0.0.0.0 means that any IP either from a local system or from anywhere on the internet can access. It is everything else other than what is already specified in routing table.
Inclusion of $0$ in the natural numbers is a definition for them that first occurred in the 19th century. The Peano Axioms for natural numbers take $0$ to be one though, so if you are working with these axioms (and a lot of natural number theory does) then you take $0$ to be a natural number.
127.0.0.1 is normally the IP address assigned to the "loopback" or local-only interface. This is a "fake" network adapter that can only communicate within the same host. It's often used when you want a network-capable application to only serve clients on the same host. A process that is listening on 127.0.0.1 for connections will only receive local connections on that socket. "localhost" is ...
@Arturo: I heartily disagree with your first sentence. Here's why: There's the binomial theorem (which you find too weak), and there's power series and polynomials (see also Gadi's answer). For all this, $0^0=1$ is extremely convenient, and I wouldn't know how to do without it. In my lectures, I always tell my students that whatever their teachers said in school about $0^0$ being undefined, we ...
As we all know the IPv4 address for localhost is 127.0.0.1 (loopback address). What is the IPv6 address for localhost and for 0.0.0.0 as I need to block some ad hosts.
If you only have to show that such bijection exists, you can use Cantor-Bernstein theorem and $ (0,1)\subseteq (0,1] \subseteq (0,2)$. See also open and closed intervals have the same cardinality at PlanetMath.
11 \0 is the NULL character, you can find it in your ASCII table, it has the value 0. It is used to determinate the end of C-style strings. However, C++ class std::string stores its size as an integer, and thus does not rely on it.
The product of 0 and anything is $0$, and seems like it would be reasonable to assume that $0! = 0$. I'm perplexed as to why I have to account for this condition in my factorial function (Trying to learn Haskell).
Your title says something else than "infinity times zero". It says "infinity to the zeroth power". It is also an indefinite form because $$\infty^0 = \exp (0\log \infty) $$ but $\log\infty=\infty$, so the argument of the exponential is the indeterminate form "zero times infinity" discussed at the beginning.
