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Mathematics
The deductive study of numbers, geometry, and various abstract constructs, or structures; the latter often abstract the features common to several models derived from the empirical, or applied, sciences, although many emerge from purely mathematical or logical considerations. Mathematics is very broadly divided into foundations, algebra, analysis, geometry, and applied mathematics, which includes theoretical computer science.
Development of Mathematics
Applied Mathematics
The earliest records of mathematics show it arising in response to practical needs in agriculture, business, and industry. In Egypt and Mesopotamia, where evidence dates from the 2d and 3d millennia b.c. , it was used for surveying and mensuration; estimates of the value of π (pi) are found in both locations. There is some evidence of similar developments in India and China during this same period, but few records have survived. This early mathematics is generally empirical, arrived at by trial and error as the best available means for obtaining results, with no proofs given. However, it is now known that the Babylonians were aware of the necessity of proofs prior to the Greeks, who had been presumed the originators of this important step. Greek Contributions
A profound change occurred in the nature and approach to mathematics with the contributions of the Greeks. The earlier (Hellenic) period is represented by Thales (6th cent. b.c. ), Pythagoras, Plato, and Aristotle, and by the schools associated with them. The Pythagorean theorem, known earlier in Mesopotamia, was discovered by the Greeks during this period.
During the Golden Age (5th cent. b.c. ), Hippocrates of Chios made the beginnings of an axiomatic approach to geometry and Zeno of Elea proposed his famous paradoxes concerning the infinite and the infinitesimal, raising questions about the nature of and relationships among points, lines, and numbers. The discovery through geometry of irrational numbers, such as √2, also dates from this period. Eudoxus of Cnidos (4th cent. b.c. ) resolved certain of the problems by proposing alternative methods to those involving infinitesimals; he is known for his work on geometric proportions and for his exhaustion theory for determining areas and volumes.
The later (Hellenistic) period of Greek science is associated with the school of Alexandria. The greatest work of Greek mathematics, Euclid\'s Elements (c.300 b.c. ), appeared at the beginning of this period. Elementary geometry as taught in high school is still largely based on Euclid\'s presentation, which has served as a model for deductive systems in other parts of mathematics and in other sciences. In this method primitive terms, such as point and line, are first defined, then certain axioms and postulates relating to them and seeming to follow directly from them are stated without proof; a number of statements are then derived by deduction from the definitions, axioms, and postulates. Euclid also contributed to the development of arithmetic and presented a geometric theory of quadratic equations.
In the 3d cent. b.c. , Archimedes in addition to his work in mechanics, made an estimate of π and used the exhaustion theory of Eudoxus to obtain results that foreshadowed those much later of the integral calculus, and Apollonius of Perga named the conic sections and gave the first theory for them. A second Alexandrian school of the Roman period included contributions by Menelaus (c. a.d. 100, spherical triangles), Heron of Alexandria (geometry), Ptolemy ( a.d. 150, astronomy, geometry, cartography), Pappus (3d cent., geometry), and Diophantus (3d cent., arithmetic). Chinese and Middle Eastern Advances
Following the decline of learning in the West after the 3d cent., the development of mathematics continued in the East. In China, Tsu Ch\'ung-Chih estimated π by inscribed and circumscribed polygons, as Archimedes had done, and in India the numerals now used throughout the civilized world were invented and contributions to geometry were made by Aryabhata and Brahmagupta (5th and 6th cent. a.d. ). The Arabs were responsible for preserving the work of the Greeks, which they translated, commented upon, and augmented. In Baghdad, Al-Khowarizmi (9th cent.) wrote an important work on algebra and introduced the Hindu numerals for the first time to the West, and Al-Battani worked on trigonometry. In Egypt, Ibn al-Haytham was concerned with the solids of revolution and geometrical optics. The Persian poet Omar Khayyam wrote on algebra. Western Development from the 12th to 18th Century
Word of the Chinese and Middle Eastern works began to reach the West in the 12th and 13th cent. One of the first important European mathematicians was Leonardo da Pisa (Leonardo Fibonacci), who wrote on arithmetic and algebra (Liber abaci, 1202) and on geometry (Practica geometriae, 1220). With the Renaissance came a great revival of interest in learning, and the invention of printing made many of the earlier books widely available. By the end of the 16th cent. advances had been made in algebra by Niccol Tartaglia and Geronimo Cardano, in trigonometry by Franois Vite, and in such areas of applied mathematics as mapmaking by Mercator and others.
The 17th cent., however, saw the greatest revolution in mathematics, as the scientific revolution spread to all fields. Decimal fractions were invented by Simon Stevin and logarithms by John Napier and Henry Briggs; the beginnings of projective geometry were made by Grard Desargues and Blaise Pascal; number theory was greatly extended by Pierre de Fermat; and the theory of probability was founded by Pascal, Fermat, and others. In the application of mathematics to mechanics and astronomy, Galileo and Johannes Kepler made fundamental contributions.
The greatest mathematical advances of the 17th cent., however, were the invention of analytic geometry by Ren Descartes and that of the calculus by Isaac Newtonand, independently, by G. W. Leibniz. Descartes\'s invention (anticipated by Fermat, whose work was not published until later) made possible the expression of geometric problems in algebraic form and vice versa. It was indispensable in creating the calculus, which built upon and superseded earlier special methods for finding areas, volumes, and tangents to curves, developed by F. B. Cavalieri, Fermat, and others. The calculus is probably the greatest tool ever invented for the mathematical formulation and solution of physical problems.
