@MISC{Milewski_trigonometricform, author = {Robert Milewski}, title = {Trigonometric Form of Complex Numbers MML Identifier: COMPTRIG.}, year = {} }

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Abstract

The scheme Regr without 0 concerns a unary predicate P, and states that: P [1] provided the parameters meet the following requirements: • There exists a non empty natural number k such that P [k], and • For every non empty natural number k such that k � = 1 and P [k] there exists a non empty natural number n such that n < k and P [n]. The following propositions are true: (3) 1 For every element z of C holds ℜ(z) ≥ −|z|. (4) For every element z of C holds ℑ(z) ≥ −|z|. (5) For every element z of CF holds ℜ(z) ≥ −|z|. (6) For every element z of CF holds ℑ(z) ≥ −|z|. (7) For every element z of CF holds |z | 2 = ℜ(z) 2 + ℑ(z) 2. (8) For all real numbers x1, x2, y1, y2 such that x1 + x2iCF = y1 + y2iCF holds x1 = y1 and x2 = y2. (9) For every element z of CF holds z = ℜ(z) + ℑ(z)iCF. (10) 0CF = 0 + 0iCF. (12) 2 For every unital non empty groupoid L and for every element x of L holds power L (x, 1) = x. (13) For every unital non empty groupoid L and for every element x of L holds power L (x, 2) = x · x. (14) Let L be an add-associative right zeroed right complementable right distributive unital non empty double loop structure and n be a natural number. If n> 0, then power L (0L, n) = 0L. 1 The propositions (1) and (2) have been removed. 2 The proposition (11) has been removed.