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On the Dual Hyperbolic Numbers and the Complex Hyperbolic Numbers

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On the Dual Hyperbolic Numbers and the Complex Hyperbolic Numbers Journal of Computer Science & Computational Mathematics, Volume 8, Issue 1, March 2018 DOI: 10.20967/jcscm.2018.01.001 On the Dual Hyperbolic Numbers and the Complex Hyperbolic Numbers Mutlu Akar*, Salim Yüce, Serdal Şahin Yildiz Technical University, College of Arts and Sciences, Department of Mathematics, Davutpasa Campus, 34210 Esenler, Istanbul, Turkey *Corresponding author email: [email protected] Abstract: In this study, we will introduce arithmetical operations on numbers and hyperbolic dual numbers, complex dual dual hyperbolic numbers w x y ju jv and hyperbolic numbers and dual complex numbers. complex numbers w x iy ju ijv which forms a commutative In this paper, we are going to focus on dual hyperbolic ring. Then, we will investigate dual hyperbolic number valued numbers, hyperbolic complex numbers and their functions. functions and hyperbolic complex number valued functions. One can see that these functions have similar properties. Keywords: hypercomplex numbers, dual hyperbolic numbers, 2. Dual Hyperbolic Numbers hyperbolic complex numbers, dual numbers, hyperbolic numbers, A dual hyperbolic number is defined as below: complex numbers. w d12, d x y j u v and the set of these numbers denoted by 1. Introduction 2 w d1 jd 2| d 1 , d 2 , j 1 . It is well known that the hypercomplex numbers systems [1], A hyperbolic dual number is defined as below: are extensions of real numbers. Complex numbers w h12, h x ju y jv z x iy| x , y , i2 1 , and the set of these numbers denoted by 2 hyperbolic (double, split-complex) numbers w h1 h 2| h 1 , h 2 , 0 . h x jy| x , y , j2 1, j 1 A dual hyperbolic number or hyperbolic dual number can [2, 3] and dual numbers be written in terms of its base elements 1, ,jj , as d x y| x , y , 2 0, 0 w x y ju jv where x is called the real part, y the dual part, u the are commutative instances of hypercomplex numbers systems hyperbolic part, and v the dual hyperbolic part. Both dual which are defined in two-dimensions [4]. Also, the quaternion which was introduced by Hamilton [5] is an hyperbolic numbers and hyperbolic dual numbers are instance of hypercomplex numbers systems and defined in identical. Clearly, w d jdfor every w four dimensions, but non-commutative [4]. Hyperbolic 12 numbers with complex coefficients are introduced by J. x y j u v Cockle in 1848, [6] and called 4-dimensional tessarines, [7]. x y ju jv Furthermore, Rochon and Shapiro presented, in a unified x ju y jv manner, a variety of algebraic properties of both bicomplex hh , for everyw . numbers and hyperbolic numbers, [8]. H. H. Cheng 12 introduced dual numbers with complex coefficients and For consistency, we will call w a dual hyperbolic number called complex dual numbers which are also commutative, in the remaining part of this paper. The base elements [9]. Similarly, we introduced hyperbolic numbers with dual 1, ,jj , of dual hyperbolic numbers satisfy the following coefficients and called dual hyperbolic numbers. We can properties: show that this number system is commutative. Also, in 1. ,1.j j ,. 0,. j j 1 analogy to complex dual numbers, we called complex .j j . , . j 0, j . j , hyperbolic numbers which are the hyperbolic numbers with complex coefficients (4-dimensional tessarines). It is clear where designates the pure dual unit, j designates the that there is no differences between complex hyperbolic hyperbolic unit, and j designates the dual hyperbolic unit. numbers and hyperbolic complex numbers, dual hyperbolic If x 0 , u 0 , and yv0 , xu is called a proper 2 On the Dual Hyperbolic Numbers and the Complex Hyperbolic Numbers dual number. If all values of x,, y u and v are not zero, * ** 1. w1 w 2 w 1 w 2 x y ju jv is called a proper dual hyperbolic number. * 2. ww* 1 1 3. w.. w* w** w 