The history of mathematics in the 18th cent. is dominated by the development of the methods of the calculus and their application to such problems, both terrestrial and celestial, with leading roles being played by the Bernoulli family (especially Jakob, Johann, and Daniel), Leonhard Euler, Guillaume de L\'Hpital, and J. L. Lagrange. Important advances in geometry began toward the end of the century with the work of Gaspard Monge in descriptive geometry and in differential geometry and continued through his influence on others, e.g., his pupil J. V. Poncelet, who founded projective geometry (1822). In the 19th Century
The modern period of mathematics dates from the beginning of the 19th cent., and its dominant figure is C. F. Gauss. In the area of geometry Gauss made fundamental contributions to differential geometry, did much to found what was first called analysis situs but is now called topology, and anticipated (although he did not publish his results) the great breakthrough of non-Euclidean geometry. This breakthrough was made by N. I. Lobachevsky (1826) and independently by Jnos Bolyai (1832), the son of a close friend of Gauss, whom each proceeded by establishing the independence of Euclid\'s fifth (parallel) postulate and showing that a different, self-consistent geometry could be derived by substituting another postulate in its place. Still another non-Euclidean geometry was invented by G. F. B. Riemann (1854), whose work also laid the foundations for the modern tensor calculus description of space, so important in the general theory of relativity.
In the area of arithmetic, number theory, and algebra, Gauss again led the way. He established the modern theory of numbers, gave the first clear exposition of complex numbers, and investigated the functions of complex variables. The concept of number was further extended by W. R. Hamilton, whose theory of quaternions (1843) provided the first example of a noncommutative algebra (i.e., one in which ab ≠ ba). This work was generalized the following year by H. G. Grassmann, who showed that several different consistent algebras may be derived by choosing different sets of axioms governing the operations on the elements of the algebra.
These developments continued with the group theory of M. S. Lie in the late 19th cent. and reached full expression in the wide scope of modern abstract algebra. Number theory received significant contributions in the latter half of the 19th cent. through the work of Georg Cantor, J. W. R. Dedekind, and K. W. Weierstrass. Still another influence of Gauss was his insistence on rigorous proof in all areas of mathematics. In analysis this close examination of the foundations of the calculus resulted in A. L. Cauchy\'s theory of limits (1821), which in turn yielded new and clearer definitions of continuity, the derivative, and the definite integral. A further important step toward rigor was taken by Weierstrass, who raised new questions about these concepts and showed that ultimately the foundations of analysis rest on the properties of the real number system. In the 20th Century
In the 20th cent. the trend has been toward increasing generalization and abstraction, with the elements and operations of systems being defined so broadly that their interpretations connect such areas as algebra, geometry, and topology. The key to this approach has been the use of formal axiomatics, in which the notion of axioms as self-evident truths has been discarded. Instead the emphasis is on such logical concepts as consistency and completeness. The roots of formal axiomatics lie in the discoveries of alternative systems of geometry and algebra in the 19th cent.; the approach was first systematically undertaken by David Hilbert in his work on the foundations of geometry (1899).
The emphasis on deductive logic inherent in this view of mathematics and the discovery of the interconnections between the various branches of mathematics and their ultimate basis in number theory led to intense activity in the field of mathematical logic after the turn of the century. Rival schools of thought grew up under the leadership of Hilbert, Bertrand Russell and A. N. Whitehead, and L. E. J. Brouwer. Important contributions in the investigation of the logical foundations of mathematics were made by Kurt Gdel and A. Church.
Branches of Mathematics Foundations
The term foundations is used to refer to the formulation and analysis of the language, axioms, and logical methods on which all of mathematics rests. The scope and complexity of modern mathematics requires a very fine analysis of the formal language in which meaningful mathematical statements may be formulated and perhaps be proved true or false. Most apparent mathematical contradictions have been shown to derive from an imprecise and inconsistent use of language. A basic task is to furnish a set of axioms effectively free of contradictions and at the same time rich enough to constitute a deductive source for all of modern mathematics. The modern axiom schemes proposed for this purpose are all couched within the theory of sets, originated by Georg Cantor, which now constitutes a universal mathematical language. Arithmetic
A branch of mathematics commonly considered a separate branch but in actuality a part of algebra. Conventionally the term has been most widely applied to simple teaching of the skills of dealing with Numbers for practical purposes, e.g., computation of areas, proportions, costs, and the like. The four fundamental operations of this study are addition, subtraction, multiplication, and division. In advanced study the concept of number is greatly generalized to include not only complex numbers, but also quaternions, tensors, and abstract entities with no other meaning than that they obey certain laws. The division of arithmetic into the practical and the theoretical dates back to classical Greek times, when the term logistic referred to elementary arithmetic and the term arithmetic was reserved for the theory. Algebra
Historically, algebra is the study of solutions of one or several algebraic equations, involving the polynomial functions of one or several variables. The case where all the polynomials have degree one (systems of linear equations) leads to linear algebra. The case of a single equation, in which one studies the roots of one polynomial, leads to field theory and to the so-called Galois theory. The general case of several equations of high degree leads to algebraic geometry, so named because the sets of solutions of such systems are often studied by geometric methods.
Modern algebraists have increasingly abstracted and axiomatized the structures and patterns of argument encountered not only in the theory of equations, but in mathematics generally. Examples of these structures include groups (first witnessed in relation to symmetry properties of the roots of a polynomial and now ubiquitous throughout mathematics), rings (of which the integers, or whole numbers, constitute a basic example), and fields (of which the rational, real, and complex numbers are examples). Some of the concepts of modern algebra have found their way into elementary mathematics education in the so-called new mathematics.
