By Garret Sobczyk

Modular quantity platforms -- complicated and hyperbolic numbers -- Geometric algebra -- Vector areas and matrices -- Outer product and determinants -- structures of linear equations -- Linear differences on R[n superscript] -- constitution of a linear operator -- Linear and bilinear kinds -- Hermitian internal product areas -- Geometry of relocating planes -- illustration of the symmetric team -- Calculus on m-surfaces -- Differential geometry of curves -- Differential geometry of k-surfaces -- Mappings among surfaces -- Non-euclidean and projective geometries -- Lie teams and lie algebras

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**Example text**

The hyperbolic numbers, which have also been called the “perplex numbers” [23], serve as coordinates in the Lorentzian plane in much the same way that the complex numbers serve as coordinates in the Euclidean plane. An event X that occurs at time t and at the place x is specified by its coordinates t and x. If c is the speed of light, the product ct is a length, and the coordinates of the event X can be combined into the hyperbolic number X = ct + ux. By the space-time distance between two events X1 and X2 , we mean the hyperbolic modulus |X1 − X2 |h which is the hyperbolic distance of the point X1 − X2 to the origin.

Do the same calculations for the conjugate product w− 2 w1 of w2 with w1 . What is the relationship between the two conjugate products? (c) What is the area of the parallelogram with the sides z1 and z2 ? (d) What is the area of the parallelogram with the sides w1 and w2 ? (e) What is the Euclidean angle between the vectors z1 and z2 ? (f) What is the hyperbolic angle between the vectors w1 and w2 ? 2. Given the complex numbers z1 = r1 eiθ1 and z2 = r2 eiθ2 , and the hyperbolic numbers w1 = ρ1 euφ1 and w2 = ρ2 euφ2 , (a) Calculate the conjugate product z1 z2 of z1 with z2 .

Do the same calculations for the conjugate product w− 2 w1 of w2 with w1 . What is the relationship between the two conjugate products? (c) What is the area of the parallelogram with the sides z1 and z2 ? (d) What is the area of the parallelogram with the sides w1 and w2 ? (e) What is the Euclidean angle between the vectors z1 and z2 ? (f) What is the hyperbolic angle between the vectors w1 and w2 ? 2. Given the complex numbers z1 = r1 eiθ1 and z2 = r2 eiθ2 , and the hyperbolic numbers w1 = ρ1 euφ1 and w2 = ρ2 euφ2 , (a) Calculate the conjugate product z1 z2 of z1 with z2 .