GENERALIZED PESCAR COMPLEX NUMBERS AND OPERATIONS WITH THESE NUMBERS. Virgil Pescar

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1 Acta Universitatis Apulensis ISSN: No. 33/2013 pp GENERALIZED PESCAR COMPLEX NUMBERS AND OPERATIONS WITH THESE NUMBERS Virgil Pescar Abstract. In this paper, the author introduced the generalized Pescar complex numbers. For these new numbers, are presented operations with generalized complex numbers, the module of a generalized complex numbers, the properties of generalized complex numbers (C Gx \{0}, ), the trigonometric form of these numbers as well. On introducing the generalized complex number, the author defines the general complex numbers. The presented findings are the results of the author s original research Mathematics Subject Classification: 30C45. Key words and phrases: Complex numbers. 1. Definition of generalized Pescar complex numbers Let V 3 be the free vector space and E 3 the Euclidean real punctual space. We denote by dim V 3 the vector space dimension V 3 and by dim E 3, the dimension of the Euclidean real punctual space E 3.We have dim V 3 = dim E 3. We assume that R 3 is the three-dimensional vector space over the real numbers and C the set of complex numbers, C = {z = x + iy x, y R, i 2 = 1.} Since V 3 is a three-dimensional vector space, we define an orthonormal Cartesian frame R ON 1 3 = {O, e 1, e 2, e 3 }, where O is the origin of the Euclidean real punctual space E 3 and B ON 1 = { e 1, e 2, e 3 } is an orthonormal basis of the vector space V 3. Any free vector of V 3 can be specified through a class representative free vector with its origin in O, determining a position vector v with its origin in O, its extremity in any point M, denoted OM and having a unique decomposition with respect to basis B ON 1. v = OM = x e1 + y e 2 + t e 3, (x, y, t) R 3. (1) 23

2 The size of the vector OM is given by If point M is on a sphere of radius r, then OM = x 2 + y 2 + t 2. (2) OM = r, (r > 0, r = constant) (3) and the sphere centered at O of radius r is described by the equation (Σ r ) : x 2 + y 2 + t 2 r 2 = 0. (4) If M is any point, then this point is found on the spheres of variable radius λ, λ > 0. We have OM = λ and the equation of the spheres of variable radius λ, centered at O is (Σ λ ) : x 2 + y 2 + t 2 λ 2 = 0. (5) Let Oxyt be the orthogonal axes system corresponding to the orthonormal Cartesian frame R ON 1 3. We assume axis Ox, the real axis with the real unity 1, and axes Oy, Ot the imaginary axes with the imaginary units i, respectively j, i 2 = j 2 = 1. We denote this orthogonal axes system (Oxyt) x, thus specifying the real axis of the axes system. Definition 1.1. The product of complex numbers x, a, b, ai, bj with x, a, b R, denoted by is defined by x x = (1 x) (1 x) = x 2, (6) x a = (1 x) (1 a) = xa, x b = (1 x) (1 b) = xb. ai ai = ai ai cos 0 0 = a ai 2 = a 2, bj bj = bj bj cos 0 0 = b bj 2 = b 2, ai bj = ai bj cos 90 0 = abij cos 90 0 = 0, x ai = (1 x) (ai) = xai, x bj = (1 x) (bj) = xbj. From (6), for a = b = 1, we obtain i i = i i cos 0 0 = i 2 = 1, 24

3 j j = j j cos 0 0 = j 2 = 1, i j = i j cos 90 0 = 0. The correspondent with respect to the axes system (Oxyt) x of the position vector OM referred to the frame R ON 1 3 = {O, e 1, e 2, e 3 } is a generalized complex number with the image in point M (Σ λ ), which has the Cartesian coordinates (x, y, t) R 3 with respect to the Cartesian axes system. Definition 1.2. We designate as generalized Pescar complex number, with the real part x and the imaginary parts y, t, the number of the form w = 1 x + iy + jt, (x, y, t) R 3, (7) referred to the axes system (Oxyt) x. The projections of the generalized complex number w defined by (7), on the coordinate axes Ox, Oy, Ot are denoted by by x = Rew, y = Im 1 w, t = Im 2 w, (8) The generalized complex number w, with the real part x can be denoted w = Rew + iim 1 w + jim 2 w. (9) We denote the set of generalized Pescar complex numbers, with the real part x, by C Gx = { w w = x + iy + jt, (x, y, t) R 3, i 2 = j 2 = 1 }. (10) 25

