dividing rational infinity

[AntonioLao]

To divide infinity, it must, first of all, be defined as a number, no matter how large or how small this number is. If infinity is unity then half of infinity in a base-10 number system is represented by ½, a third by 1/3, etc. If infinity is 2 then half is 1 and a third is 2/3. If infinity is 3 then half is unity plus ½ and a third is simply unity. If infinity is 4 then half is 2, a third is unity plus 1/3 and a quarter of infinity is unity again.

By simply rearranging these progressive divisions into a matrix, the row indices can represent the successive dividends and the column indices would then represent successive divisors. In this unique matrix representation an interesting and a surprising pattern of finite elements emerged. The elements of column one are simply a copy or one to one mapping of the row indices to itself. Column two starts with ½ then 1+1/2 then 2, the set of positive integers and their halves. Column three is again the set of positive integers and their thirds. Column four is again the set of positive integers and their fourths. It can be observed that the counts of mixed elements (wholes plus fractions) between wholes increase as the divisors become larger and larger. The largest element of each column is the one with the biggest row index. On the other hand, the smallest element of each row is the one with the largest column index. Unity is always located at equal indices of row and column. For a particular matrix of infinite order, zero would be located at the uppermost right-hand corner while the largest number approaching true infinity would be located at the lowermost left-hand corner. Surprisingly, since the elements along the main diagonal are always unity, it occupies both the uppermost left-hand corner and lowermost right-hand corner. Thus gave the idea that half of this particular rational infinite matrix universe is bounded by twice of unity. Fortunately, these four bounded corners of numbers: {0, 1, 1, ¥ } can be used to define that infinity divided infinity is unity (¥/¥=1). Moreover, it is stated without proof that a sieve of Diophantus is a proper subset of the same order but not of the same element arrangements or copies. All positive rational solutions of infinite orders algebraic polynomials can be found within the matrix as a quantum universe of real numbers excluding negative integers and irrationals. However, simply multiply the matrix by -1 gives a new matrix of negative elements of the same order.

[SteveA]

This sounds related http://www.cut-the-knot.org/blue/Stern.shtml.

There was another way of doing this beginning with positive and negative infinity in the form +1/0 and -1/0 and in that case we have two of the binary layers being symmetries of subtraction (with the symmetry around 0 for positive and negative values on the first layer) and division (with the symmetry around 1 reaching to 0 and infinity).

I've come across the idea a few times that physical processes may more fundamentally be in terms of division and subtraction though mentally we tend to use their inverses addition and multiplication to invert these processes.

Notice that we can create addition from a double subtraction, whereas we can't create a subtraction from addition and also we can construct multiplication by a double division, whereas we can't create division via multiplication. So subtract and division, in this sense, are more fundamental than addition or multiplication.

--------------------------------------------------------

(This is a bit of a sidetrack, but you might find it interesting)

If we begin with 0 and just use subtraction and division, we can construct the natural numbers (if we define 0/0 to be equal to 1, hence 0 would effectively be an infinitesimal and 0=1/Infinity or Infinity=1/0).

1=0/0
-1=0-0/0
-2=0-0/0-0/0
2=0-(0-0/0-0/0)
1/2=(0/0)/(0-(0-0/0-0/0))

If we had a stack language, similar to FORTH (you mentioned PROLOG before so I assume you're familiar with some less than common languages) we could write the number 1 (similar to a strand of DNA with 3 symbols) as:

1= 00/
2= 0000/-00/--
1/2= 00/0000/-00/--/

(There may be simpler ways of writing 2 and 1/2, but that's just an idea)

As just an interesting sidelight, imagine a triangular surface constructed as a number plane by moving toward each dimension. Moving toward each vertex is similar to a motion toward 1 of 3 directions, which over time can fill a 2-D surface. Moving toward 0000.... would be creating a space filled with nothing. Moving toward /////... would be a process of comparing different perspectives and moving toward ------... would be comparing differences between these.

For example, if we began with 0 and then constructed another 0 and then performed a division, /, this would be the same as nothing seeing itself as unity.

We can similarly find trajectories in this space that represent various irrational processes, such as the golden mean (if we had a fourth dimension, similar to a timelike "iterate until convergent" instruction)

00/{00/+00//

Where we infinitely nest everything after the brace, so this ends up computing the equivalent of:

1 (1+ 1/(1+ 1/(1+ 1/....)))