Some important abstractions recently introduced in algebra are the notions of category and functor, which grew out of so-called homological algebra. arithmetic and number theory, which are concerned with special properties of the integerse.g., unique factorization, primes, equations with integer coefficients (Diophantine equations), and congruencesare also a part of algebra. Analytic number theory, however, also applies the nonalgebraic methods of analysis to such problems. Analysis
The essential ingredient of analysis is the use of infinite processes, involving passage to a limit. For example, the area of a circle may be computed as the limiting value of the areas of inscribed regular polygons as the number of sides of the polygons increases indefinitely. The basic branch of analysis is the calculus. The general problem of measuring lengths, areas, volumes, and other quantities as limits by means of approximating polygonal figures leads to the integral calculus. The differential calculus arises similarly from the problem of finding the tangent line to a curve at a point. Other branches of analysis result from the application of the concepts and methods of the calculus to various mathematical entities. For example, vector analysis is the calculus of functions whose variables are vectors. Here various types of derivatives and integrals may be introduced. They lead, among other things, to the theory of differential and integral equations, in which the unknowns are functions rather than numbers, as in algebraic equations. Differential equations are often the most natural way in which to express the laws governing the behavior of various physical systems. Calculus is one of the most powerful and supple tools of mathematics. Its applications, both in pure mathematics and in virtually every scientific domain, are manifold. Geometry
The shape, size, and other properties of figures and the nature of space are in the province of geometry. Euclidean geometry is concerned with the axiomatic study of polygons, conic sections, spheres, polyhedra, and related geometric objects in two and three dimensionsin particular, with the relations of congruence and of similarity between such objects. The unsuccessful attempt to prove the parallel postulate from the other axioms of Euclid led in the 19th cent. to the discovery of two different types of non-Euclidean geometry.
The 20th cent. has seen an enormous development of topology, which is the study of very general geometric objects, called topological spaces, with respect to relations that are much weaker than congruence and similarity. Other branches of geometry include algebraic geometry and differential geometry, in which the methods of analysis are brought to bear on geometric problems. These fields are now in a vigorous state of development. Applied Mathematics
The term applied mathematics loosely designates a wide range of studies with significant current use in the empirical sciences. It includes numerical methods and computer science, which seeks concrete solutions, sometimes approximate, to explicit mathematical problems (e.g., differential equations, large systems of linear equations). It has a major use in technology for modeling and simulation. For example, the huge wind tunnels, formerly used to test expensive prototypes of airplanes, have all but disappeared. The entire design and testing process is now largely carried out by computer simulation, using mathematically tailored software. It also includes mathematical physics, which now strongly interacts with all of the central areas of mathematics. In addition, probability theory and mathematical statistics are often considered parts of applied mathematics. The distinction between pure and applied mathematics is now becoming less significant.
See Richard Courant and Herbert Robbins, What Is Mathematics? (1941); E. T. Bell, The Development of Mathematics (2d ed. 1945) and Men of Mathematics (1937, repr. 1961); J. R. Newman, ed., The World of Mathematics (4 vol., 1956); E. E. Kramer, The Nature and Growth of Mathematics (1970); Morris Kline, Mathematical Thought from Ancient to Modern Times (1973); D. J. Albers and G. L. Alexanderson, eds., Mathematical People (1985).
Algebra
The branch of mathematics concerned with operations on sets of Numbers or other elements that are often represented by symbols. Algebra is a generalization of arithmetic and gains much of its power from dealing symbolically with elements and operations (such as addition and multiplication) and relationships (such as equality) connecting the elements. Thus, a+a=2a and a+b=b+a no matter what numbers a and b represent.
Principles of Classical Algebra
In elementary algebra letters are used to stand for numbers. For example, in the equation ax2+bx+c=0, the letters a, b, and c stand for various known constant numbers called coefficients and the letter x is an unknown variable number whose value depends on the values of a, b, and c and may be determined by solving the equation. Much of classical algebra is concerned with finding solutions to equations or systems of equations, i.e., finding the roots, or values of the unknowns, that upon substitution into the original equation will make it a numerical identity. For example, x=−2 is a root of x2−2x−8=0 because (−2)2−2(−2)−8=4+4−8=0; substitution will verify that x=4 is also a root of this equation. Equation
In mathematics, a statement, usually written in symbols, that states the equality of two quantities or algebraic expressions, e.g., x+3=5. The quantity x+3, to the left of the equals sign (=), is called the left-hand, or first, member of the equation, that to the right (5) the right-hand, or second, member. A numerical equation is one containing only numbers, e.g., 2+3=5. A literal equation is one that, like the first example, contains some letters (representing unknowns or variables). An identical equation is a literal equation that is true for every value of the variable, e.g., the equation (x+1)2=x2+2x+1. A conditional equation (usually referred to simply as an equation) is a literal equation that is not true for all values of the variable, e.g., only the value 2 for x makes true the equation x+3=5. To solve an equation is to find the value or values of the variable that satisfy it. polynomial equations, containing more than one term, are classified according to the highest degree of the variable they contain. Thus the first example is a first degree (also called linear) equation. The equation ax2+bx+c=0 is a second degree, or quadratic, equation in the unknown x if the letters a, b, and c are assumed to represent constants. In algebra, methods are evolved for solving various types of equations. To be valid the solution must satisfy the equation. Whether it does can be ascertained by substituting the supposed solution for the variable in the equation. The simultaneous solution of two or more equations is a set of values of the variables that satisfies each of the equations. In order that a solution may exist, the number of equations (i.e., conditions) must generally be no greater than the number of variables. In chemistry an is used to represent a reaction. Roots
In mathematics, number or quantity r for which an equation f (r)=0 holds true, where f is some function. If f is a polynomial, r is called a root of f; for example, r=3 and r=−4 are roots of the equation x2+x−12 = 0, because (3)2+(3)−12 = 0 and (−4)2+(−4)−12 = 0. In the special case where f (x) = xn − a for some number a, a root of f is called an nth root of a, denoted by n √a or a1/n. For example, 2 is the third, or cube, root of 8 ( 3 √8 = 2), since it satisfies the equation x3 − 8 = 0. Every number has n different (real or complex) nth roots; e.g., there are two square roots of 9 (3 and −3) since (3)(3) = 9 and (−3)(−3) = 9.
