title page: Chapter 1: A Modular Perspective "That which hath been is that which shall be, And that which hath been done is that which shall be done; And there is nothing new under the sun." Ecclesiastes 1.9 Purpose: to define division by zero and infinity by first discussing their identities. The problem is not one that lacks a solution, but one that lacks an interpretation. Up until world travel took place, there was no reason why anyone need believe the earth was round; it was sufficient to accept a flat square earth. Likewise, it is not necessary to believe that magnitude differences are looped at the extreme ends. Without physical evidence to encourage our opinions in any direction, we might just as well go on believing that magnitude is a straight line without endpoints and straight----or that it is looped. We may continue to go on believing there is no solution to division by zero; or we may accept interpretive solutions, to fit each circumstances, in place of a definite one. Let's start the discussion with a number line: Infinity >>> & & <<<< +3, +2, +1, 0, -1, -2, -3 >>>> -& But this is really this: ? <<<<<<<<<<, (-)0, -1, -2, -3 >>>> -& +(,-)0 & <<<< +3, +2, +1, +0, >>>>>>>>>>>> ? Two vectors, linked at zero. Taken seperately, each tempts us to go beyond zero in a direction opposite to the other vector. We solve this by linking them. Normally, we don't allow ourselves to admit to a negative zero. When all we had were positive numbers, it wasn't a problem. We skirted the issue of negative zero by immediately linking both vectors and claiming that zero has no sign. This is correct; number has no sign intrinsically, but when immersed within a field of sign it has direction of magnitude according to its sign relative to a magnitude reference of zero. Negative zero is no different from positive zero. It is the direction of movement along the number line that affects the sign attached to any number. But at the moment, we are not moving along the number performing mathematical operations, we are just noting the arrangement of things as they are dispersed. So consequently, positive and negative infinity may also be linked by equality, making: A Closed Loop Scale of Magnitude Linked at Zero and Infinity >>>>>>>>>>>>>>>>>>>>>>>>>>>>> +0, +1, +2, +3, ........., +& 0 & -0, -1, -2, -3, ........., -& >>>>>>>>>>>>>>>>>>>>>>>>>>>>> What is sign? Is it an operation that is about to be performed? But against what, if a number stands by itself on a number line?: against the limits of magnitude, zero or infinity. Zero was imported from India, via Arabia, to hold place value within a number. But what do numbers, for that matter, really do? Surprisingly, simple innocent looking numbers are really short hand for writing polynomials in one unknown, one of whose factors is base ten. Even before we do a simple addition between two numbers we are invoking a complicated series of instructions involving summation, multiplication, and raising a base with an exponent, just to evaluate a number: 789 = 7 * 100 + 8 * 10 + 9 * 1 = 7 * 10^2 + 8 * 10^1 + 9 * 10^0 = 7 * x^2 + 8 * x^1 + 9 * x^0 789 = 7x^2 + 8x + 9 >>>>>>> This is the polynomial for +789 -789 + 789 = 7x^2 + 8x + 9 - 789 0 = 7x^2 + 8x - 780 0 = ax^2 + bx + c Using the quadratic formula to find the values of x: x = [-b +- sqr(b^2 - 4ac)] / 2a = [-8 +- sqr(8^2 - 4(7)(-780)] / 2(7) = [-8 +- sqr(64 + 21840)] / 14 = [-8 +- 148] / 14 x = -11 - 1/7, or, x = 10 Now let us follow the addition of two polynomial numbers: 789 = 7x^2 + 8x + 9 + 14 = 1x + 4 --------------------- 803 = 7x^2 + 9x + 13, but 13 = 1x + 3, so substituting: = 7x^2 + 9x + 1x + 3 = 7x^2 + 10x + 3, but 10 = 1x + 0, so substituting: = 7x^2 + (1x + 0)x + 3 = 7x^2 + 1x^2 + 0x + 3 803 = 8x^2 + 0x + 3 -803 + 803 = 8x^2 + 0x + 3 - 803 0 = 8x^2 + 0x - 800 0 = ax^2 + bx + c x = [-b +- sqr(b^2 - 4ac)] / 2a = [-0 +- sqr(0^2 - 4(8)(-800)] / 2(8) = [+- sqr(25600)] / 16 x = +- 10 If we want to depict zero as a polynomial, we could write: 0 = ... + 0x^3 + 0x^2 + 0x + 0 But that would appear to be unnecessary. But it is correct, because that is what we do when we want to add or subtract another number with