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 . 2 1 22 dd12 log w log d12 d jlog , 2 dd 12 3. Complex Hyperbolic Numbers
22 dd12 dd120 , dd120 , 0 , The complex hyperbolic number w is defined with aid of dd 12 hyperbolic numbers 1 2 sin w sin d1 d 2 sin d 1 d 2 h h1 jh 2∣ h 1, h 2 , j 1, j 1 , 2 1 which also called split-complex or double numbers [10-12], j sin d1 d 2 sin d 1 d 2 , 2 with complex coefficients as 2 1 w z1 jz 2, j 1, j 1, z 1 , z 2 cos w cos d1 d 2 cos d 1 d 2 2 w x iy ju ijv x,,,, y u v 1 where j cos d1 d 2 cos d 1 d 2 , 2 2 i22 1, j 1, ij 1, j 1 1 tan w tan d1 d 2 tan d 1 d 2 ij ji,, i ij ij i j j ij ij j i . 2 Hence, the set of complex hyperbolic numbers can be 1 j tan d1 d 2 tan d 1 d 2 , obtained as 2 w z jz∣ z, z , j2 1, j 1 . 1 1 2 1 2 asin w asin d1 d 2 asin d 1 d 2 2 4 On the Dual Hyperbolic Numbers and the Complex Hyperbolic Numbers
Similarly, we can define the hyperbolic complex numbers 3.3 The Conjugations and Modulus in Complex set via complex numbers with hyperbolic coefficients as Hyperbolic Numbers 2 d h1 ih 2 ∣ h 1, h 2 , i 1 The conjugation is very important for algebraic properties, where, is the set of the hyperbolic numbers. Clearly, since geometric properties of and for complex hyperbolic these number systems and are commutative, we valued functions. We defined three kind of conjugations on . Furthermore, if we multiplied a complex number with can see there is no differences for and . its conjugate we obtain the square of the modulus (Euclidean w z jzfor every w 12 metric in 2 ) for the complex number. In analogy to x iy j u iv complex numbers we can define the modulus in . Clearly, x iy ju ijv because of three different conjugate we can define three x ju i y jv different modulus.
h12 ih,for every w . 3.3.1 The first kind of conjugation 3.1 Algebraic Properties and Operations We define the first kind of conjugation of complex
The complex hyperbolic number system is an Abel group hyperbolic number w z12 jz as w z12 jz , where under addition and the multiplication has left and right zz, are complex conjugations of zz, . distribution with respect to addition, 12 12 Properties. For all ww12, : w1 w 2 w 3 w 1 w 2 w 1 w 3 , w2 w 3 w 1 w 2 w 1 w 3 w 1 ,
1. w1 w 2 w 1 w 2 w1,, w 2 w 3 . Also, complex hyperbolic numbers system is commutative 2. ww11 and associative under multiplication,
3. w1.. w 2 w 1 w 2 w1 w 2 w 2 w 1 , w1 w 2 w 3 w 1 w 2 w 3 , w1,, w 2 w 3 . 2 Thus, we showed that this numbers system is a 4. w1/ w 2 w 1 / w 2 , for w 2 0 commutative ring. We can define the arithmetic operations as following table, 3.3.2 The second kind of conjugation We define the second kind of conjugation of complex * Table 2. Complex Hyperbolic Operations hyperbolic number w z12 jz as w z12 jz .