The equations of elementary algebra usually involve polynomial functions of one or more variables. The equation in the preceding example involves a polynomial of second degree in the single variable x (see quadratic). One method of finding the zeros of the polynomial function f (x), i.e., the roots of the equation f (x)=0, is to factor the polynomial, if possible. The polynomial x2 − 2x−8 has factors (x+2) and (x−4), since (x+2) (x−4)=x2−2x−8, so that setting either of these factors equal to zero will make the polynomial zero. In general, if (x−r) is a factor of a polynomial f(x), then r is a zero of the polynomial and a root of the equation f (x)=0. To determine if (x−r) is a factor, divide it into f(x); according to the Factor Theorem, if the remainder f(r) (found by substituting r for x in the original polynomial) is zero, then (x−r) is a factor of f(x). Although a polynomial has real coefficients, its roots may not be real numbers; e.g., x2−9 separates into (x+3)(x−3), which yields two zeros, x=−3 and x=+3, but the zeros of x2+9 are imaginary numbers.
The Fundamental Theorem of Algebra states that every polynomial f(x)=anxn+an−1xn−1+ +a1x+a0, with an≠0 and n≥1, has at least one complex root, from which it follows that the equation f(x)=0 has exactly n roots, which may be real or complex and may not all be distinct. For example, the equation x4+4x3+5x2+4x+4=0 has four roots, but two are identical and the other two are complex; the factors of the polynomial are (x + 2)(x + 2)(x + i)(x − i), as can be verified by multiplication.
Principles of Modern Algebra
Modern algebra is yet a further generalization of arithmetic than is classical algebra. It deals with operations that are not necessarily those of arithmetic and that apply to elements that are not necessarily numbers. The elements are members of a set and are classed as a group, a ring, or a field according to the axioms that are satisfied under the particular operations defined for the elements. Among the important concepts of modern algebra are those of a matrix and of a vector space.
See Michael Artin, Algebra (1991).
Geometry
[Gr.,=earth measuring], branch of mathematics concerned with the properties of and relationships between points, lines, planes, and figures and with generalizations of these concepts.
Types of Geometry
Euclidean geometry, elementary geometry of two and three dimensions (plane and solid geometry), is based largely on the Elements of the Greek mathematician Euclid (fl. c.300 b.c. ). In 1637, Ren Descartes showed how numbers can be used to describe points in a plane or in space and to express geometric relations in algebraic form, thus founding analytic geometry, of which algebraic geometry is a further development (see Cartesian coordinates). The problem of representing three-dimensional objects on a two-dimensional surface was solved by Gaspard Monge, who invented descriptive geometry for this purpose in the late 18th cent. differential geometry, in which the concepts of the calculus are applied to curves, surfaces, and other geometrical objects, was founded by Monge and C. F. Gauss in the late 18th and early 19th cent. The modern period in geometry begins with the formulations of projective geometry by J. V. Poncelet (1822) and of non-Euclidean geometry by N. I. Lobachevsky (1826) and Jnos Bolyai (1832). Another type of non-Euclidean geometry was discovered by Georg Riemann (1854), who also showed how the various geometries could be generalized to any number of dimensions.
Their Relationship to Each Other
The different geometries are classified and related to one another in various ways. The non-Euclidean geometries are exactly analogous to the geometry of Euclid, except that Euclid\'s postulate regarding parallel lines is replaced and all theorems depending on this postulate are changed accordingly. Both Euclidean and non-Euclidean geometry are types of metric geometry, in which the lengths of line segments and the sizes of angles may be measured and compared. Projective geometry, on the other hand, is more general and includes the metric geometries as a special case; pure projective geometry makes no reference to lengths or angle measurements.
The general metric geometry consisting of all of Euclidean geometry except that part dependent on the parallel postulate is called absolute geometry; its propositions are valid for both Euclidean and non-Euclidean geometry. Another type of geometry, called affine geometry, includes Euclid\'s parallel postulate but disregards two other postulates concerning circles and angle measurement; the propositions of affine geometry are also valid in the four-dimensional geometry of space-time used in the theory of relativity. Ordered geometry consists of all propositions common to both absolute geometry and affine geometry; this geometry includes the notion on intermediacy (betweenness) but not that of measurement.
An important step in recognizing the connections between the different types of geometry was the Erlangen program, proposed by the German Felix Klein in his inaugural address at the Univ. of Erlangen (1872), according to which geometries are classified with respect to the geometrical properties that are left unchanged (invariant) under a given group of transformations. For example, Euclidean geometry is the study of properties unchanged by similarity transformations, affine geometry is concerned with properties invariant under the linear transformations (affine collineations) that preserve parallelism, and projective geometry studies invariants under the more general projective transformations (collineations and correlations). topology, perhaps the most general type of geometry although often considered a separate branch of mathematics, is concerned with properties invariant under continuous transformations, which carry neighborhoods of points into neighborhoods of their images.
The Axiomatic Approach to Geometry
Euclid\'s Elements organized the geometry then known into a systematic presentation that is still used in many texts. Euclid first defined his basic terms, such as point and line, then stated without proof certain axioms and postulates about them that seemed to be self-evident or obvious truths, and finally derived a number of statements (theorems) from the postulates by means of deductive logic. This axiomatic method has since been adopted not only throughout mathematics but in many other fields as well. The close examination of the axioms and postulates of Euclidean geometry during the 19th cent. resulted in the realization that the logical basis of geometry was not as firm as had previously been supposed. New axiom and postulate systems were developed by various mathematicians, notably David Hilbert (1899).
See H. G. Forder, The Foundations of Euclidean Geometry (1927); H. S. M. Coxeter, Introduction to Geometry (2d ed. 1969). Calculus
The branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limitthe notion of tending toward, or approaching, an ultimate value. The English physicist Isaac Newton and the German mathematician G. W. Leibniz, working independently, developed the calculus during the 17th cent. The calculus and its basic tools of differentiation and integration serve as the foundation for the larger branch of mathematics known as analysis. The methods of calculus are essential to modern physics and to most other branches of modern science and engineering.
