The set of numbers that include the natural numbers, their opposites and 0 is called the set of integers.
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These are known as integers.
The answer depends on what is meant by "their opposites". If you mean additive opposites then the set is of all non-zero integers.
Zero is a member of the set of whole numbers. Some people include it in the set of natural numbers, some people don't.
The set of rational numbers includes the set of natural numbers but they are not the same. All natural numbers are rational, not all rational numbers are natural.
Another name for a set of natural numbers is counting numbers.