Triples: Applications of Pythagorean Triples

190 Pages·2009·6.451 MB·English

by  Kenneth Williams

TRIPLES
Applications of Pythagorean Triples
♦ Kenneth Williams ♦
Triples shows applications of
Pythagorean Triples (like 3, 4, 5).
A simple, elegant system for
combining these triples gives
unexpected and powerful general
methods for solving a wide range of
mathematical problems, with far
less effort than conventional
methods. The easy text fully
explains how a vast range of
problems can be solved by this
single triple method. There are
applications in trigonometry (you
do not need any of those
complicated formulae), coordinate
geometry (2 and 3 dimensions)
transformations (2 and 3
dimensions), simple harmonic
motion, astronomy, complex
numbers etc. etc.
TRIPLES
Applications of Pythagorean Triples
(general (Editor
Dr S N Bhavsar
Guest (Editor
Dr Murli ManoharJoshi
5
^Editorial Cartel
Abhijit Das
Andrew Nicholas
Ashutosh Urs Strobel
Bal Ram Singh
B D Kulkami
David Frawley
David Pingee
James T Glover
Kenneth R Williams
Keshavdev Varma
M A Dhaky
Mark Gaskell
Navaratna S Rajaram
P K Srivathsa
R K Tiwari
Radheshyam Shastri
Subhash Kak
Toke Lindegaard Knudsen
VV Bedekar
Vijay Bhatkar
Vithal Nadkami
W Bradstreet Stewart
TRIPLES
Applications of Pythagorean Triples
Kenneth Williams
Foreword by
L.M. SlNGHVI
Formerly High Commissioner for India in the UK
MOTILAL BANARSIDASS PUBLISHERS
PRIVATE LIMITED • DELHI
5 th Revised Edition: 2009
First Indian Edition: Delhi, 2003
(First Published by Inspiration Book in 1984)
© 1984 K.R. WILLIAMS
All Rights Reserved.
ISBN: 978-81-208-1957-3 (Cloth)
ISBN: 978-81-208-1958-0 (Paper)
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To
RUTH

PREFACE
The unchanging laws of number have always been a source of delight and inspiration.
Particularly attractive are the Pythagorean triples which have so many elegant and interesting
properties. But these triples are also of great practical use: through the theorem of Pythagoras
the triples link the three main branches of mathematics: number, algebra and geometry.
This appears to be the first time that these triples have been developed into a useful structure,
having applications in trigonometry, transformations in 2 and 3 dimensions, coordinate
geometry in 2 and 3 dimensions, solution of triangles and equations, complex numbers,
hyperbolic functions, simple harmonic motion, astronomy etc. Many more applications are
also likely to appear. This book shows how the triples (and their 3-dimensional equivalent,
quadruples) can be developed and applied and how they form a unifying thread linking many
areas of mathematics.
Several chapters from previous editions of this book have been replaced or rewritten.
Spherical trigonometry and prediction of planetary positions have gone and chapters on
applied mathematics applications, the triple method and hyperbolic functions are now
included. Topics which have been extended from previous editions include trigonometric
equations and rotation of curves in 3-dimensional space. The chapter on angles in perfect
triples has been improved and the chapter on sine, cosine, tangent and their inverses has been
completely changed.
This book also serves as an illustration of Vedic Mathematics: a mathematical system which
has been rediscovered by Sri Bharati Krsna Tirthaji (1884-1960) from ancient Vedic texts and
is expounded in his book (see Reference 1). This system is based on sixteen formulae which
are said to give one line answers to all mathematical problems. Being based on fundamental
principles these Vedic formulae are therefore conspicuous in any structure that is developed in
a simple and natural way. As the triples idea is introduced and extended in this book the
operation of these formulae is evident. The formulae are expressed in word form (for
example, By One More than the One Before) and as they arise in the text they are indicated by
italic type. An index of these formulae will be found at the end of the book.
The diagram below gives a guide as to how the chapters in this book depend upon each other,
so that Chapter 8 for example, can be understood by first reading only Chapters 1,2 and 6.
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