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