4 Fig. 1.1 Consequences. p 1 ). The generalized complex number of the form w 0 = 0 + i 0 + j 0, (0, 0, 0) R 3, is the generalized null complex number corresponding to the set C Gx. p 2 ). The image of the null complex number w 0 with respect to the axes system (Oxyt) x is the complex point O. p 3 ). The images of the complex numbers w C Gx \ {w 0 } belong to the variable spheres centered in the complex point O. p 4 ). The sphere centered in O is called complex sphere centered in complex point O. 26

5 p 5 ). For t = 0, we obtain the complex number w = x + iy, with the image in the complex plane (Oxy), i 2 = 1, w C. p 6 ). For y = 0, we obtain the complex number w = x + jt, with the image in the complex plane (Oxt), j 2 = 1. p 7 ). For x = 0, we obtain the imaginary generalized complex number w = iy + jt, i 2 = j 2 = 1, with the image in the imaginary complex plane (Oyt). Consider the orthogonal Cartesian axes system Oytx, that corresponds to the orthonormal Cartesian frame R ON 2 3 = {O, e 2, e 3, e 1 } and let Oy be the real axis, with the real unity 1, and Ot, Ox, the imaginary axes, with the imaginary units j, k respectively so that j 2 = k 2 = 1. We denote by (Oytx) y this orthogonal axes system, thus specifying the real axis of the axes system. Definition 1.3. The product of complex numbers y, a, b, aj, bk with y, a, b R, denoted by is defined by y y = (1 y)(1 y) = y 2, (11) y a = (1 y)(1 a) = ya, y b = (1 y)(1 b) = yb, y aj = (1 y)(aj) = yaj, y bk = (1 y)(bk) = ybk, aj aj = aj aj cos 0 0 = a a j 2 = a 2, bk bk = bk bk cos 0 0 = b bk 2 = b 2, aj bk = aj bk cos 90 0 = abjk cos 90 0 = 0, aj bj = aj bj cos 0 0 = abj 2 = ab, ak bk = ak bkcos0 0 = abk 2 = ab. From (11), for a = b = 1, we obtain j j = j j cos 0 0 = j 2 = 1, k k = k k cos 0 0 = k 2 = 1, j k = j k cos 90 0 = 0. 27

6 The correspondent with respect to the axes system (Oytx) y of the position vector OM referred to the frame R ON 2 3 = {O, e 2, e 3, e 1 } is a generalized complex number with the image in point M (Σ λ ), which has the Cartesian coordinates (y, t, x) R 3 with respect to the Cartesian axes system. Definition 1.4. We designate as generalized Pescar complex number, with the real part y and the imaginary parts t, x, the number of the form w = 1 y + jt + kx, (y, t, x) R 3, (12) referred to the axes system (Oytx) y. The projections of the generalized complex number w defined by (12), on the coordinate axes Oy, Ot, Ox are denoted by y = Rew, t = Im 1 w, x = Im 2 w. (13) The generalized complex number with the real part y can be denoted by w = Rew + jim 1 w + kim 2 w. (14) We denote the set of generalized Pescar complex numbers, with the real part y, by C Gy = { w w = y + jt + kx, (y, t, x) R 3, i 2 = j 2 = 1 }. (15) Consequences. m 1 ). The generalized complex number of the form w 0 = 0+0j+0k, (0, 0, 0) R 3, is the generalized null complex number corresponding to the set C Gy. m 2 ). The image of the null complex number w 0 C Gy axes system (Oytx) y is the complex point O. with respect to the m 3 ). The images of the complex numbers w C Gy \ {w 0 } belong to the variable spheres, referred to (Oytx) y and centered in the complex point O. m 4 ). For x = 0, we obtain the complex number w = y + jt, j 2 = 1, with image in the complex plane (Oyt). 28

