Document Sample

UNIVERSITY FOR DEVELOPMENT STUDIES FACULTY OF COMPUTATIONAL AND DEVELOPMENTAL MATHEMATICS HISTORY OF MATHS (MTH300) TOPIC: THE HISTORY OF ALGEBRA PRESENTED BY: NAME: OSAM SARFO ESTEPHAN ID: FAS/1857/07 SUBMITTED TO THE DEPARTMENT OF APPLIED STATISTICS DATE: MARCH 2010 CONTENT PAGES -Presentation ………………………………………………………………………………. i -Preface ………………………………………………………………………………. ii -Acknowledgement ………………………………………………………………………… iii -Dedication ……………………………………………………………………………… iv -Introduction ……………………………………………………………………………… v-ix History of mathematics ………………………………………………………………… v-ix CHAPTER 1: -DESCRIPTION OF ALGEBRA -Introduction……………………………………………………………………………………… 1 -Symbols and special terms………………………………………………………………… 1 -Operation symbols……………………………………………………………………………. 1 -Order of operation and grouping………………………………………………………. 2 -Classification of Algebra……………………………………………………………………. 2-3 -Some Special definitions…………………………………………………………………… 3 CHAPTER 2: -History of Algebra……………………………………………………………………………. 4 -Meaning of the word Algebra…………………………………………………………… 4-5 -Ancient origins………………………………………………………………………………… 5-6 -Algebra in the middle age renaissance………………………………………………… 6-7 -Advances in Algebra in the 19th century……………………………………………… 7-8 CHAPTER 3 DEVELOPMENT OF ALGEBRA………………………………………………………………… 9 -Egyptian Algebra…………………………………………………………………………………. 9 -Babylonian Algebra…………………………………………………………………………… 9-10 -Greek Geometrical Algebra………………………………………………………………… 10 -Diophantine Algebra…………………………………………………………………………. 10 -Arabic Algebra…………………………………………………………………………………… 11 -Abstract Algebra………………………………………………………………………………… 12 CHAPTER4 A BREIF HISTORY OF DIOPANTUS -Biography……………………………………………………………………………………………… 13 -Arithmetica………………………………………………………………………………………… 13-14 -History…………………………………………………………………………………………………. 14 -Influence…………………………………………………………………………………………… 14-15 -The father of algebra…………………………………………………………………………… 15 -Diophantine Analysis…………………………………………………………………………… 15 CHAPTER 5 CONTRIBUTIONS MADE BY OTHER MATHEMATICIANS -Al-khwarizimi (780?-850?)……………………………………………………….. 16 -Leonardo Fibonacci or Leonardo of Pisa(1170?- 1240?)…………………………… 16-17 -Hero of Alexandria (as20?-62?)…………………………………………………………….. 17 -Rene Descartes (1596-1950)…………………………………………………………… 17-18 -THE DEBATE OVER THE TITLE “FATHER OF ALGEBRA”………………………… 18 -MORDEN ALGEBRA……………………………………………………………………………… 18-19 -CONCLUTIONS…………………………………………………………………………………….. 19-21 -RECOMMENDATIONS………………………………………………………………………… 21 -REFERENCES………………………………………………………………………………….. 21 PRESENTATION This work is presented to the Department of Statistic of the Faculty of Computational and Developmental Mathematics, University for Development Studies (U D S) as part of Second Trimester Course in level 300. History of Mathematics I (MTH 300) PREFACE This report gives an account of all aspect of the history of algebra be it ancient origins, Greek algebra, Babylonian algebra, Arabic algebra, Egyptian algebra and algebra in the middle age renaissance, it is written in simple and concise manner for easy comprehension and interpretation. Information as obtain from sources such as Encarta encyclopedia, internet searches and other books of algebra. The report is categorized into five (5) main chapters, the introductory part talks about the history of mathematics from the 1700-1900 century. Chapter 0ne (1) mainly describes algebra as a branch of mathematics, its classification, symbols and special terms, chapter two (2) deals with the main concept history of algebra from the ancient origins through the middle age renaissance and advances made in algebra in the 19th century, chapter (3) outline the development of algebra under the following headings: Egyptian algebra, Babylonian algebra, Diophantine algebra and abstract algebra, chapter four (4) gives a brief history of Diophantus the father of algebra and lastly chapter five (5) talks about contributions made by other mathematicians to the history of algebra, the debate over the title "father of algebra" modern algebra, conclusion, recommendation and references. ACKNOLEDGMENT In fact if not for the grace and guidance of the almighty God it would have been very difficult if not impossible to produce this document, I therefore wishes to express my profound gratitude to God almighty for his guidance. I will like to give credit to the course instructor Mr. Innocent Zebaze for his invaluable contribution towards the success of this work. I also express my gratitudes to the following students of UDS: AddoBaffoefrancis (Financial maths), AmpomahBoafoe Frederic (Financial maths), TwumasiBoatengHayford (Biology), Adinkra Bright (Earth science) and Agyemang Denis (Statistics) for their unflinging support. DEDICATION Well an effective work of this nature cannot be obtained without the grace and total guidance of our Almighty God who saw me through this period of my research, I hereby dedicate this work to the Almighty God, my parents and all those who contributed towards the success of this work. INTRODUCTION HISTORY OF MATHEMATICS Mathematics a way of describing relationships between numbers and other measurable quantities. Mathematics can express simple equations as well as interactions among the smallest particles and the farthest objects in the known universe. Mathematics allows scientists to communicate ideas using universally accepted terminology. It is truly the language of science. Until the 17th century, arithmetic, algebra, and geometry were the only mathematical disciplines, and mathematics was virtually indistinguishable from science and philosophy. Developed by the ancient Greeks, these systems for investigating the world were preserved by Islamic scholars and passed on by Christian monks during the middle Ages. Mathematics finally became a field in its own right with the development of calculus by English mathematician Isaac Newton and German philosopher and mathematician Gottfried Wilhelm Leibniz during the 17th century and the creation of rigorous mathematical analysis during the 18th century by French mathematician Augustin Louis Cauchy and his contemporaries. Until the late 19th century, however, mathematics was used mainly by physicists, chemists, and engineers. At the end of the 1800s, scientific researchers began probing the limits of observation, investigating the parts of the atom and the nature of light. Scientists discovered the electron in 1897. They had learned that light consisted of electromagnetic waves in the 1860s, but physicist Albert Einstein showed in 1905 that light could also behave as particles. These discoveries, along with inquiries into the wavelike nature of matter, led in turn to the rise of theoretical physics and to the creation of complex mathematical models that demonstrated physical laws. Einstein mathematically demonstrated the equivalence of mass and energy, summarized by the famous equation E=mc2, in his special theory of relativity in 1905. Later, Einstein’s general theory of relativity (1915) extended special relativity to accelerated