The paper presents addition of fuzzy numbers realised with the application of the multidimensional RDM arithmetic and horizontal membership functions (MFs). Fuzzy arithmetic (FA) is a very difficult task because operations should be performed here on multidimensional information granules. Instead, a lot of FA methods use α-cuts in connection with 1-dimensional classical interval arithmetic that operates not on multidimensional granules but on 1-dimensional intervals. Such approach causes difficulties in calculations and is a reason for arithmetical paradoxes. The multidimensional approach allows for removing drawbacks and weaknesses of FA. It is possible thanks to the application of horizontal membership functions which considerably facilitate calculations because now uncertain values can be inserted directly into equations without using the extension principle. The paper shows how the addition operation can be realised on independent fuzzy numbers and on partly or fully dependent fuzzy numbers with taking into account the order relation and how to solve equations, which can be a difficult task for 1-dimensional FAs.

The paper presents difficulties connected with fuzzy and interval division. If operations such as fuzzy addition, subtraction and multiplication provide as a result one compact, multidimensional granule, then a result of the fuzzy division can consists of few separated granules. Such results are more difficult to use in next calculations. The paper shows that the number of solution granules can be higher than 2 and that in certain problems division does not occur explicitly. In certain problems, separation of particular solution granules can be considerable. The paper also shows how to realize the fuzzy division when its denominator contains zero. Most types of fuzzy arithmetics forbid such operation. However, the paper shows that it is possible. Multidimensional fuzzy RDM arithmetic and horizontal membership functions which facilitate detecting of solution granules are also described. The considered problems are visualized by examples.

Computing with words is a way to artificial, human-like thinking. The paper shows some new possibilities of solving difficult problems of computing with words which are offered by relative-distance-measure RDM models of fuzzy membership functions. Such models are based on RDM interval arithmetic. The way of calculation with words was shown using a specific problem of flight delay formulated by Lotfi Zadeh. The problem seems easy at first sight, but according to the authors' knowledge it has not been solved yet. Results produced with the achieved solution were tested. The investigations also showed that computing with words sometimes offers possibilities of achieving better problem solutions than with the human mind.

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In the last few years, the number of applications of the multidimensional fuzzy arithmetic (MFA) and the multidimensional interval arithmetic is expanding. Authors of new papers about applications of MFA are often faced with comments from other researchers, especially the gradual arithmetic (GA) proponents, that the horizontal membership function (HMF) used in MFA is the same as the inverse membership function (InvMF) used in GA, and that MFA itself adds nothing new to the fuzzy arithmetic. This view leads to unfair evaluations of scientific papers about MFA applications submitted to scientific journals and to unnecessary disagreements between MFA and GA proponents. The purpose of this paper is to carefully analyze the two types of functions (HMF and InvMF) and to demonstrate their important differences. The basic and decisive difference is the dimensionality of both functions, which is illustrated by examples. It should also be added that HMF has proven its usefulness in solving difficult problems such as: systems of fuzzy equations or fuzzy differential equations, which is confirmed by numerous publications. The paper enable the reader to have a deeper understanding of the multidimensional fuzzy arithmetic.