
XXI
- The zero denominators
-Parallel Results
XXII
Discussion:
XXIII
Older methods
Validity of arithmetic infinite operations
Conclusion&Recommendations
Summery
Appendix
Paradoxical Results
Finite-Infinite ordinal scale
The zero denominators
XXIV
XXV
XXVI
XXIX
XXX
XXXI
III
Introduction
A numerical* (nc.)
is a visual representation of a number. For example 2 and
10(binary) and II are three different numericals of number 2.
Numerical systems:
a) The Alpha numericals(blind nc.; nominal nc. ; non deductive nc.,non counting
nc.,infinitary nc.):
Translation from decimals:
-n,……,-3,-2,-1,…..,-0.5, -0.25,0,1,2,3,(6) 0.5,4.6,………….,n,x+y(-1)0.5 .
a,…….,b ,c, d,……,k,
m ,e, p,I,h,g
,v,…………….,t,z .
Each numerical is absolutely different in shape from the other(i.e. no deduction).
b)The Beta numericals(semicounting nc., semideductive nc., n-periodic nc.):
These are the well-known binary, ternary,quaternary,…….,decimal,…..,etc.
nc., here 2<n<∞ .
Translation rules:
x1x2x3……….xm = x1n m-1 + x2n m-2 + x3n m-3 +…… + xi n m-i + ……+ xmn m-m………(1
y.x1 x2 x3 …..xm = y + x1 n -1 +x2 n –2 +x3 n –3 +……+ xi n –i +……+ xm n-m………….….(2
were xi = 0,1,….,n-1
i=1,2,3,……,m
characteristics:
Each n-periodic nc.system contains n nominal nc. as the first members of their
non –negantive whole numerical set and the rest of the set is made of nc. that are
derived in shape(deduced) from the n nominal numericals. For example the decimal
nc. is made of 10 nominal nc. (0,1,2,3,4,5,6,7,8,9),the rest of the natural decimal
numerical set is made from numericals that are totally derived in shape from these 10
nominal numericals.
They are semideductive because the 1st n-1 natural numericals are
non-deductive;semicounting because the 1st n-1 natural numericals are
non-counting,also these numericals don’t provide a simple representation of infinite
numbers and therefore cannot easily count infinite numbers.
c)The Gamma numericals(simple nc.,primitive nc.,philosophic nc.,fully deductive
nc.,fully counting nc.,unary nc.): These are the keys of the infinite numerical world .
Note: The alpha nc. and the gamma nc. are the n= ∞ and n=1 periodic nc.
respectively.
*

usually called a “numeral”.
0 is not a natural numerical
IV
Materials
The Gamma Numericals(the unary nc.):
Main translations from decimals:
Natural finite decimal nc.
1, 2 , 3 ,
Natural finite basic gamma nc.
I, II , III , IIII , IIIII , ……………., IIII…..nI
Natural finite Ii gamma nc.
I , I2 , I3 ,
-ve whole finite decimal nc.
4
,
5
I4 ,
- 1 , -2 , -3 , - 4
, ……………, n
, …………….,
I5
,
In
, ………, -n
-5
-ve whole finite basic gamma nc. - I ,- II , - III , - IIII , - IIIII , ……….,- IIII…..nI
-ve whole finite Ii gamma nc.
, ……….,
- I ,- I2 , - I3 , - I4 , - I 5
- In
zero finite basic gamma nc. -000…n0, …… , -000 ,- 00, -0 , + , 0 , 00 , 000 ,…,000…n0
zero finite Ii gamma nc.
-0n
Rational finite decimal nc.
, ……….,
-03 , -02 , -0 , 00 , 0 , 02 , 03
,……….,
1/2 , 1/3 , 1/4 , ………, 1/n
n-1
0
0
0
Rational finite basic gamma nc.
Rational finite Ii gamma nc.
0
0
0
0
I ,
0
I ,
I
02
I ,
,
03
I
0
0
I
, ………..….,
, ……….…...,
I
0n-1
I
Irrational finite gamma numericals are approximated by the rational finite gamma
numericals.
Imaginary finite decimals
x+y(-1 )1/2
Imaginary finite Ii gamma nc.
Ix +Iy(-I)I
0
V
0n
Comparative finite gamma nc. notations.
Whole rationales: Ix 0y…………n. =
Whole roots:
iI
I nx/(x+y)
0
I
02(i-1) ..……. n. = ( In)
Derivation of and counting by the gamma nc.
I + I = II , I + I + I = III , I + I + I + ………….+nI = III………..nI =In
+ = Interspace between two “I” . = 00
Ii notation : The counter “i” is a natural decimal abbreviating the repetitions of “I” .
iI
notation : The rank “i” is a natural decimal marking the ith“I”. so nI means the nth “I”
Operations on whole finite gamma numericals:
± Ix ± IY = ± Ix+y
+
±Ix
x >y
Iy = ± Ix-y
± Ii
+ Ii = 0i
± Ii
+
0j = ± Ii
± Ix × ± IY = + Ix×y
± Ix × +
Iy = - Ix×y
± Ii × + 0j = - 0i×j
± Ii × ± 0j = 0i×j
(- Ix)
Iy
(Ix)
(-0i)
(0i)
Iy
Iy
Iy
= +Ixy if y is even
= - Ixy if y is odd
= Ixy
= - 0iy
= 0iy
if y is odd
if y is even
= 0iy
VI
Operations on rational finite gamma numericals:
k0
.
.
.
0
0
nI
.
.
.
I
I
0k
In
=
= In / I n+k
Note: do not be confused between the small “I”(font size 10) which represent a part of the large “I”
( font size 12) which represent the number one. Similarly the small zero “0” and the large zero “ 0”.
0y
0u
0y 0u
0y-z 0u
0y (z+u) 0u (x+y)
0y (z+u) – z (x+y)
0u (x+y)
Ix + I z
= Ix Iz = Ix+z 0z ;
Ix(z+u) Iz (x+y) = Ix (z+u)+ z (x+y) 0z (x+y)
(x+y)=(z+u)
(x+y)≠(z+u)
0y-z
0y (z+u) – z (x+y)
z≤y
= Ix+z
= Ix (z+u)+ z (x+y)
;
0u (x+y) + y (z+u)
If z (x+y) > y (z+u)
then: = I
0y
0u
Ix - Iz
(x+y)≠(z+u)
0u (x+y)
- Iz (x+y)
=
0y (z+u)
Ix (z+u)
Iz (x+y) - y (z+u)
0u (x+y) + x (z+u)
= - I │ x (z+u) _ z (x+y) │
=
0y
0u
Ix × Iz
= Ixz
0y
0u
Ix ÷ Iz
0y
= Ix ×
0y (z+u) + z (x+y)
Ix (z+u) - z (x+y)
; x (z+u) <
z (x+y)
; x (z+u) > z (x+y)
0(x+y) (z+u)- xz
0(x+y) – R[x(z+u) / (x+y)]
=I ING[x(z+u) / (x+y)] IR[x(z+u) / (x+y)]
Iz+u ÷ Iz
÷ Iz
0(x+y) – R[x(z+u) / (x+y)]
= I ING[x(z+u) / (x+y)] IR[x(z+u) / (x+y)]
=
=
×
0z-1
I
0z-1 0z (x+y) – R[x(z+u) / (x+y)]
I ING[x(z+u)/ (x+y)] × I + IR[x(z+u) / (x+y)]
0z -R [x(z+u) / (x+y) / z ] 0z (x+y) – R[x(z+u) / (x+y)]
I ING[x(z+u) / (x+y) / z] IR [ x(z+u) / (x+y) / z ]
IR[x(z+u) / (x+y)]
Note : ING means integer ; R means remainder for example ING[ 14/3] = 4 ; R [14/3] = 2 .
