infinite roots of unity

[AntonioLao]

The ��-th root of a number �� is a number �� such that ��ⁿ = ��. If �� = 1 then its two square roots are 1 and -1, since 1² = 1 and (-1)² = 1. The number 1 has only one real cube root, which is simply 1 itself and 1³ = 1. However, extending this process into the complex domain, the three cube roots of unity are 1, ��, and ��² where �� = ��(2��/3) + ��������(2��/3) = -½ + ½√3��, and ��² = ������(4��/3) + ��������(4��/3) = -½ - ½√3�� such that ��² = conjugate of �� and ��² + �� + 1 = 0. By the same process, the four fourth roots of unity are: 1, -1, ��, and –��. The first two numbers are real, while the last two numbers are pure imaginary. There seem to be only half of the measureable real roots of unity.

Unfortunately, extending this process to the infinite roots of unity, it can be shown that at most only two real roots exist, namely 1 and -1. Moreover, if �� is an odd integer then there is only one real ��-th root of unity that is simply 1 by itself. The others (��-1) roots are complex roots; at most two of these are pure imaginary roots, namely �� and –��. Nonetheless, the sum of all possible ��-th roots of unity whether real, complex, or pure imaginary is always zero. Logically, the infinite roots of unity represent a one-to-one correspondence with the totality of the space-time continuum, while each root represents a quantum of space-time charges as a square of zero-point energies of the quantum vacuum fluctuations such that the total of positive and negative squares of energy added to zero for the whole universe. This implies that the total number of H-pluses of squares of energy is exactly equal to the total number of H-minuses of squares of energy and the total squares of energy (adding both H-pluses and H-minuses) of the whole universe is exactly zero.