2.1 Algebraic Properties and Operations 1 2 1 2 * 4. w// w w** w , for w 2 0 . The arithmetic operations of dual hyperbolic numbers are 1 2 1 2 2 * defined for every w1 d 11 jd 12, w 2 d 21 jd 22 as following table: 2.3.3 The third kind of conjugation We define the third kind of conjugation of complex Table 1. Dual Hyperbolic Operations hyperbolic number w d12 jd as * * ww12 d11 d 21 j d 12 d 22 w d1 jd 2 d 1 jd 2 d 1 jd 2 . ww 12 d11 d 21 j d 12 d 22 Properties. For ww12, , ww12* d11 d 21 d 12 d 22 j d 11 d 22 d 12 d 21 1. w1 w 2 w 1 w 2 ww12 22 2. ww w1* conj w 2 w 2 , w 2 0 11 conj w2 is anyone of defined 3. w1.. w 2 w 1 w 2 2 conjugations w2 in next subsection 4. , for w 0 . w1// w 2 w 1 w 2 2 The dual hyperbolic numbers form a commutative ring, a 2.3.4 The Modulus real vector space and an algebra. But it is not field. Because We define the different moduli with respect to three kinds of every element of doesn't have an inverse. Also, one can conjugation as follows: be confused with quaternions. Even though the dual w2 w. w hyperbolic numbers are commutative, quaternions are non- d jd d jd (1) commutative. 1 2 1 2 d22 d 2, jRe d d 2.2 Basis and Dimension 1 2 1 2 1, ,jj , is a basis of . Because 1, ,jj , is linear w2 w., w* d jd d jd d 2 d 2 (2) * 1 2 1 2 1 2 independent and Sp 1, ,jj , . Hence, dim 4 . 2 w w. w d1 jd 2 d 1 jd 2 (3) 2.3 The Conjugations and Modulus in Dual Hyperbolic d22 d 2, jDu d d Numbers 1 2 1 2 We shall define three kinds of conjugations in dual where Re d12 d is real part of dual number dd12 and hyperbolic numbers. Du d12 d is dual part of dual number dd12. 2.3.1 The first kind of conjugation We define the first kind of conjugation of dual hyperbolic It is clear that, if we define a linear transformation number w d jd as w d jd , where d , d 12 12 1 2 f : and obtain the matrix A , which is the are dual conjugations of d , d . 1 2 corresponding matrix to linear transformation f according Properties. For all w , w : 1 2 to basis {1,jj , , } as, 1. w1 w 2 w 1 w 2 xu00 2. ww u x0 0 x ju 0 j 0 11 A , y v x u y jv x ju 3. w.. w w w 1 2 1 2 v y u x 4. w/ w w / w ,for w 2 0 . 1 2 1 2 2 where w d12 jd and d12 x y, d u v . Clearly we can rewrite Eq. (1) with aid of matrix A by the 2.3.2 The second kind of conjugation formula We define the second kind of conjugation of dual hyperbolic 2 w w. w detA . (4) number w d jd as w* d jd . 12 12 In a similar way, we can obtain the corresponding matrix to ww, Properties. For 12 , linear transformation f according to basis {1, ,jj , } , Mutlu Akar et al. 3 xu00 1 , j asin d1 d 2 asin d 1 d 2 y x v u x y u v 2 B , 1 u00 x u v x y acos w acos d d acos d d 1 2 1 2 v u y x 2 Eq. (2) can be obtained again in terms of matrix B as, 1 j acos d1 d 2 acos d 1 d 2 , 2 2 w w. w* detB . (5) * 1 atan w atan d1 d 2 atan d 1 d 2 2.4 Algebraic Properties and Operations 2 1 f : j atan d d atan d d , (6) 1 2 1 2 d jd f d jd 2 1 2 1 2 1 is called dual hyperbolic numbers valued function. In analogy sinh w sinh d1 d 2 sinh d 1 d 2 2 to Sobczyk, [2] this function can be written for dual 1 hyperbolic numbers as j sinh d1 d 2 sinh d 1 d 2 , 2 w w e w e (7) 12 1 f w f w e f w e (8) cosh w cosh d1 d 2 cosh d 1 d 2 12 2 1 j 1 j 1 where w d12 d , w d12 d , e1 , e2 . j cosh d1 d 2 cosh d 1 d 2 , 2 2 2 Thus, with aid of Eq. (8) we can get useful results for dual 1 tanh w tanh d1 d 2 tanh d 1 d 2 hyperbolic numbers valued functions in a similar way to 2 Cheng and Thompson [9]. Let w d12 jd in this case 1 j tanh d1 d 2 tanh d 1 d 2 , we can write the following properties, 2 21 2 2 1 w d d d d 1 2 1 2 asinh w asinh d1 d 2 asinh d 1 d 2 2 2 1 22 j d d d d , 1 1 2 1 2 j asinh d1 d 2 asinh d 1 d 2 , 2 2 1 sqrt w d d d d 1 1 2 1 2 acosh w acosh d1 d 2 acosh d 1 d 2 2 2 1 j d d d d , 1 1 2 1 2 j acosh d1 d 2 acosh d 1 d 2 , 2 2 d1 d 2 0, d 1 d 2 0 , 1 atanh w atanh d1 d 2 atanh d 1 d 2 1 2 exp w ed1 e d 2 e d 2 j e d 2 e d 2 , 2 1 j atanh d1 d 2 atanh d 1 d 2 .
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