Download this PDF book: A Gateway to Modern Mathematics: Adventures in Iteration I
This book is about iterative processes, i.e., processes that are done repetitively (“reiterated”); such processes often result in sequences which are rich in patterns, making them ideal for investigation.
It is written for the student who loves to tinker around with numbers and equations, and with mathematical ideas in general; tinker around just as a child, does when he or she wanders into a wooded region and begins to explore the many marvels and mysteries that are to be found in undisturbed places.
The world of mathematics too is full of wonders but, sadly, many children never see them at all, or even suspect that they exist. The fact that mathematics is a living science and that one can make discoveries and actually contribute to the growth of the subject is hardly ever pointed out at the school level.
And yet, the actual fact is that mathematics is alive, and not merely a collection of formulas and theorems dating from Greek and Vedic times. New discoveries are constantly being made in the subject every year; new areas of research develop constantly. Problems dating from many centuries get solved; alongside this, new questions get asked, new problems posed.
The culture of the subject is such that mathematicians can speak to one another across barriers of space and time—not just decades, but centuries —with ease and familiarity, with no fear of not being understood.
To give just a few instances: in the last decade of the twentieth century, the Princeton mathematician Andrew Wiles dramatically laid to rest a question that had been lying unresolved (not for want of trying!) for more than three hundred years—the famous Fermat conjecture
A Gateway to Modern Mathematics: Adventures in Iteration I is the first of a two-volume work on iterations in the RMS series Little Mathematical Treasures. The books in this series address senior secondary school students who are interested in exploring mathematics a little beyond what the school curriculum offers.
Iterations is an exciting topic of study and should interest both the amateur as well as the professional. Many of the iterations in elementary mathematics offer scope for extended investigation. They are like a gateway to important themes of modern mathematics such as fractals and chaos and offer a route for experiencing the experimental and visually aesthetic side of mathematics.
This book, which is at an elementary level, introduces the idea of iteration. It also explores various associated notions like fixed points, orbits, cycles, limit points, convergence, solution of equations and cobwebbing. It contains a large number of illustrative examples from the world of arithmetic, algebra and geometry.
Students preparing for the mathematical Olympiads will benefit from a study of the book, and teachers who run mathematics clubs will find here a rich source of material.
Contents:
1 What are Iterations?
2 Terms and Symbols
3 Examples of Iterations
4 The Four-Numbers Game
5 Sums of Squares
6 Fixed Points and Cycles
7 The Kaprekar Constant
8 Self-Born Numbers
9 Self Referential Numbers
10 Cyclical Functions
11 A Miscellany-I
12 A Miscellany-II
13 Geometric Iterations
14 Techniques of Proof
15 Solving Equations
16 Cobwebbing
17 Memorable Problems from the IMOs
18 Miscellaneous Problems
About the book:
Publisher : Universities Press (October 19, 2009)
Publication date : October 19, 2009
Language : English
Pages : 289
File : PDF, 9MB
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