## Complex Numbers in Geometry – Yaglom

In this post, we will see the book Complex Numbers in Geometry by I. M. Yaglom.

This book is intended for pupils in the top classes in high schools and for students in mathematics departments of universities and teachers’ colleges. It may also be useful in the work of mathematical societies and may be of interest to teachers of mathematics in junior high and high schools.
The subject matter is concerned with both algebra and geometry. There are many useful connections between these two disciplines. Many applications of algebra to geometry and of geometry to algebra were known in antiquity; nearer to our time there appeared the important subject of analytical geometry, which led to algebraic geometry, a vast and rapidly developing science, concerned equally with algebra and geometry. Algebraic methods are now used in projective geometry, so that it is uncertain whether projective geometry should be called a branch of geometry or algebra. In the same way the study of complex numbers, which arises primarily within the bounds of algebra, proved to be very closely connected with geometry; this can be
seen if only from the fact that geometers, perhaps, made a greater contribution to the development of the theory than algebraists.

The book is intended for quite a wide circle of readers. The early sections of each chapter may be used in mathematical classes in secondary schools, and the later sections are obviously intended for more advanced students (this has necessitated a rather complicated system of notation to distinguish the various parts of the book).

The book was translated from Russian by Eric Primrose and was published in 1968 from a 1963 Russian edition.

You can get the book here.

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# Contents

Translator’s Note v

Preface vii

Chapter I: Three Types of Complex Numbers

1. Ordinary Complex Numbers 1
2. Generalized Complex Numbers 7
3. The Most General Complex Numbers 10
5. Dual Numbers 14
5. **Double Numbers 18
6. **Hypercomplex Numbers 22

Chapter II: Geometrical Interpretation of Complex Numbers

7. Ordinary Complex Numbers as Points of a Plane 26
8. *Applications and Examples 34
9. Dual Numbers as Oriented Lines of a Plane 80
10. *Applications and Examples 95
11. **Interpretation of Ordinary Complex Numbers in the Lobachevskii Plane 108
12. **Double Numbers as Oriented Lines of the Lobachevskii Plane 118

Chapter III: Circular Transformations and Circular Geometry

13. Ordinary Circular Transformations (Mobius Transformations) 130
14. *Applications and Examples 145
15. Axial Circular Transformations (Laguerre Transformations) 157
16. *Applications and Examples 161
17. **Circular Transformations of the Lobachevskii Plane 179
18. **Axial Circular Transformations of the Lobachevskii Plane 188

Appendix: Non-Euclidean Geometries in the Plane and Complex Numbers

A1. Non-Euclidean Geometries in the Plane 195
A2. Complex Coordinates of Points and Lines of the plane Non-Euclidean Geometries 205
A3. Cycles and Circular Transformations 212