The Integral Calculus
The second important kind of limit encountered in the calculus is the limit of a sum of elements when the number of such elements increases without bound while the size of the elements diminishes. For example, consider the problem of determining the area under a given curve y = f(x) between two values of x, say a and b. Let the interval between a and b be divided into n subintervals, from a = x0 through x1, x2, x3, xi − 1, xi, , up to xn = b. The width of a given subinterval is equal to the difference between the adjacent values of x, or &Dgr;xi = xi − xi − 1, where i designates the typical, or ith, subinterval. On each &Dgr;xi a rectangle can be formed of width &Dgr;xi, height yi = f(xi) (the value of the function corresponding to the value of x on the right-hand side of the subinterval), and area &Dgr;Ai = f(xi)&Dgr;xi. In some cases, the rectangle may extend above the curve, while in other cases it may fail to include some of the area under the curve; however, if the areas of all these rectangles are added together, the sum will be an approximation of the area under the curve.
This approximation can be improved by increasing n, the number of subintervals, thus decreasing the widths of the &Dgr;x\'s and the amounts by which the &Dgr;A\'s exceed or fall short of the actual area under the curve. In the limit where n approaches infinity (and the largest &Dgr;x approaches zero), the sum is equal to the area under the curve.
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The integral of f(x), and f(x) itself is called the integrand. This method of finding the limit of a sum can be used to determine the lengths of curves, the areas bounded by curves, and the volumes of solids bounded by curved surfaces, and to solve other similar problems.
An entirely different consideration of the problem of finding the area under a curve leads to a means of evaluating the integral. It can be shown that if F(x) is a function whose derivative is f(x), then the area under the graph of y = f(x) between a and b is equal to F(b) − F(a). This connection between the integral and the derivative is known as the Fundamental Theorem of the Calculus. Stated in symbols: ∫&ba;f(x)dx=F(b)−F(a), where F′(x)=f(x).The function F(x), which is equal to the integral of f(x), is sometimes called an antiderivative of f(x), while the process of finding F(x) from f(x) is called integration or antidifferentiation. The branch of calculus concerned with both the integral as the limit of a sum and the integral as the antiderivative of a function is known as the integral calculus. The type of integral just discussed, in which the limits of integration, a and b, are specified, is called a definite integral. If no limits are specified, the expression is an indefinite integral. In such a case, the function F(x) resulting from integration is determined only to within the addition of an arbitrary constant C, since in computing the derivative any constant terms having derivatives equal to zero are lost; the expression for the indefinite integral of f(x) is ∫f(x)dx=F(x)+C.The value of the constant C must be determined from various boundary conditions surrounding the particular problem in which the integral occurs. The calculus has been developed to treat not only functions of a single variable, e.g., x or t, but also functions of several variables. For example, if z = f(x,y) is a function of two independent variables, x and y, then two different derivatives can be determined, one with respect to each of the independent variables. These are denoted by ∂z/∂x and ∂z/∂y or by Dxz and Dyz. Three different second derivatives are possible, ∂2z/∂x2, ∂2z/∂y2, and ∂2z/∂x∂y = ∂2z/∂y∂x. Such derivatives are called partial derivatives. In any partial differentiation all independent variables other than the one being considered are treated as constants.
The Differential Calculus
The differential calculus arises from the study of the limit of a quotient, &Dgr;y/&Dgr;x, as the denominator &Dgr;x approaches zero, where x and y are variables. y may be expressed as some function of x, or f(x), and &Dgr;y and &Dgr;x represent corresponding increments, or changes, in y and x. The limit of &Dgr;y/&Dgr;x is called the derivative of y with respect to x and is indicated by dy/dx or Dxy.
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The symbols dy and dx are called differentials (they are single symbols, not products), and the process of finding the derivative of y = f(x) is called differentiation. The derivative dy/dx = df(x)/dx is also denoted by y, or f(x). The derivative f(x) is itself a function of x and may be differentiated, the result being termed the second derivative of y with respect to x and denoted by y˝, f˝(x), or d2y/dx2. This process can be continued to yield a third derivative, a fourth derivative, and so on. In practice formulas have been developed for finding the derivatives of all commonly encountered functions. For example, if y = xn, then y = nxn − 1, and if y = sin x, then y = cos x trigonometry. In general, the derivative of y with respect to x expresses the rate of change in y for a change in x. In physical applications the independent variable (here x) is frequently time; e.g., if s = f(t) expresses the relationship between distance traveled, s, and time elapsed, t, then s = f(t) represents the rate of change of distance with time, i.e., the speed, or velocity.
Everyday calculations of velocity usually divide the distance traveled by the total time elapsed, yielding the average velocity. The derivative f(t) = ds/dt, however, gives the velocity for any particular value of t, i.e., the instantaneous velocity. Geometrically, the derivative is interpreted as the slope of the line tangent to a curve at a point. If y = f(x) is a real-valued function of a real variable, the ratio &Dgr;y/&Dgr;x = (y2 − y1)/(x2 − x1) represents the slope of a straight line through the two points P (x1,y1) and Q (x2,y2) on the graph of the function. If P is taken closer to Q, then x1 will approach x2 and &Dgr;x will approach zero. In the limit where &Dgr;x approaches zero, the ratio becomes the derivative dy/dx = f(x) and represents the slope of a line that touches the curve at the single point Q, i.e., the tangent line. This property of the derivative yields many applications for the calculus, e.g., in the design of optical mirrors and lenses and the determination of projectile paths.
See Richard Courant and Fritz John, Introduction to Calculus and Analysis, Vol. I (1965); Morris Kline, Calculus: An Intuitive and Physical Approach (2 vol., 1967); G. B. Thomas and R. L. Finney, Calculus and Analytic Geometry (7th ed. 2 vol., 1988).