VII
0y ^ Im
Ix
0y 0y 0 y
= Ix × Ix × Ix ×…………..×
0y
m Ix
0y
∏ Ix
m
=
=
0{(y+x)m - xm }
Ixm
Algebra :
Gamma nc. + Gamma nc. = Gamma nc. Gamma nc.
So “+” is a vanishing sign.
I
I I
III
II
I
II
Accordingly IIII = I = II = I = I
Therefore: every number can have infinite gamma numerical representations.
A) Whole numericals algebra:
The main two gamma nc. representations for each number are the horizontal(H) and
the vertical(V) , and the rest are mixed(M) representations of these two.
I
I
II
Example :
III = I = I
mI
Type of represent.
H
.
.
.
; V ; M
H
V
Horizontal and vertical summations( + and + ):
H H H
H
V V V
V
I + I + I + ………….+mI = III……mI ; I + I + I + …….+ mI =
V
I H
I
Note: I + I = 0 + I = I
.
.
.
I
I
I
Arithmetic and Geometric summations ( ∑A , ∑G ) :
Arithmetic summation : horizontal summation of horizontal representations or
vertical summation of vertical representations .
Geometric summation: horizontal summation of vertical representations or vertical
summation of horizontal representations.
I
I
I IIII II
H
H
H
∑A ( II , I , I , I ) = II + III + IIIII + IIII = IIIIIIIIIIIIII
I
I
I
I
I
I
I
I
I
I
I I
I IIII
II
I HI H I H I
I I I
∑G ( II , I , I , I ) = I + I + I + I = I I I
VIII
I
I
I
I
xIxIxI….…. xnI
In general:
n
∑A Ix = IIIIII ……xnI
,
I I I……. .nI
n
∑G Ix = I I I……. .nI
A
Arithmetic subtraction ( - ):
A
III……nI – III…….mI = III…..n-mI0……m0 = III…..n-mI
if n>m
= - III……m-nI0…..n0 = - III….m-nI if m>n
G
Geometric subtraction ( - ):
G
0 n-1 0 n-1
0
0
III….nI - I = III…nI - I…….nI
= In-1………..nI n-1
In general :
G
0m
0m
III…nI - III….mI = In-m………n In-m
0n
if n>m
0n
= - Im-n………m Im-n
if n<m
B) Rational numerical algebra:
I is neither vertical nor horizontal as below:
n
n
∑A I = III….nI =∑G I = III….nI
H
I ÷ II….xI = I00…x-10
H
For example I0 + I0 = II = I
xI
V
I÷
I 0x-1
I=I
0 V 0 I
For example I + I = I = I
Also I0x and
0x
I are neither horizontal nor vertical .
n
See that if
n
∑ I0 n-1 = VERTICAL ONE , while ∑
v
H
IX
0 n-1
I
= HORIZONTAL ONE.( the zeros stay).
A
G
AI 0
G I 0
I ÷ II = I0 = I ÷ II = I0; I ÷ I = I = I ÷ I = I because the rationales of “I”are neither H norV.
Note: although
I and its parts (ratios) are neither horizontal nor vertical yet the combinations of them
V
are according to the type of summation that made them . For example I0 + I0 results in a vertical
0H0
00
numerical representing one of the forms of vertical one, likewise I + I = I I which is a horizontal
numerical representing one of the forms of horizontal one. Also 0 is neither V nor H.
A
III…..xI / III…yI = I ING x/y I R x/y 0 y-Rx/y ( Note : if x<y then ING x/y =0 and R x/y = x)
G
0 y-1
0 y-1
III…..xI / Iy = I ……….xI
xI
.
.
0 y-Rx/y
. A
I R x/y
I ÷ Iy = I ING x/y
xI
xI
0 y-1
.
.
.
Iy = I 0 y-1
.
.
. G
I ÷
(Note:IIx-1
A
xI
Ix-1
= III….xI , I = I )
IIx-1 / Ix= I
G
0x-1
0x-1
IIx-1 / Ix = I or I…….xI
Ix-1 G
(I0x-1)x-1
I / Ix = I0x-1
or I
≈
A
/
≈
=
I
C) The zero rational set:
n
What applies to “I” also applies to “0” exactly. So for example ∑A 0x = 000…..xn0
0x-1……n0x-1
n
G
And ∑ 0x = 0 ……..n0
Accordingly what applies to the rational “I” also applies to the rational “0” so for
example 0 / x = ++…….x-1+0 , the vertical form can be derived in a similar way.
Note: there are also rules for the summation of these vertical and horizontal forms of
these rationales that can be derived easily from working according to the formulae set
above . Also the setting of neither V nor H is optional to ease matters , alternatively
one can set “I” to be of vertical representation and its I0x and
X
0x
I as
vertical also , and an alternative whole algebra of rationales can be formed.
D)Mixed representations and operations:
I Ix-2 III……..x-1I
Mixed representations like I
=I
and others, and also mixed
n
H
H
H
V
V
V
V
operations like ∑ApGq IIx-1 = IIx-1+ IIx-1+ ….p+ IIx-1 + IIx-1 + IIx-1 +…..q+ IIx-1…….n IIx-1
,these issues are outside the scope of the current study.
Comparative notations and the role of zero:
Since the gamma nc. are unary nc. then from Eq.1 (introduction) zero will have no
differential effect, such that I0III0=II00I0I= I000I0II=I0I0I0I0=IIII
because I ^ i = I for -∞ <i<∞ . Since there is no differential effect then zero should
vanish from the whole non-zero gamma nc. set .
In a similar way; from Eq.2(introduction) the dot notation present in the Beta nc. has
no differential role in the gamma nc. so I.IIII=II.III=III.II=IIII.I=IIIII .Therefore there
is no dot notation in the rational gamma nc. .
But fractional zero is present in the rational gamma nc.?!
The real role of zero is purely comparative it doesn’t vanish from a numerical unless
its presence is not differential . so for example I00 + I00 =II0 it is seen that one
fractional zero stayed,this is because it has a differential role.This leads us to the
below conclusion:
The exact role of zero is purely comparative , its vanishing or staying doesn’t
have any relation to the type of operations but to its differential effect on
comparisons.
