DESIGN AND IMPLEMENTATION OF COMPUTER-AIDED SYSTEM THAT SOLVES ALGEBRAIC EQUATIONS

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PROJECT TOPIC AND MATERIAL ON DESIGN AND IMPLEMENTATION OF COMPUTER-AIDED SYSTEM THAT SOLVES ALGEBRAIC EQUATIONS

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• Name: DESIGN AND IMPLEMENTATION OF COMPUTER-AIDED SYSTEM THAT SOLVES ALGEBRAIC EQUATIONS
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• Length: [125] Pages

ABSTRACT

Traditionally, the concept of teaching mathematics has always been a teacher – student relationship; in which the teacher explains the concept of the topic to the student and illustrates it with some examples. The student is then left to understand the topic on his or her own using the tools given by the teacher. A problem often results when the student needs a guide while practicing and the teacher is not available. In that case, learning becomes slow and hindered. As we are in a digital age, where computers have been built to emulate most services usually offered by a human, it is believed that the computer can also stand in the gap for the teacher in his / her absence. With respect to algebra, the objective of this project is the design and implementation of a computer aided system that algebraic equations with limitations to simultaneous equations, quadratic equations and cubic equations (involving real numbers only). The system is designed using Java, CSS (Cascading Style Sheet) and MathTex as the programming languages. The methodology used is the Object-Oriented Analysis and Design method. It is expected that this software would be able to stand in the gap in the absence of the teacher and help students solve algebraic equations on their own, using their own examples and at their own pace and also help teachers in getting versatile knowledge of a algebraic equations by testing them with their own variables. It could also help teachers understand the most optimal methods for solving an algebraic equation to avoid errors in the process of teaching and learning.

TABLE OF CONTENTS

Title page                                                                                                                                 i

Certification                                                                                                                             ii

Approval Page                                                                                                                                    iii

Dedication                                                                                                                              iv

Acknowledgement                                                                                                                  v

Abstract                                                                                                                                  vi

Table of contents                                                                                                                    vii

List of Figures                                                                                                                         ix

List of Tables                                                                                                                          x

CHAPTER ONE:     INTRODUCTION

• Background of the study                                                                                            1
• Statement  of the problem                                                                                          2
• Objective  of the study                                                                                               3
• Significance of the study 3

1.4       Scope of the study                                                                                                      4

CHAPTER TWO:    REVIEW OF RELATED LITERATURE

2.0       Introduction                                                                                                                5

2.1       Theoretical Background                                                                                              7

2.2       Review of Related Literature                                                                                     9

2.3       Summary                                                                                                                     24

CHAPTER THREE:                        SYSTEM ANALYSIS AND DESIGN

• Introduction 26
• Analysis of Existing System 28
• Design of the Proposed System 29

3.2.1    Input Design                                                                                                               31

3.2.2    Output Design                                                                                                                        31

• Database Design 32

3.2.4    System Architecture                                                                                                   33

CHAPTER FOUR:              SYSTEM IMPLEMENTATION

4.0       Introduction                                                                                                                35

4.1       Choice of Programming Environment                                                                        35

4.2       Implementation Architecture                                                                                      37

4.3       Software Testing                                                                                                         37

4.4       Documentation                                                                                                           40

4.4.1    User Manual                                                                                                                40

4.4.2    Source Code listing                                                                                                     41

CHAPTER FIVE:    SUMMARY AND RECOMMENDATION

5.0       Summary                                                                                                                     42

5.1       Recommendation                                                                                                        43

REFERENCES                                                                                                                    44

APPENDIX                                                                                                                           46

List of Figures

Fig 1: Graph of Quadratic Equation showing Real and Distinct roots                                  17

Fig 2: Graph of Quadratic Equations showing Imaginary roots                                             18

Fig 3: Graph of a Cubic Equation                                                                                           24

Fig 4: Software Design Methodology using Object Oriented Analysis and Design              28

Fig 5: Use Case Diagram showing the design of the proposed system                                  30

Fig 6: Input Design sample using Quadratic Equations                                                         31

Fig 7: Output Design sample using Quadratic Equations                                                       32

Fig 8: System Architecture                                                                                                     34

Fig 9: Eclipse Integrated Development Environment Interface                                             36

