Sunday, May 3, 2020

History Of Math Essay Example For Students

History Of Math Essay Mathematics, study of relationships among quantities, magnitudes, and properties and of logical operations by which unknown quantities, magnitudes, and properties may be deduced. In the past, mathematics was regarded as the science of quantity, whether of magnitudes, as in geometry, or of numbers, as in arithmetic, or of the generalization of these two fields, as in algebra. Toward the middle of the 19th century, however, mathematics came to be regarded increasingly as the science of relations, or as the science that draws necessary conclusions. This latter view encompasses mathematical or symbolic logic, the science of using symbols to provide an exact theory of logical deduction and inference based on definitions, axioms, postulates, and rules for combining and transforming primitive elements into more complex relations and theorems. This brief survey of the history of mathematics traces the evolution of mathematical ideas and concepts, beginning in prehistory. Indeed, mathematics is nearly as old as humanity itself; evidence of a sense of geometry and interest in geometric pattern has been found in the designs of prehistoric pottery and textiles and in cave paintings. Primitive counting systems were almost certainly based on using 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. Ancient Mathematics The earliest records of advanced, organized mathematics date back to the ancient Mesopotamian country of Babylonia and to Egypt of the 3rd millennium BC. There mathematics was dominated by arithmetic, with an emphasis on measurement and calculation in geometry and with no trace of later mathematical concepts such as axioms or proofs. The earliest Egyptian texts, composed about 1800 BC, reveal a decimal numeration system with separate symbols for the successive powers of 10 (1, 10, 100, and so forth), just as in the system used by the Romans. Numbers were represented by writing down the symbol for 1, 10, 100, and so on as many times as the unit was in a given number. For example, the symbol for 1 was written five times to represent the number 5, the symbol for 10 was written six times to represent the number 60, and the symbol for 100 was written three times to represent the number 300. Together, these symbols represented the number 365. Addition was done by totaling separately the units-10s, 100s, and so forth-in the numbers to be added. Multiplication was based on successive doublings, and division was based on the inverse of this process. The Egyptians used sums of unit fractions (a), supplemented by the fraction B, to express all other fractions. For example, the fraction E was the sum of the fractions 3 and *. Using this system, the Egyptians were able to solve all problems of arithmetic that involved fractions, as well as some elementary problems in algebra. In geometry, the Egyptians calculated the correct areas of triangles, rectangles, and trapezoids and the volumes of figures such as bricks, cylinders, and pyramids. To find the area of a circle, the Egyptians used the square on U of the diameter of the circle, a value of about 3.16-close to the value of the ratio known as pi, which is about 3.14. The Babylonian system of numeration was quite different from the Egyptian system. In the Babylonian system-which, when using clay tablets, consisted of various wedge-shaped marks-a single wedge indicated 1 and an arrowlike wedge stood for 10 (see table). Numbers up through 59 were formed from these symbols through an additive process, as in Egyptian mathematics. The number 60, however, was represented by the same symbol as 1, and from this point on a positional symbol was used. That is, the value of one of the first 59 numerals depended henceforth on its position in the total numeral. For example, a numeral consisting of a symbol for 2 followed by one for 27 and ending in one for 10 stood for 2 ? 602 + 27 ? 60 + 10. This principle was extended to the representation of fractions as well, so that the above sequence of numbers could equally well represent 2 ? 60 + 27 + 10 ? († ), or 2 + 27 ? († ) + 10 ? († -2). With this sexagesimal system (base 60), as it is called, the Babylonians had as convenient a numerical system as the 10-based system. The Babylonians in time developed a sophisticated mathematics by which they could find the positive roots of any quadratic equation (Equation). They could even find the roots of certain cubic equations. The Babylonians had a variety of tables, including tables for multiplication and division, tables of squares, and tables of compound interest. They could solve complicated problems using the Pythagorean theorem; one of their tables contains integer solutions to the Pythagorean equation, a2 + b2 = c2, arranged so that c2/a2 decreases steadily from 2 to about J. The Babylonians were able to sum arithmetic and some geometric progressions, as well as sequences of squares. They also arrived at a good approximation for ?. In geometry, they calculated the areas of