1.
Calculus
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Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two branches, differential calculus, and integral calculus, these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the notions of convergence of infinite sequences. Generally, modern calculus is considered to have developed in the 17th century by Isaac Newton. Today, calculus has widespread uses in science, engineering and economics, Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, Calculus has historically been called the calculus of infinitesimals, or infinitesimal calculus. Calculus is also used for naming some methods of calculation or theories of computation, such as calculus, calculus of variations, lambda calculus. The ancient period introduced some of the ideas that led to integral calculus, the method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, indian mathematicians gave a non-rigorous method of a sort of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, Cavalieris work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieris infinitesimals with the calculus of finite differences developed in Europe at around the same time, pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, in other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were considered disreputable. These ideas were arranged into a calculus of infinitesimals by Gottfried Wilhelm Leibniz. He is now regarded as an independent inventor of and contributor to calculus, unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. Leibniz and Newton are usually credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the used in calculus today
2.
Mathematical analysis
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Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions. These theories are studied in the context of real and complex numbers. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis, analysis may be distinguished from geometry, however, it can be applied to any space of mathematical objects that has a definition of nearness or specific distances between objects. Mathematical analysis formally developed in the 17th century during the Scientific Revolution, early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, a geometric sum is implicit in Zenos paradox of the dichotomy. The explicit use of infinitesimals appears in Archimedes The Method of Mechanical Theorems, in Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. Zu Chongzhi established a method that would later be called Cavalieris principle to find the volume of a sphere in the 5th century, the Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolles theorem in the 12th century. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and his followers at the Kerala school of astronomy and mathematics further expanded his works, up to the 16th century. The modern foundations of analysis were established in 17th century Europe. During this period, calculus techniques were applied to approximate discrete problems by continuous ones, in the 18th century, Euler introduced the notion of mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the definition of continuity in 1816. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity required a change in x to correspond to an infinitesimal change in y. He also introduced the concept of the Cauchy sequence, and started the theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations, the contributions of these mathematicians and others, such as Weierstrass, developed the -definition of limit approach, thus founding the modern field of mathematical analysis. In the middle of the 19th century Riemann introduced his theory of integration, the last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the epsilon-delta definition of limit. Then, mathematicians started worrying that they were assuming the existence of a continuum of numbers without proof. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the size of the set of discontinuities of real functions, also, monsters began to be investigated
3.
Limit (mathematics)
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In mathematics, a limit is the value that a function or sequence approaches as the input or index approaches some value. Limits are essential to calculus and are used to define continuity, derivatives, the concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory. In formulas, a limit is usually written as lim n → c f = L and is read as the limit of f of n as n approaches c equals L. Here lim indicates limit, and the fact that function f approaches the limit L as n approaches c is represented by the right arrow, suppose f is a real-valued function and c is a real number. Intuitively speaking, the lim x → c f = L means that f can be made to be as close to L as desired by making x sufficiently close to c. The first inequality means that the distance x and c is greater than 0 and that x ≠ c, while the second indicates that x is within distance δ of c. Note that the definition of a limit is true even if f ≠ L. Indeed. Now since x +1 is continuous in x at 1, we can now plug in 1 for x, in addition to limits at finite values, functions can also have limits at infinity. In this case, the limit of f as x approaches infinity is 2, in mathematical notation, lim x → ∞2 x −1 x =2. Consider the following sequence,1.79,1.799,1.7999 and it can be observed that the numbers are approaching 1.8, the limit of the sequence. Formally, suppose a1, a2. is a sequence of real numbers, intuitively, this means that eventually all elements of the sequence get arbitrarily close to the limit, since the absolute value | an − L | is the distance between an and L. Not every sequence has a limit, if it does, it is called convergent, one can show that a convergent sequence has only one limit. The limit of a sequence and the limit of a function are closely related, on one hand, the limit as n goes to infinity of a sequence a is simply the limit at infinity of a function defined on the natural numbers n. On the other hand, a limit L of a function f as x goes to infinity, if it exists, is the same as the limit of any sequence a that approaches L. Note that one such sequence would be L + 1/n, in non-standard analysis, the limit of a sequence can be expressed as the standard part of the value a H of the natural extension of the sequence at an infinite hypernatural index n=H. Thus, lim n → ∞ a n = st , here the standard part function st rounds off each finite hyperreal number to the nearest real number. This formalizes the intuition that for very large values of the index. Conversely, the part of a hyperreal a = represented in the ultrapower construction by a Cauchy sequence, is simply the limit of that sequence
4.
