The number i equals -1^0.5 {imaginary number}|. By DeMoivre's theorem, any power of i is expressible as a + b*i. For example, i^2 = -1, i^3 = -i, and i^4 = 1. i^0.5 = 1/(2^0.5) + (1/(2^0.5))*i. i^0.5 = -1/(2^0.5) - (1/(2^0.5))*i. i^0.333 = (3^0.5)/2 + (1/2)*i. Therefore, complex numbers need only a real part and an imaginary part, with no other components.
i^i = e^-pi / 2, for log i = 0.5 * pi * i. All i^i are real numbers. z = r * e^i*A. log z = log r + i*A. e^i*A = cos A + i*sin A.
All polynomial roots are expressible by at least one complex number (though not as radicals, by the Abel-Ruffini theorem [1824]).
Perhaps, a new complex-number type can use a factor of reals and imaginaries, but not be a hypercomplex number.
Mathematical Sciences>Number Theory>Number Types>Complex Number
3-Number Theory-Number Types-Complex Number
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Date Modified: 2022.0224