Table Of ContentWHAT IS MATHEMATICAL ANALYSIS?
DIMENSIONS OF
MATHEMATICS
Whatls
Mathematical Analysis?
lohn Baylis
Nottingham Polytechnic
M
MACMILLAN
© John Baylis 1991
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First published 1991 by
MACMILLAN EDUCATION LTD
Houndmills, Basingstoke, Hampshire RG21 2XS
and London
Companies and representatives
throughout the world
ISBN 978-0-333-54064-0 ISBN 978-1-349-12063-5 (eBook)
DOI 10.1007/978-1-349-12063-5
A catalogue record for this book is available
from the British Library.
To Cathy
CONTENTS
Preface ix
I NUMBERS, LINES AND HOLES I
The aim of this chapter is to arrive at adefinition of the set
of real numbers, the set on which calculus and all
subsequent developments are based. We have a brief look
at some historical background, and give a fairly detailed
account of the shortcomings of the rational number system.
Finally, the incompleteness of the ration als is formulated
explicitly and the reals - modelIed by the set of infinite
decimals - are shown to be complete
2 CURVES - CONTINUOUS, DISCONTINUOUS AND
UNIMAGINABLE 17
This chapter examines various ways in wh ich a function can
be discontinuous, eventually arriving at the 'official' e-D
definition of continuity. The usual pathological cases are
considered and there is a discussion about the usefulness of
continuous functions
3 ADDING UP FOREVER - PARADOXES AT INFINITY 55
Sequences and numerical series. The historically important
examples. The Riemann rearrangement theorem for
conditionally convergent series. A brief look at series
convergence tests
4 SMOOTH OR SPIKY? - DIFFERENTIATION 77
Differentiability approached through continuity of slope.
Desirable properties of functions differentiable over closed
vii
CONTENTS
intervals. Graphical interpretation using the derivative.
More pathological examples
5 PUTTING IT ALL TOGETHER - INTEGRATION 96
Integration as area, integration as backwards
differentiation: the connection. The integral as an 'easier'
concept than area. History of integration as a rigorous
theory. Setting up the Riemann integral. Conditions for
integrability and some paradoxical examples
6 A BRIEF LOOK AT FURTHER DEVELOPMENTS OF
ANALYSIS, AND SUGGESTIONS FOR FURTHER
READING 125
Index 128
viii
PREFACE
To teachers and learners
Analysis has a bad image. It is a hard subject which students have always
found difficult. Schools (justifiably) and many colleges (wrongly, I believe)
have taken the view: analysis is hard so let's not do it. Many universities
now seem to modify this slightly to: analysis is hard but unavoidable so let's
delay it.
The first view produces a generation of students which has avoided so me
difficulties but missed out on the considerable satisfactions too. The second
runs the risk that when analysis is finally tackled it is done in a great rush,
though it is surely a necessary condition for student enjoyment that a long
'sink in' time is given.
By way of justification for this book, here are three defensible claims: (i)
analysis is interesting, (ii) exposure to analytic ideas should come 'earlier
rather than later than is currently the fashion, and (iii) there is a need for
sympathetic texts at a level somewhat lower than the first 'formal' course in
analysis. A book which is thorough and sufficiently complete to be a basis
for a first year course is Mary Hart's Guide to Analysis in the Macmillan
Mathematical Guides series. It is also very sympathetic to student needs.
What is Mathematical Analysis? aims to put students into a confident and
positive frame of mind for tackling such a course. It provides a taste of
wh at analysis is about, and therefore must contain so me fairly hard proofs.
But these are embedded in far more chat than is customary in more formal
texts. I hope that prospective students emerge from this book with a clear
idea of why analysis is necessary and of the satisfaction it can provide.
Important message to learners
You will find the text regularly interrupted by mysterious things called
PFrA s. These are 'Pauses For Thought and Action'! They are included in
the firm belief that the best way to learn anything is to grapple with it
ix
PREFACE
yourself rather than passively absorb what others tell you. My recommen
dation is that you spend a minimum of a few minutes on each one as you
co me to them, before turning to the solution which generally appears on
the same or the following page.
As for previous knowledge required to appreciate this book, a nodding
acquaintance with calculus - just basic differentiation and integration -
would be useful background. There were several inconsistencies in early
treatments of the calculus and one of the motivations for the development
of anlysis was to remove these and render calculus 'respectable'.
x
1 NUMBERS, LINES AND
HOLES
In school we learn about numbers, connections between numbers, and how
to manipulate them to solve problems - that's arithmetic and algebra. At
the same time we learn about shapes and space, shapes being made from
straight and curved lines - that's geometry. Then at so me future stage
(earlier rather than later according to current trends) these two major
threads are woven together with the realisation that equations can describe
lines - and we have coordinate geometry. How's that for a whirlwind
summary?
This fusion of numbers and lines was achieved in the 17th Century by the
French philosopher and mathematician Rene Descartes, and later, at the
end of the 17th Century, combined with ideas of continuous change, it
flowered into the calculus of Isaac Newton in England and Gottfried
Leibniz in Germany. It is fair to say that the calculus is the single most
powerful mathematical tool we have for understanding the world we live
in. Newton quite quickly tamed the universe with a single differential
equation and in more recent times the calculus has given invaluable
insights into the biological sciences and social sciences like economics.
But something was seriously wrong in the state of mathematics. Bishop
Berkeley in 1734 noticed that what Newton was doing was basically
unsound. Today people gene rally admit that Berkeley was right and
Newton wrong, but at the time Berkeley's voice was largely unheeded
because the calculus's surge of success was unstoppable. Later, in the 19th
Century people like Cauchy (more of hirn later) realised that even the
work of Descartes could be seriously flawed. People took more notice then
because Cauchy was able to turn his criticisms into positive action, to
repair the flaws and make brilliant use of the newly improved system.
So what was the nature of these flaws? Were they philosophical, math
ematical, logical, aesthetic or just plain factual? A bit of all of these really,
but before we explore some of these aspects let us briefly remind ourselves
just how useful the geometry/algebra interplay iso
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