rectangles, triangles, and trapezoids, as well as the volumes of simple shapes such as bricks and cylinders. However, the Babylonians did not arrive at the correct formula for the volume of a pyramid. Greek Mathematics The Greeks adopted elements of mathematics from both the Babylonians and the Egyptians. The new element in Greek mathematics, however, was the invention of an abstract mathematics founded on a logical structure of definitions, axioms, and proofs. According to later Greek accounts, this development began in the 6th century BC with Thales of Miletus and Pythagoras of Samos, the latter a religious leader who taught the importance of studying numbers in order to understand the world. Some of his disciples made important discoveries about the theory of numbers and geometry, all of which were attributed to Pythagoras. In the 5th century BC, some of the great geometers were the atomist philosopher Democritus of Abdera, who discovered the correct formula for the volume of a pyramid, and Hippocrates of Chios, who discovered that the areas of crescent-shaped figures bounded by arcs of circles are equal to areas of certain triangles. This discovery is related to the famous problem of squaring the circle-that is, constructing a square equal in area to a given circle. Two other famous mathematical problems that originated during the century were those of trisecting an angle and doubling a cube-that is, constructing a cube the volume of which is double that of a given cube. All of these problems were solved, and in a variety of ways, all involving the use of instruments more complicated than a straightedge and a geometrical compass. Not until the 19th century, however, was it shown that the three problems mentioned above could never have been solved using those instruments alone. In the latter part of the 5th century BC, an unknown mathematician discovered that no unit of length would measure both the side and diagonal of a square. That is, the two lengths are incommensurable. This means that no counting numbers n and m exist whose ratio expresses the relationship of the side to the diagonal. Since the Greeks considered only the counting numbers (1, 2, 3, and so on) as numbers, they had no numerical way to express this ratio of diagonal to side. (This ratio, ?, would today be called irrational.) As a consequence the Pythagorean theory of ratio, based on numbers, had to be abandoned and a new, nonnumerical theory introduced. This was done by the 4th-century BC mathematician Eudoxus of Cnidus, whose solution may be found in the Elements of Euclid. Eudoxus also discovered a method for rigorously proving statements about areas and volumes by successive approximations. Euclid was a mathematician and teacher who worked at the famed Museum of Alexandria and who also wrote on optics, astronomy, and music. The 13 books that make up his Elements contain much of the basic mathematical knowledge discovered up to the end of the 4th century BC on the geometry of polygons and the circle, the theory of numbers, the theory of incommensurables, solid geometry, and the elementary theory of areas and volumes. The century that followed Euclid was marked by mathematical brilliance, as displayed in the works of Archimedes of Syracuse and a younger contemporary, Apollonius of Perga. Archimedes used a method of discovery, based on theoretically weighing infinitely thin slices of figures, to find the areas and volumes of figures arising from the conic sections. These conic sections had been discovered by a pupil of Eudoxus named Menaechmus, and they were the subject of a treatise by Euclid, but Archimedes writings on them are the earliest to survive. Archimedes also investigated centers of gravity and the stability of various solids floating in water. Much of his work is part of the tradition that led, in the 17th century, to the discovery of the calculus. Archimedes was killed by a Roman soldier during the sack of Syracuse. His younger contemporary, Apollonius, produced an eight-book treatise on the conic sections that established the names of the sections: ellipse, parabola, and hyperbola. It also provided the basic treatment of their geometry until the time of the French philosopher and scientist Ren? Descartes in the 17th century. After Euclid, Archimedes, and Apollonius, Greece produced no geometers of comparable stature. The writings of Hero of Alexandria in the 1st century AD show how elements of both the Babylonian and Egyptian mensurational, arithmetic traditions survived alongside the logical edifices of the great geometers. Very much in the same tradition, but concerned with much more difficult problems, are the books of Diophantus of Alexandria in the 3rd century AD. They deal with finding rational solutions to kinds of problems that lead immediately to equations in several unknowns. Such equations are now called Diophantine equations (see Diophantine Analysis). Career Development through International Mobility EssayThe greatest mathematician of the 18th century was Leonhard Euler, a Swiss, who