mwace wrote:Thus, it would be impossibe for the value of ∞*0 to be anything except zero.
Why couldn't it be infinity?
After all, 3*∞ = ∞, 5.3*∞ = ∞, etc. So why not 0*∞ = ∞, as well?
You've got two trends going here. Zero times most things is zero, and infinity times most things is (+/-) infinity. So infinity times zero is either zero or infinity (or perhaps something in between). This is (a simplified explanation of) why it's indeterminate.
Here's a rundown of the definition of multiplication (which can from here be extended to other sets of numbers, of course)
For n and m integers:
n * 0 = 0 * n = 0
n * (m + 1) = n * m + n
n * -m = -(n * m)
1/m = that fraction which, when multiplied by m, gives 1 (I'm not going to go all the way into the construction of the rationals from the integers here.)
m * 1/n = m/n
for q rational
q * (m/n) = (q * m) / n (From the first couple properties above, you can prove associativity, so I'm getting perhaps a bit lazy here)
Finally, for r and s real
r * s = Lim(q in Q -> r) q * s
The problem is that nowhere in any of these properties does it say anything about infinity. Infinity is not, as generally used, a specific number. As such, even while we usually treat infinity * n as infinity, even this isn't exactly determinite.
Now, you can define multiplication on sets that include infinity in various ways, some of which are described in links other people have posted. Thing is, you have to define multiplication by infinity before you can use it. It isn't something that pops out of the more basic definitions and properties of multiplication.
So sure, you could just say that infinity * nonzero = infinity and infinity * 0 = 0, but then you've made infinity a specific number and would have to start using another symbol for the usual limit-based concept that most mathematicians are talking about when they say infinity.