systems and showed gravity to be an effect of acceleration. These mathematical models marked the creation of modern physics. Their success in predicting new physical phenomena, such as black holes and antimatter, led to an explosion of mathematical analysis. Areas in pure mathematics—that is, theory as opposed to applied, or practical, mathematics—became particularly active A similar explosion of activity began in applied mathematics after the invention of the electronic computer, the ENIAC (Electronic Numerical Integrator and Calculator), in 1946. Initially built to calculate the trajectory of artillery shells, ENIAC was later used for nuclear weapons research, weather prediction, and wind- tunnel design. Computers aided the development of efficient numerical methods for solving complex mathematical systems. Without mathematics to describe physical phenomena, we might be living in a world with beautiful art, literature, and philosophy, but no technology. Even the medical advances of the last 50 years might not have occurred. Science and technology, in their turn, have provided many of the problems that motivated progress in mathematics. Such problems include the behavior of weather systems, the motion of subatomic particles, and the creation of speedier and smaller computers that can perform multiple tasks simultaneously. ANCIENT MATHEMATICS Counting was the earliest mathematical activity. Early humans needed counts to keep track of herds and for trade. Primitive counting systems almost certainly used the fingers of one or both hands, as evidenced by the predominance of the numbers 5 and 10 as the bases for most number systems today. The first advances in arithmetic were the conceptualization of numbers and the invention of the four fundamental operations: addition, subtraction, multiplication, and division. The earliest advances in geometry dealt with simple concepts such as the line and the circle. Further progress in mathematics had to await the Babylonians and Egyptians of about 2000 bc. MATHEMATICS FROM 1700 TO 1900 The scientific revolution of the 17th century spurred advances in mathematics as well. The founders of modern science—Nicolaus Copernicus, Johannes Kepler, Galileo, and Isaac Newton—studied the natural world as mathematicians, and they looked for its mathematical laws. Over time mathematics grew more and more abstract as mathematicians sought to establish the foundations of their fields in logic. 17th CENTURYMATHEMATICS The 17th century saw the greatest advances in mathematics since the time of the ancient Greeks. The major invention of the century was calculus. Although two great thinkers—Sir Isaac Newton of England and Gottfried Wilhelm Leibniz of Germany—have received credit for its invention, they built on the work of others. As Newton noted, “If I have seen further, it is by standing on the shoulders of giants.” Major advances also were made in numerical calculation and geometry. 18 –CENTURY MATHEMATICS During the 18th century calculus became the cornerstone of mathematical analysis on the European continent. Mathematicians applied the discovery to a variety of problems in physics, astronomy, and engineering. In the course of doing so, they also created new areas of mathematics. In geometry French mathematician Gaspard Monge developed the field of descriptive geometry. Monge received his opportunity when as a draftsman he was asked to draw up a fortification plan that could be defended no matter what the enemy’s position. Monge used techniques of geometry he had developed on his own to determine firing lines and produce the fortress plan. His system of descriptive geometry has applications for solving engineering and construction problems. Also in France, Joseph Louis Lagrange made substantial contributions in all fields of pure mathematics, including differential equations, the calculus of variations, probability theory, and the theory of equations. In addition, Lagrange put his mathematical skills to work in the solution of practical problems in mechanics and astronomy. The greatest achievement of his career came in 1788 with the publication of his Mécaniqueanalytique (Analytical Mechanics). In this work Lagrange used the calculus of variations to deduce from one simple assumption the mechanics of fluids and solids. The greatest mathematician of the 18th century, Leonhard Euler of Switzerland, was also the most prolific writer on mathematical subjects of all time. His treatises covered essentially the entire fields of pure and applied mathematics. He wrote major works on mechanics that preceded Lagrange’s work. He won a number of prizes for his work on the orbits of comets and planets, a field known as celestial mechanics. But Euler is best known for his work in pure mathematics. His Introductio in analysininfinitorum (Introduction to the Analysis of Infinites), published in 1748, approached calculus in terms of functions rather than the geometry of curves. Other works by Euler contributed to number theory and differential geometry (the application of differential calculus to the study of the properties of curves and curved spaces). 19- CENTURY MATHEMATICS The 19th century was a period of intense mathematical activity. It began with German mathematician Carl Friedrich Gauss, considered the last complete mathematician because of his contributions to all branches of the field. The century saw a great effort to place all areas of mathematics on firm theoretical foundations. The support for these foundations was logic—the deduction of basic propositions from a limited set of assumptions and definitions. Mathematicians succeeded not only in firming the foundations of analysis, as the techniques of calculus were by then known, but also in making great strides in the field. Mathematicians also discovered the existence of additional geometries and algebras, and more than one kind of infinity. PURE AND APPLIED MATHEMATICS Mathematics, the language of science, has two dialects: pure mathematics and applied mathematics. Both kinds of mathematics are used to solve problems. Pure mathematics is the study of abstract relationships, whereas applied mathematics applies mathematical analysis to real-world problems, such as the rate of global warming. The relationship between pure and applied mathematics is a complex one, and the boundary between the two is constantly shifting. Branches of mathematics: .Arithmetic .Geometry .Algebra .Trigonometry .Calculus .Probability and Statistics .Set Theory and Logic Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials equations and algebraic structures. Algebra uses symbols (usually letters) to represent unknown numbers in mathematical equations; it allows the basic operations of arithmetic such as addition, subtraction and multiplication to be performed without using specific numbers. Together with geometry, analysis, topology, combinatorics and number theory, algebra is one of the main branches of pure mathematics. CHAPTER 1 DISCRIPTION OF ALGEBRA INTRODUCTION Algebra