Journal Nonlinear Analysis and Application 2013 (2013) 1-8 Available online at www.ispacs.com/jnaa Volume 2013, Year 2013 Article ID jnaa-00111, 8 Pages doi:10.5899/2013/jnaa-00111 Research Article Arithmetic Operations on Trapezoidal Fuzzy Numbers J. Vahidi 1∗ , S.Rezvani 2 (1) Iran University of Sciences and Technology, Behshahr Branch, Iran (2) Department of Mathematics, Imam Khomaini Mritime University of Nowshahr, Nowshahr, Iran Copyright 2013 c ⃝ J. Vahidi and S.Rezvani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, several new algebraic mathematics for positive fuzzy numbers of type ( a, a, a, a) are devised and do not need the computation of α -cut of the fuzzy number. Direct mathematical expressions to evaluate exponential, square root, logarithms, inverse exponential etc. of positive fuzzy numbers of type ( a, a, a, a) are obtained using the basic analytical principles of algebraic mathematics and Taylor series expansion. At the end, Various numerical examples are also solved to demonstrate the use of contrived expressions. Keywords: Inverse exponential, logarithms, Trapezoidal fuzzy numbers. 1 Introduction Algebraic equations involving fuzzy numbers are an important application of fuzzy set theory. Fuzzy numbers can be used for expert system reasoning, fuzzy process modeling and control, and so forth. Previous studies on fuzzy numbers have shown that there are no opposite and reverse fuzzy numbers in the sense of group structure [1]-[3]. Fuzzy systems including fuzzy set theory and fuzzy logic have many successful applications. Sophisticated fuzzy set theoretic methods have been applied to various areas ranging from fuzzy topological spaces to quantum optics, medicine and so on [4, 5]. Singh [16] introduced a Common Fixed Point Theorems in Non-archimedean Fuzzy Metric Spaces. But Most of the recent research work on special behavior of fuzzy numbers like factorial, exponential, logarithmic and trigonometric is limited and much of the work is concentrated on linear fuzzy system of equations and their applications [6, 7]. A good and a detailed representation of fuzzy numbers in the domain of combinatory and their related arithmetic behavior is described by Kauffman and Gupta [8] but the method of explanation is based upon the α -cut approach and a clear view of its application on triangular fuzzy numbers is not adequately comprehendible. Rezvani [10]-[15] evaluated the system of Fuzzy Numbers. Moreover, Rezvani [13] proposed a new method for ranking in perimeters of two generalized trapezoidal fuzzy numbers. Also Some of the interesting arithmetic work on fuzzy numbers can be found in Abhinav Bansal in [9]. In this paper, several new algebraic mathematics for positive fuzzy numbers of type ( a, a, a, a) are devised and do not need the computation of α -cut of the fuzzy number. Direct mathematical expressions to evaluate exponential, square root, logarithms, inverse exponential etc. of positive fuzzy numbers of type ( a, a, a, a) are obtained using the basic analytical principles of algebraic mathematics and Taylor series expansion. At the end, Various numerical examples are also solved to demonstrate the use of contrived expressions. ∗ Corresponding author. Email address: jvahidi@iust.ac.ir