Number jaunts for intellectual joy

COUNTING NUMBERS as everybody knows, form a sequence starting from 1. The sequence of whole numbers starts from 0. The sequence of integers has negative and positive number on either side of 0.

With this backdrop, an enchanting world of number sequences opens up for exploration to highlight their varied behavioural patterns.

Is it not incredible that man should continue to find himself engaged in feverish research in this regard right from ancient times?

Very few scholars in India have written books of general interest in mathematics.

The author of the book under review is himself a great scholar fascinated by number behaviour. Driven by an urge to share his experience with persons having school mathematics background up to ninth standard, he has come out with this exciting book.

To make the reader aware of the vastness of this rewarding intellectual enterprise, the author has been careful enough in titling the book as primer, that too dealing with sequences only.

As it is written in this computer age, the role of the computer as a powerful tool for speedy computation is taken into account.

Just as telescope and microscope have deepened and widened the vision of man and hence his perception, the computer has brought within access numbers of mind boggling length and remoteness.

Number theory is the only branch of mathematics where anyone with schooling can understand problems solved and unsolved and come to know of conjectures unproved or disproved with counter examples. Some can be seen intuitively making some discoveries connecting powers of numbers and feeling elated as to go about seeking recognition. This is not possible in other branches of mathematics.

The book is presented in two parts: part A (six chapters, 70 pages) on ''Method of Differences'', a powerful scanning tool in examining and identifying number sequence behaviour and part B (nine chapters, 140 pages) on a ``Gallery of Sequences'' portraying their individualistic behavioral patterns.

There are two appendices: `A' dealing with the ``Method of Proof'' by mathematical induction and `B' giving solutions to exploratory problems given at the end of each chapter and finally an Index.

Choosing classroom expository style, the author writes with passion as could be witnessed by humour and warmth with which he strives to take the reader along to become familiar with the nitty-gritty of fashioning proofs without which mathematical satisfaction would be missing.

He does not fail to warn against hastiness in drawing conclusions while at the same time mentioning explicitly that he is alive to the disappointment awaiting readers in not being provided with proofs of some assertions made, as they are tough. It would have been helpful if references with chapter or article numbers are suggested for those who feel compelled to know and appreciate the proofs, as finding them would be beyond their competence.

This book will see second edition and so while revising the book, certain aspects need to be given reconsideration and if need be rewriting:

The numerator and the denominator of the sum of the reciprocals of two consecutive odd numbers are shown to suggest Pythagorean triples. Two consecutive even numbers as well as two consecutive numbers too suggest Pythagorean triples, on being treated likewise. The proofs are simple and cute (7.4).

The ending digits 2,3 and 7,8 which are conspicuous by their absence among the ending digits in squares have alone ten complement behaviour in cubes, unlike other units place digits in cubes (8.2)

Introduction of negative digits in place value notation through use of bar notation as in logarithms facilitates elegant representation: (10.4)

e.g 729 {frac12} 243 + 27 {frac12} 9 {frac12}3 = 1101110 in base three.

If, in the Tower of Hanoi puzzle, the condition of not skipping the intermediate peg in every move is imposed, the least number of moves to be made for transfer of disks from the first pole to the last pole via the middle pole turns out to be 3 {+n} {frac12} 1

There is another arresting pattern in the transfer of disks if the pole is specified for getting the disks transferred.

If the number of disks is odd, the first move should begin at the pole specified; if even, the other pole is to be chosen. (10.6)

The abruptness in citing the series

1 {frac12} x {frac12} x{+2} + x{+5} + x{+7} - {hellip}..

could be smoothened by considering the expansion of the infinite product

(1 {frac12} x) (1 {frac12} x{+2}) (1 {frac12} x{+3}) (1 {frac12} x{+4}) {hellip}.. (12.2)Simpler and manageable version in giving a fraction as a sum of unit fractions in required number can be had through the property of equivalence in fractions (13.2)

Eg 3/5 = + 1/5 + 1/10 + 1/25 + 1/100 instead of

= + 1/11 + 1/111 + 1/1221 + 1/149096310

When publishers in our country fight shy of undertaking publication of books of general interest in mathematics by Indian authors on the plea that they are slow movers, Universities Press deserves the gratitude of lovers of mathematics and interested maths educators for catering to this need, while bringing out Indian editions of popular books of American organisations.

While looking for reference books in organising maths camps for high achievers, selecting prize books for maths contest winners, or giving mementos to guests in maths fairs, this book will be a good choice.

In short, it is a book for potential Ramanujans in VIII-IX classes and senior citizens.

P. K. SRINIVASAN

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