is the branch of mathematics that uses symbols to represent arithmetic operations. One of the earliest mathematical concepts was to represent a number by a symbol and to represent rules for manipulating numbers in symbolic form as equations. For example, we can represent the numbers 2 and 3 by the symbols x and y. From observation we know that it does not matter in which order we add the numbers (2 + 3 = 3 + 2), and we can represent this equivalence as the equation x + y = y + x. The equation is valid no matter what numbers x and y represent. Because algebra uses symbols rather than numbers, it can produce general rules that apply to all numbers. What most people commonly think of as algebra involves the manipulation of equations and the solving of equations. SYMBOLS AND SPECIAL TERMS The symbols of algebra include numbers, letters, and signs that indicate various arithmetic operations. Numbers are constants (values that do not change), but letters can represent either unknown constants or variables (values that vary). Letters that are used to represent constants are taken from the beginning of the alphabet; those used to represent variables are taken from the end of the alphabet. OPERATION SYMBOLS The basic operational signs of algebra are familiar from arithmetic: addition (+), subtraction (-), multiplication (×), and division (÷). The multiplication symbol × is often omitted or replaced by a dot, as in a · b. A group of consecutive symbols, such as abc, indicates the product (the result of multiplication) of a, b, and c. Division is commonly indicated by a horizontal bar. A slash (/), may also be used to indicate division: a/c. ORDER OF OPERATIONS AND GROUPING Algebra relies on an established sequence for performing arithmetic operations. This ensures that everyone who executes a string of operations arrives at the same answer. Multiplication is performed first, then division, followed by addition, then subtraction. For example: 1+2·3 equals 7 because 2 and 3 are multiplied first and then added to 1. Exponents and roots have even higher priority than multiplication: 3 · 22 = 3 · 4 = 12 Grouping symbols override the order of operations. All operations within a group are carried out first. Grouping symbols include parentheses ( ), brackets [ ], braces { }, and horizontal bars that are used most often for division and roots. Adding parentheses to a previous example: (1 + 2) · 3 Indicates that 1 should be added to 2 first, and then the result multiplied by 3 for a total of 9 rather than 7. Brackets and braces are used in more complicated combinations that require multiple nested (one inside the other) groups. Operations within the innermost group are carried out first. Classification of algebra Algebra may be divided roughly into the following categories: Elementary algebra, in which the properties of operations on the real number system are recorded using symbols as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied. This is usually taught at school under the title algebra (or intermediate algebra and college algebra in subsequent years). University-level courses in group theory may also be called elementary algebra. Abstract algebra, sometimes also called modern algebra, in which algebraic structure such as groups, rings and fields are axiomatically defined and investigated. Linear algebra, in which the specific properties of vector space are studied (including matrices). Universal algebra, in which properties common to all algebraic structures are studied. Algebraic number theory, in which the properties of numbers are studied through algebraic systems. Number theory inspired much of the original abstraction in algebra. SOME SPECIAL DEFINITIONS A prime number is any integer (the counting numbers: 1, 2, 3, …; their negatives; and zero) that can be evenly divided only by itself and by the number 1 or the number -1. Thus, 2, 3, 5, 7, 11, 13, 17, and 19 are all prime numbers. A factor of a number is any integer, by which the number can be divided evenly, with no remainder. The factors of 6, for example, are 1, 2, 3, and 6, because 6 ÷ 1 = 6, 6 ÷ 2 = 3, 6 ÷ 3 = 2, and 6 ÷ 6 = 1. A linear equation with one variable is a polynomial equation of degree one—that is, of the form ax + b = 0. These are called linear equations because graphing these equations results in straight lines. A quadratic equation in one variable is a polynomial equation of degree two—that is, of the form ax2 + bx + c = 0. An indeterminate equation, such as x2 + y2 = z2, involves multiple unknowns. Any statement that contains the equality relation (=), such as 3x = 9, is called an equation. An equation is called an identity if the equality is true for all values of its variables; if the equation is true for some values of its variables and false for others, the equation is conditional. CHAPTER TWO HISTORY OF ALGEBRA The word algebra comes from the Arabic al-jabr, meaning restoration. During the 9th century Islamic mathematicians systemized algebra, which they called the science of restoration and balancing. Algebraic techniques, however, have been employed to solve simple equations for thousands of years. Because the subject was studied and written about in something like the modern sense, by scholars who spoke Arabic in what is now the Middle East, in the 9th century CE. Although classical Greeks and several of their predecessors and contemporaries had investigated problems we now call "algebraic", these investigations became known to speakers of European languages not from the classical sources but from Arabic writers. So that is why we use a term derived from Arabic. Muhammad ben Musa al-Khwarizmi seems to have been the first person whose writing uses the term al-jabr. As he used it, the term referred to a technique for solving equations by performing operations such as addition or multiplication to both sides of the equation – just as is taught in first-year high school algebra. al- Khwarizmi, of course, didn't use our modern notation with Roman letters for unknowns and symbols like "+", "×", and "=". Instead, he expressed everything in ordinary words, but in a way equivalent to our modern symbolism. MEANING OF THE WORD ALGEBRA The word al-jabr itself is a metaphor, as the usual meaning of the word referred to the setting or straightening out of broken bones. The same metaphor exists in Latin and related languages, as in the English words "reduce" and "reduction". Although they now usually refer to making something smaller, the older meaning refers to making somethng simpler or straighter. The Latin root is the verb ducere, to lead – hence to re-duce is to lead something back to a simpler from a more convoluted state. In elementary algebra still one talks of "reducing" fractions to lowest terms and simplifying equations. The essence of the study of algebra, then, is solving or "reducing" equations to the simplest possible form. The emphasis is on finding and describing explicit methods for performing this simplification. Such methods are