Historically true, but glad it's there so ℤ is a group. The way to think about 0 for a child is what you add to anything so it stays the same, ie nothing.
Hate to be pedantic, but ℤ︎ is a set (not a group) and even with 0 adjoined it only forms a group under the operators of addition, subtraction, and multiplication.. If you want a set that also forms a group under division you need to ‘upgrade’ to the set ℚ︎.
ℤ︎ (…, -2, -1, 0, 1, 2, …) and + forms a group. It is also a ring with * and +. ℚ︎ with * and + forms a field.
Yes, that's what qubex said, in correcting the error in your previous comment.
ℤ︎ with * is not a group, there is no inverse for 0. When being pedantic it's good to be correct.
You are perfectly correct, I was wrong. Upon closer reflection it appears impossible to have a nontrivial set (meaning other than {0}) that forms a group both under addition and under multiplication because the identity element of the former does not have an inverse in the latter. However, take this with a grain of salt, because I haven't had time to prove this to my own satisfaction yet (and I am loathe to go googling for answers in my own field of expertise).
My abstract algebra professor would be horrified by my misstatement, and I shall go to bring flowers as a sign of atonement and rememberance to his grave at the earliest available opportunity.
Also interesting to try to show "zero" of something. Zero is fun because it took longer to exist as a number than one & two, and there was some debate over whether it should be a number. "How can nothing be something?"