Complex Numbers in Geometry focuses on the principles, interrelations, and applications of geometry and algebra. The book first offers information on the types and geometrical interpretation of complex numbers. Topics include interpretation of ordinary complex numbers in the Lobachevskii plane; double numbers as oriented lines of the Lobachevskii plane; dual numbers as oriented lines of a plane; most general complex numbers; and double, hypercomplex, and dual numbers. The text then takes a look at circular transformations and circular geometry, including ordinary circular transformations, axial circular transformations of the Lobachevskii plane, circular transformations of the Lobachevskii plane, axial circular transformations, and ordinary circular transformations.The manuscript is intended for pupils in high schools and students in the mathematics departments of universities and teachers' colleges. The publication is also useful in the work of mathematical societies and teachers of mathematics in junior high and high schools.

Translator's NotePrefaceChapter I: Three Types of Complex Numbers 1. Ordinary Complex Numbers 2. Generalized Complex Numbers 3. The Most General Complex Numbers 4. Dual Numbers 5. **Double Numbers 6. **Hypercomplex NumbersChapter II: Geometrical Interpretation of Complex Numbers 7. Ordinary Complex Numbers as Points of a Plane 8. *Applications and Examples 9. Dual Numbers as Oriented Lines of a Plane 10. *Applications and Examples 11. **Interpretation of Ordinary Complex Numbers in the Lobachevskii Plane 12. **Double Numbers as Oriented Lines of the Lobachevskii PlaneChapter III: Circular Transformations and Circular Geometry 13. Ordinary Circular Transformations (Möbius Transformations) 14. *Applications and Examples 15. Axial Circular Transformations (Laguerre Transformations) 16. *Applications and Examples 17. **Circular Transformations of the Lobachevskii Plane 18. **Axial Circular Transformations of the Lobachevskii PlaneAppendix: Non-Euclidean Geometries in the Plane and Complex Numbers A1. Non-Euclidean Geometries in the Plane A2. Complex Coordinates of Points and Lines of the Plane Non-Euclidean Geometries A3. Cycles and Circular TransformationsAddendaIndex