New Formula for Interval Numbers with Powers of Positive Rational Numbers and Its Properties
Rasi Adishamita, Mashadi and Muliana

Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Riau, Indonesia.

Abstract

Development of interval numbers have shown various beliefs regarding the arithmetics and formulas that apply to interval numbers. One of them is regarding interval numbers with powers, it has previously been shown that general formula of an interval number with power \(k\) where \(k\) is a positive integer is that the lower element and upper element of the interval number are both raised to the power \(k\) and this formula can be written as \(\widetilde{a}^k=[\underline{a}^k,\overline{a}^k]\). However, there is a weakness regarding further use for this formula, that is the basic characteristic of exponents do not apply. For an example is that the the powers of the product of the same interval numbers cannot be added or it can also be written as \(\widetilde{a}^k \otimes \widetilde{a}\neq \widetilde{a}^{k+1}\). There are some formulas given for multiplication operation for interval numbers, but most of them faced similar problem regarding of their formulas such as the result of multiplication of an interval number with its invers is not equal to identity of interval number \(\widetilde{I}=[1,1]\). Therefore, this article will establish a new formula for interval numbers with powers of positive integers using the new formula of multiplication operation for interval numbers and show the basic characteristic such as \(\widetilde{a}^k \otimes \widetilde{a} = \widetilde{a}^{k+1}\) apply. Based on this new formula of interval numbers with positif integer powers, a new formula for interval numbers with fractional powers will also be construct along with the properties that apply to this formula.

Keywords: Interval arithmetic, Powers of interval numbers

Topic: Mathematics and Its Applications