7 m 5 ). For t = 0, we obtain the complex number w = y + kx, k 2 = 1, with image in the complex plane (Oyx). m 6 ). For y = 0, we have the imaginary generalized complex number w = jt + kx, j 2 = k 2 = 1, with image in the imaginary complex plane (Otx). Fig. 1.2 Consider the orthogonal Cartesian axes system Otxy, that corresponds to the orthonormal frame R ON 3 3 = {O, e 3, e 1, e 2 }, and let Ot be the real axis, with the real unity 1, and Ox, Oy, the imaginary axes with the imaginary units k, i respectively so that k 2 = i 2 = 1. We denote by (Otxy) t this orthogonal axes system, thus specifying the real axis of the axes system. Definition 1.5. The product of complex numbers t, a, b, ak, bi with t, a, b R, 29

8 denoted by is defined by t a = (1 t) (1 a) = ta, (16) t b = (1 t) (1 b) = tb, t ak = (1 t) (ak) = tak, t bi = (1 t) (bi) = tbi, ak ak = ak ak cos 0 0 = aak 2 = a 2, bi bi = bi bi cos 0 0 = bbi 2 = b 2, ak bi = ak bi cos 90 0 = abki cos 90 0 = 0. For a = b = 1, from (16) we obtain k k = k k cos 0 0 = k 2 = 1, i i = i i cos 0 0 = i 2 = 1, k i = k i cos 90 0 = 0. The correspondent with respect to the axes system (Otxy) t of the position vector OM referred to the frame {O, e 3, e 1, e 2 } is a generalized complex number with the image in point M (Σ λ ), which has the Cartesian coordinates (t, x, y) R 3 with respect to the Cartesian axes system. Definition 1.6. We designate as generalized Pescar complex number, with the real part t and the imaginary parts x, y, the number of the form w = 1 t + kx + iy, (t, x, y) R 3, (17) it referred to the axes system (Otxy) t. The projections of the generalized complex number w defined by (17), on the coordinate axes Ot, Ox, Oy are denoted by t = Rew, x = Im 1 w, y = Im 2 w, (t, x, y) R 3, (18) The generalized complex number, with t as the real part can be written as w = Rew + kim 1 w + iim 2 w. (19) We denote the set of generalized Pescar complex numbers with the real part t, by C Gt = { w w = t + xk + yi, (t, x, y) R 3, k 2 = i 2 = 1 }. (20) 30

9 Fig Consequences. q 1 ). The generalized complex number of the form w 0 = 0+0k+0i, (0, 0, 0) R 3 is the generalized null complex number corresponding to the set C Gt. q 2 ). The image of the complex number w 0 C Gt system (Otxy) t is the complex point O. with respect to the axes q 3 ). The images of the complex numbers w C Gt \ {w 0 } belong to the variable spheres referred to (Otxy) t and centered in the complex point O. q 4 ). For x = 0, we obtain the complex number w = t + iy, with the imagine in the complex plane (Oty), i 2 = 1. 31

10 q 5 ). For y = 0, we obtain the complex number w = t + kx, with image in the complex plane (Otx), k 2 = 1. q 6 ). For t = 0, we have the imaginary generalized complex number w = kx +iy, k 2 = i 2 = 1, with the imagine in the imaginary complex plane (Oxy). Definition 1.7. The generalized complex number w 1, w 2 C Gx, w 1 = x 1 + iy 1 + jt 1, w 2 = x 2 + iy 2 + jt 2, x i, y i, t i R 3, i = 1, 2, are equal, w 1 = w 2, if x 1 = x 2 y 1 = y 2 (21) t 1 = t 2 The generalized complex numbers w 1, w 2 C Gx, w 1 = ai, w 2 = bj, a, b R, i = j = 1, are equal, w 1 = w 2, if a = b. (22) Analogous, we define the equality of two generalized complex numbers in C Gy, C Gt. The set of generalized complex numbers Pescar, with the notation C Gp is defined by C Gp = C Gx CGy CGt. (23) 2. Addition of generalized complex numbers Let be the set of generalized complex numbers C Gx. We define the intern addition operation, with the notation + such that + : C Gx C Gx C Gx, + by to rule (w 1, w 2 ) C Gx C Gx w 1 + w 2 C Gx, then w 1 = x 1 + iy 1 + jt 1, w 2 = x 2 + iy 2 + jt 2, w 1 + w 2 definition = x 1 + x 2 + (y 1 + y 2 ) i + (t 1 + t 2 ) j. (24) (C Gx, +) is commutative group, because are satisfied the conditions: 1) w 1 + w 2 = w 2 + w 1, w 1, w 2 C Gx (commutativity). 32