Trigonometry
[Gr.,=measurement of triangles], a specialized area of geometry concerned with the properties of and relations among the parts of a triangle. Spherical trigonometry is concerned with the study of triangles on the surface of a sphere rather than in the plane; it is of considerable importance in surveying, navigation, and astronomy.
The Basic Trigonometric Functions
Trigonometry originated as the study of certain mathematical relations originally defined in terms of the angles and sides of a right triangle, i.e., one containing a right angle (90). Six basic relations, or trigonometric functions, are defined.
g638;trig-angl638
Extension of the Trigonometric Functions
The notion of the trigonometric functions can be extended beyond 90 by defining the functions with respect to Cartesian coordinates. Let r be a line of unit length from the origin to the point P (x,y), and let θ be the angle r makes with the positive x-axis. The six functions become sin θ =y/r=y, cos θ=x/r=x, tan θ=y/x, cot θ=x/y, sec θ=r/x=1/x, and csc θ=r/y=1/y. As θ increases beyond 90, the point P crosses the y-axis and x becomes negative; in quadrant II the functions are negative except for sin θ and csc θ. Beyond θ=180, P is in quadrant III, y is also negative, and only tan θ and cot θ are positive, while beyond θ=270 P moves into quadrant IV, x becomes positive again, and cos θ and sec θ are positive. g639;trigonome639
Mathematics Megasites The Abacus
http://www.ee.ryerson.ca:8080/~elf/abacus/ Construction and anatomy of an abacus. Basics of how an abacus works. Ask Dr. Math: Elementary School Level
http://forum.swarthmore.edu/dr.math/drmath.elem.html Answers to lots of elementary math questions. Ask Dr. Math: High School Level
http://forum.swarthmore.edu/dr.math/drmath.high.html Answers to lots of high school math questions. Ask Dr. Math: Middle School Level
http://forum.swarthmore.edu/dr.math/drmath.middle.html Answers to lots of middle school math questions. Calculating Machines
http://www.webcom.com/calc/ From an 1885 adding machine and an abacus to history of the machines from 1623 to 1970. Interactive Mathematics Miscellany and Puzzles
http://www.cut-the-knot.com/ Interactive activities from all areas of math. Interactive Mathematics Online
http://tqd.advanced.org/2647/ This site is great for helping in high school math, especially in geometry. Math League Help Topics
http://www.mathleague.com/help/help.htm This is a help resource for 4th through 8th grades. Math Share Shop
http://tqd.advanced.org/2897/ Check out the SMART page for a list of math topics, then explore the subject you want to learn more about. Mathematics Lessons
http://math.rice.edu/~lanius/Lessons/ Alegbra, geometry, fractions, and more. MegaMathmematics!
http://www.cs.uidaho.edu/~casey931/mega-math/menu.html Games on graphs, algorithms, infinity and more. Monster Math
http://www.lifelong.com/lifelong_universe/AcademicWorld/
MonsterMath/default.html This arithmetic site explores math interactively and in both English and Spanish. Some plug-ins may be required (links provided) for the Macintosh audio/visual version, but a PC version without sound is available. Plane Math
http://www.PLANEMATH.COM/ Math activities for elementary level learners and educators using airplanes as a theme. The Proof
http://www.pbs.org/wgbh/nova/proof/ "For over 350 years, some of the greatest minds of science struggled to prove what was known as Fermat\'s Last Theorem -- the idea that a certain simple equation had no solutions. Now hear from the man who spent seven years of his life cracking the problem, read the intriguing story of an 18th century woman mathematician who hid her identity in order to work on Fermat\'s Last Theorem, and demonstrate that a related equation, the Pythagorean Theorem, is true." Topology Games
http://www.northnet.org/weeks/ Interactive games to learn topology. WebMath
http://www.webmath.com/ Everyday Math, numbers, units conversion, fractions, graphing, simplifying expressions (powers, products, like terms), polynomials, factoring polynomials (using GCF, trinomials), quadratic equations (solve by factoring, The quadratic formula.)
ArithmeticThe Abacushttp://www.ee.ryerson.ca:8080/~elf/abacus/ The abacus is a calculator whose earliest known use is circa 500 B.C. by the Chinese civilization. Addition, subtraction, division and multiplication can be performed on a standard abacus.
Arithmetichttp://hometown.aol.com/iongoal/mathlessons.htm This Site Contains: 200+ pages of Basic Arithmetic Skills; Interactive Practice on every page; An Explanation of the math topic on each page; Several Challenge Games on every page; Math Problems are randomly created.
Math Forum: Multiplication Tipshttp://www.mathforum.com/k12/mathtips/multiplication.tips.html From Math Forum, here are some quick and easy tricks for multiplying by five, nine, and eleven.
Math Problems of the Weekhttp://www.mbnet.mb.ca/~jfinch/math.html There is one problem a week, for both grades 3 and 4\'s and grades 5 and 6\'s. The answers will be found in a separate section at this site.
Monster Mathhttp://www.lifelong.com/lifelong_universe/AcademicWorld/MonsterMath/default.html Designed to introduce and review a variety of basic math concepts such as counting, addition and multiplication. These activities allow the player to be problem solvers.
Stanley Park Chase http://schoolcentral.com/willoughby5/default.htm Great interactive and visual site to practice basic multiplication while following a story.
