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By Charles G. Moore

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A x b = b x a (commutativity). � the neutral element is an Abelian group. � the operation, with I as the neutral element; here o would not have an inverse element. The definition of group rules also makes it possible to have other objects than numbers as members of a group, as long as they satisfy the requested properties. An example for such a group is the set of symmetry transformations rotation, reflec­ tion and inversion, through which a topological object such as a polygon is mapped to itself; compositions are then transformations that are applied consecutively.

1 6 + 1 6 + 1 6 16 1 6 16 1 6 1 6 I I = - + - + - + - + - + " ' _ 00 2 2 2 2 2 Shanno nic > Scamp"ri",n => Shanno nic _ 00 . Thus, the terms of the harmonic sequence do not converge sufficiently strongly to zero to ensure convergence. 3 Fibonacci sequence 40 A sufficient criterion for convergence is that the ratio of successive tenns of the sequence is smaller than I for n .... 00 (quotient criterion of d'Alembert). For the two series we have: "-+00 n + n lim __ = I I hannonic series geometric series A ,, + I -- A" ,,+ a 1 = -- = a ; a" %-+00 lim a = a < I for a < I .

I� 0 for b 2 > 4ac for b 2 < 4a c . If a, b and c are themselves complex, the general formula is still valid, but not the distinction between two cases since the order relations > and < are not applicable for complex numbers. What is the situation in the complex number space for the cube/third root and, in general, for odd root exponents? In the space of real numbers, there is always one real negative solution ( Fc = - Vc) for negative radicands. In the space of complex numbers, however, we obtain the following: Z 3 = e; c real (a + ib) (a + ib)(a + ib) = (a 2 - b 2 + i2ab) (a + ib) = a 3 3ab 2 + i (3a 2 b - b 3 ) = c b (3a 2 - b2 ) = 0 since e is real either b = 0 or ( 3a 2 - b 2 ) = 0 1 st solution b = 0 a3 = e a = Vc ....

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