The present book is intended as a text in basic mathematics. As such,
it can have multiple use: for a one-year course in the high schools during the
third or fourth year (if possible the third, so that calculus can be taken
during the fourth year); for a complementary reference in earlier high school
grades (elementary algebra and geometry are covered); for a one-semester
course at the college level, to review or to get a firm foundation in the basic
mathematics necessary to go ahead in calculus, linear algebra, or other topics.
Years ago, the colleges used to give courses in “ college algebra” and
other subjects which should have been covered in high school. More recently,
such courses have been thought unnecessary, but some experiences I have had
show that they are just as necessary as ever. What is happening is that the
colleges are getting a wide variety of students from high schools, ranging
from exceedingly well-prepared ones who have had a good first course in
calculus, down to very poorly prepared ones. This latter group includes both
adults who return to college after several years’ absence in order to improve
their technical education, and students from the high schools who were not
adequately taught. This is the reason why some material properly belonging
to the high-school level must still be offered in the colleges.
The topics in this book are covered in such a way as to bring out clearly
all the important points which are used afterwards in higher mathematics.
I think it is important not to separate arbitrarily in different courses the
various topics which involve both algebra and geometry. Analytic geometry
and vector geometry should be considered simultaneously with algebra and
plane geometry, as natural continuations of these. I think it is much more
valuable to go into these topics, especially vector geometry, rather than to
go endlessly into more and more refined results concerning triangles or
trigonometry, involving more and more complicated technique. A minimum
of basic techniques must of course be acquired, but it is better to extend these
techniques by applying them to new situations in which they become motivated, especially when the possible topics are as attractive as vector
geometry.
In fact, for many years college courses in physics and engineering have
faced serious drawbacks in scheduling because they need simultaneously
some calculus and also some vector geometry. It is very unfortunate that the
most basic operations on vectors are introduced at present only in college.
They should appear at least as early as the second year of high school. I
cannot write here a text for elementary geometry (although to some extent
the parts on intuitive geometry almost constitute such a text), but I hope
that the present book will provide considerable impetus to lower considerably
the level at which vectors are introduced. Within some foreseeable future,
the topics covered in this book should in fact be the standard topics for the
second year of high school, so that the third and fourth years can be devoted
to calculus and linear algebra.
If only preparatory material for calculus is needed, many portions of
this book can be omitted, and attention should be directed to the rules of
arithmetic, linear equations (Chapter 2), quadratic equations (Chapter 4),
coordinates (the first three sections of Chapter 8), trigonometry (Chapter 11),
some analytic geometry (Chapter 12), a simple discussion of functions
(Chapter 13), and induction (Chapter 16, §1). The other parts of the book
can be omitted. Of course, the more preparation a student has, the more
easily he will go through more advanced topics.
You can download here: https://www.ulm.edu.pk/departments/admin/upload/downloads/202110030921.pdf