And as a sequence over time:

1
2
.5
1.5
~.667
~1.667
...
~.618033...
~1.618033...

So you can map the golden mean and its reciprocal as 2 phases of time heading in a specific direction within this space and we can do the same for various roots and by nesting the { instruction we can construct various orders of infinity etc.

(Now all we need is a universe emulator and then run some simulations until we find a nice place and time to visit, correlate the direction with local space and then head off into space to meet up with it ... yes, that's a few layers of almost unimaginable difficulties deep, but I can't think of anything much better than that - a roadmap to the universe, at least within the limits of sampling precise and chaotic divergences over time)

[AntonioLao]

I dont agree that division and subtraction are more fundamental operations than multiplication and addition. Without any proof I considered addition as the only fundamental operation derived from well defined mathematical logic. Addition held all the properties of commutativity, associativity, additive identity and existence of an inverse. When applied to Hadamard matrices, the operation of matrix additions give the concept of electric charges, weak charges, as well as color charges while the applications of matrix multiplication to Hadamard matrices give the concept of physical mass and also preserve commutativity and associativity. Unfortunately, symmetric Hadamard matrices are singular therefore inverses do not exist and matrix division is meaningless.

[SteveA]

I dont agree that division and subtraction are more fundamental operations than multiplication and addition. Without any proof I considered addition as the only fundamental operation derived from well defined mathematical logic. Addition held all the properties of commutativity, associativity, additive identity and existence of an inverse. When applied to Hadamard matrices, the operation of matrix additions give the concept of electric charges, weak charges, as well as color charges while the applications of matrix multiplication to Hadamard matrices give the concept of physical mass and also preserve commutativity and associativity. Unfortunately, symmetric Hadamard matrices are singular therefore inverses do not exist and matrix division is meaningless.

If we begin with natural numbers, then we'll remain with natural numbers if we only use multiplication or addition. So in order to extend mathematics beyond natural numbers and create zero and negative integers we replace addition with its inverse subtraction.

Notice that we no longer need addition if we have subtraction because x+y=x-(y-y-y). (I used y-y to generate 0 if we're to begin with only positive integers)

So once we have subtraction, we no longer need addition as it is not as fundamental an operation as subtraction.

Also when we move from integers to rational numbers, we cannot do this using multiplication but instead create division and once we do this, we no longer need multiplication as a fundamental operation either because x*y=x/((y/y)/y).

So in this sense subtract and division can do everything multiplication and addition can do as well as more and so addition and multiplication are only subsets of subtraction and division.

Notice again that we can remove the parenthesis using reverse polish notation and rewrite these as xy* -> xyy/y// as well as xy+ -> xyy-y-- so we can use space to order the operations instead of an additional symbol. So we can denote all arithmetic computations (ignoring the quantities, but we can just add 0 for this) using just a binary set of operations and linear space. Not bad for a fundamental physical process

Also we can denote positive infinite as 1/0, which would be (0/0)/0 or 00/0/ and negative infinity as 0-1/0 or 0-(0/0)/0 or 000/0/-

With a 4th dimension we can add recursions over time (though this might more naturally be the first dimension in the representation).

Counting could be similar to computing the sequence 1+1+1+..., which would be (0/0)-(0-0/0)-(0-0/0)-... or 00/{000/-

And we're approaching the ability to compute structures in calculus using this. If we used addition, multiplication and 1 as our symbols, we'd still be limited to generating natural numbers only.

In fact we might already be able to even interprete these values as computing exponents of multidimensional matrices (let me think about that - now that would be cool to pull atomic orbitals out of this! and with division and recursion we could likely iterate roots also ... your comment gives some definite food for thought. I'll have to get back to this later ... imagine if it was possible to effectively create most any form of mathematical structure using just a few symbolic dimensions and time ... and we've got a structure similar to DNA ... just add mutations and space or count through the possibilities).

[SteveA]

Here are some interesting additional considerations:

Regarding matrix operations or operations on sets of numbers, recognize that multiplication or division can be seen similar to convolution (filtering or a spectral response) over a digital space.

