A Polar Representation for Complex Interval Numbers Aar˜ ao Lyra Information Systems and Engineering of Computation Courses Potiguar University, UnP Natal, RN, Brazil aarao@unp.br Adri˜ ao Duarte D´ oria Neto Department of Computation and Automation Engineering Federal University of Rio Grande do Norte Natal, RN, Brazil adriao@dca.ufrn.br Benjam´ ın Ren´ e Callejas Bedregal Department of Informatic and Applied Mathematics Federal University of Rio Grande do Norte Natal, RN, Brazil. bedregal@dimap.ufrn.br Roque Mendes Prado Trindade Department of Thecnologics an Exacts Sciences State University of Southwest Bahia - (UESB) Vit. Conquista, Ba, Brazil. roquetrindade@uesb.edu.br . Abstract—The present work defines the basic elements for the introduction to the Study of Complex variables under the mathematical interval context with the goal of using it as a foundation for the understanding of pure mathematical problems, associating the mathematical interval to support the results. The present article contributes to the complex interval theory taking into consideration Euler’s Identity and redefining the polar representation of interval complex numbers. In engineer- ing, the present article could be used as a subsidy for many applications where complex variable theory is applicable and requires accurate results. Index Terms—Complex Interval, Complex Interval Numbers, Complex Interval Variable I. I NTRODUCTION In pure and applied mathematics there are problems in- volving continuous sets as real and complex numbers. This knowledge can be employed in engineering sciences, such as numerical analysis, dynamic systems, fazorial analysis [7], computational geometry [21], signals processing and digital images [6], [12] as well as the theory of optimization among others. However, in most cases there is no numerical error analysis, thus taking for granted the uncertainty and insecurity of the results. Thus there is a necessity for an approach to solve continuous problems using mathematical intervals. Under the topic of real numbers, several approaches on intervals are well known [13], [14], [15] [1], [17] and [10] however, there is little work done on or few references are available on complex numbers and this field needs to be enriched. In this work, there are definitions of some basic elements on complex interval theory, which is a mathematical theory on interval of complex numbers that serve as a foundation for problems on mathematical applications but at the same time associating it with mathematical interval to offer reliability to the results that are being introduced. The authors believe that the present work contributes to the use of complex data number type that needs precision and accuracy on complex numerical data treatment as well as an addition to the mathematical literature. Definitions are suggested in this article, under the understanding of complex numbers and introducing Euler’s identity solutions for the interval case and defining a polar representation. For this, it was necessary to evaluate the current information available on the main definitions of interval complex number and their influence on some of the jobs selected in this area [4] and [16]. This work also attenps to fill in a gap in mathematical modelling in the research area of robotics, sensors fusion, autonomous tracking, innacurate robot lacalization, such as in the work by Kieffer, Jaulin and Walter [9]. Jaulin uses interval analysis for planning free-collision paths for mobile robots [8]. Interval analysis was also used for vehicle localization to solve problems of telemetric imprecision data by Leveque in [11] and it is used as fundamentation of interval digital signal processing in [18]. II. THE COMPLEX NUMBERS In the XV Century mathematicians like Cardano and Bombelli, among others, carried out studies on negative num- bers. Two centuries later, Wesses, Argand and Gauss continued this work and they are being considered as the creators of the theory of complex numbers, which became an important tool in engineering. A. Basic Definitions Definition 1 (Complex numbers): Let a, b ∈ R and z = a + bi, where i = √ −1, then, z is called a complex number. The set of all complex numbers is called complex plan and it is denoted by C.