The discussion revolves around the coexistence of real numbers and infinity within various number systems. Participants explore the implications of performing operations involving infinity and the distinctions between different mathematical frameworks that incorporate infinity.
Participants generally disagree on the validity of operations involving infinity, with some asserting that such operations are not permissible in the standard real number system, while others argue that they can be valid in alternative number systems. The discussion remains unresolved regarding the specific contexts in which these operations may or may not hold true.
Participants highlight the importance of specifying the number system in question when discussing operations involving infinity, as different systems yield different results and interpretations.
HallsofIvy said:If you are talking about the "usual" real number system, the are NO "operations with infinity" because "infinity" is not a real number. And there are several different ways to create number systems which include "infinity" as a number. Which are you talking about?
HallsofIvy said:(Not "exp(0)= 1". I have no problem with that and I don't know why you included it here.)
Jhenrique said:Your answer complicated more the things... look, exp(-∞) = 0, exp(0) = 1, exp(∞) = ∞... appears make sense make operation with ∞...
Jhenrique said:For shows that infinity acts like a number...
micromass said:They make sense in some number systems equipped with infinity, but not in others. For example, on the affine real line, the above operations are true, see http://en.wikipedia.org/wiki/Extended_real_number_line
But on the projective real line, they are false: http://en.wikipedia.org/wiki/Real_projective_line
There are many other systems which allow an infinity and where the above might make sense or not, so you need to specify.
I don't see what ##e^0 = 1## has to do with infinity.
And infinity is certainly not a real number. It might act like on in some ways, but not in others.