In reality in the gamma nc. philosophy zero is not merely absence, it should be
further specified as absence of a certain amount of “I” or “I” ( a whole one or a
rational one) , so 0i is absence of i of “I” and 0i is absence of i of “I” .
So zero emerged in the rational gamma nc. in order to make exact comparison
between the whole I and the smaller fractional one. The below example were
horizontal splitting of the whole one results in fractional one that is compared with
,make matters clearer:
I
║I
( the
“║” means “ as compared to”).
I
I is a fraction from
, but what is the exact ratio between them, a possible
solution would have been a lengthwise solution, but this length measuring solution is
not practical ,therefore putting one fractional zero “0” on top of I signifies the
XI
absence of exactly one I on top of I in order to complete it to
I , while putting x-1
fractional zeros on top of I means that it needs x-1 of I on top of I to complete it to
therefore
0x-1
I
I
= I ÷Ix . ( similarly I0x-1 is formed by x-1 vertical splitting of the whole I ).
So the exact role of zero is comparative.
From the above basis the comparative notations are formed. These notations are
whole gamma nc. that have zeros in them, they are intermediate notations in the
sense that they will eventually be converted to the zeroless whole gamma nc. after the
comparison is finished.
In the finite field the comparative gamma nc notations is useful in finding whole
rationales and whole roots as in the below examples:
What is half of II…….xI ? were x is an even natural.
Answer: I0I0I0………x. = Ix/2 (Note : the dot “.” is a gamma nc. variable ( I or 0 ).).
What is the yth fraction of II….xI ? were ( Ix/Iy ) is a whole nc.
Answer: I0y-1I0y-1……… x. = Ix/y
What is the square root of II……xI ? were (Ix1/2) is a whole nc.
Answer: II00I0000I000000…………x. =Ix1/2
0
Or in general iI02(i –1)……….. x. =
Ix1/2
= ( Ix )I
0
0
What is the cubic root of II……xI ? were (Ix1/3)= ( Ix ) I is a whole nc.
Answer: II06I018I036I060I090………..x. = 3√ Ix
Or in general iI06d ……….x. = 3√ Ix = I x1/3
i
d=( ∑ i-j )
j=1
These comparative notations also have different algebraic representations for example
see the mixed vertical horizontal(zigzag) representation below:
x.
x.
I000I000
I0I00000 = √I16,
02 ( i-1)
02 ( i-1)
= √Ix
i I……….….x i I
Also these comparative notations can be summed , subtracted , multiplied and divided
in a way quite resembling the rational operations presented in the previous
pages, consequently their algebraic representations also can have rules for these
operations ,the details of these aspects will not be presented in this summery.
XII
A final word !
These numericals and their different algebraic representations and the
intermediate comparative notations doesn’t make so much difference on
the finite calculus but they have tremendous effect on the infinite calculus
as it will be shown in Results.
XIII
Results
The infinite calculus
Representation of infinity by gamma numericals.
I+I+I+I+I+I+…………. = IIIIIIIIIIIIII……=III…….
Characteristics of unit infinity gamma nc. representation:
1)
2)
3)
It has a start.
It doesn’t have an end.
It’s full of ones in between.
1.Summations and Products:
A)
Arithmetic sum.
H
H
H
III….. + I = III….. , III….. + II = III….. , III….. + III = III…..
H
H
III….. + IIx-1 = III….. ; III… + III….. = III…
∞
∞
∞
∑A Ix = III…… , ∑A III…… = III……. , ∑A Ix + III……=III….
x
x
The arithmetic summation is non-discriminative for whole infinite gamma nc.
B)Geometric sum.
V
I
III….+ I = III…. ,
V
II
III…. + II = III….
,
V
III
III….+ III = III….
V
IIx-1
III…. + II x-1 =III…….
V
III…….
III….. + III……. = III…….
xIII…….
∞
…….
∑G Ix = III……. =
Ix-1 …..
I ……..
xIII…….
…….
x
= ∑G III……. = III……. = I Ix-1
……..
III……..
∞
III……..
∑G III……. = III…….. = (III…..)2
( like a square with one angle static and the others ever-thrusting outwards)
……..
…….
∞
(III…..)2
G
2
∑ (III…..) = (III…..)2 ( an ever-enlarging cube with one angle static)
XIV
V
IIx-1
(III…..)2 + IIx-1 = III……..^ 2
(III…..)∞
…….
……..
∞
(III…..)∞
∑G (III…..)∞ = (III…..)∞
(an ∞+1 ever-enlarging equilateral figure with one angle fixed)
and so forth…………..
( The total number of coordinates in space is also subject to the same rules of
geometric summation , as a result they are never-ending).
From the above account , The natural infinite gamma numericals are formed from
the geometric summation , product and exponent operations.
2.Subtraction
A)Arithmetic sub.
H
H
III…. - IIx-1 = 0x IIIII……… , III…. – III….. = 000…….
H
H
H
III….. –(III….. - Ix ) = III…… - 0x III…….. = Ix 000………. = Ix
So Ix000………. Represent the infinite representation of finite x.
H
∞
∞
0-III……. = -III……., ∑A - Ix = ∑A - III…..= -III……
Arithmetic subtraction is discriminative only if it was from a positive numerical.
B)
Geometric sub.
…………….
I………………
I ………………
V
0x-1 ……………
III….. - IIx-1 = 0 ……………
…….
V
0…….
, III….. – III…… =0……..
I x-y-1…….
I y-x-1…….
I ………
I ………
Ix-1….. V Iy-1….. 0y-1……….
I x-y-1…….
0x-1……….
I y-x-1…….
I…… - I…… = 0………. = I ……… ,if x<y = - 0……….= - I ………
x>y
In that way the whole set of whole infinite gamma numericals is formed, the whole
infinite gamma nc.zero set is formed likewise.
XV
3. Division
A)
Arithmetic div.
H
III….. / IIx-1 = I0x-1 I0x-1 I0x-1……………
Ix-1 V
I / I =
B)
0
0
0
Ix-1
I ,
,
H
III…./ III… = I
0
0
Geometric div.
0
0
V 0x-1
V 0
0
III….. / Ix= I …………. , IIx-1 / I∞= I……..xI ,
V
III….. / I∞ =
H
IIx-1 / III….. = IIx-1 00000………
0
0
0
I……….
Ix-1 H
Ix-1 0x-1………..
I / I∞ = I 0000………
Ix-1…… V Iy-1……
V
, I ……. / I ……. = IIx-1 / IIy-1
In that way The infinite rational gamma numericals are formed.
C)
Infinite rational sum.
C.1:Arithmetic sum.
H
I0x-1 I0x-1 …+ IIy-1 =II…I y+10…I0x-1 I0x-1 …=I ING[ y/ x-1]+1+ y 0 x-1- R[ y / x-1] I0x-1 I0 x-1
H
I0x-1 I0x-1 I0x-1………+ I0x-1 I0x-1 I0x-1………= I20x-2 I20x-2 I20x-2………
y
∑A I0x-1 I0x-1 I0x-1……… = Iy0x-y Iy0x-y Iy0x-y………..
if y > x then the summation becomes Indiscriminative.
