Exploring the Extension of Natural Operations on Intervals, Matrices and Complex Numbers W. B. Vasantha Kandasamy Florentin Smarandache ZIP PUBLISHING Ohio 2012 This book can be ordered from: Zip Publishing 1313 Chesapeake Ave. Columbus, Ohio 43212, USA Toll Free: (614) 485-0721 E-mail: [email protected] Website: www.zippublishing.com Copyright 2012 by Zip Publishing and the Authors Peer reviewers: Professor Paul P. Wang, Ph D, Department of Electrical & Computer Engineering, Pratt School of Engineering, Duke University, Durham, NC 27708, USA Prof. Catalin Barbu, V. Alecsandri National College, Mathematics Department, Bacau, Romania Florentin Popescu, Facultatea de Mecanica, University of Craiova, Romania. Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm ISBN-13: 978-1-59973-179-7 EAN: 9781599731797 Printed in the United States of America 2 CONTENTS Dedication 4 Preface 5 Chapter One INTRODUCTION 7 Chapter two EXTENSION OF NATURAL OPERATIONS TO INTERVALS 9 Chapter Three FINITE COMPLEX NUMBERS 41 Chapter Four NATURAL PRODUCT ON MATRICES 81 FURTHER READING 147 INDEX 148 ABOUT THE AUTHORS 150 3 D EDICATION PROF. IQBAL UNNISA (21.11.1936 – 12.06.2011) We dedicate this book to Prof. Iqbal Unnisa on her first death anniversary which falls on 12 June 2012. She was the first Muslim woman in India to receive her doctorate degree in the Mathematical sciences. As a brilliant researcher, she was the first woman to become director of the Ramanujan Institute for Advanced Study in Mathematics, University of Madras. Today we are proud of her mathematical legacy and we are inspired by her courageous spirit that never compromised and always challenged injustice. 4 PREFACE This book extends the natural operations defined on intervals, finite complex numbers and matrices. The intervals [a, b] are such that a ≤ b. But the natural class of intervals [a, b] introduced by the authors are such that a ≥ b or a need not be comparable with b. This way of defining natural class of intervals enables the authors to extend all the natural operations defined on reals to these natural class of intervals without any difficulty. Thus with these natural class of intervals working with interval matrices like stiffness matrices finding eigen values takes the same time as that usual matrices. Secondly the authors introduce the new notion of finite complex modulo numbers just defined as for usual reals by using 5 the simple fact i2 = −1 and −1 in case of Z is n – 1 so = n − 1 n where i is the finite complex number and i ’s value depends on F F the integer n of Z . Using finite complex numbers several n interesting results are derived. Finally we introduced the notion of natural product × on n matrices. This enables one to define product of two column matrices of same order. We can find also product of m × n matrices even if m ≠ n. This natural product × is nothing but the usual n product performed on the row matrices. So we have extended this type of product to all types of matrices. We thank Dr. K.Kandasamy for proof reading and being extremely supportive. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE 6 Chapter One INTRODUCTION In this chapter we just give references and also indicate how the arithmetic operations ‘+’, ×, ‘–’ and ÷ can be in a natural way extended to intervals once we define a natural class of intervals [a, b]; to be such that a > b or a < b or a = b or a and b cannot be compared. This has been studied and introduced in [2 books]. By making this definition of natural class of intervals it has become very easy to work with interval matrices; for working for interval eigen values or determinants or products further, it takes the same time as that of usual matrices (we call all matrices with entries from C or R or Q or Z or Z as usual n matrices). The operations the authors have defined on the natural class of intervals are mere extensions of operations existing on R. Thus this had made working with intervals easy and time saving. Further the authors have made product on column matrices of same order, since column matrices can be added what prevents one to have multiplication so we have defined this sort of product on column matrices as natural product. Another reason is if the transpose of a row vector (matrix) is the column matrix so it is natural, one can take the transpose of a column matrix and find the product and then transpose it. Thus the introduction of the natural product on matrices have paved way for nice algebraic structures on matrices and this is also a natural extension of product on matrices. We call the existing product on matrices as usual product. On the row matrices both the natural product and the usual 8 Exploring the Extension of Natural … product coincide. Further the natural product is like taking max (or min) of two matrices of same order. Finally this natural product permits the product of any two matrices of same order. This is another advantage of using natural product. Thus if two rectangular array of numbers of same order are multiplied the resultant is again a rectangular array of numbers of the same order. Natural product on square matrices of same order is commutative where as the usual product on square matrices of same order is non commutative. Next we have from the definition of complex number i, where i2 = –1 developed to the case of finite modulo integers. For if Z = {0, 1, 2, …, n – 1} the role of –1 is played by n – 1 n so we define finite complex number i to be such that i2 = n – 1 F F so that i3 = (n – 1)i and i4 = 1 and so on. F F F One of the valid observations in this case is that the square value of the finite complex number depends on n for the given Z . Thus i2 = 7 for Z , i2 = 11 for Z and i2 = 18 for Z . n F 8 F 12 F 19 When we have polynomial ring C(Z )[x] where C(Z ) = {a + bi n n F | a, b ∈Z and i2 = n – 1}, we see the number of roots for any n F polynomial in C(Z )[x] can be only from the n2 elements from n C(Z ); otherwise the equation has no root in C(Z ). For instance n n in C(Z )[x]; x2 + x + 1 = p(x) has no root in C(Z ). Thus we 2 2 cannot be speaking of algebraically closed field etc as in case of reals. Introduction of finite complex numbers happen to be very natural and interesting [4]. Finally authors have constructed matrices using the Boolean algebra P(X); the power set of a set X. Study in this direction is also carried out and these matrices of same order with entries from P(X) happen to be a lattice under min and max operations. Thus this book explores the possibilities of extending natural operation on matrices, construction of natural class of intervals and employing all the existing operations on reals on them and finally defining a finite complex modulo number [4]. Chapter Two EXTENSION OF NATURAL OPERATIONS TO INTERVALS In this chapter we just give a analysis of why we need the natural operations on intervals and if we have to define natural operations existing on reals to the intervals what changes should be made in the definition of intervals. Here we redefine the structure of intervals to adopt or extend to the operations on reals to these new class of intervals. Infact authors of this book often felt that the operations on the intervals (addition, subtraction multiplication and division) happen to be defined in such a way that compatability of these operations alone is sought. But it is suprising to see why we cannot define an interval [a, b] to be such that a > b or a < b or a = b. If operations can be so artificially defined to cater to the compatability what is wrong in accepting intervals of the form [a, b] where a > b; so if we make a very small change by defining an interval [a, b] in which a > b can also occur, certainly all the four operations defined on the reals can be very naturally extended.