A Gateway to Modern Mathematics Adventures in Iteration I (Little Mathematical Treasures) by Shailesh Shirali Universities Press

272 Pages·2019·2.84 MB·English

by  Shailesh Shirali

A Gateway to
MODERN
MATHEMATICS
Adventures in Iteration I
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Ramanujan Mathematical Society
Little Mathematical Treasures
Editorial Policy
In times past, mathematics was studied and researched by a select few—an
elite class who valued the subject for its intellectual elegance and perhaps also
for its philosophic connotations. But today it is studied and practised by an
extremely large number of individuals, for mathematics has acquired a pivotal
role in the sustenance of a technologically advanced society.
Broadly speaking, there are two distinct groups of students to whom
expository material in mathematics may be addressed:
a. students who have a deep interest in mathematics and wish to pursue a
career involving research and teaching of mathematics;
b. students who are good at mathematics but wish to pursue a career in
some other discipline.
Clearly such students require suitable material (in addition to contact with
inspiring teachers and practising mathematicians) to nurture their talent in
the subject.
To this list we may add professionals in industry and government and
teachers in mathematics and other disciplines who require a good
understanding of certain areas of mathematics for their professional work.
The proposed series is addressed to mathematically mature readers and to
bright students in their last two years of school education. It is envisaged that
the books will contain expository material not generally included in standard
school or college texts. New developments in mathematics will be presented
attractively using mathematical tools familiar at the high school and
undergraduate levels. There will be problem sets scattered through the texts
which will serve to draw the reader into a closer hands-on study of the
subject. Readers will be invited to grapple with the subject, and so experience
the creative joy of discovery and contact with beauty.
A thing of beauty is a joy forever So it is with mathematics. The discoveries of
Archimedes, Apollonius and Diophantus have been sources of joy from very
ancient times. It is our hope that these books will serve our readers in a
similar manner.
Proposals for publication of a book in the series is invited with a draft of the
proposed book. The decision of the Editorial Board will be final.

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© Universities Press (India) Private Limited 2009
First published 2009
Reprinted 2010
eISBN 978-81-7371-960-8
e-edition:First Published 2015
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Contents
Cover
Half Title
Title Page
Copyright
Contents
Preface
1 What are Iterations?
1.1 A card trick
1.2 The riffle shuffle
1.3 Feedback
2 Terms and Symbols
3 Examples of Iterations
3.1 The GCD algorithm
3.2 Squares
3.3 Square roots
4 The Four-Numbers Game
4.1 The iteration
4.2 Exercises
4.3 Concluding remarks
5 Sums of Squares
5.1 The iteration
5.2 Exercises
5.3 Further comments
6 Fixed Points and Cycles
6.1 Fixed points
6.2 Cycles
7 The Kaprekar Constant
7.1 The Kaprekar iteration
7.2 Exercises
7.3 Concluding remarks
7.4 Recent results
7.5 Biographical details
8 Self-Born Numbers
8.1 The iteration
8.2 Exercises
8.3 Self-born numbers
8.4 Junction numbers
9 Self Referential Numbers
9.1 The iteration
9.2 Exercises
10 Cyclical Functions
10.1 Iterations on functions
10.2 Constructing cyclical functions
10.3 Fractional iterates
10.4 Iterated square root of the negative identity function
10.5 Exercises