H
H
Iy0x-y Iy0x-y ……+ Iz0u-z Iz0u-z …… = Iyu0(x-y)u Iyu0(x-y)u ..…+ Izx0(u-z)x Izx0(u-z)x ……
=Iyu+zx0 (x-y)u - zx Iyu+zx0 (x-y)u – zx… if zx >(x-y)u then the summation is Indiscriminative.
0x-y
H 0u-z
Iy …………. + Iz ………….
= Indiscriminative .
C.2: Geometric sum.
V
III….yI
Iy0x-y Iy0x-y ………+ III….yI = Iy0x-y Iy0x-y ……
V
Iz0u-z Iz0u-z ……
Iy0x-y Iy0x-y …… + Iz0u-z Iz0u-z …… = Iy0x-y Iy0x-y ……
XVI
V
III…….
Iy0x-y Iy0x-y …… + III……. = Iy0x-y Iy0x-y ……
Effect on finite calculus: for example see that
I I I I I I ……
I 0I 0 I 0……=2+1+2+1+…..=1.5∞
Also 1+2+3+4+…...+ i + (i+1) +…… = [∞2 / 2 ] + [ ∞ / 2 ] ! Prove it yourself.
0x-y
Iy ………….
V 0u-z
0(x-y) u
V 0(u-z) x
0(x-y) u-z x
+ Iz …………. = Iy u ..…… + Iz x …….. = Iy u+z x ..…… if zx ≤ (x-y)u
0(u-z) x+(x-y) u
Iz x - (x-y) u ……..
= III…………. if zx >(x-y)u
0x-y
Iy ………….
0x-y
V
Iy …………. +
III…….= III……….
0x-y
V
Iy …………. +
0x-y 0x-y
Iy … . .Iy 0x-y
III……xI = I….. xI Iy ………….
Geometric summation is always discriminative.
D)
Main line infinite rational subtraction:
H
H
Iy0x-y Iy0x-y …… - Iz0u-z Iz0u-z …… = Iyu0(x-y)u Iyu0(x-y)u ..…- Izx0(u-z)x Izx0(u-z)x ……
=Iyu-zx0 (x-y)u + zx Iyu-zx0 (x-y)u + zx ….…if yu>zx
= - I│yu-zx│0 (u-z)x + yu I│yu-zx│0 (u-z)x + yu …….if yu<zx
= 0000………. if yu=zx
0x-y
Iy ………….
V 0u-z
0(x-y) u
V 0(u-z) x
0(x-y) u+z x
- Iz …………. = Iy u ..…… - Iz x …….. = Iy u - z x ..…… if zx < yu
0(u-z)x + yu
= - I│y u - z x │ ..…… if zx > yu
0(x-y) u
= 0y u ………………………. if zx = yu
E)
Main line infinite rational multiplication:
H
Iy0x-y Iy0x-y …… × Iz0u-z Iz0u-z …… = Iy z 0ux-yz Iy z 0ux-yz…………
0x-y
Iy ………….
V 0u-z
0ux-yz
× Iz …………. = Iy z ..…………….
XVII
F)
Main line infinite rational division:
H
Iy0z Iy0z …… ÷ Ix = I0x-1 I0x-1……y I0x-100x-1 00x-1 ……z 00x-1 …………………….
H/V
H/V
Iy0x Iy0x ……÷ Iz0u Iz0u ……..= Iy0x ÷ Iz0u
H/V
0x
Iy0x Iy0x ……÷ III……= Iy0x , Iy
V
0x-1 0x-1 0x-1 0x-1
0x-1
0x-1
Iy0z Iy0z …… ÷ Ix = Iy 0 z Iy 0 z ……….= Iy 0 z Iy 0 z ………..
4.Roots and Irrational gamma nc.
The general form of infinite irrational gamma nc.
Ix0yIz0uIp0q ………….
A)
were x /y ≠ z/u ≠ p/q ≠ ………..
Roots:
Arithmetic roots:
A
√III…. = II00I0000I000000I…………= iI02(i-1) ………….
A
i
3
√III…. = II06I018I036I060I090I………. = iI06d ……….; d=( ∑ i -j )
j=1
A
III…. I 0 0 0 I 0 0 0………
III…. = I 0 I 0 0 0 0 0……… ( the zigzag root) = √2∞
and so forth….
B)
Geometric roots:
G
02(i-1) 02(i-1)
√III…. = iI
iI ……………..= (1/√∞)∞ = √∞
In a similar way the set of roots of infinite gamma nc. can be formed.
Also addition ,subtraction, mult. and div. rules can be set easily.
Complex infinite gamma nc.
y
The general form:
Ix-1….
Iy-1….
…. I3 . (I …, I3)
I……. + I……. √-III…...
…….. I2
……. I
0 I I2I3 x
MIXED INFINITE ALGEBRA
Example :
H V HV
I
I+I+I+I+………..= I II I ……= 2∞ -1
The details of mixed infinite calculus is outside the scope of this preliminary study.
XVIII
Comparatives of infinite gamma nc.and the role of zero
Zero is essentially a comparative tool , it is visually a spacer, so I0I0I0……. in reality
means I + + I + + I + + I +………….. , so if infinity is consistent(i.e.. every time we
mention infinity it means the same thing) then the above numerical surely has less
amount of ones in it than the numerical III…… for example; accordingly the comparison
should tell us exactly the value of I0I0I0….. ,this is the role of zero,see below:
I 0 I 0 I 0 ……….
I I I I I I ………..
It is obvious that for each two ones in the lower numerical , there is only one one in the
upper numerical. Then even if these numericals are infinitely progressing, the ratio is
still the same that is 2 to 1, accordingly the upper numerical should be half the lower
one.
This comparative role of zero is exactly the same role it has in comparing between whole
and rational finites presented earlier in page XI,XII . Zero do only disappear when it is
of no differential effect on comparison. As an example III – I = 0II but since 0II and II
means the same thing ,then III – I = II , otherwise zero would be only an additional
unnecessary numerical, while in III….. – I = 0III…… the zero don’t disappear because
0III…. ≠ III….. , this is similar to I – I00 = II0 , in reality finites do behave as ratios of
infinities , so for example I is the one infinith (1/∞) of single infinite nc. III…..,the
intermediate comparative representation of that is I0000……… that ultimately converts
into I. The infinite simile of II –I = 0I=I is as below:III….
III….
III…. – III…. = 000….. = III…..
It is obvious that the numerical I0I0I0…… is the same intermediate comparative
notation discussed under finite comparatives in pages VI,XII . but here when it is
extended into infinity, it is no longer an intermediate numerical, the zeros should stay in
order to show the comparison clearly.