made basic contributions to calculus and to all other branches of mathematics, as well as to the applications of mathematics. He wrote textbooks on calculus, mechanics, and algebra that became models of style for writing in these areas. The success of Euler and other mathematicians in using calculus to solve mathematical and physical problems, however, only accentuated their failure to develop a satisfactory justification of its basic ideas. That is, Newtons own accounts were based on kinematics and velocities, Leibnizs explanation was based on infinitesimals, and Lagranges treatment was purely algebraic and founded on the idea of infinite series. All these systems were unsatisfactory when measured against the logical standards of Greek geometry, and the problem was not resolved until the following century. 19th Century In 1821 a French mathematician, Augustin Louis Cauchy, succeeded in giving a logically satisfactory approach to calculus. He based his approach only on finite quantities and the idea of a limit. This solution posed another problem, however; that of a logical definition of ?real number.? Although Cauchys explanation of calculus rested on this idea, it was not Cauchy but the German mathematician Julius W. R. Dedekind who found a satisfactory definition of real numbers in terms of the rational numbers. This definition is still taught, but other definitions were given at the same time by the German mathematicians Georg Cantor and Karl T. W. Weierstrass. A further important problem, which arose out of the problem-first stated in the 18th century-of describing the motion of a vibrating string, was that of defining what is meant by function. Euler, Lagrange, and the French mathematician Jean Baptiste Fourier all contributed to the solution, but it was the German mathematician P eter G. L. Dirichlet who proposed the definition in terms of a correspondence between elements of the domain and the range. This is the definition that is found in texts today. In addition to firming the foundations of analysis, as the techniques of the calculus were by then called, mathematicians of the 19th century made great advances in the subject. Early in the century, Carl Friedrich Gauss gave a satisfactory explanation of complex numbers, and these numbers then formed a whole new field for analysis, one that was developed in the work of Cauchy, Weierstrass, and the German mathematician Georg F. B. Riemann. Another important advance in analysis was Fouriers study of infinite sums in which the terms are trigonometric functions. Known today as Fourier series, they are still powerful tools in pure and applied mathematics. In addition, the investigation of which functions could be equal to Fourier series led Cantor to the study of infinite sets and to an arithmetic of infinite numbers. Cantors theory, which was considered quite abstract and even attacked as a ?disease from which mathematics will soon recover,? now forms part of the foundations of mathemat ics and has more recently found applications in the study of turbulent flow in fluids. A further 19th-century discovery that was considered apparently abstract and useless at the time was non-Euclidean geometry. In non-Eculidean geometry, more than one parallel can be drawn to a given line through a given point not on the line. Evidently this was discovered first by Gauss, but Gauss was fearful of the controversy that might result from publication. The same results were rediscovered independently and published by the Russian mathematician Nikolay Ivanovich Lobachevsky and the Hungarian J?nos Bolyai. Non-Euclidean geometries were studied in a very general setting by Riemann with his invention of manifolds and, since the work of Einstein in the 20th century, they have also found applications in physics. Gauss was one of the greatest mathematicians who ever lived. Diaries from his youth show that this infant prodigy had already made important discoveries in number theory, an area in which his book Disquisitiones Arithmeticae (1801) marks the beginning of the modern era. While only 18, Gauss discovered that a regular polygon with m sides can be constructed by straightedge and compass when m is a power of 2 times distinct primes of the form 2n + 1. In his doctoral dissertation he gave the first satisfactory proof of the fundamental theorem of algebra. Often he combined scientific and mathematical investigations. Examples include his development of statistical methods along with his investigations of the orbit of a newly discovered planetoid; his founding work in the field of potential theory, along with the study of magnetism; and his study of the geometry of curved surfaces in tandem with his investigations of surveying. Of more importance for algebra itself than Gausss proof of its fundamental theorem was the transformation of the subject during the 19th century from a study of polynomials to a study of the structure of algebraic systems. A major step in this direction was the invention of symbolic algebra in England by George Peacock. Another was