Table Of ContentI. M. YAGLOM
COMPLEX NUMBERS
IN GEOMETRY
Complex Numbers
in Geometry
by I. M. YAGLOM
U.S.S.R.
Translated by ERIC j. F. PRIMROSE
UNIVERSITY OF LEICESTER
LEICESTER, ENGLAND
1968
ACADEMIC PRESS
New York and London
CoPYRIGHT © 1968, BY AcADEMIC PREss INc.
ALL RIGHTS RESERVED.
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM,
BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT
WRITTEN PERMISSION FROM THE PUBLISHERS.
Originally published in the Russian language under the title
"Kompleksnye Chisla i Ikh Primenenie v Geometrii" by
Fizmatgiz, Moscow, 1963.
ACADEMIC PRESS INC.
111 Fifth Avenue, New York, New York 10003
United Kingdom Edition published by
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LIBRARY OF CONGRESS CATALOG CARD NUMBER: 66-26269
PRINTED IN THE UNITED STATES OF AMERICA
Translator's Note
In the original book the references are all to Russian literature;
wherever possible I have replaced these either by references to
the corresponding English translations or to what, I hope, are
equivalent references to books written in English and, occasion
ally, German.
A number of minor errors in the original edition have been
corrected in this translation. Additions supplied by the author
that could not be incorporated in the text have been placed at
the end of the book, with references to them in the text.
I should like to acknowledge the great help which my wife
has given in preparing the typescript.
Preface
This book is intended for pupils in the top classes in high
schools and for students in mathematics departments of universi
ties 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 geom
etry. 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.
Today different types of complex numbers are studied intens
ively; their study is connected with important unsolved problems,
in which scientists from many countries are engaged. This book,
naturally, does not aim to acquaint the reader with contemporary
problems. Here only one of the threads that link the study of
complex numbers with geometry will be shown and even in such a
vii
viii Preface
limited field we shall not claim completeness. The range of
questions touched on in this book, however, is sufficiently wide.
In particular, we do not limit ourselves to the introduction of
basic concepts, but in all cases we try to use these ideas to prove
interesting geometrical theorems.
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 main line of the exposition is contained in Sections 1-4, 7,
9, 13, and 15, not denoted by asterisks. Parallel to this, and intended
chiefly for present and future teachers of mathematics, there is
a fairly wide selection of illustrations of an elementary geo
metrical character. The way to apply the apparatus of complex
numbers to elementary geometry is demonstrated in Sections 8,
10, 14, and 16, which are denoted by one asterisk. Each of these
four sections contains applications and examples and various
geometrical theorems proved by using complex numbers. The
theorems collected here are, as a rule, of purely illustrative
significance; somewhat closer to the main line of the exposition
are the theorems about the power of a point and a line with
respect to a circle (Sections 8 and 10 ), which are used in Section 16
for a new geometrical definition of axial (Laguerre) inversion,
which plays an essential role in the content of Section 15. The
omission of Sections 8, 10, 14, and 16 will not affect the under
standing of the rest of the book, and we recommend that the reader
not spend very much time on them on a first reading; when
he has mastered the basic material, the reader who is interested
in elementary geometry can return to these sections.
Sections 5, 6, 11, 12, 17, and 18, denoted by two asterisks,
are of a quite different character. Here we extend the bounds of
the exposition and go beyond the material which, sometimes
rather conventionally, is regarded as elementary geometry.
The fact is that the chief application of complex numbers
to geometry is still to Euclidean geometry, which is studied
Preface ix
in high schools, and to the so-called non-Euclidean geometries,
of which the best known is that of Lobachevskii. Even in
a popular book devoted to complex numbers, it seemed to
us absolutely inadmissible to ignore completely this geomet
rical application of complex numbers. Although there is no
possibility of dealing with this question in a short space, we
have still thought it necessary to include in the book a short
exposition of the role played by complex numbers in the geom
etry of Lobachevskii. The corresponding sections are naturally
intended for the reader who has some familiarity with the
content of this remarkable geometry. However, his preparation
in this respect can be very little: it need not go beyond the
limits of the material presented in popular science books and
pamphlets on non-Euclidean geometry (some of these will be
referred to in the footnotes). In accordance with the special
character of the sections denoted by two asterisks, even the
exposition there has a character rather different from the
other parts of the book; for example, the proofs here are some
times not completely carried out, and the filling in of some of
the details is left to the reader. Thus, the omission of Sections 5,
6, 11, 12, 17, and 18 will not prevent the understandingofthe
rest of the book, which in its elementary parts (that is, those
not connected with non-Euclidean geometry) forms a complete
whole. We note that here and there in Sections 5, 6, 11, 12, 17,
and 18 the account is somewhat more concise than that in the
other parts of the book, and these sections contain hardly any
concrete examples, similar to those contained in Sections 8, 10,
14 and 16, of the application of the techniques developed there;
the independent construction of all the details of the proofs,
and the carrying over to the non-Euclidean geometry of
Lobachevskii of some of the results of Sections 8, 10, 14, and 16
may be regarded as problems which may properly be recommen
ded to the student-reader.
The origin of this book was a lecture on the subject given by
the author in 1958 to members of the school mathematical society
attached to the Moscow state university. A broad account of
this lecture was published in volume 6 of the collection Mathe-
X Preface
matical Education (Fizmatgiz, Moscow) in 1961. A considerable
part of the material was presented also to the society for students
of the first course of the mathematical faculty of the Moscow
state pedagogical institute. The author expresses his thanks to
A. M. Yaglom, whose advice was taken during work on the
manuscript, to his pupils D. B. Persits, M. M. Arapova, and
F. M. Navyazhskii, who have provided some of the proofs
carried out in the book, and to the editors of the book, M. M.
Goryachaya and I. E. Morozova, who made a number of useful
observations. Finally, he is grateful to R. Deaux, professor at
the Polytechnic Institute of Mons (Belgium), who obligingly sent
the latest edition of his book on complex numbers.
This English edition of the book differs from the original
Russian edition in respect of a small number of corrections and
additions, and also a brief Appendix, the basis of which is an
article (in Russian) by the author: "Projective metrics in the plane
and complex numbers" (Proceedings of a Seminar on Vector and
Tensor Analysis at Moscow State University, Part 7, pp. 276-318,
1949).
The author thanks D. B. Persits for his constructive criticism
of one part of the original Russian book; he is also grateful to
the translator, Eric J. F. Primrose, for the great care with which
he has carried out the work of preparing the English edition,
and to Academic Press for their considerate treatment of all the
author's requests.
I. M. YAGLOM
Contents
Translator's Note v
Preface vii
Chapter 1: Three Types of Complex Numbers
I. Ordinary Complex Numbers
2. Generalized Complex Numbers 7
3. The Most General Complex Numbers 10
4. 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
I I. **Interpretation of Ordinary Complex Numbers
in the Lobachevskii Plane 108
12. **Double Numbers as Oriented Lines of the
Lobachevskii Plane I I 8
xi
xii Contents
Chapter Ill: 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 171
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
AI. 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
Addenda 220
Index 241