zero: we pair off terms of x that are equal in exponent to add or subtract their coefficients. This is called parity; whenever we want two or more numbers to mathematically interact, we group coefficients of similar base powers. If a number is missing a coefficient for an x term, we make one up by assigning a 0x^? for that number's missing digit: 79 + 4 = 83 79 = 7x + 9 + ?4 = 0x + 4 ------------- 83 = 7x + (13 = 1x + 3) 83 = 7x + 1x + 3 83 = 8x + 3 This way, all the children (digits of a number) get to play---even though it may be with a non-existent partner. Or is it non-existent? We could pad out those zeros to the left of four, indefinitely: 4 = ... + 0x^3 + 0x^2 + 0x + 4 We know the sign of this four is positive----that has been given. When I say: "I have no apples", it normally does not imply that I have a negated state: negative apples, but that I have zero apples, as if to imply the apples' very existence in my possession has been negated, not their relative number. For existence to be negated, we have to know how much existence is---- infinite, all possibilities? What is the simplest way to depict infinite existence as a polynomial, whose coefficient must be less than the number's chosen base?: & = ax^&, 0 =< |the absolute value of a| =< 9, x = ten And what would be the simplest way to depict its negation, non-existence?: 0 = -& = -ax^&, 0 =< |a| =< 9, x = ten Notice that the only things that has been changed in the second of the above two lines are: existence as infinity >>>>changed into>>>> non-existence as zero the sign of infinity changed from positive >>>>into>>>> negative the sign of the coefficient changed from positive >>>>into>>>> negative It is as if negation compliments infinity with an opposite value of zero. The same is accomplished in the opposite direction when zero is negated into infinity. Incomplete Table of Complimentary Values ---------------------------------------- 0` >>>>> to read: zero's compliment -0 = 0` = & -& = &` = 0 There is another way to illustrate this with modular division. Modular division is division without any quotient. The quotient answer is ignored leaving a remainder. This makes different problems share the same answer because of the cyclic nature of modular division: 4 = 4 mod 9 = 13 mod 9 = 67 mod 9 -2 = -2 mod 9 = 7 mod 9 = 70 mod 9 0 = 0 mod 9 = 9 mod 9 = 18 mod 9 0 = 0 mod & = & mod & +1 = 1 mod & -1 = -1 mod & = (& - 1) mod & ? = -0 mod & ? = -& mod & Because of this, we may reevaluate the closed loop scale of magnitude: A Closed Loop Scale of Magnitude Linked at Zero and Infinity >>>>>>>>>>>>>>>>>>>>>>>>>>>>> +0, +1, +2, +3, ........., +& 0 & -0, -1, -2, -3, ........., -& >>>>>>>>>>>>>>>>>>>>>>>>>>>>> As: A Closed Loop Scale of Magnitude Linked at its Complimentary Limits and Everywhere Else in Between >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< +0=+0+0, +1=+0+1, +2=+0+2, +3=+0+3, ........, -3=+&-3, -2=+&-2, -1=+&-1, +&=+&-0 +0/-& +&/-0 -&=-&+0, +1=-&+1, +2=-&+2, +3=-&+3, ........, -3=-0-3, -2=-0-2, -1=-0-1, -0=-0-0 >><<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< Isn't this a contradiction, that zero should equal infinity and infinity equal zero? Or that some very large positive number one less than infinity equals negative one; or that positive one equals a very large negative number equal to negative infinity plus one? It is an irony----not a contradiction. Duality implies relativity, and relativity implies multiplicity of interpretations depending on orientation. At the limit of magnitude, it is ambiguous what that limit is unless orientation of direction is added. Dependent on the orientation, the starting point of magnitude is always zero and the finish is always infinity. This leads to an interesting effect: every number has its complimentary value---even limits do. It is actually possible to take a limit and interpret its sign: 0 = [minimum positive; maximum negative] = [+0, -&] & = [maximum positive; minimum negative] = [+&, -0] To put it another way: |0| = |&| -0 = & -& = 0 To say that zero is maximum negative is to say that it is the embodiment of negative---as good as the real thing. Since the