i see things like this all the time.
it's not so illogical...we learn that negative numbers are an enlargement of natural numbers that allow for us to evaluate things like the formerly "forbidden" 3 - 5, and then we learn that rational numbers are an enlargement of the integers that allow us to say things like 1 is divisible by 4, so it seems like a natural progression to wonder:
"why can't we enlarge the real numbers, so as to allow for expressions like 1/0 having meaning?"
and, in point of fact, we can do this, in several ways. the trouble is, doing so changes the rules of the game, unlike the other enlargements. and you can't just apply the "old rules" to the "new system".
this doesn't mean we have to throw all of the rules "out the window". from a spatial point of view, for example, there is no problem with regarding ∞ as just another point.
but there are different ways of regarding ∞ as the extension of a real number, since real numbers have so many interesting properties. and keeping some properties comes at the expense of others.
for example: naively, it seems that 2*∞ ought to equal 1*∞ (or else we need a 2nd ∞).
but if that's true, then we can no longer say that ax = bx implies a = b, our number system loses its cancellative property. which in turn, is going to make equations MUCH harder to solve. of course, ax = bx doesn't imply a = b if x is 0, so it's not surprising that we feel intuitively as if ∞ should in some sense be the "opposite" of 0, and share some exceptional qualities. it turns out it is easier to just leave ∞ out of the number system entirely, than to create a lengthy list of except when... (whatever involving ∞)'s and ... (something here)'s if x ≠ ∞. making exceptions for 0 turns out to be complicated enough, without adding another "weird-acting number" to the mix.
but that hasn't stopped mathematicians from trying: cantor wrote a book on how to extend induction to certain infinite sets (and here, again, we lose a rule, for example, in his system ∞+1 ≠ 1+∞ (he did not actually use "∞", but rather an ordinal number ω), so addition isn't commutative, anymore, kind of disturbing), and perfectly well-defined extensions of real numbers exist with "transfinite" elements, which ARE reciprocals of "infinitesimals" (which have a "standard part" of 0, but are not themselves 0). in one of those systems, the equations the original poster made would be:
1/ε = ∞
1 = ε*∞
0 = 0*1 = 0*(ε*∞)
0 = (0*ε)*∞
0 = 0*∞
0 = 0 see the difference? the infinitesimal difference between ε and 0, keeps 1/ε from "cancelling out the zeros".