Times Tables Tests http://www.times-tables.com/times_tables_test/times_tables_test.htm We have selected these tables test work sheets because they mirror the type of test your child might be given in school. For example, the questions are listed randomly and each question is worded to make your child think. At the end of each page click finish to see your score. Algebra Algebra: Fun With Calendars
http://math.rice.edu/~lanius/Lessons/calen.html Math tricks to amaze your friends and family! Algebra Online
http://www.algebra-online.com/ "Algebra OnlineSM is a free service designed to allow students, parents, and educators throughout the world to communicate. This includes free private tutoring, live chat, and a message board, among many other features. Questions and discussions relating to all levels of mathematics (not just Algebra) are welcome." High School Algebra: Ask Dr. Math
http://forum.swarthmore.edu/dr.math/tocs/algebra.high.html Frequently asked questions about high school algebra answered here. Introduction to Algebra
http://www.mathleague.com/help/algebra/algebra.htm Clearly-written definitions and easy-to-understand Algebra examples. Middle School Algebra: Ask Dr. Math
http://forum.swarthmore.edu/dr.math/tocs/algebra.middle.html Frequently asked questions about middle school algebra answered here. Origins of Algebra
http://www.comlab.ox.ac.uk/oucl/users/jonathan.bowen/
algebra/section3_1.html This text-based site explains the history and origins of algebra. Algebra Academic Assistance Access
Read the FAQ-list first, and then subscribe to this free and generally
fast service. Subscribe to the Post-Secondary (that is, college) Algebra
list, or the Secondary (that is, high school) Math list.
http://www.tutoraid.org/faq
Algebra This site has a few lessons that teach functions, terminology, and
graphing.
http://tqd.advanced.org/2647/algebra/algebra.htm
algebra.com
This is an "answers only" sort of site. The webmaster has quite a few
solvers, if you only need to check that you have the correct answer.
http://www.algebra.com/
"Algebra Friendly"
This algebra lesson program will take you through two semesters of college
algebra (that is, two years of high-school algebra). You have to go through
the lessons in order, but the coverage is quite nice, so the review won\'t
hurt. When you download the six necessary files, be sure to unzip disks 2
through 5 first; then disk 1 can install the program.
http://www.dhies.com/algebra_friendly.htm
algebra.help
This site has lessons on basic algebra topics and techniques, study tips,
calculator advice, and more.
http://www.algebrahelp.com/
AMath
The "pre-algebra" software covers geometry, negatives, fractions, beginning
algebra, and more. The demo version is severely truncated.
http://www.amath.com/prealg/home.html
Ask Adam
Sign up and submit your question. Write down your "ticket number", as this
is how you can track your question and answer. Answers are available only
on-site (not through e-mail), but you can also check the archives for other
people\'s questions. Turn-around time was same-day when I tested the service.
http://www.askadam.com
All Experts
Choose from a long list of tutors. Response was same-day when I tested the
service, and the replies were very friendly and supportive.
http://www.allexperts.com/edu/math.shtml
CoolMath
Visit Karen\'s CoolMath.com site for study tips, hints on how to study for
math tests, and lessons on functions.
http://www.coolmath.com/index.html
Dan\'s Lessons
These lessons range from algebra to calculus and beyond, and are heavy on
worked examples that illustrate the "basics".
http://home.earthlink.net/~djbach/lessons.html
DAU\'s Math Modules
This school has a lengthy list of algebra lessons. You can follow their
lesson map, or else find what you want in their index.
http://www.cne.gmu.edu/modules/dau/math/dau1_frm.html(lesson map)
http://www.cne.gmu.edu/modules/dau/math_idx_frm.html(index)
Easy Algebra
These quiz-type lessons on algebra come complete with explanations,
solutions, and cross-referencing.
http://www.gcse.com/Maths/esalg.htm
Exercises in Math Readiness
EMR has lessons, examples, and short quizzes (complete with hints and
solutions). They cover only a few topics, but the coverage is excellent, and
extends from algebra to trigonometry and set theory.
http://math.usask.ca/readin/menu.html
Equation Grapher
If you don\'t have a graphing calculator, or would like to do your graphing
on your home computer, download "Equation Grapher" It works quite nicely.
This program is shareware so, if you plan to use it long-term, be sure to
return to the site and pay for it.
http://www.mfsoft.com/equationgrapher/index.html
Felicia\'s Algebra Tutorial
Felicia\'s lessons (on arithmetic through beginning algebra) are friendly,
chatty, and illustrated, with many worked examples and check-your-answer
problems covering basic techniques and problem areas.
http://algebra.freeservers.com/springridge1.html
Free-Ed (pre-algebra):
If you need lessons on fractions, percents, or negatives, Free-Ed\'s
pre-algebra page has these (and other) topics covered.
http://www.free-ed.net/fr07/lfc/course070101_01/index.html
Graph Paper Printer
This program is a small (354K) download, and is really handy if you need
specialized graph paper, such as log, log-log, or polar. The settings are
customizable.
http://perso.easynet.fr/~philimar/graphpapeng.htm
IEP Math
The "Arithmetic Review" software covers fractions, and the "Pre-Algebra"
software covers negatives. The interface is very nice. The demoware versions
are limited-use, but the "real" programs are quite affordable.
http://www.iepmath.com/
Interactive Learning Network
This fee-based service includes some nice freebie lessons. The site runs
kinda slow, and the lessons require the Real Audio plug-in. But if you have
a fast (cable?) connection (or lots of time), these lessons could be
helpful. The subjects run from Arithmetic to Calculus and Statistics.
http://www.iln.net/main/
Introduction to Algebra
This run-down of the basics is heavy on word-problem examples, and stresses
the how-to\'s of "translating" between math and English.
http://www.mathleague.com/help/algebra/algebra.htm
Professor Symancyk\'s algebra lessons
Professor Symancyk has written some great lessons, which include
illustrations and worked examples. Pick your topic from his menu.
http://www.aacc.cc.md.us/dfsymancyk/web131/wmenu1311.html
Stroh\'s Algebra Help Page
Eric Stroh has pored over many algebra sites, and has arranged links to
lessons according to topic. If you are needing help in one specific area,
this is a good place to look.
http://www.homestead.com/stroh/algebrahelp.html
"Understanding Algebra"
James Brennan has put his algebra book online. This site covers many
standard algebra topics, complete with worked examples.