maximum here is infinity, it doesn't get any better than this. Every positive number has an infinite number of potential zeros written out to the left or right of the number, like this: +1 = .....0000001.0000000..... What we have here is a positive something in the one's place representing the presence of something along with the lack everywhere else. If we had plunked a zero down into a field of zeros instead of a one, it would have disappeared. Although one, up above, is positive, its field is its opposite: negative. Otherwise, it would have disappeared into a field of one's, infinity, and serve no purpose as a relative number: .....111111(1).1111111..... So, the sign of a positive number is numerically coded as a negative field of zeros surrounding that number. Since the field infinitely dwarfs the magnitude of the number, it is more significant than the number----especially if the field contains no number. The opposite would be the case with a negative number: -1 = .....&&&&&&(&-1).&&&&&&&..... with infinity minus 1 in the one's place to show a slight depression there. Now we are ready to define division by either zero or infinity. There are four possible situations: 0 <= |a| <= & 1] +a / 0 = & 2] +a / & = 0 +a = & * 0 +a = 0 * & 0 = & * 0 0 = 0 * & - = + * - - = - * + 3] -a / 0 = 0 4] -a / & = & -a = 0 * 0 -a = & * & & = 0 * 0 & = & * & + = - * - + = + * + This is a system of four options, either one of which may be transformed into either one of two compliments by negation: 1] -(+a / 0 = &) 2] -(+a / & = 0) 2] +a / & = 0 1] +a / 0 = & 3] -a / 0 = 0 4] -a / & = & Everything of the above looks as fine to you as I am sure it does to me accept for: & = 0 * 0. That looks about as foreign as it can get. But I am trying to be consistent with the discussion here. A corrolary to this would be: (AN UPPER OR LOWER) MAGNITUDE LIMIT = {0, &} LIMIT^(-1) = 1 / LIMIT = -(LIMIT) -[LIMIT^(-1)] = -(1 / LIMIT) = LIMIT Incomplete Table of Compliments ------------------------------- 0` >>>>> to read: zero's compliment -0 = 0` = & -1 = 1` = ? -2 = 2` = ? etc... -& = &` = 0 The remainder of the table depends on what base number system is being used: We assume that the number of terms to a polynomial's expression of a number is infinitely long in potential, but the simplest way to depict infinity as a polynomial number requires only one term for x: & = ?x^&. For Base One ------------ 1` = 1 In base one, there is no zero or infinity, since unity represents the unit value of a module, 1: x = 0 x = 0 x = & x = 1 x/x = 0/x 2x = 2*0 x/x = &/x x-1 = 0 = & I'm getting 1 = 0 2x/x = 0/x 1 = & ahead of this 2 = 0 discussion; 2/2 = 0/2 see below. 1 = 0 Diversity of value doesn't exist in unity. But why isn't zero or infinity representing unity? Because one has no opposite, but both zero and infinity do. Zero and infinity are byproducts of base two reasoning. Hence, it is best to use numbers of at least the same base as the sign system with at least one of the factors of the number's base to be the number of signs in the math system. For example: base ten is divisible by a two sign system; base two, as used by computers. Otherwise, trying to perform base two math operations would be a hopeless case----the duality of base two operations would get swallowed up by the unity of a base one number system; subtraction, multiplication, and everything else would leave unity unaffected. For Base Two ------------ 0 = 0` = 1 = -0 1 = 1` = 0 = -1 2 = 10` = 01 = -2 3 = 11` = 00 = -3 4 = 100` = 011 = -4 5 = 101` = 010 = -5 6 = 110` = 001 = -6 7 = 111` = 000 = -7 etc................ & = ....111111` = ....000000 = -& But we could have said this, just as easily: For Base Two ------------ 0 = ....000000` = ....111111 = -0 1 = ....000001` = ....111110 = -1 2 = ....000010` = ....111101 = -2 3 = ....000011` = ....111100 = -3 4 = ....000100` = ....111011 = -4 5 = ....000101` = ....111010 = -5 6 = ....000110` = ....111001 = -6 7 = ....000111` = ....111000 = -7 etc................ & = ....111111` = ....000000 = -& Not only does zero equal infinity, but zero is interpreted as being negative infinity, while infinity is negative zero. Negative zero confused computer scientists in the early days of computing. Whenever a number was subtracted from its own likeness its result was a negative zero. You'll never find that happening anymore, even though the internal operations of computing are predominantly still the same. More on this later. Notice the string of zero's in front of all positive numbers up to everything just shy of infinity. Or for that matter, the string of one's in front of all negative numbers up to everything just shy of zero. These numbers are known in the computer trade as padding. They code for sign. Notice also that the interpretation of a negative number's value is the opposite to interpreting a positive number's value: one means zero and zero means one---a zero in the two's place means two, a one in the two's place means zero two's. For complimenting base three: For Base Three -------------- 0 = 0` = 2 = -0 1 = 1` = 1 = -1 2 = 2` = 0 = -2 3 = 10` = 12 = -3 4 = 11` = 11 = -4 5 = 12` = 10 = -5 6 = 20` = 02 = -6 7 = 21` = 01 = -7 8 = 22` = 00 = -8 etc................ & = ....222222` = ....000000 = -& Or: For Base Three -------------- 0 = 0` = 2 = -0 1 = 1` = 1 = -1 2 = 2` = 0 = -2 3 = 10` = 12 = -3 4 = 11` = 11 = -4 5 = 12` = 10 = -5 6 = 20` = 02 = -6 7 = 21` = 01 = -7 8 = 22` = 00 = -8 etc................ & = ....222222` = ....000000 = -& The only thing missing from making this a looped scale is by making & = -&. Let's do another number line, only this time, let's make the above scale the exponents of a variable, x with a coefficient c in the range: 0 =< c < x. Let's also let each c vary indepedently of one another by giving each c a subsript in the range: & <<<< 0 >>>> -& c(?) & = c(&)*x^&, c( &-1)*x^(&-1), <<<< c(1)*x^1, c(0)*x^0, Postulate 1] every relative magnitude has its compliment -? >>> negation of a value >>> ?` >>> inverting a value into its compliment -1 * y = y` example: if y = 0, then -y = & if y = 0, then y` = & -0 = 0` = 9 -1 = 1` = 8 -2 = 2` = 7 -3 = 3` = 6 -4 = 4` = 5 -5 = 5` = 4 -6 = 6` = 3 -7 = 7` = 2 -8 = 8` = 1 -9 = 9` = 0 Zero and infinity are relatively polarized ways of qualifying the transcendent. Their framework of perspective is duality: binary complimentation is a means of defining the undefinable in terms of some form of multiplicity. Zero would mean something else by itself, but when paired with infinity it gets a relative meaning. Zero and infinity are limits to magnitude. It is unknown where they are located. Since the day when zero came into our life we have assumed it is a quantifiable value---something we can relate to; but it is not. Example: "I have no alibi. How many alibies do I have? Zero." Correction: "I have no alibi. But what I have in its place is a good excuse." The lack of alibies doesn't mean that I have zero alibies, but that I have the negation of alibi, the state of not-alibi. If I don't have any alibies, then I probably have replaced it with something else: the desire for an alibi, maybe. In other words, we have mistakenly redefined a not-condition as a zero-condition when in reality it is an inverse-condition. To say that I "have" something already implies that it is a quantity greater than zero. Even if it is a not- something, it is still countable since we are accustomed by now to negative numbers. Zero is as immeasurable as is infinity with the only distinctions of being the pure embodiment of emptiness in an abstract sense and of being infinity's opposite. Infinity is likewise unbounded and the opposite to zero, but is the pure abstract embodiment of fullness. The silly thing to all this is that for all the distinction we make between the two, our distinction is arbitrary according to context: infinity and zero are one and the same ineffable, but according to need it polarizes. Remember how I said in the beginning that duality is a framework for perspective to operate within? Reality is still the same unbounded fullness of potentiality for relative frameworks of operation, but the actual expression of its potential does not in any way limit its uninvolvement. Modular arithmetic makes our universe of magnitude a closed loop. Infinity/zero makes it a loop