Quote Originally Posted by phantomotap View Post
How would anything equal multiplied by 0 manage to be equal to 1?
Oh, it wouldn't, I'm trying to imply that maybe infinitesimal != 0.

Quote Originally Posted by Mario F. View Post
I seriously doubt that infinitesimal is considered anything resembling 0. While math is not my area, I think there is a huge difference between something infinitely small and nothing at all and this difference somehow has been expressed in maths. What I remember learning is that an infinitesimal is a quantity approaching zero but not reaching it. It's distance to zero is infinitesimally small.
Yes this is as far as I know too, but I found some compelling arguments showing that because infinitesimal is so small, that is, if one were to search for it's 'end', they could never find it, thus making it nothing.
Well, I can't find the pages with the proofs anymore. But here's a good reference nonetheless: 0.999... - Wikipedia, the free encyclopedia...
Now, I hope my assuming that saying that 0.(999) = 1 is the same as saying that 0.(000)1 = 0, isn't wrong

Quote Originally Posted by Mario F. View Post
Infinite quantities may be applied to maths, but you already realize that rules are different for them when for instance we learn that any operation on infinity returns infinity. So, you shouldn't just assume good old algebra is the way to solve your question. 0 * 1 is not what you should be looking for. Infinite quantities aren't numbers at all.
True, and in real life calculations, they're not used. To me, this seems to lead to holes in the mathimatical system, though, for example the infamous divide by 0 problem. The idea is too see how infinity and numbers would co-exist in the same mathimatical system. Formal math seems to say they don't/can't.

Quote Originally Posted by Mario F. View Post
- We can simply establish that infinitesimal quantities are infinite quantities in fact. Which they are. There's no direction for infinity. So an infinitesimally small quantity is an infinite quantity. Infinite quantities aren't big or small. They are simply infinite.
We know that infinity is bigger than 1, and that infinitesimal is smaller than 1, so, when a difference in relationship to tangible real numbers can be shown, doesn't that prove that they aren't the same thing? On other words: Surely something can't be smaller and bigger than something else at the same time

Quote Originally Posted by Mario F. View Post
- Or we can remember that infinitesimal is not the same as 0, as such -- and because any operation on infinity results in infinity -- the answer to your question is infinity.
The problem with this is the assumption that any operation on infinity results in infinity, for example, 0 * infinity, is of course, 0.
Just as we have 1000, and 0.001, we have infinity, that is, 1(00)0, and infinitesimal, that is, 0.(00)1, I purposed 1 as an answer because if 1000 * 0.001 = 1, then why not 1(00)0 * 0.(00)1 = 1?

Quote Originally Posted by Mario F. View Post
- If we were to represent infinitesimals differently than infinitely large quantities, it would probably make sense to establish a rule based on the position of them in the equation. That is to say an infinitesimally small number multiplied infinite times is still an infinitesimal number, while an infinitely large quantity multiplied by an infinitesimal amount is still an infinitely large quantity. This holds true for all operations.
This breaks the rules, though.