there is no such thing as an infinite number
Not in the context you're thinking of...we're not talking about the real numbers though - it's a superset of them though. Robinson proved in the 50s that they're logical to work with, so they do exist.
A very big number is not infinity now is it?
Depends what you mean by number. The cardinality of finite sets have finite value, for instance. The cardinality of countably infinite sets of a "smaller" infinite value than uncountably infinite sets. We're not talking sets though, so I apologize for that aside. A very big number isn't infinity in the reals, but again we're not limiting ourselves to the reals...there's not enough structure in them.
The 0 multiplication rule applies to a number x, not infinity
It's not limited in this algebra...in this algebra infinity is a number. Have you taken introductory complex analysis before? Do you know what points at infinity are? The Riemann sphere is something you may want to look up.
A number that is small and there is no smaller number does not exist if it's not 0 if you're using division
"If you're using
real number division"
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Skip this part if you want until after the stars
infinity-infinity not the cardinality, though I don't understand the concept very well.
If you have an infinite set of [x,x,x,.....] minus [y,y,y,y....] you still have an infinite set of [x-y,x-y,x-y....]
In that last part I'm not sure what minus you're using. Are those ordered sets of are they tuples of numbers? If it's a tuple then it's got nothing to do with subtraction in our case. If they're sets, then that's not how you "subtract" them.
If you have triples S = (1,2,3) and T = (4,5,6) and define subtraction as you have above, then
(1,2,3) - (4,5,6) = (1-4 , 2-5 , 3-6) = (-3,-3,-3)
which makes sense. But we're not talking tuples, we're talking sets...so even if we used your subtraction definition (which doesn't exist in set theory the way you're thinking of it), we'd have
{1,2,3} - {4,5,6} = {1-4 , 2-5 , 3-6} = {-3,-3,-3} = {-3}
since we're talking sets now. The size of that set is 1, not 3, so even if that subtraction were the way we do it, it would come up with a different sized set in the end, possibly.
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But as I mentioned before, the infinity - infinity thing doesn't follow the rules you're used to because we're not dealing with the same sets. This subtraction is different in the same sense that matrix multiplication is different than real number multiplication. They share the same name, but aren't the same operation.
I talk too much
