ABSTRACT
This project gives a basic thing that is required to develop this application. It also provide a guideline for running the binomial theory using Pascal triangle up to power 9 as well as giving instruction to the user on how to expand the binomial theory from power of 1,2,3 to the power also expand input box for the values and button command for the calculate, refresh and exit.
TABLE OF CONTENT
COVER PAGE – – – – – – – – – – – I
TITLE PAGE – – – – – – – – – – – II
APPROVAL PAGE – – – – – – – – – – III
CERTIFICATION PAGE – – – – – – – – – IV
DEDICATION PAGE – – – – – – – – – V
ACKNOWLEDGMENT – – – – – – – – – VI
ABSTRACT – – – – – – – – – – – VII
TABLE OF CONTENT – – – – – – – – – VIII
CHAPTER ONE
1.1 INTRODUCTION – – – – – – – – – 1
1.2 STATEMENT OF THE PROBLEMS – – – – – – – 3
1.3 PURPOSE OF THE STUDY – – – – – – – – 4
1.4 SIGNIFICANCE OF THE STUDY – – – – – – – 4
1.5 SCOPE AND LIMITATION OF THE STUDY – – – – – – 4
1.6 DEFINITION OF TERMS – – – – – – – – 4
CHAPTER TWO
2.0 LITERATURE REVIEW – – – – – – – – – 6
2.1 THE BINOMIAL THEOREM – – – – – – – – – 9
2.2 STATEMENT OF BINOMIAL THEOREM – – – – – – – 11
2.3 PASCAL TRIANGLE AND THE BINOMIAL THEOREM – – – – -11
2.4 USING PASCAL’S TRIANGLE TO EXPAND A BINOMIAL EXPRESSION – 12
2.5 DIAGONALS OF PASCAL TRIANGLE – – – – – – – 13
2.6 NEWTON’S GENERAL BINOMIAL THEOREM – – – – – – 13
2.7 GENERALIZATION – – – – – – – – – – 14
2.8 THE MULTINOMIAL THEOREM – – – – – – – – 14
2.9 THE MULTINOMIAL THEOREM – – – – – – – – 15
CHAPTER THREE
3.0 RESEARCH METHODOLOGY – – – – – – – 16
3.16 SYSTEM ANALYSIS AND DESIGN – – – – – – – 16
3.2 ANALYSIS OF THE EXISTING SYSTEM – – – – – – – 16
3.3 ANALYSIS OF THE PROPOSE SYSTEM – – – – – – – 16
3.4 SYSTEM DESIGN – – – – – – – – – 17
3.4.1 INPUT DESIGN – – – – – – – – – – 17
3.4.3 OUTPUT DESIGN – – – – – – – – – – 18
3.5 HARDWARE AND SOFTWARE REQUIREMENT – – – – – – 19
3.6 SOFTWARE REQUIREMENT – – – – – – – – – 19
3.7 THE TYPE OF PEOPLE REQUIRED TO OPERATE:- – – – – – 19
3.8 PROTOTYPE – – – – – – – – – – 19
3.9 FLOW CHART – – – – – – – – – – – 20
CHAPTER FOUR
4.0 INTRODUCTION – – – – – – – – – – 21
4.1 IMPLEMENTATION – – – – – – – – – – 21
4.2 DOCUMENTATION – – – – – – – – – – 21
4.2 PROGRAMMING PARADIGM – – – – – – – – 21
4.2.1 CHOICE OF PROGRAMMING LANGUAGE – – – – – – 21
4.2.2 PROGRAM MODULES – – – – – – – – – 22
4.3 WELCOME SCREEN – – – – – – – – – – 22
4.4 PROGRAM TESTING – – – – – – – – – – 22
4.4.1 ERROR HANDLING – – – – – – – – – – 23
4.5 PROGRAM OUTPUT – – – – – – – – – – 23
4.6 USER MANUAL – – – – – – – – – -23
4.7 PROGRAM CODE – – – – – – – – – – 23
CHAPTER FIVE
5.0 SUMMARY, CONCLUSION, RECOMMENDATION – – – – – 32
5.1 SUMMARY – – – – – – – – – – – 32
5.2 CONCLUSION – – – – – – – – – – – 32
5.3 RECOMMENDATION – – – – – – – – – – 32
REFFERENCE – – – – – – – – – – 33
CHAPTER ONE
1.1INTRODUCTION:
Computer are becoming widely use in an increasing number of application and the growth is taking place at such a rate in the next decade only very institution in affected by the computations of Binomial Expansion using Pascal triangle.
The professional status of computing in development countries examine many devices application of computer own everyday term in own language to impress on the radio and TV and they appear in films and books or without science fiction.
Many of the routine activities in today society are perform by computers, some people may be exhilarated at the prospect of an Automation of Binomial Expansion using Pascal triangle.
Finally, computer as the latest innovation has improved the standard of every field of human endeavor.
A Pascal triangle is the name after the seventeenth country by a mathematician Blasé.
Several after mathematicians knew about that, and have applied their knowledge of triangle hundreds of years ago before the birth of Pascal in 1623. Binomial Expansion in general, when a Binomial like X+Y is raised to a positive integer power.
Blaise Pascal versions of the triangle is the set of number that form Pascal triangle were known before Pascal’s.
However, Pascal developed many user of it and was the first one to organize all information together in his treatise trait the triangle Arithmetic 1653. The number originally arise from Hindu studies of combinatory and Binomial number and the Greeks study of figurate number. The earliest explicit depiction of a triangle of binomial coefficient occur in the 10th century in commentaries on the Charles Shactra, and Indian book on Shanslarit prosody written by Pingalas work only in or before 2nd century be will Prigals work only survives in fragments, the commentator Halanetle, around 1975, use the triangle to explain observe references to Moruprastara, the staircase of mount, many it was also realized that the shallow diagonals of the triangle sum to the Fibonacci number in 1068, four columns of the first sixteen rows were given by the mathematician Bhatfotpala, who realized the combinatorial significant.
At the same time it wasdiscenosed in Persia (Iran) by the person mathematician Al-Karaji (953-2029) it was later repeated by the person poetastronomer mathematician Omar Khayyan (1048-1131) thus the triangle is referred to as the Khayyan-pascal triangle or simply the Khayyan were known including the Binomial theorem Khayyan used a method of finding roots based on the binomial expansion and therefore an binomial coefficients.
In Handy it is referred to as tartaglia is credited with the general formula for solving cabic polynomials (which may in fact be from scipioned delferro but was published by Gerolamo cordon 1545).
Pascal’s trait do triangle was published posthumously in 1661. In this Pascal collected several result’s then known about the triangle and employed them to solve problems in probability theorem the triangle was later name after Pascal by prime roumond to moutuort (1708) who called it “table de Pascal pour les combination” (French table or Mr. Pascal for combination) and Abraham de move (1780) who called it triangle Arithmetical pascalianum (Latin Pascal’s arithmetic triangle), which become the modern wax term name Binomial Expansion.
In this research work attention is mainly going to be focused on the applications of computer in binomial (Pascal Triangle) that can be automated. Pascal triangle is a number arranged in staggered rows such as:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 1 1
As it is well known that (a + b) 2=a2+2ab+b2 at a time many student find it difficult to write down similar expression for (a+b) 3 without doing some work on papers, however, it has been understood that such solution has some problems that may affect its effectiveness. Lack of orientation on how to calculate and write down the binomial Coefficient (Pascal Triangle) and entering the number in the computer to display the result is one of the basic problems most students do face and it is the objective of this research to know how power of (a+b) can be expanded with the computer.
1.2 STATEMENT OF THE STUDY
Despite the fact that Binomial Expansion, using Pascal triangle has been in use for long, people still find it difficult to understand the power and the Expansion of two numbers (a+b)6 in almost all the discipline.
At time student finds it difficult to writer down the Expansion of (a+b) 3 and (a+b) 4 without doing some work on paper. However it has been understand that such solution has some problem that affect the effectiveness. Lack of orientation on how to calculate and write the binomial expansion.
1.3 PURPOSE OF THE STUDY
The purpose of the study is to design a Automated system for solving Binomial expansion using Pascal triangle. Below are some of the specific purpose of this project.
- To design software that is capable of handling the activity of finding or solving problems related to binomial expansion.
- To maintain or enhance accuracy of the process unlike manual way.
- To study problem associated with the existing manual system and replace with the automated system and to enhance student on the use of computer in solving mathematical problem.
1.4 SIGNIFICANCE OF THE STUDY
The project was design to make student learn computerization of solving binomial expansion using Pascal triangle.
This topic Binomial Expansion is very important for student in all fields of study more especially Science and Engineering. Students will benefit from this project by reading through the project or text books on how to solve the problem topic, at the end this project will be able to solve binomial expansion using Pascal Triangle.
1.5 SCOPE AND DELIMITATION OF THE STUDY
This program is to expand the product of (A+B)9 the program will produce input box, command button for the user to enter number of the product and compute the result and also see how the binomial expansion relate to the simple construction of Pascal triangle, where it will display an error message if the user enter more than 9 power value. It only convert expansion loss than (A+B)9
1.6 DEFINITION OF TERMS
BINOMIAL EXPANSION: is used to expand binomial expansions (a+b) raised to given power without direct multiplication.
PASCAL TRIANGLE: Is the formed by connecting three size polygon points not in a straight line by straight line segments.
COEFFICIENT: Is number in an algebra expansion or constant quantity placed above and multiplying the variable.
POWER:Is a number that represent by an exponent or index, denoted by a superior numeral number which is use to multiply the whole number to the given power. In each term the first number decrease and the last increase
(a+b) 6=a6+6a5b1+15a4b2+20a3b3+15aab4+6a1b5+b6
FLOWCHART: Is a presentation of the each symbol has a new function. The algorithm steps determine which symbol to be used to represents it in the flow. Each step link to another step by using the directions arrows.
DESIGNING THE INTERFACE: In side class you define variable which are the properties of class. Theoretically you could declare all variable as public so that other classes could set and retrieve their values.
CODE: Is a source written in an acceptable computer language, the code may be a modification of an existing source or something completely new.
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