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Gradual Complex Numbers 16th World Congress of the International Fuzzy Systems Association (IFSA) 9th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT) Gradual Complex Numbers Emmanuelly Sousa 1 Regivan H N Santiago2 1,2Group for Logic, Language, Information, Theory and Applications - LoLITA∗ Abstract Finally, in order to achieve the results, this paper proposes a notion of distance for GCNs. This paper aims to introduce the concept of Grad- ual Complex Number (GCN) based on the ex- 2. Gradual Real Number(GCN) isting definition of gradual numbers and to show that the algebraic and polar forms, as well as some Gradual real numbers were introduced by J. Fortin, algebraic properties, of complex numbers can be di- Didier Dubois e H. Fargier in [4]. rectly extended from the crisp case. Furthermore, a kind of distance is established in order to deal with Definition 2.1 A gradual real number r˜ is de- such numbers. fined by a total assignment function Ar˜ : (0, 1] → R such that Ar˜(1) = r. Keywords: Fuzzy Numbers, Gradual Numbers, Gradual Complex Numbers. Consider R˜ as the set of all gradual real num- bers. A real number r can be seen as a gradual real 1. Introduction number r˜ such that, for all α ∈ (0, 1], ar˜(α) = r. The operations defined in R can be extended to the In 1965, Lofti Zadeh [4] introduced the concept of set R˜. It is easy to see that the operations of ad- fuzzy sets to quantify subjective linguistic terms dition and multiplication defined on R˜ inherits the such as: “approximately”, “around” etc. Fuzzy sets algebraic properties of real numbers. generalize classical sets, the classical characteristic functions are generalized to membership functions — i.e. the values of pertinence are generalized from 2.1. Fuzzy number represented by gradual {0, 1} to [0, 1]. numbers A fuzzy subset under certain conditions is called Observation 2.1 (Fortin et. al. [4]) An or- a fuzzy number. Fuzzy numbers were introduced dered pair of gradual real numbers can be seen as a in order to extend usual numbers. Many authors fuzzy real number. To describe a fuzzy number F purposed to equip such entities with arithmetics, as a pair of gradual real numbers (˜r1, r˜2), it is nec- which usually inherits the properties of the arith- essary that their assignment function Ar˜1 and Ar˜2 metic on intervals and does not provide inverses for be total, increasing and decreasing (respectively), addition and multiplication. Considering this prob- and for all 0 < α ≤ 1, Ar˜1 (α) ≤ Ar˜2 (α). lem, Didier Dubois and Henri Prade [3] introduced two new concepts: (1) Gradual Elements and (2) thus, the membership function of F is defined as: Real Gradual Numbers. 1 This article introduces Gradual Complex Numbers (GCN) endowed with a full compatible arithmetic with usual complex numbers. It shows Sup{α/A (α) ≤ x}, if x ∈ Im(A ) how is the algebraic and the polar form of such num- r˜1 r˜1 bers. The paper is structured as follows: Sections 1, if Ar˜1 (1) ≤ x ≤ Ar˜2 (1) µF (x) = 2 provide the required concepts to the development Sup{α/Ar˜ (α) ≥ x}, if x ∈ Im(Ar˜ ) 2 2 of this paper. Section 3 introduces the numbers 0, otherwise proposed here together with its algebraic and polar (2) forms. Section 4 provides the final remarks. It shows how gradual complexes can be charac- The next section proposes a definition for Grad- terized in terms of a fuzzy complex number as well ual Complex Numbers. as how Zhang complex fuzzy numbers [5] can be 1Note that when A and A are invertible functions, characterized in terms of gradual complex numbers. r˜1 r˜2 the fuzzy membership function, µF , is composed by such inverses: ∗This paper is to be presented in the Special Session Fuzzy Numbers and Applications. All authors are with the Department of Informatics and Applied Mathemat- ics, Federal University of Rio Grande do Norte (UFRN), −1 A (x), if x ∈ Im(Ar˜ ) Natal – RN, 59.072-970, Brazil. E-mail addresses: em- r˜1 1 1, if Ar˜1 (1) ≤ x ≤ Ar˜2 (1) [email protected], [email protected]. This re- µF (x) = −1 (1) A (x), if x ∈ Im(Ar˜ ) search was supported by the Brazilian Research Council r˜2 2 0, otherwise (CNPq) under the process 306876/2012-4. © 2015. The authors - Published