1 = .9999.....

[I Took the Red Pill]

Infinity/infinity = infinity

infinity - infinity = infinity

These two can't be evaluated, actually. The problem with using infinity interchangeably with large numbers is that it doesn't work. You can;'t use it like that.

Well, let's say you have an infinite line of red and yellow sticks (yes, sticks. Because sticks are cool). Now, take away all the yellow ones. Since the amount of coloured sticks total is infinite, there's an infinite amount of yellow sticks and also an infinite amount of red sticks. The amount you take away (all the yellow ones), is infinite, yet what remains (the red ones) is also infinite. There, just took away an infinite from an infinity, yet infinity remains. In other words, infinity - infinity = infinity is 100% possible.

As for dividing, it's exactly the same. There's just an infinite amount of "different coloured sticks" that you divide into an infinite amount of infinite lines.

If you do it that way, it's possible.

No. It's undefined. Lrn2Cardinality. The problem arises because there can be different "kinds" of infinities. For example, are there more whole numbers or perfect squares? The answer, to both, is infinite, even though that would defy intuition; there has to more whole numbers than perfect squares, right? Nah, they're just different infinities. That's one reason why ∞ - ∞ is undefined. Not to mention the fact that infinity is not a number and doing operations on it is silly.

If you want something more concrete and with an actual example, check this out.

edit: what you're saying is basically the same as the example in the link I posted.

Assume ∞ - ∞ = 0

Add infinity to both sides: (∞ + ∞) - ∞ = 0+ ∞

Since infinity plus infinity is infinity, and 0 plus infinity is infinity, you get

∞ - ∞ = ∞

In the same way that you can assume that ∞ - ∞ = 0, add pi to both sides, getting (∞ + pi) -∞ = pi, and since ∞ + pi = infinity, you get ∞ - ∞ = pi.

It all has to do with the Cardinality of infinites.