Augustin-Louis Cauchy
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Baron Augustin-Louis Cauchy FRS FRSE was a French mathematician who made pioneering contributions to analysis. He was one of the first to state and prove theorems of calculus rigorously and he almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. A profound mathematician, Cauchy had an influence over his contemporaries. His writings range widely in mathematics and mathematical physics, more concepts and theorems have been named for Cauchy than for any other mathematician. Cauchy was a writer, he wrote approximately eight hundred research articles. Cauchy was the son of Louis François Cauchy and Marie-Madeleine Desestre, Cauchy married Aloise de Bure in 1818. She was a relative of the publisher who published most of Cauchys works. By her he had two daughters, Marie Françoise Alicia and Marie Mathilde, Cauchys father was a high official in the Parisian Police of the New Régime. He lost his position because of the French Revolution that broke out one month before Augustin-Louis was born, the Cauchy family survived the revolution and the following Reign of Terror by escaping to Arcueil, where Cauchy received his first education, from his father. After the execution of Robespierre, it was safe for the family to return to Paris, there Louis-François Cauchy found himself a new bureaucratic job, and quickly moved up the ranks. When Napoleon Bonaparte came to power, Louis-François Cauchy was further promoted, the famous mathematician Lagrange was also a friend of the Cauchy family. On Lagranges advice, Augustin-Louis was enrolled in the École Centrale du Panthéon, most of the curriculum consisted of classical languages, the young and ambitious Cauchy, being a brilliant student, won many prizes in Latin and Humanities. In spite of successes, Augustin-Louis chose an engineering career. In 1805 he placed second out of 293 applicants on this exam, one of the main purposes of this school was to give future civil and military engineers a high-level scientific and mathematical education. The school functioned under military discipline, which caused the young, nevertheless, he finished the Polytechnique in 1807, at the age of 18, and went on to the École des Ponts et Chaussées. He graduated in engineering, with the highest honors. After finishing school in 1810, Cauchy accepted a job as an engineer in Cherbourg. Cauchys first two manuscripts were accepted, the one was rejected
5.
Moigno
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François-Napoléon-Marie Moigno, known in his later life as the Abbé Moigno, was a French Catholic priest and one time Jesuit, as well as a physicist and author. He considered himself a student of Cauchy, Moigno was born at Guémené-sur-Scorff, Morbihan, in Brittany, on 15 April 1804. He received his education at the Jesuit college at Sainte-Anne-dAuray. He did his studies at Montrouge, devoting his leisure to mathematics and physics. On the outbreak of the Revolution of 1830, he left with his fellow Jesuits for Brieg in Switzerland, there he acquired several languages, including Hebrew and Arabic. In 1836 he was appointed professor of mathematics at the College of Sainte-Geneviève, Rue des Postes, here he became known not only as a scholar, but also as a preacher and writer. Shortly afterwards he undertook a tour of Europe, contributing numerous letters to the journal LEpoque and he acted as chaplain of the Lycée Louis-le-Grand from 1848 to 1851. He became scientific editor of the Presse in 1850 and of the Pays in 1851, in 1862 he founded Les Mondes and became associated with the clergy of Saint-Germain-des-Prés. In 1873 he was appointed a canon of the chapter of the Basilica of Saint-Denis. He died there on 14 July 1884, Moigno was a prolific writer, an expositor of science rather than an original investigator. He also translated numerous English and Italian memoirs on science into French, list of Roman Catholic scientist-clerics Attribution This article incorporates text from a publication now in the public domain, Herbermann, Charles, ed. François-Napoléon-Marie Moigno. The entry cites, Cosmos, 3rd series, VIII,443
6.