of unknown number of magnitude markings along its perimeter. While a perspective of duality influences our mathematics in every way and flattens this conceptual loop to a two-way magnitude street with a U-turn connector at every marking. One direction of the street is positive, the other negative. Zero and infinity have the added distinction of sharing the same location on the loop: they overlap each other. The other numbers have conceptual distance placed between each others' compliment. That is where the traffic takes place: along lines of relative value. At the end-points only cross over occurs, but the instant one immeasurable value is crossed the other is also. example: 0 + 1 = 1 1 + 1 = 2 ......... 8 + 1 = 9 1>>>>carry around 9 + 1 --- 0 + 1<<>>>out-of-bounds examples: 2 = 2 carry over is - 1 = 1` = + 8 --- --- carried back 1 = 0 + 1>>>to here and added. --- 1 1 = 1 - 1 = 1` = + 8 --- --- 0 = 9 Now we need a way to numerically code the equivalent for sign. Number plays a dual role, another example of duality, in that it can be interpreted as a numeric value or as a sign value. Sign is coded as a field of the same digits. A zero code signifies that a number is positive when it is attached to the number's left. Nine (or one less than base) signifies negative status and assumes that the number has already been complimented. This is why subtraction need not occur since it's nature is being imbedded into the very fabric of a number. The other digits from one to eight do not serve any coding purpose. Here is how it looks: Sign + >>> 0000....0000number - >>> 9999....9999anumber'scompliment +0 = ...00000000 -0 = ...99999999 +& = ...99999999 -& = ...00000000 +1 = ...00000001 -1 = ...99999998 +9 = ...00000009 -9 = ...99999990 +492 = ...00000492 -492 = ...99999507 etc....... We try to use enough padding of repeated digits to the left of a number so that it won't be confused as being a very large positive number. example: 9991 is negative eight, not nine thousand nine hundred and ninety-one; a poor choice of using too few nines to the left of negative eight. Continuing our examples of subtracting one from zero/infinity (0/9) coded for sign: 1>>>>>>>>>>>>>> 111111 0 = 0000000 = 9 = 9999999 - 1 = 1` = + 9999998 = - 1 = 1` = + 9999998 --- --------- --- --------- -1 = 9999998 = 8 9999997 + 1<<<<<<< --------- = 9999998 1>>>>>>>>>>>>>>> 111111 -1 = 9999998 - 1 = + 9999998 --- --------- -2 9999996 + 1<<<<<<<< --------- = 9999997 Now for another postulate: 0 <= |y| <= & {0, &} >>> a, b, c a, b, c are the limits to y Postulate 2] zero and infinity have no appreciable affect on one another: for: a * b = c or: a / b = c, a, b, and c vary independently of one another Although we can conjecture certain solutions when multiplying or dividing one magnitude by another, complimentation of any segment of the situation undoes any solidity to their relationships. examples: And lastly: Postulate 3] y / a = a` or: a * a` = (y >>>> b) (y tends toward b, but does not equal b) A hyperbola and its asymptotes, is a perfect example for this postulate. All of this has been an excursion into base two logistical thinking. What about other bases? Historical Precedence This thinking isn't new. The "I Ching", the Chinese "Book of Changes" states: whenever anything goes to far in the extreme, it changes into its opposite. This forms the basis for a closed, looped vision of the universe (the dragon biting its tail is a traditional Oriental image). The capacity of a closed system to embody relatively absolute limits (parameters of magnitude), and their similarity toward each other, is suggested by an idea stated as the topic of Maharishi Mahesh Yogi's 18th course? on The Science of Creative Intelligence: "...two fullnesses: fullness of fullness and fullness of emptiness". Miscelany Somebody gave me a rational for not dividing by zero that I would like to pick apart. **************** end of chapter Padding >>> an indeterminate length (from one to infinity) of similar digits, from set {z}, to the immediate left of a number; a numeric way to code a number for sign. Thus, digits play dual roles depending on context: they either