So the only thing you need to know about entropy is that it's a real-valued number you can associate with a probability distribution? And that's it? I disagree. There are several numbers that can be associated with probability distribution, and entropy is an especially useful one, but to understand why entropy is useful, or why you'd use that function instead of a different one, you'd need to know a few more things than just what you've written here.
Exactly, saying that's all there is to know about entropy is like saying all you need to know about chess are the rules and all you need to know about programming is the syntax/semantics.
Knowing the plain definition or the rules is nothing but a superficial understanding of the subject. Knowing how to use the rules to actually do something meaningful, having a strategy, that's where meaningful knowledge lies.
In particular, the expectation (or variance) of a real-valued random variable can also be seen as "a real-valued number you can associate with a probability distribution".
Thus, GP's statement is basically: "entropy is like expectation, but different".