known as algorithms – in honor of al-Khwarizmi. Different types of methods can be used. Guessing at solutions, for instance, is a method. One can often, by trying long enough, guess the exact solution of a simple equation. And if one has a guess that is close but not exact, by changing this guess a little one can get a better solution by an iterative process of successive approximation. This is a perfectly acceptable method of "solving" equations for many practical purposes – so much so that it is the method generally used by computers (where irrational numbers can be specified only approximately anyhow). Some approximation methods are fairly sophisticated, such as "Newton's method" for finding the roots of polynomial equations – but they're still based essentially on guessing an initial rough answer. ANCIENT ORIGINS The history of algebra began in ancient Egypt and Babylon, where people learned to solve linear (ax = b) and quadratic (ax2 + bx = c) equations, as well as indeterminate equations such as x2 + y2 = z2, whereby several unknowns are involved. The ancient Babylonians solved arbitrary quadratic equations by essentially the same procedures taught today. They also could solve some indeterminate equations. The Alexandrian mathematicians Hero of Alexandria and Diophantus continued the traditions of Egypt and Babylon, but Diophantus's book Arithmetica is on a much higher level and gives many surprising solutions to difficult indeterminate equations. This ancient knowledge of solutions of equations in turn found a home early in the Islamic world, where it was known as the "science of restoration and balancing." (The Arabic word for restoration, al-jabru, is the root of the word algebra.) In the 9th century, the Arab mathematician al-Khwarizmi wrote one of the first Arabic algebras, a systematic exposé of the basic theory of equations, with both examples and proofs. By the end of the 9th century, the Egyptian mathematician Abu Kamil had stated and proved the basic laws and identities of algebra and solved such complicated problems as finding x, y, and z such that x + y + z = 10, x2 + y2 = z2, and xz = y2. Ancient civilizations wrote out algebraic expressions using only occasional abbreviations, but by medieval times Islamic mathematicians were able to talk about arbitrarily high powers of the unknown x, and work out the basic algebra of polynomials (without yet using modern symbolism). This included the ability to multiply, divide, and find square roots of polynomials as well as knowledge of the binomial theorem. The Persian mathematician, astronomer, and poet Omar Khayyam showed how to express roots of cubic equations by line segments obtained by intersecting conic sections, but he could not find a formula for the roots. A Latin translation of Al-Khwarizmi's Algebra appeared in the 12th century. In the early 13th century, the great Italian mathematician Leonardo Fibonacci achieved a close approximation to the solution of the cubic equation x3 + 2x2 + cx = d. Because Fibonacci had traveled in Islamic lands, he probably used an Arabic method of successive approximations. ALGEBRA IN THE MIDDLE AGES AND RENAISSANCE Early in the 16th century, the Italian mathematicians Scipione del Ferro, Niccolò Tartaglia, and Gerolamo Cardano solved the general cubic equation in terms of the constants appearing in the equation. Cardano's pupil, Ludovico Ferrari, soon found an exact solution to equations of the fourth degree (see quartic equation), and as a result, mathematicians for the next several centuries tried to find a formula for the roots of equations of degree five, or higher. Early in the 19th century, however, the Norwegian mathematician Niels Abel and the French mathematician Evariste Galois proved that no such formula exists. An important development in algebra in the 16th century was the introduction of symbols for the unknown and for algebraic powers and operations. As a result of this development, Book III of La géometrie (1637), written by the French philosopher and mathematician René Descartes, looks much like a modern algebra text. Descartes's most significant contribution to mathematics, however, was his discovery of analytic geometry, which reduces the solution of geometric problems to the solution of algebraic ones. His geometry text also contained the essentials of a course on the theory of equations, including his so-called rule of signs for counting the number of what Descartes called the "true" (positive) and "false" (negative) roots of an equation. Work continued through the 18th century on the theory of equations, but not until 1799 was the proof published, by the German mathematician Carl Friedrich Gauss, showing that every polynomial equation has at least one root in the complex plane. By the time of Gauss, algebra had entered its modern phase. Attention shifted from solving polynomial equations to studying the structure of abstract mathematical systems whose axioms were based on the behavior of mathematical objects, such as complex numbers, that mathematicians encountered when studying polynomial equations. Two examples of such systems are algebraic groups (see Group) and quaternions, which share some of the properties of number systems but also depart from them in important ways. Groups began as systems of permutations and combinations of roots of polynomials, but they became one of the chief unifying concepts of 19th-century mathematics. Important contributions to their study were made by the French mathematicians Galois and Augustin Cauchy, the British mathematician Arthur Cayley, and the Norwegian mathematicians Niels Abel and Sophus Lie. Quaternions were discovered by British mathematician and astronomer William Rowan Hamilton, who extended the arithmetic of complex numbers to quaternions while complex numbers are of the form a + bi, quaternions are of the form a + bi + cj + dk. Immediately after Hamilton's discovery, the German mathematician Hermann Grassmann began investigating vectors. Despite its abstract character, American physicist J. W. Gibbs recognized in vector algebra a system of great utility for physicists, just as Hamilton had recognized the usefulness of quaternions. The widespread influence of this abstract approach led George Boole to write The Laws of Thought (1854), an algebraic treatment of basic logic. Since that time, modern algebra—also called abstract algebra—has continued to develop. Important new results have been discovered, and the subject has found applications in all branches of mathematics and in many of the sciences as well. ADVANCES MADE IN ALGEBRA IN THE 19th CENTURY Algebra underwent a transformation during the 19th century, moving from the solution of polynomial equations to a study of the structure of algebraic systems. A first step in this direction was the publication of Treatise on Algebra (1830) by George Peacock, an English mathematician. Peacock attempted to provide algebra with the logical foundation Euclid had given geometry. The creation of different