11 We have x 1 + x 2 + i (y 1 + y 2 ) + j(t 1 + t 2 ) = x 2 + x 1 + i (y 2 + y 1 ) + j (t 2 + t 1 ) and from (21) we obtain x 1 + x 2 = x 2 + x 1 y 1 + y 2 = y 2 + y 1 t 1 + t 2 = t 2 + t 1. (25) Since (R, +) is commutative group, the relations (25) are true. 2) w 1 + (w 2 + w 3 ) = (w 1 + w 2 ) + w 3, w 1, w 2, w 3 C Gx (associativity). We obtain x 1 + (x 2 + x 3 ) = (x 1 + x 2 ) + x 3 (26) y 1 + (y 2 + y 3 ) = (y 1 + y 2 ) + y 3 t 1 + (t 2 + t 3 ) = (t 1 + t 2 ) + t 3, true relations, because (R, +) is group. 3) e C Gx, e = e 1 + e 2 i + e 3 j, (e 1, e 2, e 3 ) R 3 ( e neutral element), such that w + e = e + w = w, w C Gx, w = x + iy + jt. Let w + e = w x + e 1 + i (y + e 2 ) + j (t + e 3 ) = x + iy + jt and using (21), we have obtaining x + e 1 = x y + e 2 = y t + e 3 = t. e 1 = 0, e 2 = 0, e 3 = 0, e = 0 + i0 + j0 C Gx. (27) The relation w + e = e + w is true by the condition of commutativity 1). Neutral element of the set C Gx is e and we have e = 0+i0+j0, ( 0, 0, 0 ) R 3. Neutral element e C Gx, is the zero generalized complex number, e = w 0 C Gx. 33

12 4) w C Gx, w s C Gx such that w + w s = w s + w = e ( w s symmetrical element for the element w C Gx ). We determine w s = x s + iy s + jt s, (x s, y s, t s ) R 3 and w + w s = e x + x s + i (y + y s ) + j (t + t s ) = 0 + i0 + j0. We obtain x + x s = 0 y + y s = 0 t + t s = 0 (28) and hence x s = x, y s = y, t s = t, w s = x + i ( y) + j ( t), ( x, y, t) R 3, w s C Gx. The relation w + w s = w s + w is true by the commutativity 1). Definition 2.1. The symmetrical generalized complex number w s is called generalized complex number opposite for w G x, w s = w, w s C Gx. 3. Multiplication of generalized complex numbers Definition 3.1. The multiplication of real number a R by w C Gx, w = x + yi + tj, with the notation, is defined by: Remark 3.2. From the relations (6) we obtain a w = a x + a yi + a tj. (29) a w = a x + a yi + a tj. (30) Definition 3.3. Let be generalized complex numbers Pescar, w 1 C Gx, w 2 C Gx, w 1 = x 1 + y 1 i + t 1 j, w 2 = x 2 + y 2 i + t 2 j. The multiplication of complex numbers w 1, w 2 having notation is the generalized complex number w C Gx, w = w 1 w 2 = x 1 x 2 y 1 y 2 t 1 t 2 + (x 1 y 2 + x 2 y 1 ) i + (x 1 t 2 + x 2 t 1 ) j. (31) 34