Fig 10: Implementation Architecture                                                                                      37

Fig 11: Screenshot showing the Homepage                                                                            38

Fig 12: Screenshot showing Insertion of Variables                                                                39

Fig 13: Screenshot showing Choice Method                                                                          39

Fig 14: Screenshot showing Output                                                                                       40

List of Tables

Table 1: Database Structure for Simultaneous Equations Design                              33

CHAPTER ONE

INTRODUCTION

• Background of the study

Algebra is a field of mathematics that together with number theory, geometry and analysis, is the study of mathematical symbols and the rules for manipulating those symbols. It is a unifying thread of almost all mathematics. As a result, it includes everything from elementary equations, to the study of abstractions such as groups, rings and fields. Algebra is divided into two main parts; elementary and abstract or modern algebra. Elementary algebra encompasses some of the basic concepts of algebra, and is often used to build one’s understanding of arithmetic (dealing with specified numbers) by introducing quantities without fixed values (called variables). Elementary algebra is mostly concerned with structures within the realm of real and complex numbers. Abstract or modern algebra is the study of algebraic structures such as groups, rings, fields, modules, vector spaces, lattices etc.

Algebraic equations are needed in many aspects of life such as engineering, industry, medicine etc. As a result, we end up solving algebraic equations almost every day as we have to make decisions about specific quantities such as the amount of food to last a week, amount of materials needed for construction of a block in a site, amount of money needed to follow up a project from start to finish etc. As solving algebraic equations manually can be tiring or time-consuming; which is a consequence of the bulky steps one has to pass through that increases as the complexity of the equation increases, this project illustrates the construction of a desktop application that simulates and solves systems of algebraic equations and shows the user the algorithm followed by the computer in solving such equations.

1.1 Statement of the Problem

The following problems are observed in the manual solution of systems of algebraic equations;

• Time Conservation: Manually solving an algebraic equation from start to finish can be time consuming especially in cases where the equation is as complex as a quadratic or cubic equation or an exponential equation with long procedures.
• Cost: One who wants to solve an algebraic equation prefers manual solution using a calculator, a pen, a piece(s) of paper as well as four-figure tables in order to get his / her facts right. Thus, a project analyst in an industry would need a stand-by calculator, stacks of paper as well as a writing pen, all of which are costly to constantly supply and exhausts space.
• Accuracy: Man always has the tendency to make errors as a result of extensive approximation. Example, an average individual tends to solve mathematical problems with values not more than 3-4 decimal places. This can cause significant errors when used in the long-run.

1.2 Aims and Objectives

The aim of this project is to develop a computerized solution for a system of equations, the specific objectives are to;

• Reduce the time and energy exhumed in the process of solving algebraic equations manually.
• Help students solve algebraic equations on their own without the constant presence of a school teacher or the constant usage of physical textbooks, as well as alleviate the stress of having to carry too many learning materials while going for studies.
• Minimize the cost of analytic materials; in a data analyst’s office one workstation or desktop could carry as many mathematical problem solving applications as possible, which reduces the cost of buying, writing and solving materials such as the calculator, papers, pen etc.
• Aid teachers and examination bodies in the preparation of questions and the construction of error-free marking schemes.

1.3 Significance of the study

The beneficiaries of this project are;

• Science students involved in mathematics
• Mathematics teachers

Every science student needs an in-depth understanding of mathematics for any significant goal is to be achieved in his / her study. This project would provide a reliable means of sourcing for help in mathematical problems involving algebraic equations. It would help the student to solve algebraic equations with high degree of accuracy, sighting norms and exceptions, as well as rules to be followed.

Mathematics teachers would benefit widely because they no longer have to rely on the limited examples textbooks offer them, but they can try as many problems as possible to expand their understanding of the algebraic equation to be solved and hence increase their efficiency while teaching.

1.4 Scope of the Project

This project covers three main types of algebraic equations, namely;

• Simultaneous Equations which could involve;
  • Two Linear equations
  • One Linear and one quadratic equation
  • One quadratic and one cubic equation
  • One Linear and one cubic equation
• Quadratic Equations
• Cubic Equations

It is limited to real numbers, meaning that complex numbers, trigonometric functions and exponential functions are not considered in this context.