0i is absence of Ii , 00 is absence of zero , so I 00 I = I+I=II, accordingly
00 = + .The effect of that and of -0i shall be shown later.
One additional note about comparison is that the infinite nc. representation by gamma
nc. should be understood well, their should be no numerical before the start of the
numerical , for example I0I0I0….. + I0I0I0….. ≠ 000000….. IIII……. ;
I0I0I0….. + I0I0I0…..= IIII……. 000000….. =III…… since all zeros will eventually
be invaded by the ones of the III….. infinite nc.
All changes should occur visually on the starting nc. of the infinite nc. and the following
nc. no changes take place before the start of the infinite gamma nc. since that is
contradictive.
00
V00
000………
I I…….. + I I…….. =III………. = III……….
The above example is clear enough to be explained.
XIX
Additional results
1.Exponents:
xI…….xI
H
Ix ^ Iy = II xy-1
G
xI……..xI xI….....xI
Ix ^ I3 = Fig1 = xth 3 dimensional equilateral
gamma nc. figure.
G
I……..xI
Ix ^ Iy = xth y dimensional equilateral gamma nc. figure.
H
G
Ix ^ III….. = III…..(x>1) , Ix ^ III….. = xth ∞ dimensional equilateral gamma nc. figure.
G
III…..^ Ix = ∞th x dimensional equilateral gamma nc. figure.
2.Roots of finites in an infinite gamma nc. Representation: !!!
Though roots of finites can be translated from the approximated nth decimal rational
finite representation like saying y√x = z. u1u2u3u4………..≈ z .u1u2u3u4………un
as
010- u1 0102- u2 0103- u3
010n- u n
Iz Iu1 Iu2 Iu3 …………….. Iu n
and even √III……. = I0x∞-1 I0x∞-1…………. x∞I0 x∞-1 (=the rational infinite gamma
nc.representation of the square root of single infinity.).
The above approximate translation for roots of finites is incomplete since it is rounded to
the n-th decimal , it is useful for finite calculus but not for infinite calculus. Therefore the
complete infinite form of the root is needed in infinite calculus!
What is the complete gamma nc. representation of √2 ?
.............
............
.............
............
.............
............
000……………… = 2/∞ × ∞ = 2 ; 000…………….. = 2/√2∞= √2 /√∞ =
000………………
I I I……………..
I I I……………..
.............
.............=
000………………
000………………
000………………
√2=
(√2 /√∞)√∞=
√2/√∞
√2 /√∞
000……………..
I000I000………
I0I00000………
0
√2/√∞
0
√2/√∞
2
0
4
XX
√2/√∞
0
6
3. 00 and complementary gamma nc.
Ix Ix
Ix Ix
Ix……….
Since 00 = + then I0y I0y…….. + I-0y I-0y……..=I ……….
Ix Ix
Ix………. Ix Ix
I-0y I-0y……..=I ………. - I0y I0y……..
Questionable Results
-The zero denominators
What is w/0?
Possible highly questionable answer!
∞
∑ w/xi = w/(x-1) ; x = 2,3,4,………….(derived from the n-periodic systems were
i=1
1/(n-1)=0.1111……..= 1/n + 1/n2 + 1/n3 +………..+1/nIII…… ).
if x =1 then w∞ = w/0 (because 1x = 1; ∞ ≥ x ≥ - ∞
)
G
G G G
G
G G G
w/0 = w + w + w +…………….. ≠ z / 0 (were z ≠ w ) = z + z + z+…………..
A A
but w/0=z/0= ± III……….. ; │w,z│ ≥ 1 and │w│/w =│z│/z .
G
0/0 = 0 + 0 + 0 + ……………. =000…. ; w/02 = w∞2
A
w/0 = ±III….. (│w│≥ 1 )
A
(1/w)/0 = ±I0w-1 I0w-1…………(│w│≥ 1 )
G
Iw-1…….
w/0 = ± I ……….
G
0w-1 0w-1
(1/w)/0 = ± I I …………
Addition ,subtraction ,multiplication & division of w/0 can be simply made after
translating it to the gamma nc. representation.
∞
See that ∑ w/xi = w/(x-1) cannot be applied for x =0 !Why?if so it gives contradictive
i=1
results!
Also a possible highly questionable result is the explanation of the infinite nature of
the transcedental number “e” :
e =[ 1+{1/(1/0)}](1/ 0) = [1+ (1/∞)]∞
XXI
Parallel results
Parallel to the unary nc. the entire beta nc. can also be redefined as possessing
algebraic forms like vertical , horizontal & mixed vertical- horizontal representations.
The general nc. notation of the n periodic beta nc. is as below:
x1mx2mx3m……………..xpm
. . . …………….. .
. . . …………….. .
p m
. . . …………….. .
= xij
= ∑( ∑ xij × n j-1) n p-i
x13x23x33……………..xp3
i = 1,2,3,……..,p
i j
x12x22x32……………..xp2
j= 1,2,3,……...,m
x11x21x31……………..xp1
xij = 0,1,……,n-1
The above nc. is called the two-dimensional matrix beta numerical of size pm. Of
course the customary beta nc. are in reality the horizontal vector beta nc.. There can
q
be multidimensional matrix beta nc. or the q matrix beta nc. of size ∏ pk as below:
pq
p3
k
p2 p1
x (i1,i2,i3,…….,iq) = ∑ ( ….…∑ ( ∑( ∑ x (i1,i2,i3,…….,iq) n(p1) – i1) n(i2) – 1) n(i3) – 1……..) n(iq) – 1
i1= 1,2,3,………,p1
iq
i3
i2
i1
i2=1,2,3,………,p2
i3 =1,2,3,………,p3
.
.
.
iq=1,2,3,………,pq
x (i1,i2,i3,…….,iq)=0,1,…….., n-1
The above nc. is of course a q dimensional beta nc.
111
As an example: 111 (decimal = n =10) = (1 ×100 + 1 ×10) 102 +(1 ×100 + 1 ×10)10
+(1 ×100 + 1 ×10)100
= 1100 +110+11=1221
x (i1,i2,i3,…….,iq) ÷ nm = x (i11 ,id) . x (i12 ,id)
i11 = 1,2,3,………,(p1) – m
i12 = (p1) – m+1, (p1) – m+2, ………,p1
d = 2,3,4,………..,q
10000………….
Solve : 00000…………. ?
111
11.1
1
for example: 111 ÷ n = 11.1
111
= 111
0 000……..00
0.000……..00
and 0.000……..01
?
Answer: 10∞ from 10∞ -1 × 101 =10 ∞ !! and 10-∞-1 from 10 - ∞ +1 × 10-2= 10 -∞ -1 !!
In a similar way to what was mentioned in the unary system algebra, an algebra of
these matrix beta nc. can be formed. Also we can have the multi-infinite dimensional
matrix beta nc., like ∞ +x or ∞x matrix beta nc..
XXII
Discussion
The full analytic proof of this method is the job of mathematical philosophers and
professional mathematicians , and it is beyond the scope of this account.
