Download PDF by Perez E.: Foundations of transcomplex numbers: An extension of the

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Extra resources for Foundations of transcomplex numbers: An extension of the complex number system to four dimensions (2008)

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COMPLEX NUMBERS From the above result we deduce that for a real number a and another complex number C = (a′ , c′ i): ′ a ∗ C ′ = (aa′ , ac′ ). 17) Hence, for complex numbers the scalar product needs no new definition. 1. The unit element under multiplication of the complex numbers field is the complex number 1 + 0i = 1. That is, for every complex number C: C ∗ 1 = C. 18) Proof. 2. The inverse of the complex number C = a + ci = (a, ci) is the complex number C −1 given by: c a − 2 2 +c a + c2 a c = 2 , − .

To overcome this we also define the imaginary scalar product as (ai, bi) = (a, b)i. Definition 33. The imaginary scalar product of the real numbers ordered pair (a, b) and the unit imaginary number i is defined to be: (a, b)i = (ai, bi). 13) Absolute value of an imaginary ordered pair Definition 34. The absolute value of the imaginary ordered pair (ai, bi) is defined to be: (ai, bi) = (a, b) i. 14) That means that the absolute value of an imaginary ordered pair is another imaginary ordered pair.

Therefore, the XY ˜plane and the II˜plane must be perpendicular planes and that poses us a visualization problem since with the exception of the origin nothing else is coincidental among those two planes. 1: The coordinate plane of the imaginary ordered pairs No number that belongs to the I-axis can concurrently belong to the I˜-axis, and vice versa; the exception is the zero, or origin O. 3) I ∩ I˜ = (0, 0) = {0} = 0. 4) O = (0, 0) = {0}. 5) or Note that The set {0} is a one-element set; that means that the I- and I˜-axes have nothing in common except the single member zero.

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