systems of algebra began with Irish mathematician William Rowan Hamilton. In searching for general properties of complex numbers, Hamilton in 1843 discovered quaternions, a class of complex numbers that break the commutative law in algebra. This law states that a x b = b x a. Hamilton’s quaternions paved the way for the study of new algebraic systems. Immediately after Hamilton’s discovery, German mathematician Hermann Grassmann and American mathematician and physicist J. Willard Gibbs began the analysis of three-dimensional vectors. From his investigations, Grassmann developed what is now called exterior algebra, which he applied to spaces of n (indefinitely many) dimensions. Gibbs used ideas of Grassmann to produce a system of vector analysis that could be applied to physics. He published his Elements of Vector Analysis in three parts from 1881 to 1884. Another major step in algebra during the 19th century was the development of the theory of groups, which had its beginnings in the work of Lagrange. Norwegian mathematician Niels Henrik Abel demonstrated that it was impossible to solve by elementary algebra any equation of degree greater than four. Évariste Galois of France introduced the group concept to the solution of algebraic equations, showing that equations have associated groups of substitutions that govern their solubility. Galois’s work signaled a new direction in mathematics. Just as Descartes had applied the algebra of his time to the study of geometry, so too did German mathematician Felix Klein and Norwegian mathematician Marius Sophus Lie apply the new algebra of the 19th century to geometry. Klein continued the group theory work of Galois, studying the properties that remained constant in a geometry when it underwent a group of transformations. Lie, too, worked in group theory, applying it not only to geometry but also to differential equations and other areas of mathematics. Among the mathematicians who used the discoveries of Hamilton and Grassman was George Boole in England. Boole claimed mathematics could be investigated in terms of logic and provided symbolic notation for mathematical operations. Boole’s major contribution to mathematics is Boolean algebra, an algebra of sets that later formed the basis of symbolic logic and computer technology. CHAPTER 3 DEVELOPMENT OF ALGEBRA The development of algebra is outlined in these notes under the following headings: Egyptian algebra, Babylonian algebra, Greek geometric algebra, Diophantine algebra, Hindu algebra, Arabic algebra, European algebra since 1500, and modern algebra. Since algebra grows out of arithmetic, recognition of new numbers - irrationals, zero, negative numbers, and complex numbers - is an important part of its history. The development of algebraic notation progressed through three stages: the rhetorical (or verbal) stage, the syncopated stage (in which abbreviated words were used), and the symbolic stage with which we are all familiar. Egyptian Algebra Much of our knowledge of ancient Egyptian mathematics, including algebra, is based on the Rhind papyrus. This was written about 1650 B.C. and is thought to represent the state of Egyptian mathematics of about 1850 B.C. They could solve problems equivalent to a linear equation in one unknown. Their method was what is now called the "method of false position." Their algebra was rhetorical, that is, it used no symbols. Problems were stated and solved verbally. The Cairo Papyrus of about 300 B.C. indicates that by this time the Egyptians could solve some problems equivalent to a system of two second degree equations in two unknowns. Egyptian algebra was undoubtedly retarded by their cumbersome method of handling fractions. Babylonian Algebra The mathematics of the Old Babylonian Period (1800 - 1600 B.C.) was more advanced than that of Egypt. Their "excellent sexagesimal [numeration system]. . . led to a highly developed algebra" [Kline]. They had a general procedure equivalent to solving quadratic equations, although they recognized only one root and that had to be positive. In effect, they had the quadratic formula. They also dealt with the equivalent of systems of two equations in two unknowns. They considered some problems involving more than two unknowns and a few equivalent to solving equations of higher degree. There was some use of symbols, but not much. Like the Egyptians, their algebra was essentially rhetorical. The procedures used to solve problems were taught through examples and no reasons or explanations were given. Also like the Egyptians they recognized only positive rational numbers, although they did find approximate solutions to problems which had no exact rational solution. Greek Geometrical Algebra The Greeks of the classical period, who did not recognize the existence of irrational numbers, avoided the problem thus created by representing quantities as geometrical magnitudes. Various algebraic identities and constructions equivalent to the solution of quadratic equations were expressed and proven in geometric form. In content there was little beyond what the Babylonians had done, and because of its form geometrical algebra was of little practical value. This approach retarded progress in algebra for several centuries. The significant achievement was in applying deductive reasoning and describing general procedures. Diophantine Algebra The later Greek mathematician, Diophantus (fl. 250 A.D.), represents the end result of a movement among Greeks (Archimedes, Apollonius, Ptolemy, Heron, Nichomachus) away from geometrical algebra to a treatment which did not depend upon geometry either for motivation or to bolster its logic. He introduced the syncopated style of writing equations, although, as we will mention below, the rhetorical style remained in common use for many more centuries to come. Diophantus' claim to fame rests on his Arithmetica, in which he gives a treatment of indeterminate equations - usually two or more equations in several variables that have an infinite number of rational solutions. Such equations are known today as "Diophantine equations". He had no general methods. Each of the 189 problems in the Arithmetica is solved by a different method. He accepted only positive rational roots and ignored all others. When a quadratic equation had two positive rational roots he gave only one as the solution. There was no deductive structure to his work. Arabic Algebra In the 7th and 8th centuries the Arabs, united by Mohammed, conquered the land from India, across northern Africa, to Spain. In the following centuries (through the 14th) they pursued the arts and sciences and were responsible for most of the scientific advances made in the west. Although the language was Arabic many of the scholars were Greeks, Christians, Persians, or Jews. Their most valuable contribution was the preservation of Greek learning through the middle ages, and it is through their translations that much of what we know today about the Greeks became available. In addition they made original