by Atlantis Press 1493 3. Gradual Complex Numbers • Identity: The identity element for addition is a gradual complex number e¯, such that for Definition 3.1 Let C be the set of complex num- all α ∈ (0, 1], Az¯ +¯e(α) = Az¯ (α). Consider ˜ 1 1 bers , a˜ and b gradual real numbers with assignment e¯ defined by the assignment function: Ae¯(α) = functions A ,A : (0, 1] → , correspondingly. A def a˜ ˜b R (A (α),A (α)), A (α) = A (α)+A (α) = gradual complex number z¯ = (˜a, ˜b) is defined x˜ y˜ z¯1+¯e z¯1 e¯ Az¯1 (α). Therefore, Ae¯ = A0¯. by an assignment function Az¯ : (0, 1] → C, s.t. Az¯(α) = (Aa˜(α),A˜(α)). Aa˜ called real part of the b Therefore A0¯ is the identity element for the sum complex gradual z¯ and is denoted by Rez¯, A˜b called of gradual complex numbers. Similarly the remain- imaginary part, and is denoted by Imz¯. The set ing properties of a field can be easily verified. of all gradual complex numbers is denoted by Observe that in contrast to fuzzy numbers, the ¯ C. assignment of uncertainty to the complex numbers does not result the loss of algebraic properties. Definition 3.2 A set of gradual complex num- bers G = {z¯i : i ∈ I} is defined by its assign- Observation 3.1 Note that there is an algebraic ment function: AG : (0, 1] → P(C), s.t. AG(α) = injection R˜ in C¯, given by the function I : R˜ → C¯, {a (α): i ∈ I}. z¯i I(Ar˜) = (Ar˜,A0˜), where A0˜ = 0, ∀ α ∈ (0, 1]. Proposition 3.1 For all gradual complex number 3.2. Algebraic form of a Gradual Complex z¯, Az¯(1) ∈ C. Number Proof : The Gradual Imaginary Unit is the the gradual ¯ ˜ complex number ¯i characterized by the assignment Given z¯ ∈ C, since z¯ = (˜a, b), Aa˜(1) = a, A˜b(1) = b def function A¯i(α) = (A0˜,A1˜), where A0˜(α) = 0 and ∈ R. Then, Az¯(1) = (Aa˜(1),A˜b(1)) = (a, b) ∈ C. A1˜(α) = 1, for all α ∈ (0, 1]. QED Proposition 3.4 All gradual complex number z¯ = (˜a, ˜b), defined by the assignment function A (α) = Proposition 3.2 All complex number z = (a, b) ∈ z¯ (A (α),A (α)), can be written as A (α) = can be seen as a gradual complex number z¯ defined a˜ ˜b z¯ C A (α). This representation is called algebraic by the assignment function A (α) = (a, b) for all α a˜+˜b·¯i z¯ form of a gradual complex number, it can also be ∈ (0, 1]. seen as z¯ =a ˜ + ˜b · ¯i. 3.1. Arithmetic on C¯ Proof ¯ If Az¯(α) = (Aa˜(α),A˜b(α)), then Definition 3.3 (equality) Given z¯1 and z¯2 ∈ C, then z¯1 =z ¯2 ⇔ Az¯1 (α) = Az¯2 (α), ∀α ∈ (0, 1]. (Aa˜(α),A˜(α)) = (Aa˜(α),A˜(α)) + (A˜(α),A˜(α)) Definition 3.4 Given z¯ , z¯ z¯ ∈ ¯, α ∈ (0, 1] and b 0 0 b 1 2 3 C = (A (α),A (α)) + (A (α),A (α)) · A (α) assuming the complex arithmetic: “+, −, · and /”, a˜ 0˜ ˜b 0˜ ¯i ¯ the operations defined on the set C are given by:. = Aa˜(α) + A˜b(α) · A¯i(α) = Aa˜+˜b(α) · A¯i(α) def • addition: (Az¯1 + Az¯2 )(α) = Az¯1+¯z2 (α) = = Aa˜+˜b·¯i(α) Az¯1 (α) + Az¯2 (α) def Definition 3.5 Let z¯ =a ˜+˜b¯i be a gradual complex • subtraction:(Az¯1 − Az¯2 )(α) = Az¯1−z¯2 (α) = number defined by the assignment function Az¯(α) = Az¯1 (α) − Az¯2 (α) def (A +A ·A )(α). The conjugate of z¯ is the gradual • multiplication: (A ·A )(α) = A (α) = a˜ ˜b ¯i z¯1 z¯2 z¯1·z¯2 complex number z¯0 defined by the assignment func- Az¯1 (α) · Az¯2 (α) tion Az¯0 (α), where for all α ∈ (0, 1], • division: Consider Az¯2 (α) 6= (0, 0), ∀ α ∈ (0, 1]. A ¯0 (α) = Aa˜(α) − A˜(α) · A¯ (3) def z b i Az¯1 (α) (Az¯1 /Az¯2 )(α) = Az¯1/z¯2 (α) = A (α) z¯2 It is not difficult to conclude that the proposed gradual complex numbers retrieves all the usual Proposition 3.3 The structure hC¯, +, −, ×, ÷, 1, 0i is a field. complex algebra, e.g. division may be obtained us- ing the conjugate concept. Proof 3.3. Trigonometric form of a gradual • Commutativity: For all α ∈ (0, 1], complex number def ¯˜ Az¯1+¯z2 (α) = Az¯1 (α) + Az¯2 (α) = Az¯2 (α) + Consider z¯ =a ˜ + ib defined by the assignment func- Az¯1 (α) = Az¯2+¯z1 (α). tion Az¯(α) = Aa˜(α) + A¯i(α) · A˜b(α). Note that 1494 for each degree α, the assignment function Az¯ asso- Called trigonometric form ou polar form ciates one complex number (a, b) and hence a point of a gradual complex number. Pα = (a, b) in the Argand-Gauss plane. Thus, each α establishes a module ρα and an angle θα.
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