Traditional ways of dealing with infinities depends so far on alpha and beta
numerical representations like ‫אּ‬0, ‫אּ‬1,‫אּ‬2 , ‫אּ‬3,………, ‫אּ‬n to represent cardinal infinities or
ω0 , ω 1, ω 2, ω 3,……, ω n to represent serial infinities. Or the customary ∞ to represent
infinity in general. However the calculation process was always extra-numerical,
something that didn’t depend on the visual characteristics of these symbols, exactly
similar to calculations with alpha numericals the ancient peoples performed. Thus I
call it alpha level of calculation which means extra-numerical calculation.
The quest of this account is to try to perform calculations of infinities by representing
them by visual symbols, the visual properties of which when manipulated in a certain
manner can provide a straight forward visual deduced calculations without the need
for an extra-numerical logico-mathematical calculation. The difference between the
aim of this account and the traditional extra-numerical method is similar to the
difference between calculations performed using beta and alpha numericals
respectively.
Unfortunately the powerful practical beta numerical systems widely used, especially
the decimal numericals,loos their calculatory visual deductive abilities at infinite
level. This called for the invention of a more visual deductive numerical
system,namely “The Gamma numerical system”. The proof of the consistency of
this system also needs mathematical philosophic work that is beyond this brief
discussion.
I should provide some elaboration on the origin of the gamma nc. and their
superficial pseudo-contradictive forms. Originally this system was invented to
achieve a numerical system that bears the closest resemblance and mimicry to the
true meaning of what a number is . From the first glance it is obvious visually that
numericals like II, III, IIII are simpler and visually more direct than numericals
2,3,4 (decimal) or 2,3,10 ( quaternary) in representing numbers two, three and four.
Therefore II,III,IIII should be nearer to the true philosophic meaning of numbers two,
three and four than the beta and alpha nc. representations . In a similar visual
deductive mimicry process the gamma rationales are derived, since small “I” with
small zeros“0” on top of it or beside it (denoting the missing small “I” s required to
complete it to full large “I” ( which represent number one) ) do represent a consistent
extension of the same visual deductive mimicry method that made the natural gamma
numericals. That’s why I originally called these numericals as the “philosophic nc.”
or the “simple nc.” or the “primitive nc”. Afterwards I discovered that the natural
gamma nc. set can be viewed as an extension of the beta numerical systems with n=1
, and therefore I discovered that the absence of zero role in forming these natural
numericals is in reality an extension of the beta numericals towards n=1, therefore I
called it the Unary system. And I presumed that the rational gamma nc. are also an
extension of that system .Why I call these numericals as fully deductive and fully
counting and thereby fully calculating needs no further explanation.
XXIII
Superficial pseudo-inconsistencies in the gamma nc. are mainly two:
1)
2)
It is called Unary while it has four different nominal numericals?
It has non-gamma nc. attached to it as the counter (i on the right )and the
rank(i on the left).
Regarding the first objection , the system is called unary because the natural gamma
nc. set ( that begins with numerical I ) are all derived in shape from nc. “I” only. How
to prove that the rational gamma nc. set is in effect an extension of the unary
system? In my present opinion the answer is mainly teleological , the priori proofs of
that issue is still obscure to me. The only priori clue that I have is the superficial
resemblance of (the visual deductive mimicry process of the meaning of natural
numbers which was originally responsible for deriving the natural gamma nc. set) to
(the visual deductive mimicry process of the meaning of rational numbers which was
responsible for deriving the rational gamma nc. set )!?
Regarding the second objection , the answer is simple .The counter and the rank
notations attached to the gamma nc. are simply helpers , they have no role in deriving
the visual deductive calculations that the gamma nc. can do outside the province of
the visual calculative abilities of the beta nc. the counter and the rank made from.In
reality they can be completely dismissed from the gamma nc.notation and one can
work only with the basic gamma nc. ,but this would be only suitable for computers
not humans. In reality the counter and the rank are nothing but abbreviators of the
basic gamma nc.
Infinities have also been regarded by mathematical philosophers as reflexive numbers
a notion that I feel un-necessary for infinite visual calculatory techniques.
By the reflexive properties of an infinite number it was meant to show that these
numbers do behave in a way that differs from the properties possessed by finites, thus
a lot of the customary operations well know in finite math. would not be applicable to
infinities, therefor the notion “ non-inductive” or reflexive.
The results of that preliminary account proves something else and something much
more specific.It tell us that infinities do possess inductive properties in certain
situations ,like in geometric processing, on the other hand they lose these properties
in other situations,like in arithmetic processing, furthermore there are mixed
situations were these infinities swing between possessing and loosing inductiveness.
The symbols ( the gamma nc.) here are simpler , more specific and more handy ,
therefore I think( if proved to be consistent )that they are superior to the previous
extra numerical methods in dealing with the problem of infinite calculus.
Weather the Indiscriminative infinite arithmetic operations represent a defect in
visualization of these operations between infinite numbers, and consequently not
reflecting the reality of infinite numerical operations and calculations?or is a property
that differentiates infinities from finites and thereby fundamental to the philosophy of
these numbers and their operations ?is a question that I cannot answer though I am
more inclined to the later probability.
XXIV
Regarding the highly questionable result of w/0 = w∞ which is derived from the
formula 1/(n-1)=0.1111……. in the n-th periodic beta numericals like for example
1/9 (decimal =10th periodic nc.), 1 / 2 (ternary) ,1/ 7 (octal) and in general:
[1/(n-1 )] ( nth periodic nc.) = 0.1111…….=1/n + 1/n2 + 1/n3 +………..+1/nIII……
.
Naw extending this derivation to n=1 will result in the conclusion that
I / 0 (unary) = 0.III…… = III…… , and from that all other translations springs.
Why we cannot extend the same above formula to n=0 ? the answer is very simple
,simply the zero periodic numericals do not exist! Because every n-periodic system
has n nominal nc. whereby all the other natural numericals are derived in shape
(deduced) from them , accordingly if n=0 then there is no nominal numericals and
therefore no deduced nc. from them , in other words there is no numerical system at
all that is representing numbers.That’s why when we apply the formula blindly we
get an odd result that defies logic,that is 1/0 + 1/02 + 1/03 +…………… = -1 , a
result that is clearly impossible since the addition of positive numbers cannot result
in a negative number .
The parallel results shows that it is the algebraic forms that count in this method more
than the type of the numerical system, yet without doubt the unary system is the
simplest and therefore it should be used first as a start to discover the infinite
numerical world.
The main conclusion of this study is that a numerical solution to the problem of
infinite calculus can be contemplated by going back to a more simpler numericals
that possess nearer relationship to the true meaning of a number. And that this study I
hope could be somehow illuminating.
I recommend trying more harder to achieve the conclusion above.
XXV
SUMMARY
The current account was conducted with the aim to find a simple, handy technical
numerical solution to the difficult problem of infinite calculus,that bypass the need
for philosopho-logico-mathematical deductive analytic thought in calculating or in
deriving the rules for calculation of infinities.
