IN FAVOR OF IMAGINARIES IN GEOMETRY
Abstract and keywords
Abstract (English):
“Complex numbers are something complicated”, as they are perceived in most cases. The expression “real numbers are also complex numbers” sounds strange as well. And for all that complex numbers are good for many areas of knowledge, since they allow solve problems, that are not solved in the field of real numbers. First and most important is that in the field of complex numbers all algebraic equations are solved, including the equation x2 + a = 0, which has long been a challenge to human thought. In the field of complex numbers, the problem solutions remain free from listing special cases in the form of "if ... then", for example, solving the problem for the intersection of the line g with the circle (O, r) always gives two points. And in the field of real numbers, three cases have to be distinguished: | Og | r → there is no intersection; | Og | = r → there is one double point. The benefit of complex numbers also lies in the fact that with their help not only problems that previously had no solutions are solved, they not only greatly simplify the solution result, but they also hold shown in this text further amazing properties in geometric figures, and open door to the amazing and colorful world of fractals.

Keywords:
imaginary, real, complex, interconnection, spherical belt, sphere, hyperboloid, conjugate, fragment, fractal
References

1. Adrian Duadi. Kompleksnyye chisla i Fraktaly [Complex numbers and Fractals]. Available at: https://www.youtube.com/watch?v=Wsr-f9a0nR8

2. Balk M.B., Balk G.D., Polukhin A.A. Real'nyye primeneniya mnimykh chisel [Real applications of imaginary numbers]. Kiyev, Radyans'ka shkola Publ., 1988. 255 p. ISBN 5-330-00379-2. (in Russian)

3. Girsh A.G. Kompleksnaya geometriya - evklidova i psevdoevklidova [Complex geometry - Euclidean and pseudoEuclidean]. Moscow, Maska Publ., 2013. 216 p. (in Russian)

4. Girsh A.G. Mnimosti v geometrii [Imaginations in geometry]. Geometriya i grafika [Geometry and Graphics]. 2014, V. 2, I. 2, pp. 3-8. - DOI:https://doi.org/10.12737/5583. (in Russian)

5. Girsh A.G. Naglyadnaya mnimaya geometriya [Visual imaginarygeometry]. Moscow: Maska Publ., 2008. 216 p. (in Russian)

6. Girsh A.G. Novye zadachi nachertatel'noj geometrii [New problems in descriptive geometry]. Geometriya i grafika [Geometry and Graphics]. 2019, V. 7, I. 4, pp. 3-8. - DOI:https://doi.org/10.12737/5583. (in Russian)

7. Grafskii O.A. Modelirovanie mnimyh elementov na ploskosti [Modeling imaginary elements on a plane]. Khabarovsk, Publishing house of dvgups Publ., 2004. 161 p. (in Russian)

8. Ivanov G.S., Dmitrieva I.M. O zadachah nachertatel'noj geometrii s mnimymi resheniyami [On descriptive geometry problems with imaginary solutions]. Geometriya i grafika [Geometry and Graphics]. 2015, V. 3, I. 2, pp. 3-8. - DOI:https://doi.org/10.12737/12163. (in Russian)

9. Kirillov A.A. Chto takoye chislo? [What is a number]. Moscow, Fizmatlit Publ., 1993. 80 p. ISBN 5-02-014942-3. (in Russian)

10. Korotkiy V.A. Sinteticheskiye algoritmy postroyeniya krivoy vtorogo poryadka [Synthetic algorithms for constructing a second-order curve]. Vestnik komp'yuternykh i informatsionnykh tekhnologiy [Bulletin of computer and information technologies]. 2014, I. 11, pp. 20-24. (in Russian)

11. Mironov V.V. Sovremennyye problemy yestestvennykh, tekhnicheskikh i sotsial'no-gumanitarnykh nauk [Modern problems of natural, technical and socio-humanitarian Sciences]. Moscow, Gardariki Publ., 2006. 639 p. (in Russian)

12. Peklich V.A. Mnimaya nachertatel'naya geometriya [Imaginary descriptive geometry]. Moscow, ASV Publ., 2007. 104 p. (in Russian)

13. Sklyarevskij E.S. Krasivaya zhizn' kompleksnyh chisel [Modern problems of natural, technical and socio-humanitarian Sciences]. Hard'N'Strajk [Hard'n'Soft]. 2002, I. 9, p. 90. (in Russian)

14. Suvorov F.M. Ob izobrazhenii voobrazhayemykh tochek i voobrazhayemykh pryamykh na ploskosti i o postroyenii krivykh liniy vtoroy stepeni, opredelyayemykh s pomoshch'yu voobrazhayemykh tochek i kasatel'nykh [On the representation of imaginary points and imaginary lines on the plane and on the construction of curved lines of the second degree, determined using imaginary points and tangents]. Kazan, Tipografiya imperatorskogo Universiteta Publ., 1884. 130 p. (in Russian)

15. Yaglom I.M. Kompleksnyye chisla i ikh primeneniye v geometrii [Complex numbers and their application in geometry]. Moscow, Editorial URSS Publ., 2004. 192 p. (in Russian)

16. Hirsch, A.: Extension of the 'Villarceau-Sektion' to Surfaces of Revolution with a Generating Conic. Jurnal for Geometriy and Graphics, 6 (2000/2). p.121-132.

Login or Create
* Forgot password?