In standard arithmetic, if we, e.g., accidentally add a wrong number y to the preliminary result x, we can undo this operation by subtracting y from the result x + y. In this paper, we prove the following two results:

• First, a similar possibility to invert (undo) addition holds for fuzzy numbers (although in case of fuzzy numbers, we cannot simply undo addition by subtracting y from the sum).
• Second, if we add a single fuzzy set that is not a fuzzy number, we lose invertibility.

Thus, invertibility requirement leads to a new characterization of the class of all fuzzy numbers.