Mathematics: Marvels and Milestones

225 Pages·2010·7.78 MB·English

by  A. Audichya

MATHEMATICS
Marvels and Milestones
(Queries and Answers)

MATHEMATICS
Marvels and Milestones
(Queries and Answers)
A.L. Audichya
Oxford
Book Company
Jaipur India
I
ISBN: 978-81-89473-40-2
First Published 2008
Oxford Book Company
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whatsoever
A nation dies when it stops asking questions .
.... Anonymous

Preface
1. What is the object of writing this book?
To transport the reader to the highest level of mathematical
awareness and to acquaint him with outstanding mathematical
achievements.
2. Which mathematical achievements are being alluded to ?
The foremost of them all is the pluralization of mathematics, i.e.,
where we had geometry, we now have geometries, and algebras rather
than algebra, and number systems rather than number system.
Some others are:
Galois' theory of algebraic equations;
Godel's incompleteness theorem;
Fourier's series and infinite sets;
Group Theory; Matrices; complex analysis;
Topology; Functional analysis;
etc.
3. For whom is the book intended?
It is intended for the intelligent layman who is in search of short
and pointed answers to his queries but is little inclined to undertake
detai led study of mathematical ideas and concepts.
4. Does it hold any appeal for young students?
Yes. Here they can have a glimpse of mathematics beyond what
they had studied at the school.
5. Is the book intended to be a substitute for some text book?
No, not at all. The aim is far too modest. It is to spur the reader to
go on to fuller accounts than given here.
6. Is the book of any interest to the mathematician ?
A mathematician is usually confined to his special but limited field
Preface
of interest. This book will provide him with an overall view of
mathematics.
This book will also assist him in seeking answers to his philosophical
uncertainties in mathematics. Incidentally every discipline has such
uncertainties.
7. The book has three broad divisions. Should they be read
in the sequence in which they are given in the book?
No, not necessarily. Any order may be adopted.
The questions also need not be read consecutively unless they
interest the reader. If something seems uninteresting or unattractive
at first glance it may be skipped.
One could come back to it if one thought it was still interesting.
8. Why has th~ question answer form of description been
adopted?
It is because long narratives soon tire down the patience of the
general reader whereas question answer form helps to sustain his
interest.
9. What pattern is followed in the sequence of questions ?
As far as possible logical pattern is followed, by which is meant
that a question is either suggested or anticipated by the previous
question or it may be a related question.
10. What is the style of presentation?
The answers are in simple, lucid and easy to understand language,
and meant to be brief as far as possible.
11. But what if sometimes detailed answers are unavoidable ?
In such cases the answers have been split into small paragraphs
after one or more of which one could skip the rest according to taste
and patience.
12. What is the basic requisite for reading the book?
Love for mathematics and for things mathematical.
13. What should be the mathematical background for reading
this book?
Not much. Knowledge of elementary mathematics is enough.
Prefac:e
14. What are the main topics in the chapter on Geometry ?
It includes the following:
(i) Euclidean geometry and allied concepts.
(ii) Lobachewskian and Riemannian geometry.
(iii) Geometry of the Earth, space and Elementary particles.
(iv) Projective Geometry.
(v) Coordinate geometry of2, 3, 4 and n dimensions.
(vi) Geometry of colour space.
(vii) Finite geometry.
(viii) Topology.
(ix) Problem of the Bridges of Koenigsberg.
(x) Four colour problem.
(xi) The axiomatic method in geometry.
(xii) Hilbert's Formalism.
(xiii) Godel's discovery.
15. What are the main topics in the chapter on Algebra?
It includes the following:
(i) Arithmetic as abstraction.
(ii) Arithmetic as the theory of numbers.
(iii) Extension of the number system.
(iv) Algebra as the Theory of Equations.
(v) Galois' Theory of Equations.
(vi) Diophantine Equations.
(vii) Abstract Algebra.
(viii) Theory of Groups and related matters.
(ix) Rings, Vectors, Matrix, Integral Domain, Field, Vector
space, Linear Algebra.
(x) Hilbert space, Banach space.
(xi) Boolean Algebra.
(xii) Russell's epigram of 1901.
(xiii) Countable and Uncountable sets.
(xiv) Continuum Hypothesis.
(xv) Barber's Paradox.
(xvi) Russell's Paradox.