As an introduction , what is a visual representation of a number is to be called a
numerical(nc.), and reclassification of the known numerical systems into non
deductive , semideductive and fully deductive numericals was achieved and they
received the titles of alpha,beta and gamma numericals respectively.
The gamma numericals constitutes the material of this study , they were originally
constructed to be the numericals that bears the nearest resemblance to the true
meaning of what a number is . They are simply the most primitive numericals man
ever thought of, they symbolize numbers like one, two, three and four simply as I, II,
III, IIII respectively ; while rationales are symbolized using small “I” and small zeros
“0” associated with them, like for example I0,I00,I000 to symbolize half, one third, and
one fourth respectively. And it was assumed that both the whole gamma nc. and the
rational gamma nc. do represent a spectrum of the same visual numerical system.
The main feature of these numericals is their algebraic forms , each number can be
represented by different equivalent gamma nc. the main two gamma algebraic forms
of a number are the horizontal and the vertical representations, for example number
three is represented as:
I
I
II
III ( the horizontal form)= I (the vertical form)= I (the mixed horizontal-vertical
form)
That feature raised the concept of Arithmetic and Geometric summation of these
forms; an arithmetic summation was defined as horizontal summation of horizontal
forms or vertical summation of vertical forms. While a geometric summation was
defined as horizontal summation of vertical forms or vertical summation of
horizontal forms.
That feature and these summations don’t make much difference in finite calculus ,
but they are the most important aspects the gamma numericals possess that enable
them to be the numericals that can be used in infinite calculus.
Results: First the gamma nc. representation of single infinity was defined as :
1+1+1+……. = III………. =1∞
Main characteristics of that representation are: 1) It has a definite start.
2) It is endless.
3) It is full of ones in between.
XXVI
The main result was that there is a difference between the arithmetic and geometric
summations of the gamma nc. when these summations are continued ad infanitum.
H H H
For example: II + II + II +………………. = III……….=1∞
I HI H IH
I I I……….
While I + I + I + ……………… = I I I………. = 2∞
So geometric summation is responsible for forming the Natural infinite gamma nc.
set. The negative counterpart is formed by infinite geometric summation of negative
gamma nc.The zero set is formed also by infinite geometric summation of zero
gamma nc.So geometric summation forms the Whole infinite gamma nc. set.
Rational infinite gamma nc. set is formed from both the arithmetic and the geometric
divisions. So I0I0I0…….. represents half infinity , this is the result of arithmetic
division of III…… by II , while the result of geometric division of III…../ II is:
000
I I I ………. .
Both arithmetic and geometric divisions are responsible for the
arithmetic and geometric rational infinite gamma nc. sets.
The finite numbers also can be represented by an infinite gamma nc., the main form
is : Ix = Ix000………, so for example number three is represented as:
III00000000000…… .
Roots of infinite gamma nc. are also derived by both arithmetic and geometric
rooting processes. For convenience the arithmetic square and cubic roots of single
infinity and the square root of double infinity are illustrated below:
A
√III…. = II00I0000I000000…………= iI02(i-1) ………….
A
i
3
√III…. = II06I018I036I060I090………. = iI06d ……….; d=( ∑ i-j )
j=1
A
III…. I 0 0 0 I 0 0 0………
III…. = I 0 I 0 0 0 0 0……… ( the zigzag root) = √2∞
The main additional result was the complete infinite representation of the roots of
finite numbers like √2 ,√3, √5,……. .
Finally results ends by the highly questionable solution for the zero denominators
w/0 , by translating them into the infinite gamma nc.as below:
G
G G G
I w-1…..
w/0 = w + w + w + ……………….., so w/0 = w + w + w +……..= I………(if w ≥ 1)
A
A A A
while w/0 = w + w + w +……..= III…… (if w ≥ 1)
= I 0u-1 I 0u-1……….= ∞w (if w <1 ; w =1/u)
G
G G G
0u-1 0u-1
also w/0 = w + w + w +……..= I I …………….. = ∞w (if w <1; w =1/u).
XXVII
Parallel results shows beta nc. algebraic forms , the general q-dimensional matrix
beta nc. is illustrated.
The results were discussed against the background of older methods of infinite
calculus that are in main an extra-numerical philosopho-logico-mathematical ways of
calculations. Also the origin of the invention of the gamma nc. and their superficial
apparent inconsistencies were illustrated and discussed briefly . The main conclusion
is that these numericals are inspiring.The general way of trying to find a numerical
solution to the infinite calculus resembling the gamma nc.fashion is recommended.
Finally the study ends at the dot at the end of this sentence
.
Reference
Russell .B. Introduction to mathematical philosophy. London, 1950
This work is completed at 4th of Eed Al- Adha, 4 /2/2004.
Completely done by Dr. Zuhair Abdul Ghafoor Al-Johar
Specialist of Neurology.
An amateur of mathematical philosophy.
Iraq.Baghdad.Al-Husseinyah . section 225.street 52.house NB. 13 .
e-mail: [email protected]
XXVIII
APPENDIX
Paradoxical results:
A)
Since: 1 + 2 + 3 +……..+ i + ( i+1 ) +……..+ n = n2/2 + n/2
Then: 1 + 2 + 3 +……..+ i + (i+1) +……….. = ∞2/2 + ∞/2 = ∞2/2 ( 1 + [1/∞] )
[1 + 2 + 3 +……..+ i + (i+1) +………. ] × 2/∞2 = 1 + [1/∞]
{[2 + 4 + 6 +……..+2i + (2i+2) +………. ] / ∞} × 1 /∞ = 1 + [1/∞]
Ae / ∞ = 1 + [1/∞] were Ae is the average of all finite natural evens .
So Ae = ∞ + 1
This is right because [2+4+6+…….+2n] / n = n+1
Therefore [2+4+6+…………] /∞ = ∞+1 !!!
This replacement of the finite variable n by ∞, if it proves consistent, then it’ll have a
great impact , see below:
If ( 1 + [1/∞] )∞ = e ( review the questionable results)
Then [Ae / ∞]∞ = e .
So this would be the infinite explanation of the well-known transcendental number
“e”.
B) from ½ + ¼ +⅛ +………… = 1
2∞-x
1+2+4+8+………..+2∞-x-1)
- (
x = 0,1,2,3,4,………..
= {[0+1+2+4+8+……+2
x-1
A
] / 2 } × ( x /x )
x
while 2n - ( 1+2+4+8+………..+2n-1) = 1
n= 1,2,3,4,……….
Now this is a paradox, since we cannot simply replace n by ∞ as in A) above?
The reason may be related to the type of the above infinite series , the series in A) are
divergent while in B) they are convergent. In divergent series ∞ do parallel n , while
in convergent series a kind of mixed algebra is possibly taking place rather than the
direct fully inductive geometric processing the divergent series has?