Function (mathematics)
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In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that each real number x to its square x2. The output of a function f corresponding to a x is denoted by f. In this example, if the input is −3, then the output is 9, likewise, if the input is 3, then the output is also 9, and we may write f =9. The input variable are sometimes referred to as the argument of the function, Functions of various kinds are the central objects of investigation in most fields of modern mathematics. There are many ways to describe or represent a function, some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function, in science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation, sometimes the codomain is called the functions range, but more commonly the word range is used to mean, instead, specifically the set of outputs. For example, we could define a function using the rule f = x2 by saying that the domain and codomain are the numbers. The image of this function is the set of real numbers. In analogy with arithmetic, it is possible to define addition, subtraction, multiplication, another important operation defined on functions is function composition, where the output from one function becomes the input to another function. Linking each shape to its color is a function from X to Y, each shape is linked to a color, there is no shape that lacks a color and no shape that has more than one color. This function will be referred to as the color-of-the-shape function, the input to a function is called the argument and the output is called the value. The set of all permitted inputs to a function is called the domain of the function. Thus, the domain of the function is the set of the four shapes. The concept of a function does not require that every possible output is the value of some argument, a second example of a function is the following, the domain is chosen to be the set of natural numbers, and the codomain is the set of integers. The function associates to any number n the number 4−n. For example, to 1 it associates 3 and to 10 it associates −6, a third example of a function has the set of polygons as domain and the set of natural numbers as codomain
7.
Limit point
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Note that x does not have to be an element of S. This concept profitably generalizes the notion of a limit and is the underpinning of concepts such as closed set, Let S be a subset of a topological space X. A point x in X is a point of S if every neighbourhood of x contains at least one point of S different from x itself. Note that it doesnt make a difference if we restrict the condition to open neighbourhoods only and this is equivalent, in a T1 space, to requiring that every neighbourhood of x contains infinitely many points of S. If every open set containing x contains infinitely many points of S then x is a type of limit point called an ω-accumulation point of S. If every open set containing x contains uncountably many points of S then x is a type of limit point called a condensation point of S. If every open set U containing x satisfies |U ∩ S| = |S| then x is a type of limit point called a complete accumulation point of S. A point x ∈ X is a point or accumulation point of a sequence n ∈ N if, for every neighbourhood V of x. If the space is Fréchet–Urysohn, this is equivalent to the assertion that x is a limit of some subsequence of the sequence n ∈ N, the set of all cluster points of a sequence is sometimes called a limit set. The concept of a net generalizes the idea of a sequence, Let n, → X be a net, where is a directed set. Cluster points in nets encompass the idea of both points and ω-accumulation points. Clustering and limit points are defined for the related topic of filters. We have the following characterisation of limit points, x is a point of S if. Proof, We use the fact that a point is in the closure of a set if, If x is in S, we are done. If x is not in S, then every neighbourhood of x contains a point of S, in other words, x is a limit point of S and x is in L. If x is in S, then every neighbourhood of x clearly meets S, If x is in L, then every neighbourhood of x contains a point of S, so x is again in the closure of S. A corollary of this gives us a characterisation of closed sets, A set S is closed if. Proof, S is closed if and only if S is equal to its closure if and only if S = S ∪ L if, another proof, Let S be a closed set and x a limit point of S
8.
