**4. Conclusions**

We regarded membership grades as the "degrees of confidence" in this chapter. In particular, Ref. [22] is well known for this same idea of literature.

Although the application aspect is important in fuzzy theory, this chapter focused on how to recognize fuzzy numbers in the first place rather than how to apply them.

This is because if the starting point of the discussion is iffy, so will the outcome. Fuzzy theory is not a theory to derive fuzzy results, but the ability to approach fuzziness mathematically (with some rigor) is the real appeal of fuzzy theory. That is, the beginning is crucial, and this mindfulness encourages the proper application of fuzzy numbers.

This chapter reviewed


These things are common to type-*n* fuzzy numbers for any *n*∈ℕ.

We have redefined the concept of fuzzy numbers by comparing them to round numbers. A round number is a number whose exact value is known and whose value is replaced by a tractable number. On the other hand, a fuzzy number is a concept that attempts to estimate its value when the exact value is (forever) unknown and determine its membership function. Even using the same word "about," they are different in concept itself, let alone approach. In summary, fuzzy numbers are a valid concept for quantities that definitely exist but whose values are difficult to obtain.

Furthermore, the discussion of fuzzy numbers can be reduced to that of interval analysis. Instead of dealing directly with fuzzy numbers, we can discuss them by dealing with the level-cut sets, which are (nonempty bounded closed) intervals. Hence, we want to treat fuzzy numbers whose level-cut sets are easily obtained. Compared to type-1 fuzzy numbers, type-2 fuzzy numbers are generally more complex and difficult to find for their level-cut sets, and in particular, type-2 fuzzy numbers we should be dealing with must be easy to compute. With this in mind, TST2FNs were introduced in Ref. [11]. The application of TST2FNs to type-2 fuzzy differential equation theory can be also seen in [11]. If we want to know about type-2 fuzzy differential equation theory, we can also see in, e.g., [9, 16, 23].

The computation of fuzzy numbers tends to be more tedious and complicated than that of crisp numbers. Therefore, it is not sufficient to make anything a fuzzy number if it is an ambiguous number. It is necessary to appropriately determine what should be regarded as a fuzzy number, even at the expense of calculation tediousness and complexity. One criterion for such a judgment was given in this chapter.
