James Knight Wrote:I think the Delta Numbers are elements of the hyperreal sets. In addition, I think that Deltation will revolutionize calculus once it has been well defined. I don't think anymore that deltation values are complex or undefined, but are either infinitely far or infinitesimally close to all real numbers.
Deltation (as you use it) has nothing to do with hyperreal numbers. If you have a constant function \( f(x)=c \) then this function is simply not injective and has hence no inverse function (moreover this function is not injective on *each* open interval). This is true also when considering the constant function on the hyperreal numbers. So it is nothing gained by escaping to the hyperreal numbes. Btw, the hyperreals calculus is translateable into the classical calculus, its just another view on the same thing, no new properties are gained.
Another thing is, that you cant extend the real numbers with infinities, without destroying really fundamental properties. For example \( 1+\infty=\infty\Rightarrow 1=0 \) by subtraction.
Quote: I think that it is rather interesting that the delta symbol was chosen to represent deltatation as it has to do with calculus and infinitesimal quantities.I think you mean chosen by Rubtsov et. al. However take into account that he defines deltation different from you. I think it anyway makes not much sense to make such a fuss about the increment function \( x+1 \) as to give the inverse, which every shool kid knows to be \( x-1 \) a new name etc.
PS: Please dont double post. Every post should exist in only one thread. So please remove your symbol/notation post in this thread.