KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY COMPUTING POWERS OF POSITIVE INTEGERS USING THE MODIFIED DETACHED COEFFICIENTS METHOD AND THE STAIRCASE METHOD. BY [AVOKA JOHN ( Bsc.)] A Thesis presented to the Department of Mathematics, Kwame Nkrumah University of Science and Technology in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE INDUSTRIAL MATHEMATICS Institute of Distance Learning JUNE 2013. DECLARATION I hereby declare that this submission is my own work towards the Master of Science (Msc) and that, to the best of my knowledge, it contains no material previously published by another person nor material which has been accepted for the award of any other degree of the university, except where due acknowledgement has been made in the text. Avoka John (PG6318311) ……….……………. ……….………… Student Signature Date Mr. K. F. Darkwah ……………..……… …….………… Supervisor Signature Date Prof. S. K. Amponsah …………………….. ………………. Head of Department Signature Date Prof. I. K. Dontwi ..………………..… ...………………. Dean, IDL Signature Date ii ABSTRACT The Modified Detached Coefficients Method is used to find power numbers ,say Q = n (d1d2…dm) to any positive nth power. The power number is first converted to a multinomial, Say 3 Q= (d1d2d3) m-1 m-2 0 n Q = (d1p +d2p +dmp ) …(1),where m is the number of digits and n is a positive power of the expansion. Then equation (1) is expanded using the multinomial expansion. The coefficients (d’s) of the p’s are extracted to become the result of the power number if and only if all the coefficients are single digits otherwise convert the coefficients which are more than 1 to p’s by replacing 10’s by p. For example, supposing 54 is a coefficient it is replaced by 5p+4,i.e. 5*10+4 =5*p+4. Where 10=p. The Staircase Method is also used to find power numbers. The procedure is the same as the Modified Detached Coefficients Method but after the expansion of the converted form of the power number. The coefficients (d’s) of the p’s are arranged in the staircase form. The result of the power number is then gotten by adding the staircase numbers arrangement column wise as done in multiplication of two numbers. Results of both methods were done manually. iii TABLE OF CONTENT DECLARATION ------------------------------------------------------------------------------- ii ABSTRACT ------------------------------------------------------------------------------------ iii TABLE OF CONTENT ------------------------------------------------------------------------ iv LIST OF TABLES ----------------------------------------------------------------------------- vii LIST OF ABBREVIATIONS --------------------------------------------------------------- viii DEDICATION: --------------------------------------------------------------------------------- ix ACKNOWLEDGEMENTS: ------------------------------------------------------------------ x CHAPTER 1 ----------------------------------------------------------------------------------- 1 INTRODUCTION ----------------------------------------------------------------------------- 1 1.1 BACKGROUND OF THE STUDY ----------------------------------------------------- 1 1.2 PROBLEM STATEMENT --------------------------------------------------------------- 3 1.3 OBJECTIVES OF THE STUDY -------------------------------------------------------- 3 1.4 METHODOLOGY ------------------------------------------------------------------------- 3 1.5 JUSTIFICATION -------------------------------------------------------------------------- 4 1.6 THESIS ORGANISATION -------------------------------------------------------------- 4 CHAPTER 2 ------------------------------------------------------------------------------------ 5 LITERATURE REVIEW ------------------------------------------------------------------- 5 CHAPTER 3 ----------------------------------------------------------------------------------- 20 METHODOLOGY --------------------------------------------------------------------------- 20 3.0 INTRODUCTION ------------------------------------------------------------------------ 20 3.1.1 Series: ------------------------------------------------------------------------------------ 20 3.1.2 Power series: ----------------------------------------------------------------------------- 20 3.1.3 Taylor series. ----------------------------------------------------------------------------- 21 3.1.4 Maclaurin series. ------------------------------------------------------------------------ 21 3.1.5 Convergence of the power series ----------------------------------------------------- 22 3.1.6 Arithmetic series.