Table of Content
1 The laws of algebra
2 The Cartesian Plane
4 Euclidean Geometry
5 Spherical Trigonometry
6 Solving Equations and Factorizing Polynomials
10 Differential Calculus
This book is intended primarily for university students. In particular, this book can be
used as a textbook or an additional reference book by university students attending a course
in algebra, trigonometry, geometry or calculus.
As a calculus textbook, this book is unique in that it contains all the mathematics (and
more) that students will need to know in order to be successful in a calculus course. For
this reason, the book will be invaluable for students who may need to fill in some “gaps” in
their mathematical background, or review certain topics, while attending a university level
calculus course. (Calculus professors will normally not have the time to do this!)
It is the author’s opinion that mathematics can be appreciated and enjoyed more when
it is presented as a story. Hence there is a strong emphasis in this book on the unfolding
and development of the concepts, and there are comments throughout the book to help
the reader trace the historical development of mathematics. Of course, students also need
to develop their skills. For this reason there are many exercises included at the ends of
the chapters and students are encouraged to do all of the exercises as an essential part
of working through the book. The answers for many of the exercises can be found at the
author’s website http://www.tamug.edu/gacd/fs/math02.html
The range of topics (geometry, trigonometry, algebra and calculus) in this book is another
unique aspect of the book that educators should find refreshing. What’s more, there
are many attempts in the book to demonstrate the interplay and interconnectedness of
these topics. This is a most rewarding aspect of learning mathematics, which university
students are typically not exposed to because of the way current–day university courses
The four chapters that make up the introduction to algebra are Chapter 1, Chapter 2,
Chapter 6 and Chapter 9. They can be read independently of the other chapters. Chapter 3
(trigonometry) and Chapter 5 (spherical trigonometry) can also be read independently (after
the first two chapters have been read, if needed). Chapter 4 (Euclidean Geometry) does not
depend on any of the other chapters except for some symbolism from Chapter 1. Chapter 7,
which is an introduction to functions, can be regarded as the beginning of calculus because
the operations of calculus are applied to functions. The concepts and methods relating
to the calculation of limits, which underlie the operations of calculus, are introduced in
Chapter 8. Chapter 10 is an introduction to differential calculus. The definition of the
derivative, the rules for computing derivatives and the formulas for derivatives of all the
standard types of function (i.e. polynomial, rational, trigonometric, root, absolute value,
exponential and logarithmic functions) are introduced in this chapter.
Chapter 9, the final chapter on algebra, is placed after Chapter 7 and Chapter 8 because
the material in these two chapters can be learned without a high level of algebraic skill. Because
many students have difficulties with algebra when they enroll in a university calculus
course, a first semester course in calculus may well consist of Chapter 7, Chapter 8, Chapter
9 and Chapter 10. It should also be mentioned that Chapter 9 includes an introduction
to partial fractions, a topic that is usually not taught in a course in algebra.
In the following two paragraphs, some comments are made regarding the presentation
of the material in Chapter 8 and Chapter 10.
It is usual, in most calculus textbooks, for the rules of limits to be taken as the starting
point for the evaluation of limits. In Chapter 8, the slightly different approach to evaluating
limits is to take as a fact the continuity of all of the standard functions and any algebraic
combinations and compositions of the standard functions. (This fact can be proved using
methods of real analysis, which is too advanced for this book.) This means that a limit to
any point in the domain of any of these functions can be evaluated simply by making a
substitution into the function (i.e. by an application of the equation of continuity). The rules
of limits are introduced, instead, at the end of the chapter, where they are used to evaluate
certain limits involving trigonometric functions.
In Chapter 10, the derivative of a function is first defined in the special case that the
graph of the function passes through the origin, and the tangent line (through the origin)
is defined in a precise way as the best approximating line to the graph near the origin.
Then, in the general case, the derivative can be defined at any point in the domain of a
function by means of an appropriate horizontal and vertical shift of the function so that
the derivative can be computed using the formula for the derivative in the special case
(at the origin). With this approach to the definition of the derivative, it is possible for
students to focus on the special case where the formula for the derivative is the simplest
possible. Furthermore, the concept of a tangent line as a “best approximating line” is a
much more flexible concept than the concept of the tangent line as a limit of secant lines,
as it is sometimes defined. Consequently, students should be able to identify a tangent
line more easily in certain situations. For example, they shouldn’t have too much difficulty
identifying the x-axis as the tangent line to the graph of x = y^3 at the origin. The definition
of the derivative in the special case (of a graph passing through the origin) also makes it
possible to derive the formula for the chain rule in the same special case, in which it is much
easier to understand why the slope of the tangent line (through the origin) to the graph of
a composition of functions is the product of the slopes of the tangent lines (through the
origin) to the graphs of each of the functions.
One of the innovative aspects of this book is the introduction of vectors at an early
stage (in Chapter 2). The concept of a vector is very natural and vectors have very practical
applications; however, the notation and terminology relating to vectors may be confusing
at first. The hope is that a gradual introduction to vectors will give students more time to
become comfortable with vectors. (Vectors are typically introduced for the first time in a
course in vector calculus, causing great alarm to the students!)
Another surprise in this book is the inclusion of Chapter 5 on spherical trigonometry.
Much of the material in this chapter was obtained from the book “Plane and Spherical
Trigonometry’’, by K. Nielsen and J. Vanlonkhuyzen, first published in 1944, which is one of
a few texts available on this topic. The purpose of including a chapter on spherical trigonometry
in this book is to give students the opportunity to acquire a deeper understanding of
trigonometry and also to give students some exposure to non–Euclidean geometry. Chapter
5 also has a section on vectors in three dimensions, including the definition of the cross
product of vectors, which students normally do not see until they take a course in vector
calculus. Spherical trigonometry is also interesting from a historical point of view: many
of the formulas relating to spherical trigonometry were discovered by Arabic and Iranian
mathematicians from the ninth to the thirteenth centuries.
A few comments regarding format, notation and mathematical language are in order:
new terminology is presented for the first time in italics, the most important definitions are
presented in highlighted definition boxes, statements that make important clarifications
are presented in highlighted remark boxes, and theorems, corollaries and lemmas are presented
in highlighted boxes. Examples are also presented in highlighted boxes in order
to distinguish them from the main text. A theorem is a general mathematical statement
which is proved on the basis of known mathematical truths. A corollary is a consequence
or a special case of a theorem that is important enough to be stated separately from the
theorem. Lemmas are statements that can be used as stepping stones toward the proofs of
more general statements (theorems). Throughout this book, the bi-conditional phrase “if,
and only if” is used when two statements are implied by each other.
This book can serve as a textbook, a reference book or a book that can be read for fun!
Please tell your friends about it, if you like it!
The author should be contacted regarding any errors in the book so that corrections can
be made for future editions. Suggestions for future editions will also be welcome.