Automation of Binomial Expansion Using Pascal Triangle
Chapter One
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.
CHAPTER TWO
LITERATURE REVIEW
Binomial theorem, familiar at least in its elementary aspect to every students of algebra, has a long and reasonably plain history. Most people associate it vagnoly in their minds with the name of Newton, either invented it or it was carved on his tomb. In some way or the other it work did make an important advance in the general theory. First trace of the binomial theorem in Euclidii, 4 “if a straight line be cut random the square on the whole is equal to the square on the segments” if the segment are A and B this means in algebra language.
- (a+b)2=a2+b2+2ab.
The corresponding formula for the square of different is found in Eudidii,, 7 “if a straight line be cut at random the square on the whole and that on one of the segment both together, are equal to twice the rectangle contained by the whole and said segment and the square on the remaining segment” here if (a) represents the whole, and (b) the first segment, have
- a2+b2=2ab+(a-b)2
It would have been perfectly easy for Eudid to go ahead and prove the formula for the cube of a binomial, but that would have broken the thread of the argument. In books ii and x was prodigiously interested in the squares of binomials, any generalizations of those does not have interest at all. The modern tendency to generalized as far as possible and stretch each binomial to its most general, form was quit foreign to the thinking of the Greek in mathematics; clearness and precision were the sovereign qualities which were always sought. A wider mathematics curiosity in diaphanons who cubed various binomials, especially (n-1) whether he had a general formula or multiplied out each time is not clear (HEIBERG,1915).
It is a curious fact that the first use beyond Eudid ‘s for root’s have a significant remark in the commentary of Eudid on Archimedes essay on the measure of the circle, modo adpropinguando radix quadratadai numeric invenienda est, dictum est ab Herone in Metrics a pappo, thoone compluribus allies, allies, qui magnum syntaxin claudii ptolemi interpretation sunt”.
- Suggests a search in ptolomy’s syntaxes. I have failed to find the passage. Tannery assure us that popups followed the general method of hero of Alexandra.
- Hero’s method is simplicity itself. We wish .to find an apart something much closer to our familiar method of finding square roots in the work of theon of Alexandra who uses our technique of adding to our correction (A-a)2/2al
- Of course it is a question merely of order of procedure, for us to add this correction we have Hero’s formular we pass cube roots. Health say on 63 in no extent Greek writer do about A.D 390 with various translation of his Arysbhstiya 1 follow that of Datta and singh, Divid the second place by thrice the square of the cube root; subtract from the first afghan place the square of the quotient multiplied by thrice the proceed (cubo root) and (subtract) the cubo (of the quotation) from the afghans place. The quotient put down at the next place (in the line of the root) gives the root. I think that this shows clearly enough that Arybahata was familiar with the binomial formula for a cube. Whether the Hindus had the curiosity to raise binomial to higher power or not I cannot say; a power higher than the important of such matter as may be judge from the following quotation, which is the significant for the whole purpose of this paper. The writer is Omar Khayyam and in speaking of a work of his own, now must unfortunately. Although Pascal triangle is named after the seventeenth century by a mathematician Blaise Pascal, several other mathematician knew about and applied their knowledge of the triangle hundreds of years before the birth of Pascal 1623. As of today, the triangle appears to have been discovered independently by both the Persians and the Chinese during the eleventh century. Although no longer in existence, the work of Chinese mathematicians Chai Hsein (Ca 1050) showed that they were using the triangle to extract square and cube roots of number (Clawson, 133). Also having method extracting roots of numbers, Omar Khayyam (1048-1113), a Persian mathematician seemed to have called Pascal’s triangle.
Pascal triangle is a geometric arrangement of Binomial coefficients in triangle. The row of Pascal is numbers in old row are usually staggered relative to the number in even row. A simple construction of the triangle proceeds in the following manner. On the zero write only 1 then to construct the element of the following row add the number directly above and to the left directly above and to the right to find the new value. If either number of right or left is not present, substitute a zero in its place. For example the number in 1 and 3 in the first row 0+1=1 whereas the number in 1 and 3 in the third row are added to produce the number fourth row. The construction is related to the binomial coefficient by Pascal’s which state.
CHAPTER THREE
RESEARCH METHODOLOGY
Research methodology refers to the systematic and carefully investigation of a particular subject or system using various materials as sources of information. These include collection of procedure, techniques, tools and documentation aids, which will help system developing effect to implement a new information system.
SYSTEM ANALYSIS AND DESIGN
System analysis and design is the process of collecting and analyzing fact about an existing system with the aid to facilitate the development of an information system and computer application by bridging communication goal that exit common non-technical system. Analyzing is mainly concerning with proper study of the current system with aim of discovery area effective it’s efficiently.
ANALYSIS OF THE EXISTING SYSYTEM
Information retrieval in the manual system and finding a particular personal problem is difficult in the sense that all problems the user has to go through all the problems manually. This lead to in consistency and usage of time. Manual solving finding of an expansion of binomial take more time than when the system is computerized i.e. the automated system can solve an equation within the smitten which many taken up to is smitten in the manual system.
The problem associated with manual system are;
- Inaccuracy
- Data redundancy
- Vulnerability
- Lack of sophisticated mechanism to keep of successful
ANALYSIS OF THE PROPOSE SYSTEM
This is concerned with the coordination of activities, job procedure and software equipment utilization in other to achieve the desire goal. The propose system is an automated binomial expansion using Pascal triangle that will allowed many teachers and student from far near to solve problems portend these. Hence, is working with stand along computer.
CHAPTER FOUR
INTRODUCTION
IMPLEMENTATION
This chapter is concerned with the implementation of the proposed system that was carried out in the preceding chapter. The design was made in the previous chapter, and this chapter gives the implementation stage of the system. This includes implementing the attributes and methods of each object and integrating all the objects such that they function as a single system.
CHAPTER FIVE
SUMMARY, CONCLUSION, RECOMMENDATION
SUMMARY
The ultimate aim of this project is to expand the binomial theorem using Pascal triangle from power of 1,2,up to the power of 9 it automatically calculate, expand and display the result on the option that selected by the user. And the program will display and error message if the user enter more than 9 powers.
CONCLUSION
In conclusion the project has archive it objective in the sense that provide a fast reliable and an efficient means of calculating and expanding the binomial theory using Pascal triangle it has the problem of obtaining wrong and slow results which are obtained from manual calculation.
RECOMMENDATION
In line with this project I recommend that the computer available at the department should be properly maintaining to avoid computer break down and failure all the necessary programming language and compilers should be installed to enable student carry out their projects successfully. Power supply should be made available to the computer at all cost. More so the present relationship assistant been rendered by the project supervisors in the department for the students during project.
REFFERENCE
- Bradley Julia case (2003), Advanced programming using visual basic Net McGraw hill Companies.
- Bradley Julia case (2002), programming in visual basic 6.0 McGraw Hill Irwin J.K
- Backhouse J.K (1999), Pure Mathematics I, SPT Hould Sucsth.
- Backhouse J.K (2000), Pure Mathematics2, SPT Hould Sucsth.
- Bondasy, B.D (1996) “Pure Mathematics for Advance level; 2nd edition, Oxford London. Bradley.
- Gorman, A. (1 998) “Addition mathematics for west Africa”, s” Edition Longman Group, UK.
- Julia case (2002), programming in visual basic 6.0 McGraw Hill Irwin J.K Bradley Julia case (2003), Advanced programming using visual basic Net McGraw hill Companies.
