Journal of Computer Science & Computational Mathematics, Volume 8, Issue 1, March 2018 DOI: 10.20967/jcscm.2018.01.001

On the Dual Hyperbolic 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 . Then, we will investigate dual hyperbolic valued numbers, hyperbolic complex numbers and their functions. functions and hyperbolic 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 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 , 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 yv0 , 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 , a 2.3.4 The Modulus real vector space and an . 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 . 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 dd120 , dd120 ,  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 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  Sp1,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 zz120 , zz120 ,  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 sinz1  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 11jj    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

1 [5] W. Hamilton, Elements of Quaternions, Chelsea  tanh w  tanh z1  z 2  tanh z 1  z 2  2 Publishing Company, New York, 1969. 1 [6] J. Cockle, “On a New Imaginary in Algebra”, j tanh z1  z 2  tanh z 1  z 2  , 2 Philosophical magazine, London-Dublin-Edinburgh, vol. 3, no. 34, pp. 37-47, 1849. 1  asinh w  asinh z1  z 2  asinh z 1  z 2  [7] D. Alfsmann, “On Families of 2N-dimensional 2 Hypercomplex Suitable for Digital Signal 1 j asinh z1  z 2  asinh z 1  z 2  , Processing”, in Proc. European Signal Processing Conf. 2 (EUSIPCO 2006), Florence, Italy, 2006. 1 [8] D. Rochon, M. Shapiro, “On Algebraic Properties of  acosh w  acosh z1  z 2  acosh z 1  z 2  2 Bicomplex and Hyperbolic Numbers”, Analele 1 Universitatii din Oradea. Fascicola Matematica, vol. j acosh z1  z 2  acosh z 1  z 2  , 2 11, 71-110, 2004. 1 [9] H. H. Cheng, S. Thompson, “Dual Polynomials and  atanh w  atanh z1  z 2  atanh z 1  z 2  2 Complex Dual Numbers for Analysis of Spatial 1 Mechanisms”, Proc. of ASME 24th Biennial j atanh z1  z 2  atanh z 1  z 2  . 2 Mechanisms Conference, Irvine, CA, August pp. 19-22, References 1996. [1] I. Kantor, A. Solodovnikov, Hypercomplex Numbers, [10] S. Yüce, N. Kuruoğlu, “One-Parameter Plane Springer-Verlag, New York, 1989. Hyperbolic Motions”, Advances in Applied Clifford [2] G. Sobczyk, “The Hyperbolic Number Plane”, The Algebras, vol. 18, no. 2, pp. 279-285, 2008. College Mathematics Journal, vol. 26, no. 4, pp. 268- [11] I.M. Yaglom, Complex Numbers in Geometry, 280, 1995. Academic Press, New York, 1968. [3] P. Fjelstad, “Extending Special Relativity via the [12] I. M. Yaglom, A Simple Non-Euclidean Geometry and Perplex Numbers”, American Journal of Physics, vol. Its Physical Basis, Springer-Verlag, New York, 1979. 54, no. 5, pp. 416-422, 1986. [4] P. Fjelstad, G. Gal Sorin, “n-dimensional Hyperbolic Complex Numbers”, Advances in Applied Clifford Algebras, vol. 8, no. 1, pp. 47-68, 1998.