If we had, for example in decimal, 302*101 or 302/(1/101) you can interpret this as an addition of copies of the same string of symbols, but offset or shifted over a space:

   302
*  101
------
302
 000
  302
------
3322

In the decimal case each element in this "space" contains one of 10 digits/symbols. (With a recurring multiplication or division by 10 we should shift these elements past a specific point in space over time as well)

But if we wanted to allow arbitrarily large values to be contained in a single location in space, we could separate them by orders of infinity, but in order to do these, we must use a specific value for infinity that's at least equal to, if not greater than, the largest quantity of an element within a location in that "space" (this specific "infinity" would then determine the maximum information storable in each location in space).

For example, if we had some integers a,b and c and we wanted to concatenate them into a single irrational quantity, we could weight these by powers of n in the form x=c/n^2+b/n+a. If we then allowed the quantity n to approach infinity and took the approximate physical quantity this would approach, it would be a. If we then iterated a subtraction of a and a multiplication by n, we then find the limit to approach b and we can continue on to isolate all elements in this irrational "number".

With convolution we can perform Fourier computations.

Notice also that we can iterate solutions to roots of a function using Newton's method which is an iteration of a division and a subtraction also. We use gradient descent and compute the derivative of a function at a point and then estimate the zero cross point from this slope:

So if we want to determine x for f(x)=0, we can iterate:

x(0)=0
x(t+1)=x(t)-f(x)/f'(x)

Notice that other than the derivative, this uses subtraction and division as well, and actually computing the derivative uses subtraction and division also:

f'(x)=(f(x+h)-f(x))/h as h->0

If we use a finite approximation and shifted over a unit distance for simplicity, this could be f'(x)~=f(x)-f(x-1)

Or for an infinitesimal form, due to the identity 0=h=1/infinity, f'(x)=(f(x)-f(x-0))/0

So we have an equivalent method of computing a root of f(x) using 0, subtract and division for Newton's Method of:

x(t)=x(t-1)-f(x)/((f(x)-f(x-0))/0)

For duplicating elements in space, we can use convolution. For reordering them we can use the non-linear division operation.

I've got to run for now, but I can post later.

Oh, and as a sidenote, consider the irrational "number" e, the base of the natural logarithm.

e=lim((1+1/n)^n) as n->infinity

Notice that in a similar form we can write this as:

1/e=lim((1-1/n)^n) as n->infinity

or

1/e=(1-0)^(1/0)

And if we similarly defined the recursive operation, {, to precisely iterate an "Infinite" number of iterations, this could simplify to a recursive multiplication like this:

1/e=1*(1-1/n)*(1-1/n)*(1-1/n)*...*(1-1/n)
e=((1/(1-1/n))/(1-1/n))/...

And so we could then, by having a more precise (first order) specification for the quantity of iterations of the recursion operation, denote 1/e as:

e=(((0/0/((0/0)-0))/((0/0)-0)/...

e=0/0{00/0-/

Look at all the things we can do with zero or "nothing"!

NOTE: There's a difference between these zero and the additive zero though. The zero being worked with is actually an infinitesimal and potentially dynamic quantity. It would be similar to viewing the influence of a Planck scale object from the perspective of a growing universe and an ideal cancellation, like 5-5 wouldn't imply such a remnant, so for a more robust system of computation, this would need to be more precisely defined as well and not left to arbitrary reinterpretation.

There's also an issue with orders of infinity. Notice that though 1/0 would be "infinite" or we could simply write this as an unknown and growing quantity or variable, similar to time, and have all computations performed over this time variable and unify all computations within a single irrational structure, if a quantity larger than this "infinity" was created, then we should really treat that new quantity as the singular, one of a kind, "largest", capital I, Infinity.

If we have more than a single infinite value, then there should always be a largest and all the other unknowns and variables or dynamic structures that evolve from it over time need to be described in terms dependent and relative to it, otherwise we get back to the indeterminant division and multiplication by 0 problem and we have additional dimensions and unknowns. At most a single unknown should be sufficient to describe indeterminancy and I think all such unknowns can united into a single varible, similar to a specific time within a potentially infinite timeline, as the largest infinite quantity in a computation and there's no need for an infinite quantity larger than the largest anyway, so all the work with alephs can be rewritten as them all being smaller than a single largest one and it's only when comparisons between a smaller, dependent, infinite quantity is compared against something closer to the source constructing quantities over time that an infinity appears larger than another one.

