∞ is like i, you (usually) can't simplify things with it. E.g.
If ∞+∞ is ∞, then we would be obliged to say ∞-∞=∞.
We can use something I like to abbreviate as P.I.G (Patterns In Graphs, which can also refer to look for formulas), which is what I'll do.
If we graph x-x=y, we always get 0.
It's a truly linear equation.
We don't expect that when it gets to ∞ it will suddenly rise to ∞ too.
Same story with ∞*∞ and ∞/∞.
What makes sense in both cases, is that neither ∞+∞ or ∞*∞ should be expressed as ∞, but as 2∞, and ∞2, respectively.
So ∞+1 should be written as it's written here, and so should 3/∞, 2∞+1, xy=y∞x, or ∞x2+∞x+∞=0, etc.
...or is it?
This would work perfectly if it wasn't for the definition of ∞, the biggest quantity that can be described, which doesn't have an end.
If ∞+1>∞, then ∞+1 should be infinity!
But if ∞ (∞+1)<∞+1 (∞+2), then ∞+2 should be ∞!
But if this is smaller than ∞∞(totally infinite!), then ∞∞ SHOULD BE ∞!!!
So my definition of ∞ is: