Conquering Mathematics: From Arithmetic to Calculus

312 Pages·1991·5.58 MB·English

by  Lloyd Motz

Conquering
Mathematics
From Arithmetic to Calculus
Conquering
Mathematics
From Arithmetic to Calculus
Lloyd Motz
and
Jefferson Hane Weaver
Springer Science+Business Media, LLC
LIbrary of Congress CatalogIng-In-PublIcatIon Data
Motz, Lloyd, 1910-
ConquerIng lathelatlcs fro. arlthletlc to calculus I Lloyd Motz
and ~efferson Hane Weaver.
p. CI.
Includes bIbliographIcal references and Index.
t. MatheaatIcs. I. Weaver, ~efferson Hane. II. TItle.
OA37.2.M68 1991
510--dc20 91-9329
CIP
10 9 8 7 6
ISBN 978-0-306-43768-7 ISBN 978-1-4899-2774-3(eBook)
DOI 10.1007/978-1-4899-2774-3
(~I1991 Lloyd Motz and Jefferson Hane Weaver
OriginallypublishedbyPlenumUSin1991.
Allrights reserved
No part ofthis book may be reproduced, stored ina retrieval system,or transmitted
inany form or byany means, electronic, mechanical, photocopying, microfilming,
recording,or otherwise, without written permission from the Publisher
To Minne and Shelley
Preface
We have designed and written this book. not as a text nor
for the professionalmathematician.but for the generalreader
who is naturally attracted to mathematics as a great intellec
tual challenge.and for the specialreaderwhose work requires
him to have a deeper understanding ofmathematics than he
acquired in school. Readers in the first group are drawn to
mental recreational activities such as chess. bridge. and
various types of puzzles. but they generally do not respond
enthusiastically to mathematics because of their unhappy
learning experiences with it during their school days. The
readersinthe secondgrouptumtomathematicsasanecessity.
but with painful resignation and considerable apprehension
regarding their abilities to master the branch ofmathematics
they need in their work. In either case. the fear of and
revulsion to mathematics felt by these readers usually stem
from their earlier frustrating encounters with it.
vii
viii PREFACE
This book will show these readers that these fears,
frustrations, and general antipathy are unwarranted, for, as
stated, it is not a textbook full of long, boring proofs and
hundreds of problems, rather it is an intellectual adventure,
to be read with pleasure. Itwaswritten to be easilyaccessible
and with concern for the mental tranquilityofthe readerwho
willexperience considerable fulfillment when he/sheseesthe
simplicity of basic mathematics. The emphasis throughout
this book is on the clear explanation of mathematical con
cepts. Wherever proofs are needed for clarity, they are
developed using the kind of reasoning one applies to the
solutionofproblemsingeneral. Wehaveincludedsuchproofs
not only to clarify the mathematics, but also to demonstrate
thatmathematicalreasoningisnot aspecial kind ofreasoning
that requires a special kind of mind but can be produced by
any mature mind.
Since all mathematics isessentially arithmetic invarious
guises, we have emphasized the importance of a thorough
understanding of arithmetic as the basis for understanding
all otherbranches ofmathematics.Accepting this asatruism,
those who understand arithmetic, which we all do to some
extent, can read this book with much more confidence,
gratification,and agreatersenseofaccomplishmentthanthey
would otherwise.With this in mind, wehave devoted the first
two chapters of our book to the basic laws of arithmetic,
since these laws apply to all branches of mathematics. The
reader, mastering these laws, willbe well prepared to under
stand and enjoy the chapters that follow.
Arithmeticcan be fully understood only ifthe properties
of numbers are known; therefore we discuss these in some
detail, pointingoutthatnumbershavetwoaspects:theordinal
(ordering) aspect, which leads to geometry, and the cardinal
(quantitative) aspect, which leads to the traditional arith-
PREFACE ix
metic. In the first two chapters we discuss the various
categories [integers, fractions, irrational numbers (algebraic
and transcendental), prime numbers, and complex numbers]
into which our number system has been divided. These
various subgroups ofthe number system are all governed by
the same arithmetic laws, but they differ from each other in
certain basic properties we describe and analyze; these
differences make arithmetic extremely interesting. We con
clude these two chapters with a discussion of the different
bases (e.g., base 10) on which a number system can be
constructed and with a discussion of logarithms.
In Chapter 3 we show that algebra is essentially the
arithmetic ofletters or symbols, pointing out that it enlarges
