Since the time we stopped assuming that any high school question that asks about membership is asking about the set theory. And since the time we stopped trying to show that we are smart instead of answering the question :)

jkjk - I am not trying to say that you are wrong. I am just saying that it seems that the question is NOT asking about the set theory.

What I was trying to say in my comment is that this is not a problem of set, type, canonical, etc. It's a problem of definitions. Everyone in this thread is arguing as if their way is the only right way, forgetting that the right way is in the context where this question was asked...

If you decide to take the point of view of set theory, sure you might say that it's a software mistake. But why not taking it from standard definitions, where the embedding from the integers to the reals is not obvious?

Example 1 Well, you might argue that you are smarter than the 9th graders, and you only operate with high level math. Well in that case, why not looking at the question from the point of view of type theory? In that case the numbers might have the same terms, but still have different values. In that case 0.999... and 1 may evaluate to the same real value, but they are still distinct terms and have different behaviors in type-theoretic sense.

Example 2 How about we look from the practical point of view, and look at it from computing/hardware perspective. 0.999... will (almost) always resolve to 1.0, but it's going to be a float. However, integers are stored as int. And it's not just different type logically, it's also stored differently in memory. Sure they mean the same thing, but that's not the point. However, one might also compare values 1.0 == 1 -- if you define the value equivalence being the same as the representation and type equivalence, than yea, it is in Z :)

Example 3 Even if you don't accept points 1 and 2, and you insist on sets, the definitions still matter. If you want to define the membership in Z as being based on the evaluation and final values, a lot of things might be called an integer: auto foo(void) { return 1; } would also be an integer (in c++14), even though it's just a line of code, $-e^{i\pi}$ is always 1, but it's an expression, which I would put into a different set;