By convention, in measure theory 0 . ∞ = 0 to handled measures with infinite value in a consistent way. But, is it natural and logical?.
It probably depends in the target processor and compiler, but for Intel FPUs and the two different compilers I tried, the result is -NAN.
Conceptually (setting aside the simple math explanation) 0 means nothing, so even if you have infinite number of nothing, you still have nothing. Likewise, if you have zero infinities, you still have nothing.
I think the question is not so clear if we have \lim_{b \goes \infty} 0.b = 0 now it should be different if we ask \lim_{b \goes \infty, c \goes 0} c.b = indeterminate, I should also say that infinite is not a number
Thanks to all, Using following notation
R : set of real numbers excluding infinity both sides = ( - ∞, + ∞)
R* : set of extended real numbers including infinity both sides ( = R U { -∞ , + ∞} )
some doubts arises regarding this question in R and R* and handling division by zero in Machines.
I realize that Miguel, but actually i was asking myself about be the sense in to make such question, because as i know $\hat{R}$ (real with infty) just have the order properties and the nested intervals property, and with that property it is possible to prove that \infty*0 should not be a number different of zero, now should be reasonable to assume that infty * 0 is zero and also that \infty.0 = indeterminate, but this I think it is part of the some necessary definitions about infty as number
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