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When we build integers from natural numbers, we're looking for additive inverse of natural numbers?
What's an additive inverse? Well, for example, the additive inverse of 2 is a numbers such that
We call this number -2. So, the real meaning behind the negative sign is "if you add me and my positive counterpart, the result is zero".
So, -5 is the additive inverse of 5, -16 is the additive inverse of 16, and so on, because
Note that this is a symmetrical relation: if -5 is the inverse of 5, it is also true that 5 is the inverse of -5.
So, when you write something like
it means that you want to add 5 and the inverse of the inverse of 4. But given what we just said, the inverse of the inverse of a number is the number itself, which is why subtracting a negative number is the same as adding it.