contributions of their own. They took over and improved the Hindu number symbols and the idea of positional notation. These numerals (the Hindu-Arabic system of numeration) and the algorithms for operating with them were transmitted to Europe around 1200 and are in use throughout the world today. Like the Hindus, the Arabs worked freely with irrationals. However they took a backward step in rejecting negative numbers in spite of having learned of them from the Hindus. In algebra the Arabs contributed first of all the name. The word "algebra" come from the title of a text book in the subject, Hisab al-jabr w'al muqabala, written about 830 by the astronomer/mathematician Mohammed ibn-Musa al-Khowarizmi. This title is sometimes translated as "Restoring and Simplification" or as "Transposition and Cancellation." Our word "algorithm" in a corruption of al- Khowarizmi's name. The algebra of the Arabs was entirely rhetorical. They could solve quadratic equations, recognizing two solutions, possibly irrational, but usually rejected negative solutions. The poet/mathematician Omar Khayyam (1050 - 1130) made significant contributions to the solution of cubic equations by geometric methods involving the intersection of conics. Like Diophantus and the Hindus, the Arabs also worked with indeterminate equations. Abstract Algebra In the 19th century British mathematicians took the lead in the study of algebra. Attention turned to many "algebras" - that is, various sorts of mathematical objects (vectors, matrices, transformations, etc.) and various operations which could be carried out upon these objects. Thus the scope of algebra was expanded to the study of algebraic form and structure and was no longer limited to ordinary systems of numbers. The most significant breakthrough is perhaps the development of non-commutative algebras. These are algebras in which the operation of multiplication is not required to be commutative. (The first example of such an algebra were Hamilton's quaternions - 1843.) Peacock (British, 1791-1858) was the founder of axiomatic thinking in arithmetic and algebra. For this reason he is sometimes called the "Euclid of Algebra." DeMorgan (British, 1806-1871) extended Peacock's work to consider operations defined on abstract symbols. Hamilton (Irish, 1805-1865) demonstrated that complex numbers could be expressed as a formal algebra with operations defined on ordered pairs of real numbers ( (a,b) + (c,d) = (a+b,c+d) ; (a,b)(c,d) = (ac-bd,ad+bc) ). Gibbs (American, 1839- 1903) developed an algebra of vectors in three-dimensional space. Cayley (British, 1821-1895) developed an algebra of matrices (this is a non-commutative algebra). The concept of a group (a set of operations with a single operation which satisfies three axioms) grew out of the work of several mathematicians. Perhaps the most important steps were by Galois (French, 1811-1832). By the use of this concept Galois was able to give a definitive answer to the broad question of which polynomial equations are solvable by algebraic operations. His work also led to the final, negative resolution of the three famous construction problems of antiquity - all were shown to be impossible under the restrictions imposed. The concept of a field was first made explicit by Dedekind in 1879. Peano (Italian, 1858-1932) created an axiomatic treatment of the natural numbers in 1889. It was shown that all other numbers can be constructed in a formal way from the natural numbers. ("God created the natural numbers. Everything else is the work of man." - Kronecker) Abstract algebra is a branch of mathematics in which researchers have been very active in the twentieth century. CHAPTER4 A BRIEF HISTORY OF DIOPHANTUS Biography Little is known about the life of Diophantus. He lived in Alexandria, Egypt, probably from between 200 and 214 to 284 or 298 AD. While most scholars consider Diophantus to have been a Greek, others speculate him to have been a non-Greek, possibly either a Hellenized Babylonian, an Egyptian, aJew, or a Chaldean. Much of our knowledge of the life of Diophantus is derived from a 5th century Greek anthology of number games and strategy puzzles. One of the problems (sometimes called his epitaph) states: 'Here lies Diophantus,' the wonder behold. Through art algebraic, the stone tells how old: 'God gave him his boyhood one-sixth of his life, One twelfth more as youth while whiskers grew rife; And then yet one-seventh ere marriage begun; In five years there came a bouncing new son. Alas, the dear child of master and sage After attaining half the measure of his father's life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.' This puzzle implies that Diophantus lived to be about 84 years old. However, the accuracy of the information cannot be independently confirmed. In popular culture, this puzzle was the Puzzle No.142 in Professor Layton and Pandora's Box as one of the hardest solving puzzles in the game, which needed to be unlocked by solving other puzzles first. Arithmetica The Arithmetica is the major work of Diophantus and the most prominent work on algebra in Greek mathematics. It is a collection of problems giving numerical solutions of both determinate and indeterminate equations. Of the original thirteen books of which Arithmetica consisted only six have survived, though there are some who believe that four Arab books discovered in 1968 are also by Diophantus. Some Diophantine problems from Arithmetica have been found in Arabic sources. It should be mentioned here that Diophantus never used general methods in his solutions. Hermann Hankel, renowned German mathematician made the following remark regarding Diophantus. “Our author (Diophantos) not the slightest trace of a general, comprehensive method is discernible; each problem calls for some special method which refuses to work even for the most closely related problems. For this reason it is difficult for the modern scholar to solve the 101st problem even after having studied 100 of Diophantos’s solutions” HISTORY Like many other Greek mathematical treatises, Diophantus was forgotten in Western Europe during the so-called Dark Ages, since the study of ancient Greek had greatly declined. The portion of the Greek Arithmetica that survived, however, was, like all ancient Greek texts transmitted to the early modern world, copied by, and thus known to, medieval Byzantine scholars. In addition, some portion of the Arithmetica probably survived in the Arab tradition (see above). In 1463 German mathematician Regiomontanus wrote: “No one has yet translated from the Greek into Latin the thirteen books of Diophantus, in which the very flower of the whole of arithmetic lies hidden . . . .” Arithmetica was first translated from Greek into Latin by Bombelli in 1570, but the translation was never published. However, Bombelli borrowed many of the problems for his own book Algebra. The editio princeps of Arithmetica was published in 1575 by Xylander. The best known Latin translation of Arithmetica was made by