13 Using the relations (6), we have w = w 1 w 2 = (x 1 + y 1 i + t 1 j) (x 2 + y 2 i + t 2 j) = = x 1 x 2 + x 1 y 2 i + x 1 t 2 j + y 1 i x 2 + y 1 i y 2 i+ +y 1 i t 2 j + t 1 j x 2 + t 1 j y 2 i + t 1 j t 2 j = = x 1 x 2 y 1 y 2 t 1 t 2 + (x 1 y 2 + x 2 y 1 ) i + (x 1 t 2 + x 2 t 1 ) j. Remark 3.4. If w 1 C Gy, w 2 C Gy, w 1 = y 1 + t 1 j + x 1 k, w 2 = y 2 + t 2 j + x 2 k, then w 1 w 2 = w C Gy and we have w = w 1 w 2 = y 1 y 2 t 1 t 2 x 1 x 2 + (y 1 t 2 + t 1 y 2 ) j + (x 1 y 2 + x 2 y 1 ) k. (32) Remark 3.5. If w 1 C Gt, w 2 C Gt, w 1 = t 1 + x 1 k + y 1 i, w 2 = t 2 + x 2 k + y 2 i, then w 1 w 2 = w C Gt and we have w = w 1 w 2 = t 1 t 2 x 1 x 2 y 1 y 2 + (x 1 t 2 + x 2 t 1 ) k + (y 1 t 2 + y 2 t 1 ) i. (33) 4. The conjugated of generalized complex number Pescar Definition 4.1. Conjugated of generalized complex number w C Gx, w = x + yi + tj, is the generalized complex number, with the notation w C Gx, w = x yi tj. (34) 5. The modulus of generalized complex number Definition 5.1. Let the complex point O and M the image of generalized complex number w C Gx, w = x + yi + tj, M(x, y, t) with respect to the Cartesian axes system (Oxyt) x. The module of generalized complex number w, with the notation w, is distance of the point O, at the point M, O(0, 0, 0) with notation dist(o, M). Remark 5.2. Let w C Gx, w = x + yi + tj. We have w = dist (O, M) and from rectangular triangle OMM, figure 1.1, we obtain w = x 2 + y 2 + t 2. (35) 35

14 Remark 5.3. Let w C Gy, w = y + tj + xk, then from rectangular triangle OMM, figure 1.2, we obtain w = y 2 + t 2 + x 2. (36) Remark 5.4. Let w C Gt, w = t + xk + yi, then from rectangular triangle OMM, figure 1.3, we obtain w = t 2 + x 2 + y 2. (37) Remark 5.5. Let be complex numbers w, w C Gx, then we have w w = w 2. (38) Proof. We have w = x + yi + tj, w = x yi tj and w w = x 2 + y 2 + t 2 = w The properties of generalized complex nhumbers (C Gx \ {0}, ) Let be the set of generalized complex numbers C Gx \{0}, 0 = 0 + 0i + 0j and the law of intern composition, multiplication of generalized complex numbers such that : {C Gx \{0}} {C Gx \{0}} C Gx \{0}, defined by rule: (w 1, w 2 ) {C Gx \{0}} {C Gx \{0}} w 1 w 2 C Gx \{0}, then w 1 = x 1 + y 1 i + t 1 j, w 2 = x 2 + y 2 i + t 2 j, w 1 w 2 definition 3.3 = x 1 x 2 y 1 y 2 t 1 t 2 + (x 1 y 2 + x 2 y 1 ) i + (x 1 t 2 + x 2 t 1 ) j. (C Gx \{0}, ) satisfies the properties: 1) w 1 w 2 = w 2 w 1, w 1, w 2 C Gx \{0} (commutativity). We have x 1 x 2 y 1 y 2 t 1 t 2 + (x 1 y 2 + x 2 y 1 ) i + (x 1 t 2 + x 2 t 1 )j = = x 2 x 1 y 2 y 1 t 2 t 1 + (y 2 x 1 + y 1 x 2 ) i + (t 2 x 1 + t 1 x 2 ) j. 36

15 From (21) we obtain x 1 x 2 y 1 y 2 t 1 t 2 = x 2 x 1 y 2 y 1 t 2 t 1 x 1 y 2 + x 2 y 1 = y 2 x 1 + y 1 x 2 x 1 t 2 + x 2 t 1 = t 2 x 1 + t 1 x 2. (39) Since (R, ) is commutative group and (R, +) is commutative group the relations (39) are true. 2) e C Gx \{0} ( e neutral element), such that w C Gx \{0}, we have: w e = e w = w. We determine e of the form e = a + bi + cj. Let w C Gx \{0}, w = x + yi + tj and w e = w. We obtain: ax by ct + (ay + bx) i + (at + cx) j = x + yi + tj and hence We obtain ax by ct = x bx + ay = y cx + at = t. a = 1, b = 0, c = 0ande = 1 + 0i + 0j C Gx \{0}. (40) The relation w e = e w is true by the condition of commutativity in (C Gx \{0}, ). The generalized complex number e = 1 + 0i + 0j is neutral element in (C Gx \{0}, ). 3) w C Gx \{0}, w C Gx \{0} (w symmetrical element), such that w w = w w = e. Let w = x + yi + tj C Gx \{0} and we determine w = p + qi + rj, w C Gx \{0}. We consider w w = e and we obtain xp yq tr + (xq + yp) i + (xr + tp) j = 1 + 0i + 0j, and hence we have xp yq tr = 1 xq + yp = 0 (41) xr + tp = 0. 37

