Hierarchic number system

[M Benesi]

  It was developed to allow division by zero, following a mistaken comment I made in another forum.  It works. 

   It may allow us to work past various "infinities" in certain physical equations.  I've already developed a WORKING exponential function extension for it, which allows the input of more than one imaginary portion (this means more than one angle can be entered into the exponential function, and it has INTERESTING behaviors, that propagate to lower hierarchy positions!!!).

  How it works is:  Any number multiplied by zero moves down in the hierarchy to a lower position, division by zero raises the hierarchical position.

  I use the  symbol to denote a lower hierarchy number if it is on the right of a number/variable. 

     note that the real portion is 0 in this case- the hierarchic system does not alter any mathematics for real or complex numbers, and I don't see how it would for other number types (quaternions, etc.).

  A higher hierarchy number times a lower hierarchy number results in a drop of hierarchy:

     y, yz, and xz are all lower hierarchy then the real x portion

  Numbers of the same hierarchy multiplied by one another stay in the same hierarchy, same with other operations within the hierarchy:

   

  When adding numbers, numbers of the same hierarchy add together:

 

  As multiplication by zero lowers the hierarchic position of a number, division by zero raises it.  For now I'll put  in front of a number or variable that is ABOVE reals in hierarchy:

      One can also increase the hierarchy of a lower hierarchy number by division by zero:

  Here is the exponential function, I also did a 6 variable one (with a second lower hierarchy) that reveals more interesting patterns.

 

 
 I'm linking to a wxMaxima script that will allow you to play with the exponential function. 

  Someone needs to put it into fragmentarium or something.


  Here is my wxMaxima script.  I use wxMaxima 0.8.3   

[hobold]

Is this, by any chance, related to the hierarchy of infinities ... wait ... a bit of googling around finds me the so called "Beth Numbers"?

http://en.wikipedia.org/wiki/Beth_number

I was initially thinking of the sequence of infinite cardinal numbers usually denoted by a small omega,

http://en.wikipedia.org/wiki/Ordinal_number#Ordinals_extend_the_natural_numbers

but the others denoted with a capital Beth seem more related to what you describe. In either case, this way of resolving a division by zero sounds interesting. You say it is consistent with the usual algebraic rules; and it seems to me it is also consistent with this vague intuitive idea of blowing the number up to some infinity.

[M Benesi]

  All right, I thought about it a bit more.  I'm refining the number system as I go along, gotta keep it tight.  Here's a word for word copy of my post on another forum:

On further thought, each zero must be unique in a non-branching hierarchical system.  This is a more well defined idea than the branching system, and it justifies the exponential function I used.

  This means that we must track zeros.  Tracking zeros allows us to preserve the associative property (and not to mention makes the exponential function I used legitimate (I just said that.. like using 2 of the same zero, doesn't alter anything, although it introduces clutter)).  




  Multiplying both sides of an equation by multiple zeros will only result in an equal change in hierarchy, which can be disregarded (like multiplying both sides of an equation by 10 a bunch of times... what's the point?).  

  In cases with multiple zeros, we have to track zeros that have changed the hierarchy of variables or numbers.  We must use a specific nomenclature to track the hierarchy of variables in these cases.

  
  

  This means that the above variable x has a unique hierarchical position.  If it is multiplied by or a variable that has those zeros associated with it, it will be a real number again.



  It can interact with reals in another scenario as well, but this has some caveats, and will result in branching in some scenarios.  In this case, the zeros must still be tracked in case they are applied again- the 1,2 zeros that have lowered this variable cannot lower it again, although they can raise it in hierarchy by division.  The opposite applies to zeros 3 and 4.  



  The zero above the 4 in y's Theta's definition indicates it only has a division by zero component.  

  So while the above can be added to real variable z:

has a real value of xy + z, if it is multiplied by zero 1 it splits:

so that the real portion is xy, and the z portion is of a lower hierarchy.  You have to track zeros, unless unique zeros are "divided out".


  Don't apply a zero twice:

  

  


   You know, even though I'm just learning something the continuum figured out a long time ago, it still seems new to me because I didn't read about it on wikipedia.

[M Benesi]

  Sorry Hobold, revisiting this thread I realize I cut out the portion of the above post that was a reply to your comment about the Beth numbers.  Looks a bit douchy since I didn't engage your comments- I think I cleaned up the above post and cut out my reply to your specific question (which I remember). 

  I wasn't really focusing on the cardinality of the set(s).  And I ran into some pretty basic problems with my attempts to integrate the reals as part of the set of hierarchic numbers- it looks like I break a few identities and basically throw the reals under the bus in my attempt to delineate specific zeros and hierarchies. 


  The one thing I did get out of this idea is new "logarithmic" and "exponential" functions.  With the new "exponential function", I ended up with a new cosine and sin that could be applied to fractals of dimensions (spatial, not Hausdorff) greater than 2.

  I'm having problems implementing the function within Fragmentarium (code below).   I've even created a regular cosine/sin exponential function that doesn't quite work.  My assumption is that the factorials that I divide by are getting too large, so I might have to split the factorials.  I already got it to work better by constraining the angles to less than 2 pi, although I should experiment a bit and find the sweet spot for number of elements that need to be added up in order to get the correct answer. 
 
  If anyone wants to check my code for a brain malfunction, I'd love your input.  I might have done something silly when I translated the code for the function from Maxima script to GPU language for Fragmentarium. 
  I just inserted the following code into the MathUtils.frag in the Include directory that is within the Examples directory.

Code:

// first one is the normal exponential function.  It SHOULD output
// cosine and sin as parts of a vec2...  It's input is a single angle (float).
vec2 special2(float an)
{
	vec2 tot=(1.0,an);
	float b2=an;
	vec2 temp=(0.0,0.0);
	vec2 work=(0.0,an);

	float fact=2.0;
	int i = 1;
	float count=2.0;

	while (i<56) {
		temp.x= -work.y*b2;
		temp.y= work.x*b2;
		work=temp;	
		
		if (count>2.5) {fact=count*fact;}

		count=count+1.0;
		tot=tot+ temp/fact;
		i++;		
	}
	
return vec2(tot);

}

// This one should output cosine and sin, then an additional cosine and sin for the second angle that varies
// a bit more than regular sin and cosine.   It's input is a 2d vector which represents 2 angles.  
vec4 special1(vec2 an)
{
	vec4 tot=(1.0,an.x,.0,an.y);
	float fact=2.0;
	int i = 1;
	float b2=an.x;
	float d2=an.y;
	vec4 temp=(0.0,0.0,0.0,0.0);

	vec4 work=(0.0,an.x,0.0,an.y);

float count=2.0;
	while (i<56) {
		temp.x= -work.y*b2;
		temp.y= work.x*b2;
		temp.z= -work.y*d2-work.w*d2-work.w*b2;
		temp.w= work.x*d2 +work.z*b2 + work.z*d2;
		work=temp;	
		
		
if (count>2.5) {fact=count*fact;}
		count=count+1.0;
	tot=tot+ temp/fact;
		i++;		
	}
	
return vec4(tot);

}