Bachet in 1621 and became the first Latin edition that was widely available. Pierre de Fermat owned a copy, studied it, and made notes in the margins. INFLUENCE Diophantus' work has had a large influence in history. Editions of Arithmetica exerted a profound influence on the development of algebra in Europe in the late sixteenth and through the seventeenth and eighteenth centuries. Diophantus and his works have also influenced Arab mathematics and were of great fame among Arab mathematicians. Diophantus' work created a foundation for work on algebra and in fact much of advanced mathematics is based on algebra. As far as we know Diophantus did not affect the lands of the Orient much and how much he affected India is a matter of debate. THE FATHER OF ALGEBRA? Diophantus is often called “the father of algebra" because he contributed greatly to number theory, mathematical notation, and because Arithmetica contains the earliest known use of syncopated notation [11]. However, it seems that many of the methods for solving linear and quadratic equations used by Diophantus go back to Babylonian mathematics. For this reason mathematical historian Kurt Vogel writes: “Diophantus was not, as he has often been called, the father of algebra. Nevertheless, his remarkable, if unsystematic, collection of indeterminate problems is a singular achievement that was not fully appreciated and further developed until much later.” DIOPHANTINE ANALYSIS Today Diophantine analysis is the area of study where integer (whole number) solutions are sought for equations, and Diophantine equations are polynomial equations with integer coefficients to which only integer solutions are sought. It is usually rather difficult to tell whether a given Diophantine equation is solvable. Most of the problems in Arithmetica lead to quadratic equations. Diophantus looked at 3 different types of quadratic equations: ax2 + bx = c, ax2 = bx + c, and ax2 + c = bx. The reason why there were three cases to Diophantus, while today we have only one case, is that he did not have any notion for zero and he avoided negative coefficients by considering the given numbers a,b,c to all be positive in each of the three cases above. Diophantus was always satisfied with a rational solution and did not require a whole number which means he accepted fractions as solutions to his problems. Diophantus considered negative or irrational square root solutions "useless", "meaningless", and even "absurd". To give one specific example, he calls the equation 4 = 4x + 20 'absurd' because it would lead to a negative value for x. One solution was all he looked for in a quadratic equation. There is no evidence that suggests Diophantus even realized that there could be two solutions to a quadratic equation. He also considered simultaneous quadratic equations. CHAPTER 5 CONTRIBUTIONS MADE BY OTHER MATHEMATICIANS Al-Khwarizmi (780?-850?), Arab mathematician. Muhammad ibn Musa al-Khwarizmi was born in Khwarizm (now Khiva, Uzbekistan). He was librarian at the court of Caliph al-Mamun and astronomer at the Baghdad observatory. His works on algebra, arithmetic, and astronomical tables greatly advanced mathematical thought, and he was the first to use for mathematical purposes the expression al jabr, from which the English word algebra is derived. The Latin version, by the Italian translator Gerard of Cremona, of his treatise on algebra (based on a Hindu work) was responsible for much of the mathematical knowledge of medieval Europe. His work on algorithm, a term derived from his name, introduced the method of calculating by use of Arabic numerals and decimal notation. Leonardo Fibonacci or Leonardo of Pisa (1170?-1240?), Italian mathematician, who compiled and supplemented the mathematical knowledge of classical, Arabic, and Indian cultures, and who made contributions to the mathematical fields of algebra and number theory. Fibonacci was born in Pisa, Italy, a commercial city, where he learned the basics of business calculation. When Fibonacci was about 20, he went to Algeria, where he began to learn Indian numerals and Arabic calculating methods, knowledge he supplemented during more extensive travels. Fibonacci used this experience to improve on the commercial computing techniques he knew and to extend the work of classical mathematical writers, such as the Greek mathematicians Diophantus and Euclid. Few works by Fibonacci still exist; he wrote on number theory, practical problems of business mathematics and surveying, advanced problems in algebra, and recreational mathematics. His writings on recreational mathematics, which were often posed as story problems, became classic mental challenges as early as the 13th century. Such problems often involved the summation of recurrent series, such as the Fibonacci series (kn = kn-1 + kn-2—for example, 1, 2, 3, 5, 8, 13,. . .), which he discovered. Each term of this series is called a Fibonacci number—the sum of the two numbers preceding it in the series. Fibonacci solved the problem of calculating the value for any entry. He was awarded a yearly salary by the Republic of Pisa in 1240, indicating the importance accorded to his work and also, possibly, public service to the city's administration. Hero of Alexandria (AD20?-62?), Greek mathematician and scientist. His name is also spelled Heron. About 18 Greek writers were named Hero, or Heron, making individual identification difficult. However, Hero of Alexandria was probably born in Egypt and accomplished his work in Alexandria, Egypt. Hero wrote at least 13 works on mechanics, mathematics, and physics. He developed various mechanical devices, and many of them had practical uses. They include the aeolipile, a rotary steam engine; Hero's fountain, a pneumatic apparatus that produces a vertical jet of water by air pressure; and the dioptra, a primitive surveying instrument. He may be best known, however, as a mathematician in both geometry and geodesy (a branch of mathematics that seeks to determine the shape and size of earth, and the location of objects or areas on the earth). Hero handled problems of measurement more successfully than anyone of his time. He also devised a method of approximating the square roots and cube roots of numbers that are not perfect squares or cubes. Hero is sometimes credited with developing the formula for finding the area of a triangle in terms of its sides, but this formula was probably developed before his time. René Descartes (1596-1650), French philosopher, scientist, and mathematician, sometimes called the father of modern philosophy. The most notable contribution that Descartes made to mathematics was the systematization of analytic geometry (see Geometry: Analytic Geometry). He was the first mathematician to attempt to classify curves according to the types of equations that produce them. He also made contributions to the theory of equations. Descartes was the first to use the last letters of the alphabet to designate unknown quantities and the first letters to designate