ww12 z11 z 21 j z 12 z 22 Properties. For ww12, ,
* ** ww12 z z j z z 11 21 12 22 1. w1 w 2 w 1 w 2 ww* * 12 z11 z 21 z 12 z 22 j z 11 z 22 z 12 z 21 * 2. ww11 ww12 22 w1* conj w 2 w 2 , w 2 0 * ** 3. w1.. w 2 w 1 w 2 conj w is anyone of defined 2 4. w/ w* w** / w , for w 2 0 conjugations w in next 1 2 1 2 2 * 2 subsection 3.3.3 The third kind of conjugation We define the third kind of conjugation of complex where w1 z 11 jz 12, w 2 z 21 jz 22 . The complex hyperbolic number w z12 jz as w z12 jz . hyperbolic numbers form a commutative ring, a real vector Properties. For ww, , space and an algebra. But it is not field. Because every 12 element of doesn't have an inverse. Also, one can be 1. w1 w 2 w 1 w 2 confused with quaternions. Even though the complex 2. ww11 hyperbolic numbers are commutative, quaternions are non- commutative. 3. w1.. w 2 w 1 w 2 4. w/ w w / w, for w 2 0 1 2 1 2 2 3.2 Basis and Dimension 1,i , j , ij is a basis of . Because 1,i , j , ij is linear 3.3.4 The Modulus independent and Sp1,i ,j , ij. Hence, dim 4 . In analogy to complex numbers, for a complex hyperbolic
number w z12 jz where, zz12, we can define three kind of modulus as follows: Mutlu Akar et al. 5
w2 w. w complex hyperbolic valued function as dual hyperbolic valued function (9) z1 jz 2 z 1 jz 2 z22 z 2, jRe z z 21 2 2 1 2 1 2 w z z z z 2 1 2 1 2 2 * 2 2 w w. w z1 jz 2 z 1 jz 2 z 1 z 2 , (10) 1 22 * j z z z z , 2 1 2 1 2 w w. w 2 1 z jz z jz (11) sqrt w z1 z 2 z 1 z 2 1 2 1 2 2 z22 z 2, ijIm z z 1 1 2 1 2 j z z z z , 2 1 2 1 2 where Re z12 z is real part of complex number zz12 and z1 z 2 0, z 1 z 2 0 , Im z z is imaginer part of complex number zz . 1 12 12 z1 z 2 z 2z 2z 2 exp w e e e j e e , On the other hand, if we define a linear transformation 2 f : and obtain the matrix A , which is the 1 22 z12 z log w log z12 z jlog , corresponding matrix to linear transformation f according to 2 z12 z basis {1,j , i , ij } as, 22 zz12 zz120 , zz120 , 0 , x u y v zz12 u x v y x ju () y jv 1 sin w sin z z sin z z A , 1 2 1 2 y v x u y jv x ju 2 v y u x 1 j sinz1 z 2 sin z 1 z 2 , where w z jz and z x iy, z u iv . 2 12 12 1 Clearly we can rewrite Eq. (9) with aid of matrix A by the cos w cos z1 z 2 cos z 1 z 2 2 formula 2 1 w w. w detA . (12) j cos z1 z 2 cos z 1 z 2 , 2 In a similar way, we can obtain the corresponding matrix to 1 tan w tan z1 z 2 tan z 1 z 2 linear transformation f according to basis {1,i , j , ij } , 2 x y u v 1 j tan z1 z 2 tan z 1 z 2 , y x v u x iy u iv 2 B , u v x y u iv x iy 1 asin w asin z1 z 2 asin z 1 z 2 v u y x 2 Eq. (10) can be obtained again in terms of matrix B as, 1 j asin z1 z 2 asin z 1 z 2 , 2 * 2 w w. w detB (13) * 1 acos w acos z1 z 2 acos z 1 z 2 2 3.4 Complex hyperbolic valued function and its 1 j acos z1 z 2 acos z 1 z 2 , Properties 2 We can define the complex hyperbolic valued function as a 1 atan w atan z1 z 2 atan z 1 z 2 function whose values are complex hyperbolic numbers, 2 f : 1 (14) j atan z1 z 2 atan z 1 z 2 , z1 jz 2 f z 1 jz 2 2 Similarly, in analogy to Sobczyk, [2] this function can be 1 sinh w sinh z1 z 2 sinh z 1 z 2 written for dual hyperbolic numbers as 2
w w e12 w e (15) 1 j sinh z1 z 2 sinh z 1 z 2 , 2 f w f w e12 f w e , (16) 1 11jj cosh w cosh z1 z 2 cosh z 1 z 2 where w z z,,, w z z e e . Let 2 1 2 1 2 122 2 1 w in this case we can write the following properties for j cosh z1 z 2 cosh z 1 z 2 , 2 6 On the Dual Hyperbolic Numbers and the Complex Hyperbolic Numbers
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