For example, if we have m=10^n and we allow n to grow unbounded, then m appears to grow larger than n, but if m truly exists, it should be constructable and the "work" or time constructing the volume of m would take exponentially more time than n (yes, I've switched context here some and am referring to time as a construction of a volume of space - if it's something real, then it should be constructable), so in order to generate both m and n, we need to create a total of m+n units, so the total volume needed to describe this relationship is the largest and it is m+n. We can create a single process that constructs both these and maintains this exponential relationship over time by simply multiplexing the addition of new "space" or quantities to both m and n dependent upon a binary decision of whether or not we see m>10^n or m=10^n. So this can be seen similar to a single thread being interleaved between two spaces with a total length of m+n that maintains a convergent ratio of volume between each space as m:10^n and so this relationship is described by an irrational binary string and there's nothing larger in this relationship. If we add more relationships that depend upon our n->infinity then we simply add a larger set of symbols to this string to denote which unit volumes in time are appended to which numeric quantities over that time and we have a growing space that can approach any irrational ratios of quantities of various objects and everything is deterministically related by a single largest infinite quantity (it makes no sense to have anything larger than this largest, unless the largest is a dynamic structure that is continually growing larger than itself).

There are also a whole bunch of correlations with most any area of mathematics or physics if we begin to analyze the statistical properties of patterns of time along this creative thread.

I know I packed a lot of ideas into a rather small post, but hopefully you'll find something of interest in there. (I still didn't show how we can muliplex and reorder elements within an irrational number using division, but basically consider that if we computed a reciprocal of 2, we get 1/2 and by computing a reciprocal again, we return to 2. If we wrote this as binary we'd have 10.0 and 00.1 if we computed the reciprocal of 1, it remains at 1 and could be seen as a stationary bit in "space", whereas bits outside this are mirror imaged by a reciprocal. This is just the beginning. If we compute x^2 we can construct a "stretching" of space as the distance from the origin is doubled (and we've inserted complex or imaginary bits in between these "real" bits) or for x^3 we triple the distance from the origin and insert 2 hidden root phases between these. Such a process could be seen to perform an equivalent of stretching or red shifting a waveform spaced across an irrational quantity. We can then convolve these using division to filter various frequencies out)

Dang, I need to go, but here's another example:

Consider taking a string of digits 101010.. If we looked at these as samples in time, it would be a value toggling at the maximal (observed) rate with a period of 2. We can construct a filter to cancel this frequency by summing adjacent positions (this would be the equivalent of an addition of a frequency to a 180 deg shifted version of itself). This would be a multiplication by 11,

...10101010
*11
------------
...10101010
....10101010
------------
...111111111

(Or similarly a division by 1/11)

If these digits are spaced by infinite powers, then it would a multiplication by (infinity+1), or (0+0/0)/0

We can actually construct arbitrary spectral responses in this manner as either finite impulse response filters or infinite impulse response filters.

If we then do a multiplication between digits, this is similar to heterodyning a radio signal or computing the sum and difference of two frequencies via the trigonometric identity [url]http://en.wikibooks.org/wiki/Trigonometry/Sum_and_Difference_Formulas[/quote] to "route" within a spectrum A and B to (A+B) and (A-B).

[SteveA]

(continued...)

So we can have a single chaotic generator, such as a logistic function treated as generating an irrational number at any instant that represents the equivalent of a white noise spectrum and then we can perform computations by routing filtering, cross interacting and and reamplifying these signals much like computations using transistors and wires for routing in space, all with just a few arithmetic operations. Of course we come up with statistical objects similar to those in classical Newtonian physics but they exist as statistical features embedded or molded from an energetic source. If we drop down another layer there could be a deterministic process, similar to pseudo-random number generation generating that source spectrum (for example, the blackbody radiation spectrum can give clues as to the properties of that fundamental process as it can show how many equivalent poles and zeroes are in the filter response by looking at the slope of this curve).

Anyway, there are just a ton of interrelationships across the map in science and basically it seems like everyone's talking about the same stuff but in different words, but there's still an unknown core to it (and maybe it never ends).