http://www.edteach.com/algebra/table_of_contents.htm
Usenet newsgroup: alt.algebra.help: This newsgroup doesn\'t seem to be too
busy, so your request for help won\'t be lost in the crowd. Fractions & Decimals Ask Dr. Math: Fractions and Decimals
http://forum.swarthmore.edu/dr.math/tocs/fractions.elem.html Read the answers to questions about fractions and decimals, or send in your own question! Decimals, Whole Numbers, and Exponents
http://www.mathleague.com/help/decwholeexp/decwholeexp.htm This site defines words and shows how to work with decimal numbers. Fraction to Decimal Conversion Tables
http://www.sisweb.com/math/general/arithmetic/fradec.htm Use this chart to convert fractions into decimals. Fractions
http://www.mathleague.com/help/fractions/fractions.htm Lots of help with fractions! This site shows how to add, subtract, multiply and divide fractions, as well as convert and reduce fractions. What are Some Properties of Fractions?
http://www.mathleague.com/help/fractions/fractions.htm Find out about some of the basic rules for working with fractions. WebMath: Fractions
http://www.webmath.com/fractions.html This site shows how to reduce, add, subtract, multiply and divide fractions. Geometry Flashcard
http://www.aplusmath.com/cgi-bin/flashcards/geoflash Flashcards for geometry online. Gallery of Interactive Geometry
http://www.geom.umn.edu/apps/gallery.html Define curves of a plane, build a rainbow, and more. Introduction to Geometry
http://tqd.advanced.org/2609/ Learn geometry from four basic lessons provided at this site then take a final quiz to test your knowledge. The Storybook of Geometry
http://library.advanced.org/3654/ Geometry story problems site that takes you through the steps or lets you work on the problems on your own. Problem Solving 21st Century Problem Solving
http://www2.hawaii.edu/suremath/home.html A site for students, teachers and parents about algebra, problem solving, physics, chemistry and more with detailed problems and examples provided. Grade 5/6 Finch Math Problems of the Week!
http://www.mbnet.mb.ca/~jfinch/math.html This site provides one problem a week, for both grades 3 and 4\'s and grades 5 and 6\'s and includes past problems and an answer table. Word Problems for Kids
http://juliet.stfx.ca/people/fac/pwang/mathpage/math1.html By Canada\'s SchoolNet, this site breaks down word problems for grades 5 to 12. Puzzles, Quizzes and Flashcards AIMS Puzzle Corner
http://www.aimsedu.org/Puzzle/PuzzleList.html 12 fun math puzzles a year including printable worksheets. 1995 to the present. Brain Teasers
http://www.eduplace.com/math/brain/index.html Brain teasers for grades 3 through 7+. The Factory
http://tqd.advanced.org/3288/ Learn about fractals at this site separated into five levels of difficulty. Flashcard
http://www.aplusmath.com/cgi-bin/flashcards/geoflash Flashcards for geometry online. Flashcards for Kids
http://www.wwinfo.com/edu/flash.html Online flashcards for kids. Mathematics Center
http://www.eduplace.com/math/index.html Math problems for kids 3rd grade and up which include brain teasers, games and other math resources categorized by grade levels. Mathmagic Activities
http://www.scri.fsu.edu/~dennisl/CMS/activity/math_magic.html Card tricks, calculational activities, and geometry curiosities. MathMagic on the Web
http://forum.swarthmore.edu/mathmagic/ Try to solve weekly math problems on the Web. Mathmania
http://csr.uvic.ca/~mmania/student.htm Interactive math puzzles from knots to graphs. The Mult Applet
http://forum.swarthmore.edu/mathmagic/ Practice your multiplication on the internet. Every time a new set of tables appears. Java required. Link to download provided. The Pi Trivia Game
http://eveander.com/~eveander/trivia/ Take this quiz to test your math knowledge about pi. Stanley Park Chase
http://schoolcentral.com/willoughby5/default.htm Great interactive and visual site to practice basic multiplication while following a story. Strawberry Macaw\'s Puzzle Page
http://www.serve.com/games/puzzles.htm Interactive puzzles and logic for elementary students from Five Gallon Cans to Sliding Number Puzzle. Tower of Hanoi
http://www.cut-the-knot.com/recurrence/hanoi.html Learn how to solve the tower of hanoi puzzle. Victoria Search
http://schoolcentral.com/necklace/vic1.htm An interactive site to practice basic division while reading a story. Unit Conversions Common Equivalent Weights and Measures
http://www.cchem.berkeley.edu/ChemResources/
Weights-n-Measures/weights-n-measures.html A comprehensive list of units along with their metric and US equivalents. A Dictionary of Units
http://www.ex.ac.uk/cimt/dictunit/dictunit.htm This provides a summary of most of the units of measurement to be found in use around the world today. Measurements Converter
http://www.mplik.ru/~sg/transl/ A conversion table.
Multiplication Chart MathematiciansIndexes of Biographies
http://www-groups.dcs.st-and.ac.uk/~history/BiogIndex.html A comprehensive guide to mathematicians. Mathographies
http://scidiv.bcc.ctc.edu/Math/MathFolks.html Biographies of 24 mathematicians, prepared by faculty at Bellevue Community College . Past Notable Women of Computing & Mathematics
http://www.cs.yale.edu/homes/tap/past-women.html Biographies of women in computing and mathematics as well as other links. Shiyali Ramamrita Ranganathan
http://www.uniroma1.it/Mathematics/I-Ranganathan.html "Indian librarian and educator (mathematician) who was considered the father of library science in India and whose contributions had worldwide influence."
http://www.hawaii.edu/suremath/intro_algebra.html
QuickMath is an automated service for answering common math problems over the internet.
Think of it as an online calculator that solves equations and does all sorts of algebra and calculus problems - instantly and automatically!
When you submit a question to QuickMath, it is processed by Mathematica, the largest and most powerful computer algebra package available today. The answer is then sent back to you and displayed right there on your browser, usually within a couple of seconds.
http://www.quickmath.com/
http://rsts.net/home/
http://www.wolfram.com/products/webmathematica/
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