16 We obtain p = x x 2 +y 2 +t 2 q = y x 2 +y 2 +t 2 r = t x 2 +y 2 +t 2, (42) x 2 + y 2 + t 2 0, since w C Gx \{0}. The symmetrical element for w C Gx \{0} is w = x x 2 + y 2 + t 2 y x 2 + y 2 + t 2 i t x 2 + y 2 + t 2 j C G x \{0}. Relation w w = w w is true, since commutativity in (C Gx \{0}, ). The generalized complex number Pescar, w = x yi tj x 2 + y 2 + t 2 is the symmetrical element for w C Gx \{0}, w = x + yi + tj. Remark 6.1. We observe that w = w w The division of generalized complex numbers Definition 7.1. Let the generalized complex numbers Pescar, w 1 C Gx, w 2 C Gx \{0}, w 1 = x 1 + y 1 i + t 1 j, w 2 = x 2 + y 2 i + t 2 j. The division of generalized complex numbers w 1, w 2 is defined by w 1 w 2 = w 1 w 2 w 2 w 2 = w 1 w 2 w 2 2. (43) 38

17 From (43) we obtain w 1 = w 1 w 2 = x 1x 2 + y 1 y 2 + t 1 t 2 + y 1x 2 x 1 y 2 i + t 1x 2 x 1 t 2 j. (44) w 2 w 2 2 x y2 2 + t 2 2 x y2 2 + t 2 2 x y2 2 + t 2 2 Remark 7.2. Analogous we define the division of generalized complex numbers in set of C Gy and C Gt. 8. Raising to power of generalized complex number Pescar Definition 8.1. Let be w C Gx, n N. By definition Remark 8.2. w n = w w... w }{{} ntimes C Gx. Analogous we define the raising to power of generalized complex numbers from C Gy and C Gt. 9. The trigonometric form of generalized complex numbers Let be the set of generalized complex numbers C Gx and w C Gx, w = x + yi + tj. From the figure 1.1 we obtain x = ρ cosu y = ρ sinu ρ = w sinv t = w cosv. (45) We have M (Σ r ) and w = r, where r is radius of sphere (Σ r ), then we obtain x = r cosu sinv y = r sinu sinv, u [0, 2π ], v [ 0, π ] t = r cosv. u = (Ox, OM ) v = (Ot, OM). (46) 39

18 For the generalized complex number w C Gx we obtain the trigonometric form w = r (cosu sinv + i sinu sinv + j cosv), w = r. (47) Remark 9.1. If u = 0, then we obtain the complex number from C with the trigonometric form [ ( π ) ( π )] [ w = z = r cos 2 v + j sin 2 v, v 0, π ]. (48) 2 Remark 9.2. If v = π, then we obtain the complex number from C with the trigonometric 2 form and the image in complex plane (Oxy). w = z = r (cosu + i sinu), u [0, 2π]. (49) Remark 9.3. Analogous we define the trigonometric form of complex numbers from C Gy and C Gt. In C Gy, for w C Gy, w = y + tj + xk we have w = r (cosu sinv + j sinu sinv + k cosv), u [0, 2π], v [ 0, π ]. (50) 2 In C Gt, for w C Gt, w = t + xk + yi we have w = r (cosu sinv + k sinu sinv + i cosv), u [o, 2π], v [ 0, π ]. (51) Properties of modulus generalized complex numbers p 1 ) Let be the generalized complex numbers w 1 C Gx, w 2 C Gx, w 1 = x 1 + y 1 i + t 1 j, w 2 = x 2 + y 2 i + t 2 j. We have w 1 + w 2 w 1 + w 2. (52) 40

