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. Decision of the Editorial Board will be final.
Editorial Board: K. R. Parthasarathy, Phoolan Prasad (editor-in Chief), E. Sampathkumar (Managing Editor), V. P. Saxena.
A Gateway to Modern
Mathematics
Adventures in Iteration -I
By
`Dr. Shilesh.
A. Shirali
About the Book
To “reiterate” anything means to say something for a second time. In mathematics, “iteration” refers to any sequence of operations that is performed repeatedly; its essential aspect is that the “output” at each stage is used as the “input” for the following step. Many actions in elementary arithmetic and algebra are iterations in hidden form; e.g., the Euclidean algorithm for finding the greatest common divisor of two integers, the division algorithm for finding the square root of a number, Newton's method for the numerical solution of equations, and the simplex method for solving linear programs.
Iterations are an exciting topic to study, for the amateur as well as the professional. Many of the iterations in elementary mathematics offer scope for extended investigation; e.g., the Kaprekar iteration that leads to the number 6174. Another example is the “four-numbers iteration.” In the process, one encounters unsolved problems that look easy but are extremely difficult, e.g., the Collatz “3n+1” problem. Concepts such as those of fixed point, limit point, convergence, orbit, etc., emerge naturally while studying iterations. They are like a gateway for learning important themes of modern mathematics, such as fractals and chaos; they offer a route for experiencing the experimental and visually aesthetic side of mathematics. The Mandelbrot set, the snowflake curve and the Sierpinski triangle, for example, are all defined through iterations.
These topics, and many more, are studied in the two-volume book Adventures in Iteration. Volume 1 is at an elementary level, and is suitable for students aged 13–18 years; Volume 2 is more challenging, and suitable for students aged 15–20 years. Teachers who run mathematics clubs in their schools will find here a rich source of material. The book will also be of value to members of the general public who have an interest in mathematics and take delight in learning a beautiful area of modern mathematics.
"I enjoyed reading this
manuscript, and I think that anyone with any curiosity about numbers and their
many strange and wonderful properties
will find it fascinating too. Writing a book around the topic of `iteration'
is an excellent idea - it is a central theme of current research in athematics (under such names as `non-linear
systems theory' . `chaos') yet there are examples easily accessible to
non-mathematicians, and easily stated problems which have resisted solution so
far. The author's treatment conveys very effectively the `culture' of
mathematics - the ideas which interest mathematicians and have done world-wide
over the millennia. ." a reviewer
About the author
Shailesh Shirali has a Ph.D. in Operations Research from the University of Texas. He has taught mathematics at Rishi Valley School (Krishnamurti Foundation India) for more than two decades. He served as Principal of the school, 1992–2004, and received the National Award for Teachers 2003 from the President of India.
He has a deep interest in the nurture of mathematically talented students, and since 1987, he has been closely involved with the Mathematical Olympiad movement in India. He has taken part in the training programmes for the Indian IMO teams each year from 1989 till the present. In July 1996, when India hosted that year's IMO in Mumbai, he served as Chairman of the Problem Committee, and as Deputy Chairman of the Jury. He has served as Team Leader of the Indian IMO team on several occasions.
He is actively involved in writing and publication activity. He serves on the Council of Editors for Resonance, an expository journal for science and mathematics education, published by the Indian Academy of Sciences, and is the Founding Editor of Samasyā, a problem-solving journal for school and college students, published by the Leelavati Trust. He has written many articles for Resonance and Samasyā. He has also written many books: A Primer on Number Sequences, First Steps in Number Theory: A Primer on Divisibility, A Primer on Logarithms, Adventures in Problem Solving, A Sourcebook of Mathematics for CBSE XIIth Standard, ISC Mathematics–11 and ISC Mathematics–12. Adventures in Iteration is his latest book.
Price: Rs. 200.00 in India, Pakistan, Sri Lanka, Nepal, Bhutan and Bangladesh and U.S $25.00 in other countries.
Order your copies from
Managing Editor, RMS Series on LMT
Department of Mathematics, University of Mysore
Mysore–57006
Karnataka, India.
e–mail: esampathkumar@eth.net