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Is there any way to cite the "opposite" of a specific number, other than its inverse on the number line?
There are additive inverses (4 and –4) and multiplicative inverses (4 and 1/4). So in a sense, they are opposites since when you add the first pair you get 0, and when you multiply the second pair you get 1.
Why are these opposites though? Well, zero is a special number in addition, since anything plus zero is still the same. So that's why the multiplicative inverse of 4 is 1/4, since their product gives 1, and anything times 1 is still the same.
If you want, you can also define inverses for subtraction and division, so a number's opposite would be itself! (do you see why?)
For another way to define an inverse, we can look at a clock. Let's call "twelve o clock" the special number, because it's like "zero o clock". (If you add twelve hours to any number, you go around and end up at the same number.) If it were 7, then its inverse would be 5, because after 5 hours it would be 12. Another inverse for 7 would be 17, since it would still be on 12 after 17 hours. Also see that –7 is also an inverse of 7 in our clock numbers! (With clock numbers, how many inverses are there? What if we used clocks with 10 hours?)
So there are lots of ways to define number opposites. And opposites come up in a lot of different ways, which you'll find out if you learn about something called group theory!
To be able to answer this question you need to provide some context:
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what set of “numbers” (or other things) are you talking about; and
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what sort of operation are you doing to the things in this set, with respect to which a pair of them can be considered “opposites”?
By providing this context you will have defined a “group”. The example you provided—the set of numbers on the number line—are a group under the operation of addition. If the “numbers” on your line include fractions and not just integers, they are also a (different) group under the operation of multiplication: the “opposite” of each member of this group is its reciprocal.
There are lots of other sorts of groups, whose members are not necessarily “numbers” in the usual sense: the set of all permutations of a collection of n objects, for example (in this case the group operation is just doing the permutations in sequence). The opposite of a permutation would be the permutation that restores the collection to its prior state.