By Nicolas Conti

ISBN-10: 2754003568

ISBN-13: 9782754003568

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M. Yaglom, "On the geometries of the simplest algebras," Matematicheskii sbornik, Vol. 28, No. 1 (1951), pp. 205-216, and the book (in Russian) by B. A. Rozenfel'd, Non-Euclidean Geometries, Chap. VI (Gostekhizdat, Moscow), 1955, where the algebraic literature referred to here is also mentioned. CHAPTER II Geometrical of Complex Interpretation Numbers §7. Ordinary Complex Numbers as Points of a Plane18a The development of the theory of complex numbers is very closely connected with the geometrical interpretation of ordinary complex numbers as points of a plane, which apparently was first mentioned by the Danish surveyor K.

3 #1 Z$ Z Z-^ Z 2 . Z2 % ^3 Zi Z% /i o\ (z - z2){z — ζ3)[{ζ± — z3){zx - z2)] = (*— *3)(* - *2)[(*1 - *2)(*1 — *s)] That is, Azz + Bz - Ëz + C = 0 where A ={z± — z3){z± - z2) — {zx — z2)(z1 — z3) B = -z3{zx C = Z2Z3{Z1 - * 3 )(*! - z2) + z2(z1 - z2){z± ^3/(^1 ^2) ^3^21*1 ^X^l z3) ^3/ Thus, the equation of every circle {or line) may be written in the following form: Azz + Bz — Bz + C = 0, A C purely imaginary (14) Conversely, the locus of points z which satisfy any equation of this form is a circle or a line (if such points exist).

18 18 See, for example, the article refered to in footnote 11 on p. 14, and also an article (in Russian) by B. A. Rozenfel'd and I. M. Yaglom, "On the geometries of the simplest algebras," Matematicheskii sbornik, Vol. 28, No. 1 (1951), pp. 205-216, and the book (in Russian) by B. A. Rozenfel'd, Non-Euclidean Geometries, Chap. VI (Gostekhizdat, Moscow), 1955, where the algebraic literature referred to here is also mentioned. CHAPTER II Geometrical of Complex Interpretation Numbers §7. Ordinary Complex Numbers as Points of a Plane18a The development of the theory of complex numbers is very closely connected with the geometrical interpretation of ordinary complex numbers as points of a plane, which apparently was first mentioned by the Danish surveyor K.

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