known ones. He also invented the method of indices (as in x2) to express the powers of numbers. In addition, he formulated the rule, which is known as Descartes's rule of signs, for finding the number of positive and negative roots for any algebraic equation. THE DEBATE OVER THE TITLE “FATHER OF ALGEBRA” The Hellenistic mathematician Diophantus has traditionally been known as the "father of algebra" but in more recent times there is much debate over whether al-Khwarizmi, who founded the discipline of al-jabr, deserves that title instead. Those who support Diophantus point to the fact that the algebra found in Al-Jabr is slightly more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully rhetorical. Those who support Al- Khwarizmi point to the fact that he introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation) which the term al-jabr originally referred to, and that he gave an exhaustive explanation of solving quadratic equations, supported by geometric proofs, while treating algebra as an independent discipline in its own right. His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems." MODERN ALGEBRA By the beginning of the 19th century, algebra had entered its modern phase. Attention shifted from numbers and solving polynomial equations to studying the structure of abstract mathematical systems whose laws are based on the behavior of mathematical objects. Groups, sets of elements and operations that take any two elements of a set and form another element of the set, are an example of such a system. Groups share some of the properties of number systems but also depart from them in important ways. Groups became one of the chief unifying concepts of 19th-century mathematics. French mathematicians Galois and Augustin Cauchy, British mathematician Arthur Cayley, and Norwegian mathematicians Abel and Sophus Lie made important contributions to the study of groups.German mathematician Hermann Grassmann laid the foundations of another important branch of modern algebra, vector analysis, during the 1840s. Vectors are mathematical quantities that have both magnitude and direction. Despite the abstract character of vector analysis, American physicist J. Willard Gibbs later recognized that it could be extremely useful for physicists. The velocity of a car, for example, can be considered a vector because it has magnitude (speed) and direction. The widespread influence of vector analysis led British mathematician George Boole to write An Investigation of the Laws of Thought (1854), an algebraic treatment of basic logic. Since that time, modern algebra— also called abstract algebra—has continued to develop. Important new results have been discovered, and algebra has found applications in all branches of mathematics as well as in many of the physical sciences. CONCLUTIONS The word algebra comes from the Arabic word al-jabr, meaning restoration. During the 9th century Islamic mathematicians systemized algebra, which they called the science of restoration and balancing. Algebraic techniques, however, have been employed to solve simple equations for thousands of years. Muhammad ben Musa al-Khwarizmi seems to have been the first person whose writing uses the term al-jebr. As he used it, the term referred to a technique for solving equations by performing operations such as addition or multiplication to both sides of the equation. Al-Khwarizmi, of course, didn't use our modern notation with Roman letters for unknowns and symbols like "+", "×", and "=". Instead, he expressed everything in ordinary words, but in a way equivalent to our modern symbolism. Diophantus looked at three types of quadratic equations ax2 + bx = c, ax2 = bx + c and ax2 + c = bx. The reason why there were three cases to Diophantus, while today we have only one case, is that he did not have any notion for zero and he avoided negative coefficients by considering the given numbers a, b, c to all be positive in each of the three cases above. Equations which would lead to solutions which are negative or irrational square roots, Diophantus considers as useless. To give one specific example, he calls the equation 4 = 4x + 20 'absurd' because it would lead to a meaningless answer. Diophantus is often regarded as the 'father of algebra' but there is no doubt that many of the methods for solving linear and quadratic equations go back to Babylonian mathematics. For this reason. Diophantus was not, as he has often been called, the father of algebra. Nevertheless, his remarkable, if unsystematic, collection of indeterminate problems is a singular achievement that was not fully appreciated and further developed until much later. Diophantus made important advances in mathematical notation. He was the first person to use algebraic notation and symbolism. Before him everyone wrote out equations completely. Diophantus introduced an algebraic symbolism that used an abridged notation for frequently occurring operations, and an abbreviation for the unknown and for the powers of the unknown. Mathematical historian Kurt Vogel states: “The symbolism that Diophantus introduced for the first time, and undoubtedly devised himself, provided a short and readily comprehensible means of expressing an equation... Since an abbreviation is also employed for the word ‘equals’, Diophantus took a fundamental step from verbal algebra towards symbolic algebra.” Although Diophantus made important advances in symbolism, he still lacked the necessary notation to express more general methods. This caused his work to be more concerned with particular problems rather than general situations. Some of the limitations of Diophantus' notation are that he only had notation for one unknown and, when problems involved more than a single unknown, Diophantus was reduced to expressing "first unknown", "second unknown", etc. in words. He also lacked a symbol for a general number n. Where we would write (12 + 6n) / (n2 − 3), Diophantus has to resort to constructions like : ... a sixfold number increased by twelve, which is divided by the difference by which the square of the number exceeds three. Algebra still had a long way to go before very general problems could be written down and solved succinctly. RECOMMENDATIONS .I recommend that the course should be giving 3 credit hours considering the broad nature of the course. .Also I recommend that this course should be an essential course for all level 300 students in the faculty so that each student will recognize the contributions made to mathematics by some personalities. .Moreover the faculty should design a guide for the course as well as provide effective internet services to make work easy for the students. REFERENCES .ENCARTA ENCYCLOPEDIA 2009 .RELATED WEBSITES - www.algebra.com/algebra/about/history - en.wikwipedia.org/wiki/algebra - en.wikipedia.org/wiki/diophantus

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 4 |

posted: | 9/28/2012 |

language: | |

pages: | 34 |

OTHER DOCS BY mrxtee

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.