u need them if ur going to make another number infinite in a x=x formula, M8, and 1/3 =/ .33............ it's 1 third of a number.

Does 0.00....1 = 0?
No

Proof: 1/0 =/= 1/0.000...1

Therefore
0.99...9 =/= 1

0.00000...1 is not the same as 1 - 0.9999.....

1/0 is undefined, so you can't use that argument.

Aside from that 0.00...1 is not well defined. That is precisely the phantom "infinitesimal" quantity used in the intuition behind calculus that cannot be well defined without doing crazy things to the real number line.

You don't really have to do anything to the reals to use it, as long as you cover your tracks well enough, right? In my head I'm thinking use it like dirac-delta, or are distributions too far removed from what you're talking about?

You called?

I LOL'd

Infinitesimals are nice heuristics that you can usually use without problem, but some issues do pop up. The biggest one is saying that dy/dx = 1/(dx/dy), which is not always true. The whole development of epsilon-delta proofs was to put the idea of infinitesimals on a rigorous footing. You'll notice, though, that some of the algebraic manipulations (like the one above) you would like to do with dy/dx don't work with limits. Anyway, everyone uses them to some extent. When I do physics problems, I tend to write them since that is the convention in physics. In higher math, I don't bother to write out the infinitesimals since they are not formally well defined. My experience is that mathematicians tend to write total derivatives with primes, partials with subscripts, and integrals without dummy variables (so no need for the infinitesimal quantity to integrate with respect to).

As for the dirac-delta, that depends on how you define it. The limit definition is formally just wrong, so I'll ignore that (the limit function integrates to 0 in the extended plane and no limit function exists in the reals). If you mean the measure theoretic definition (which IIRC is how the distribution is defined), that is different, since you are assigning measure 1 to the point 0. That functions differently from infinitesimals.

And yes, I did call. This thread needs more dancing suit cat.

Edit: Oh I see what you meant now. The difference between infinitesimals and the dirac-delta is that the dirac-delta can be well-defined without screwing with the number line, while infinitesimals can't.
Edit2: I should really think of a better reason why infinitesimals are risky. That's a really crappy exception since going through the real problem with it requires some nasty functions. I'll get back to you with a better answer on the problems of using the traditional calc notation.

This post was edited by darkfire on Apr 12 2010 06:40pm