the discovery of algebraic systems that have many, but not all, of the properties of the real numbers. Such systems include the quaternions of the Irish mathematician William Rowan Hamilton, the vector analysis of the American mathematician and physicist J. Willard Gibbs, and the ordered n-dimensional spaces of the German mathematician Hermann G?nther Grassmann. A third major step was the development of group theory from its beginnings in the work of Lagrange. Galois applied this work deeply to provide a theory of when polynomials may be solved by an algebraic formula. Just as Descartes had applied the algebra of his time to the study of geometry, so the German mathematician Felix Klein and the Norwegian mathematician Marius Sophus Lie applied the algebra of the 19th century. Klein applied it to the classification of geometries in terms of their groups of transformations (the so-called Erlanger Programm), and Lie applied it to a geometric theory of differential equations by means of continuous groups of transformations known as Lie groups. In the 20th century, algebra has also been applied to a general form of geometry known as topology. Another subject that was transformed in the 19th century, notably by Laws of Thought (1854), by the English mathematician George Boole and by Cantors theory of sets, was the foundations of mathematics (Logic). Toward the end of the century, however, a series of paradoxes were discovered in Cantors theory. One such paradox, found by English mathematician Bertrand Russell, aimed at the very concept of a set ( Set Theory). Mathematicians responded by constructing set theories sufficiently restrictive to keep the paradoxes from arising. They left open the question, however, of whether other paradoxes might arise in these restricted theories-that is, whether the theories were consistent. As of the present time, only relative consistency proofs have been given. (That is, theory A is consistent if theory B is consistent.) Particularly disturbing is the result, proved in 1931 by the American logician Kurt G?del, that in any axiom system complicated enough to be interesting to most mathematic ians, it is possible to frame propositions whose truth cannot be decided within the system. Current Mathematics At the International Conference of Mathematicians held in Paris in 1900, the German mathematician David Hilbert spoke to the assembly. Hilbert was a professor at G?ttingen, the former academic home of Gauss and Riemann. He had contributed to most areas of mathematics, from his classic Foundations of Geometry (1899) to the jointly authored Methods of Mathematical Physics. Hilberts address at G?ttingen was a survey of 23 mathematical problems that he felt would guide the work being done in mathematics during the coming century. These problems have indeed stimulated a great deal of the mathematical research of the century. When news breaks that another of the ?Hilbert problems? has been solved, mathematicians all over the world await the details of the story with impatience. Important as these problems have been, an event that Hilbert could not have foreseen seems destined to play an even greater role in the future development of mathematics-namely, the invention of the programmable digital computer (Computer). Although the roots of the computer go back to the geared calculators of Pascal and Leibniz in the 17th century, it was Charles Babbage in 19th-century England who designed a machine that could automatically perform computations based on a program of instructions stored on cards or tape. Babbages imagination outran the technology of his day, however, and it was not until the invention of the relay, then of the vacuum tube, and then of the transistor, that large-scale, programmed computation became feasible. This development has given great impetus to areas of mathematics such as numerical analysis and finite mathematics. It has suggested new areas for mathematical investigation, such as the study of algorithms. It has also become a powerful tool in areas as diverse as number theory, differential equations, and abstract algebra. In addition, the computer has made possible the solution of several long-standing problems in mathematics, such as the four-color problem first proposed in the mid-19th century. The theorem stated that four colors are sufficient to color any map, given that any two countries with a contiguous boundary require different colors. The theorem was finally proved in 1976 by means of a large-scale computer at the University of Illinois. Mathematical knowledge in the modern world is advancing at a faster rate than ever before. Theories that were once separate have been incorporated into theories that are both more comprehensive and more abstract. Although many important problems have been solved, other hardy perennials, such as the Riemann hypothesis, remain, and new and equally challenging problems arise. Even the most abstract mathematics seems to be finding applications. Mathematics

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