# Width, Absolute Value and Midpoint

The width of an interval x is defined and denoted by *w (x) = x - x.*

The absolute value of x, denoted | x |, is the maximum of the absolute values of its endpoints

The midpoint of x is given by *m(x*) = — *(x + x*).

# Interval Arithmetic

If *[x, x*] and [*y*,*y*J are two intervals, then the interval arithmetic (Neumaier 1990; Moore et al. 2014) may be written as

The traditional interval arithmetic is sometimes difficult to use. When large numbers of computations are involved, then the process becomes difficult to handle and uncertainty rises. It may also be difficult to formulate the methods in general. Here, the traditional interval arithmetics have been redefined and proposed.

Let us consider two intervals [x, x] and [*y*,yJ, then the traditional interval arithmetic may be represented in an alternate form as follows.

If all the values of the interval are in R+ or R^{-}, then the arithmetic rules may be written as

where

One may observe that if we consider an interval that includes 0, then we will have at least one solution that is undefined in the division operation. For example, if we consider two intervals [-1, 2] and [-3, 1] and divide [-1, 2] by [-3, 1], then we have a solution where 0 divides 0, which is not defined in general. Hence, 0 has not been considered in the proposed arithmetic. However, this difficulty is a great challenge to the interval/fuzzy researchers.