Notable Identities: Classical Identities in Quadratic Nodes, Rational Rotations and Linear Recurrences

Notable Identities places the classical identities of elementary algebra, the fórmulas notables or productos notables of the Spanish-language curriculum, at the steps they carry in a set of constructions: the quadratic node sequence and its odd intervals, the Geometric Matrix, the Smith chart reflection coefficient, rational rotations with Pythagorean entries, the Fibonacci and Tribonacci recurrences, and the inscribed hexagon. Every step of the algebra is shown. The paper does not present new identities or new theorems. Every identity is classical and attributed to its source. The quadratic identities are propositions II.4 to II.10 of Euclid’s Book II, with earlier Babylonian counterparts, and the Fibonacci equation is the reciprocal form of the cut in extreme and mean ratio of II.11 and VI.30. The others come from Diophantus, Brahmagupta, Fibonacci, Nicomachus, Viète, Euler and Hamilton. Each identity is also placed beside the names it takes later in algebra, engineering and physics: the conjugate, the norm, Viète’s relations, rational points on a circle, power balance on a transmission line. Two kinds of identity carry the steps. Norm relations belong to the split-complex numbers, the Gaussian integers, the Eisenstein integers, the integers of ℚ(√5) and the quaternions, and the Tribonacci roots are the conjugates of a cubic field. Polynomial expansions carry the relations among powers of these conjugates. The contribution is the map. Each step of the constructions can be checked against a named classical identity, and the map identifies the identities on which the exact computations depend, computations that use no trigonometric evaluation and no floating-point approximation.