let me begin by stating that all numbers are really infinitesimal
and any infinitesimal will do nicely as a zero

here's the website that says it is so:
which was what originally inspired me to work this theory out:
http://www.cs.princeton.edu/~kazad/resources/math/infinity/infinity.htm
on number systems and how everything is an infinitesimal

this is the other interesting website that gave me an idea:
http://www.wholeo.net/Trips/Wholeo/BooksOnline/zen0/zen6.htm
on dividing by zero

i have other links which i cant find at the moment
i will try to send later since i didnt save them on favorites

of all the numbers to pick the best choice was for first infnitesimal
since it is the smallest number, from there i just used the inverse
to figure out the first infinity

here are some infinity links to verify what i have done:
http://mathforum.org/library/drmath/view/55764.html
http://www.physicsforums.com/archive/topic/511-1.html
http://ubmail.ubalt.edu/~harsham/zero/ZERO.HTM

compared with infinity regular numbers will never reach infinity
and therefore can be regarded as 0, infinity being 1
and the rest of infinity are the other numbers
regular numbers can never reach any fraction of infinity
unless the fraction is added to the regular number ex. + 1/2
in other words is an infinitesimal
and the first such infinitesimal is 0*1
then 0*2, 0*3, and so on
are the regular counting numbers
compared to infinity if infinity is equal to 1
0*0 being a higher order infinitesimal
here are the rest of the equations:

let w = infinity = 1/0
and d = derivative or 1st degree infinitesimal = 1*0
= 1/w = (0*1)/1 = 01 = d1 = d

let n be any number or variable
0*n = 0n = dn
02/0 = 02*w = 2 = d2/d1
02/2 = 01 = = d2/2 = d1 = d
0/2*w = 1/2 = d/d2
0/0 = 1 = w/w = 0*w = d/d
0*0 = 1/ww = 1/(w^2) = 0^2 = dd =d^2
0/w = 0*0/1 = 0^2 = 1/w^2 = d^2
0+0 = (1/w)+(1/w) = 02 = d2
0-0 = (1/w)-(1/w) = 01-01 = d0 = d^2 = higher order derivative
-0-0 = -02 = -d2

d+2 = 2, d is so small it can be discarded
d99999999 + 1/99999999 = 1/99999999
(d^2)+d2 = d2, because d2 is infinitely bigger than d^2
(d^-1)+(d^-2)+2 = d^-2 = higher order infinity or ww
as opposed to d^-1(2) which is 2nd infinity or w2
inverse of d2 which when multiplied by w produces w2

to summarize, everything is an infinitesimal
when compared with a greater infinity
zero is just one of those infinitesimals
when compared to one, nothing is just an
inconsistency, numbers tend to be bigger
or smaller than the standard by an infinitesimal
amount that is practically nothing
so you can ignore it coz it cant be measured
in order to measure infinities
you must have an infinite scale
infinite scales become inaccurate
measuring less than infinite things
there is a first 0, what you cant measure
and there are zeros more zero than 0

the theory should solve the division by zero problem
by saying that multiplication by zero is really something
it should resolve that infinities can be of equal size
and therefore when divided by itself equals one
also when zero is divided by itself it would equal one too
proving the identity that 0/0 multiplied by any number is that number
it would also provide an easier means of understanding
why undefined slopes are vertical and not a hole

i think thats pretty comprehensive
and will work for all cases
if not let me know

heres an example of a derivative using the new notation:
from http://www.friesian.com/calculus.htm

let dx = 0x and dy = 0y
TAKING A DERIVATIVE

I. y = 3x^2 + 4x + 5
Given an equation,
where y is a function of x: ( y = f (x) ).

II. y + 0y = 3(x + 0x)^2 + 4(x + 0x) + 5
If the value of x changes, then value of y
will change. We add in the changes in the
values (0x and 0y) to the original values.

III. y + 0y = 3(x2 + 2x(0x) + (0x)^2) + 4(x + 0x) + 5
(x + 0x)2 is multiplied out.

IV. y + 0y = 3x2 + 6x(0x) + 3(0x)^2) + 4x + 4(0x) + 5
The constants are multiplied into all the terms.

V. y + 0y = 3x^2 + 6x(0x) + 3(0x)^2+ 4x + 4(0x) + 5
-(y = 3x^2 + 4x + 5)
= 0y = 6x(0x) + 3(0x)^2 + 4(0x)
The original equation (I) is now
subtracted from equation IV
This gives us an equation
about the change in y (or 0y).

VI. 0y/0x = (6x(0x) + 3(0x)^2+ 4(0x))/0x
= 6x + 3(0x) + 4
Now both sides of the equation are
divided by the change in x (or 0x),
giving us an expression for the ratio
between the change in y and the
change in x (i.e. 0y/0x).

VII. If 0x becomes small and approaches zero as a limit
0y also approaches zero; and (6x + 3(0x) + 4)
approaches (6x + 4). This is expressed as
(0y/0x = 6x + 4), which is the "derivative"
of the original equation (I . y = 3x2 + 4x + 5)
Note that, from the original equation, each x variable
drops one power, the constant on each variable is
multiplied by the previous power and the lone constant
is simply lost. These are general characteristics of derivates
Since constants are lost in derivatives, the opposite of derivation,
integration, always (for indefinite integrals) introduces a
constant (whose value will then be unknown, though it may = 0).

If y is in units of distance (s) and x in units of time (t)
the derivative (0s/0t) is the velocity, indeed, the "instantaneous"
velocity of a moving object, at a point in time and space
This in itself was philosophically paradoxical
hearkening back to the paradoxes of motion
described by Zeno of Elea, since an object that
does not move a finite distance might be said
to have no velocity, since it is not moving.

thus 0y/0x = dx/dy = the derivative = y/x = the slope of the equation

here's a site that also gave me ideas
http://numerometria.freewebpage.org/index.html
check it out - it is interesting

just yesterday i found this website
on the final theory of everything
i think it's worth a look
http://www.thefinaltheory.com

if you're interested in these kinds of stuff

please let me know,

moe