designate numeric magnitude or else they designate a sign state when attached to the immediate left of a number. example: padding + number = ...??signstate??number {+} is coded as {0 padding} {-} >>> {& padding}, or {base - 1 padding} example: if base = ten, then {& padding} = {9 padding} Postulate 1] -z = z` >>> (negative z is equal to the compliment of z) example: if z = 0, then z` = 9 or: if z = 000, then z` = 999, etc. Table of Complimentary Digits z, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 z`, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 or more succinctly: 0<--->9 1<--->8 2<--->7 3<--->6 4<--->5 example: z + z` = ...9 (without padding) 0 + 9 = 9 23 + 76 = 99 1849372506 + 8150627493 = 9999999999 etcetera........... examples: +0 = ...00000000 (with padding) -0 = ...99999999 +& = ...99999999 -& = ...00000000 +1 = ...00000001 -1 = ...99999998 +9 = ...00000009 -9 = ...99999990 +492 = ...00000492 -492 = ...99999507 etc....... + >>> Algebraic Addition, and Positive Sign for Both Methods - >>> Algebraic Subtraction, Negative Sign for Algebra, and Negative/Compliment Sign for Ring Method +` >>> Ring Addition (Wrap-Around CarryOver: carryover value from left-most place value summation goes to right-most digit recapitulated: Postulate 2] -0 = & = z - z = z + z` = ...??(base-1 padding)?? -& = 0 = z - z (Algebraic Subtraction, only) summation examples: 111 11111 +756 = 000756 +47 = 0047 -3 = 996 -837 = 999162 +439 = 000439 -91 = 9908 +5 = 005 -28 = 999971 ------------- ---------- -------- ------------- +1195 = 001195 -44 = 9955 = -0044 1001 1999133 >>>1 >>>>>>1 --- ------ +2 = 002 -865 = 999134 = =-000865 Our math operates within the boundaries of linear, two end-point, base two logic, so we are restricted to math operations that are geared to base two linear thinking: this equals that; from one statement we proceed to the next; linear reasoning. Even though number magnitude forms a multi-ringed circle, these rings only intersect and overlap at one point: 0/&, or positive/ negative sign designate. Thus, the circumference of these rings maintains a simple linear curvature (these rings being figurative concepts). This is traditionally called one's compliment mathematical operations in honor of the early days of computer calculation methods when the compliment of 1 always equaled zero in base two. Since computer scientists didn't like negative zero as an answer whenever a figure was subtracted from itself, they have since modified the end result to conform to our traditional style of calculation calling it two's compliment. I prefer to stick with one's compliment ----it is more honest. As a consequence to one's compliment summation: 0 + 0 = 0 +` 0 = 0 >>> by 0 + & = 0 +` & = & >>> both & + & = & +` & = & >>> methods 0 - 0 = 0, by the traditional algebraic method (Am) 0 - 0 = 0 +` 0` = 0 +` & = 0, by the ring method (Rm) 0 - & = undefined by (Am) 0 - & = 0 +` &` = 0 +` 0 = 0, (Rm) & - 0 = &, (Am) & - 0 = & +` 0` = & +` & = &, (Rm) & - & = 0, (Am) & - & = & +` &` = & +` 0 = &, (Rm) The Postulates: 1] -z = -1 * z = z` 2] z + z` = z - z = & = -0 3] z / x = x`; as a consequence: ? 3a] x * x` = z 3b] x ^ -1 = x` 3c] x * x = x **************** Conclusion Zero has traditionally been thought of as a measurable value, while infinity has not. Neither one is measurable. There is always some margin of error when giving emptiness a value. The error is of unknown magnitude. Whether creation is of limited or unlimited scope is of little practical value. As far as this style of closed-loop thinking can cope with, the extent of creation---- its magnitude----is all things to all perspectives simultaneously. So long as we can measure relative values we will have contrasted a point of reference with its point of perspective. But the minute relativity is revoked, and absolute capacity is invoked, then there is no longer any standard for reference and all opposites fuse into one singularity----which is why I call this fused value: the extent of, or the capacity for, creation to manifest. It may be a limited, or an unlimited capacity----we will never know from a commensurate stand point. ****************