Solusion: 1/n + 1/n2 + 1/n3 +……+1/nk = [1/(n-1)] - [1/{nk(n-1)}] ; n=2,3,4,….
Accordingly 1/n + 1/n2 + 1/n3 +…+1/n∞ = [1/(n-1)] - [1/{n∞ (n-1)}] ; n=2,3,4,….
XXIX
∞
m
According to this study 1/{n (n-1)}≠ 0 ↔ ∑1/n ≠ lim ∑1/ni = 1/(n-1)
∞
Therefore: ½ + ¼ +⅛ +…… = 1-(1/2∞)
i
i=1
m→ ∞
Accordingly 2∞-x- ( 1+2+4+8+………..+2∞-x-1) = 1
x = 0,1,2,3,4,……….
Which parallels 2n - ( 1+2+4+8+………..+2n-1) = 1
n = 1,2,3,4,……….
However the formula 1/n + 1/n2 + 1/n3 +……+1/nk = [1/(n-1)] - [1/{nk(n-1)}]
is not applicable for n = 1,0 when k is finite, and this fact may destroy the whole
basis for the highly questionable solution for the zero denominators problem
mentioned in this study.
Still for those who believe that 1/{n∞ (n-1)}= 0, and therefore believing that:
∞
m
∑1/ni = lim ∑1/ni = 1/(n-1)
i=1
m→ ∞
Then, there would be a chance for the questionable solution of the zero denominators
provided in this study.
Finite – Infinite Ordinal Scale:
The mathematical operations so far mentioned in this study are all cardinal in nature.
In the following lines a trial for ordinal scale is contemplated.
1)The cardinal-summation based ordinal numericals(@):
I
II
III
III……nI III……
I,II,III,………….,In , III…..,III…..,III…..,III…..,…………..,III………,III……,……
2)The cardinal-(summation-subtraction) based ordinal numericals( # ):
I
+[I,II,III,…..., In ,0nIII……,0n-1III……,……..,III……,III…….,……]
In a similar way one can form the cardinal-(summation-subtraction-divisionroot extraction ) based ordinal numericals($).
@# $
According to the above, every ordinal operation should be specified,like + ,+,+
@
#
So for example In + I = III……. ; In + I = 0nIII…… ;etc….. .
In symbolize the last finite numerical.
However I have the feeling that there is no maximum finite , and that the infinite
numericals lies in parallel to the finite scale.
XXX
The zero denominators (an insight):
The formula:
1/n + 1/n2 + 1/n3 +……+1/nk = [1/(n-1)] - [1/{nk(n-1)}]
is true for { n: - ∞ ≤ n ≤ ∞ } , { k = 1,2,3,………., ∞ }.
Now if n = 1 → (1/0) – (1/0) = 1;2;3;4;……..; ∞ { “ ;” means “or” }
Also (2/0) – (2/0) = 2 ; 4 ; 6 ; 8 ; …….. ; 2∞
And if n=0 → -1 + (1/0) = (1/0); (1/0) +(1/0);3(1/0);……….; ∞ (1/0).
This means that the zero denominators are in fact: formal- logical- categorial
numbers (i.e, numbers that denote a category or a class of numbers).
Thus: 0/0 = x : - ∞ ≤ x ≤ ∞ .
While 1/0 = X´ : X´ = ~ x ,{ “~” means “not”}.
For formal purposes it is better to symbolize the operation “or” as below:
÷
F=x
y = {x};{y} = x or y.
So (1/0) – (1/0) is in reality a variable denoting all natural numbers including“∞” .
The above shows the fact of symbolizing spesific sets of numbers by numbers rather
than by alpha nc. like x, y, z,…. .So variables are symbolized by “w/0” numbers !!
In contrast to the “or” operation , the “and” operation “+” is symbolized as below:
F = x + y = {x,y}.
So writing xij
, means
the matrix X ; while writing xij
i,j =1,2,3,……
,
means the variable x.
i,j = 1;2;3;…….
= (x1 x2 x3………..) , a vector! ; while xi = c1;c2;c3 , a variable!
similarely xi
i =1,2,3,……
i =1;2;3;……
↔
for example: xi = 2
X=( 2 2 2 2 ……… )
i =1,2,3,……
↔ x1 ; x2 ; x3 ; ………. = {2}.
while: xi = 2
i =1;2;3;……
p m
xij
= ∑( ∑ xij × n j-1) n p-i
i = 1,2,3,……..,p
j= 1,2,3,……...,m
i
j
, this is the correct definition of the
matrix present in the parallel results.
xij = 0;1;……;n-1
i = 1;2;3;……..;p
j =1;2;3; ……...;m
XXXI
According to the above the transcedental number “e” can be writtin as below:
e = [{ ∑( 2/0 –2/0 )} / ∞2 ]∞
Another interesting result is (-1 )∞ = 1
since –1+1–1+1–1+1………. = 0 from –1+ 0–1+0…..= –1/2
0+1+0+1…..= +1/2
–1+1–1+1…..= 0
∞
and since –1+1–1+1–1+1………....= ∑ 1/(-1)i = {-1/2 } - {1 / [ (-1) ∞ × -2]}
i=1
→ (-1) ∞ = +1. So single infinity “∞” behaves as an even number.
k
Applying the results of the formula: ∑ 1/ni = [1/(n-1)] - [1/{nk(n-1)}] for all values
i=1
of n including [1,0] and k=1;2;3;4;………; ∞ , resulted in all the above strange
concepts.
These results require discussion of what is the meaning of a “number”.
A number is “a differential existentialistic relationship between a pre-defined subject
( ie, Type ) and its surrounding within a specified enumeration field”.
As an example the square A inside the circular field F.
F
A
Square A exist inside F and no similar square A exist in
the surrondings of A ,inside F.
The above is the meaning of the sentence: There is “one” squareA inside F.
If we have many squares fulfilling the same criteria for square A (Typal assignement
or linquistic similarity assignement),inside F; then we are having multiple types of
existentialistic relationships of the square A ( as a type) and the surroundings within
F. Number: is the differential aspect of these types of existentialistic relationships .
The above definition of number is complex and abstract, a more practical definition is
to say that a number is anything that results from the basic processing of number one
by basic operations that have a sane logical base, and “number one” is defined by the
existentialistic way mentioned above .This is similar to saying that humans are the
offspring of “the first human:Adam”, and defining “the first human:Adam” as the
creature that have the so and so genetic code.
Now, basic operations like (+,=)are the results of the more basic logical functions
“and”, “then”. This account says that the other logical functions like “or” , “nor”
can also breed numbers, so for example (1 or 2) is also a number or ( not 1 ) is also a
number; these were usually looked at as variables that were usually symbolized by
alpha nc.,but this study says that it should not , it can be better symbolized by zero
denominators in a beta or a gamma nc. representation. These zero denominators can
have a formal algebra that denotes processing of sets of numbers weather infinite or
finite.They represent a link between logics of sets and mathematics!!!
XXXII