19 Proof. The relation (52) is equivalent with (x 1 + x 2 ) 2 + (y 1 + y 2 ) 2 + (t 1 + t 2 ) 2 x y1 2 + t x y2 2 + t 2 2 or x 1 x 2 + y 1 y 2 + t 1 t 2 x y t 2 1 x y t 2 2. (53) Case 1. If x 1 x 2 + y 1 y 2 + t 1 t 2 0 the relation (53) is true. Case 2. If x 1 x 2 + y 1 y 2 + t 1 t 2 0 from the relation (53) we obtain and hence, we have 2 (x 1 x 2 y 1 y 2 + x 1 x 2 t 1 t 2 + y 1 y 2 t 1 t 2 ) (x 1 y 2 ) 2 + (y 1 x 2 ) 2 + (x 1 t 2 ) 2 + (t 1 x 2 ) 2 + (y 1 t 2 ) 2 + (y 2 t 1 ) 2 (x 1 y 2 y 1 x 2 ) 2 + (x 1 t 2 t 1 x 2 ) 2 + (y 1 t 2 y 2 t 1 ) 2 0. So, the relation (52) is true. p 2 ) Let be w 1 C Gx, w 2 C Gx, w 1 = x 1 + y 1 i + t 1 j, w 2 = x 2 + y 2 i + t 2 j. We have w 1 w 2 w 1 w 2. (54) Proof. The relation (54) is equivalent with 2y 1 y 2 t 1 t 2 (y 1 t 2 ) 2 + (t 1 y 2 ) 2 or and the property p 2 ) is true. (y 1 t 2 t 1 y 2 ) 2 0 p 3 ) Let be w 1 C Gx, w 2 C Gx \{0}, w 1 = x 1 + y 1 i + t 1 j, w 2 = x 2 + y 2 i + t 2 j. We have w 1 w 1 w 2. (55) w 2 Proof. 41

20 We have w 1 = w 1 w 2 w 1 w 2 = w 1 w 2 2 w 2 2 w 2. w 2 p 4 ) Let be w 1 C Gx, n N. We have w n w n. (56) Proof. We have w n = w w... w w w w = w n. }{{} ntimes 11. General complex numbers Let be C Gp = C Gx CGy CGt and w 1 C Gx, w 2 C Gy, w 3 C Gt, w 1 = x 1 + y 1 i + t 1 j, w 2 = y 2 + t 2 j + x 2 k, w 3 = t 3 + x 3 k + y 3 i. We define addition of generalized complex numbers w 1, w 2, w 3 thus w 1 + w 2 + w 3 def. = x 1 + y 2 + t 3 + (y 1 + y 3 ) i + (t 1 + t 2 ) j + (x 2 + x 3 ) k. (57) Therefore w 1 + w 2 + w 3 = a + bi + cj + dk, a, b, c, d R. Definition 11.1 We usually call the set of general complex numbers, the set: V P = { w = a + bi + cj + dk, a, b, c, d R, i 2 = j 2 = k 2 = 1 }. (58) Definition 11.2 Let be w 1 C Gx, w 2 C Gy, w 3 C Gt w 1 = x 1 + y 1 i + t 1 j, w 2 = y 2 + t 2 j + x 2 k, w 3 = t 3 + x 3 k + y 3 i. The multiplication of generalized complex numbers w 1, w 2, w 3 is the ge-neral complex number w V P, w = w 1 w 2 w 3 = p + qi + rj + sk, p, q, r, s R. (59) 42

21 Definition 11.3 The modulus of general complex number w V P, w = a + bi + cj + dk is defined by w = a 2 + b 2 + c 2 + d 2. (60) Definition 11.4 The conjugate of general complex number w V P, w = a + bi + cj + dk, is the general complex number w V P, w = a bi cj dk. Remark Let be w, w V P, then w w = w 2. (61) Definition 11.6 Let be u V P, v V P \ {0}, u = a 1 + b 1 i + c 1 j + d 1 k, v = a 2 + b 2 i + c 2 j + d 2 k. The division of general complex numbers u is defined by v u v def. = u v v v = u v v 2. (62) References [1] V. Pescar, Generalized Pescar complex numbers, Pescar complex function. Braşov, Transilvania University of Braşov Publishing House, July Virgil Pescar Department of Mathematics and Computer Science Faculty of Mathematics and Computer Science, Transilvania University of Braşov 2200 Braşov, Romania virgilpescar@unitbv.ro 43

Department of Physics, Korea University

Department of Physics, Korea University Name: Department: Notice +2 ( 1) points per correct (incorrect) answer. No penalty for an unanswered question. Fill the blank ( ) with (8) if the statement is correct (incorrect).!!!: corrections to an