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Little Mathematical Treasures

RMS SERIES ON LITTLE MATHEMATICAL TREASURES
(Published jointly by the Ramanujan Mathematical Society and the Universities Press)

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 either with a note for writing a book for the series or with a draft of the proposed book. Decision of the Editorial Board will be final.

The RMS series on LMT is jointly published by RMS and Universities Press. The authors are paid a royalty of 10% of the net price of each book. The books are distributed in India and abroad by the Universities Press. 

Books Published in the Series:
1. A Gateway to Modern Mathematics: Adventures in Iteration-I by S.A. Shirali
2. A Gateway to Modern Mathematics: Adventures in Iteration- II by S. A. Shirali
3. Arithmetic and Algebra - Numbers and the beginnings of algebra by S. A. Shirali


 
 


A GATEWAY TO MODERN MATHEMATICS

Adventures in Iteration

By



Resume of "Adventures in Iteration "

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 61 74. 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 modem 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 modem mathematics.

Reviewer's Comments: "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 mathematics (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 effiectively the 'culture' of mathematics - the ideas which interest mathematicians and have done world-wide over the millennia." a reviewer.

"Arithmetic and Algebra" is the first in the set of three books, intended for students in the age group of 12 to 14 years (i.e., 7th – 9th standards) in schools, however it would be useful also to students in 11th and 12th standards and of course teachers.

One of the reviewers writes “Shirali's book is a superb introduction to some substantial material for students of mathematics in classes 7, 8 and 9 in Indian schools. If this book had come twenty years ago, there would have been an entire generation of school students who would have "journeyed into a world of pattern, power and beauty", to quote the author. This book is far from being routine and needs dedication both from the teacher and the pupil. It is time that such a book is available at a reasonable price to our students. For those students who have missed reading such a book at an earlier level, it will serve as revision material even at the plus two level. I would like to add that these volumes will inspire students to seek a career in pure or applied mathematics.

Any publisher who offers to publish this will be doing a great service to Indian Mathematics and will also find a large market from the enormous number of mathematically talented students in Indian schools.” - Alladi Sitaram