------------------------------------------------------------------------ 22 3.1.7 Geometric series. ------------------------------------------------------------------------ 23 3.1.8 Convergence of goemetric series ----------------------------------------------------- 23 3.2 THE BINOMIAL THEOREM --------------------------------------------------------- 24 iv 3.2.1 Example of the Binomial Theorem --------------------------------------------------- 24 3.3 The Multinomial Theorem --------------------------------------------------------------- 24 3.3.1 Example of The Multinomial Theorem --------------------------------------------- 24 3.3.2 Expanding numbers of different digits as Multinomials -------------------------- 25 n 3.3.3 Multinomial Expansion of the form (d1d2d3) ,where values of variables are given --------------------------------------------------------------------------------------------- 25 n 3.3.4 Expanding (d1d2d3) using the Modified Detached Coefficients Method ------ 25 n 3.3.5 Expanding (d1d2d3) using the Staircase Method. ---------------------------------- 26 3.4.1 Methodology of the modified detached coefficients method: -------------------- 27 3.4.2 Methodology of the staircase method: ---------------------------------------------- 28 3.5.1 Computing the cubes of integers, using the coefficients -------------------------- 29 method (3digits) -------------------------------------------------------------------------------- 29 3.5.2 Computing the cubes of integers, using the staircase method (3digits). -------- 31 CHAPTER 4 ----------------------------------------------------------------------------------- 33 ANALYSIS AND RESULTS --------------------------------------------------------------- 33 4.0 Introduction -------------------------------------------------------------------------------- 33 4.1.1 Computing the fifth power of integers, using the modified detached coefficients method (2digits). ------------------------------------------------------------------------------- 34 4.1.2 Computing the squares of integers, using the modified detached coefficients method (4digits ) ------------------------------------------------------------------------------- 35 4.1.3 Computing the squares of integers, using the modified detached coefficients method (5digits ) ------------------------------------------------------------------------------- 37 4.1.4 computing the cubes of integers, using the modified detached coefficients method (4digits) -------------------------------------------------------------------------------- 38 4.1.5 Computing the cubes of integers, using the modified detached coefficients method (5digits). ------------------------------------------------------------------------------- 40 4.2 Computing the fifth power of integers, using the staircase method (2digits) ----- 41 4.2.1 Computing the squares of integers, using the staircase method (4digits) ------- 42 4.2.2 Computing the squares of integers, using the staircase method (5digits ). ----- 44 4.2.3 Computing the cubes of integers, using the staircase method (4digits). -------- 45 4.2.4 Computing the cubes of integers, using the staircase method (5digits). -------- 47 4.3 Comparing the modified detached coefficients method with the staircase method49 v 4.3.1 Computing the cubes of integers, using the modified detached coefficients method ------------------------------------------------------------------------------------------- 49 4.3.2 Computing the cubes of integers, using the staircase method (4digits). -------- 51 4.4 DISCUSSION: ----------------------------------------------------------------------------- 52 CHAPTER 5 ----------------------------------------------------------------------------------- 54 CONCLUSION AND RECOMMENDATION ---------------------------------------- 54 5.1 CONCLUSION: --------------------------------------------------------------------------- 54 5.2 RECOMMENDATION ------------------------------------------------------------------ 54 REFERENCES-------------------------------------------------------------------------------- 55 vi LIST OF TABLES Table 4.1: Summary of results. ------------------------------------------------------------- 53 vii LIST OF ABBREVIATIONS IDL ----------Institute of Distance Learning KNUST-----Kwame Nkrumah University of Science and Technology viii DEDICATION I dedicate this thesis to my parents Mr. and Mrs. Donatus Avoka and to my siblings Avoka Juliana, Avoka Andrews, Avoka Jarvis, Avoka Victor and Avoka Emmanuel. Also, this piece of work is dedicated to the special individuals who have been a blessing to me: Ayine Ruth, Nabwomya John Millim, Wilfred Bormeh and Anarho Richard. ix ACKNOWLEDGEMENTS I am most grateful to God for his mercies and Guidance through graduate school and for His Grace I can still hope. I am extremely grateful to my Supervisor, Mr. K.F. Darkwah who trained me not only academically but also spiritually. I cannot repay in any form what has invested