and generalizes arithmetic, releasing it from the restraints
imposed on it by the specific quantitative features that num
bers introduce. Algebra thus carries usfrom a numericallogic
to a pure logic of entities that mayor may not represent real,
concrete, physical objects or numbers. Indeed, algebras have
been created that have their own rules ofaddition and multi
plication. Thus, in the algebra ofvectors, the law ofaddition
isquite different from thatofarithmetic,and innoncommuta
tive algebras the product ab of two factors a and b does not
havetoequal ba (multiplicationisnoncommutative;theorder
ofthe factors inaproductisimportant).Clearly,inanoncom
mutative algebra the factors of a product are not ordinary
numbers but represent some kind of operation which must
be performed in a definite order. In another algebra, called
Boolean algebra, the elements may be abstract objects, such
aslogicalpropositionsandsetsofabstractentitiesorelectrical
networks. Boolean algebra is thus the algebra of pure logic.
We do not discuss these esoteric algebras in our text but
mention them here to indicate the vast wealth of algebra.
x PREFACE
Although any algebra can be developed as a logical
system of abstract concepts and propositions without refer
ence to any kind of concrete representation, it can be under
stood more easily ifit can be described by such a representa
tion. We do this in Chapter 4 where we develop the basic
algebraic concepts graphically, which leads one quite
naturally to the function concept. The graph (the plot ofthe
dependence of one quantity on another) shows the relation
ship between the two entities being plotted and thus defines
a function. The function concept leads to the polynomial and
to the algebraic equation, which we illustrate by the contour
of an imaginary road, laid off in one direction only (east
west), but rising and falling with respect to sea level. This
contourreveals allthe properties ofafunctional relationship,
no matter how complex it may be. It also gives us a simple
understanding of the solutions of the algebraic equation
associated with a polynomial ifwepicturethe road as cutting
sea-level points to the east or west of our position. The
distance from this position (the origin of our graph) of the
sea-level cutting point is a solution ofalgebraicequation; the
number of such solutions that may exist depends on the
complexity of the functional relationship. We discuss these
from the simple to the complex.
The limitation ofthe number system to points on a line
must be removed ifwe are to go from arithmeticto geometry.
We do this in Chapter 5 where we introduce the concept of
the dimensionality of a manifold. Limiting ourselves to the
points on a plane-a two-dimensional manifold of points
we develop the geometry of triangles, which requires the
introduction of the angle concept. We complete this chapter
by showing how most theorems in geometry, such as the
theorem of Pythagoras, can be deduced with elementary
reasoning and a few geometric properties oftriangles.
PREFACE xi
Continuing with geometry, we devote Chapter 6 to the
geometry ofthe circle and to trigonometry. These two facets
of mathematics go together because all the trigonometric
properties ofangles can be deduced from the unit circle, that
is,a circle whose radius isone unit oflength (one centimeter,
one inch, one foot, etc.). This method of introducing
trigonometry removes all mystery associated with it and per
mits the reader to grasp its basic elements quite easily. Since
the circleisaveryspecial caseofaclassofgeometricfigures
called conic sections-that play a very important role in
physical phenomena from astronomy to atomic physics, as
well as in mathematics itself, wedescribe the mathematics of
conic sections in Chapter 7which we call analytic geometry.
This phase of mathematics, which was developed by
Descartes, deals with the application of algebra to geometry;
the most interesting results of this work are the algebraic
equations of conic sections-the ellipse, the parabola, and
the hyperbola. We derive these equations not by long alge
braic and geometric arguments but from simple symmetry
arguments, and then show, in a very elementary way, that
the orbits of the planets are ellipses.
In the penultimate chapter, Chapter 8, we show how
differential calculus arose simultaneously from Newton's
study of the instantaneous speed of a body and Leibnitz's
analysis of the rate of change of a function with respect to
the entity on which it depends. We also discuss how the
calculus made possible much of what we now know as
modem science and technology.
Lloyd Motz
Jefferson Hane Weaver