Limit of a function
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In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below, informally, a function f assigns an output f to every input x. We say the function has a limit L at an input p, more specifically, when f is applied to any input sufficiently close to p, the output value is forced arbitrarily close to L. On the other hand, if some inputs very close to p are taken to outputs that stay a distance apart. The notion of a limit has many applications in modern calculus, in particular, the many definitions of continuity employ the limit, roughly, a function is continuous if all of its limits agree with the values of the function. It also appears in the definition of the derivative, in the calculus of one variable, however, his work was not known during his lifetime. Weierstrass first introduced the definition of limit in the form it is usually written today. He also introduced the notations lim and limx→x0, the modern notation of placing the arrow below the limit symbol is due to Hardy in his book A Course of Pure Mathematics in 1908. Imagine a person walking over a landscape represented by the graph of y = f and her horizontal position is measured by the value of x, much like the position given by a map of the land or by a global positioning system. Her altitude is given by the coordinate y and she is walking towards the horizontal position given by x = p. As she gets closer and closer to it, she notices that her altitude approaches L, if asked about the altitude of x = p, she would then answer L. What, then, does it mean to say that her altitude approaches L. It means that her altitude gets nearer and nearer to L except for a small error in accuracy. For example, suppose we set a particular goal for our traveler. She reports back that indeed she can get within ten meters of L, since she notes that when she is within fifty horizontal meters of p, the accuracy goal is then changed, can she get within one vertical meter. If she is anywhere within seven meters of p, then her altitude always remains within one meter from the target L. This explicit statement is quite close to the definition of the limit of a function with values in a topological space. To say that lim x → p f = L, means that ƒ can be made as close as desired to L by making x close enough, the following definitions are the generally accepted ones for the limit of a function in various contexts. Suppose f, R → R is defined on the real line, the value of the limit does not depend on the value of f, nor even that p be in the domain of f
9.
Real number
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In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing
10.
Cardinal number
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In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a set is a natural number, the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite sets, cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, in the case of finite sets, this agrees with the intuitive notion of size. In the case of sets, the behavior is more complex. It is also possible for a subset of an infinite set to have the same cardinality as the original set. There is a sequence of cardinal numbers,0,1,2,3, …, n, …, ℵ0, ℵ1, ℵ2, …, ℵ α, …. This sequence starts with the natural numbers including zero, which are followed by the aleph numbers, the aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number, If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs. Cardinality is studied for its own sake as part of set theory and it is also a tool used in branches of mathematics including model theory, combinatorics, abstract algebra, and mathematical analysis. In category theory, the numbers form a skeleton of the category of sets. The notion of cardinality, as now understood, was formulated by Georg Cantor, cardinality can be used to compare an aspect of finite sets, e. g. the sets and are not equal, but have the same cardinality, namely three. Cantor applied his concept of bijection to infinite sets, e. g. the set of natural numbers N =, thus, all sets having a bijection with N he called denumerable sets and they all have the same cardinal number. This cardinal number is called ℵ0, aleph-null and he called the cardinal numbers of these infinite sets transfinite cardinal numbers. Cantor proved that any unbounded subset of N has the same cardinality as N and he later proved that the set of all real algebraic numbers is also denumerable. His proof used an argument with nested intervals, but in an 1891 paper he proved the result using his ingenious. The new cardinal number of the set of numbers is called the cardinality of the continuum. His continuum hypothesis is the proposition that c is the same as ℵ1 and this hypothesis has been found to be independent of the standard axioms of mathematical set theory, it can neither be proved nor disproved from the standard assumptions
11.
Zero to the power of zero
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Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent n. The exponent is usually shown as a superscript to the right of the base, Some common exponents have their own names, the exponent 2 is called the square of b or b squared, the exponent 3 is called the cube of b or b cubed. The exponent −1 of b, or 1 / b, is called the reciprocal of b, when n is a positive integer and b is not zero, b−n is naturally defined as 1/bn, preserving the property bn × bm = bn + m. The definition of exponentiation can be extended to any real or complex exponent. Exponentiation by integer exponents can also be defined for a variety of algebraic structures. The term power was used by the Greek mathematician Euclid for the square of a line, archimedes discovered and proved the law of exponents, 10a 10b = 10a+b, necessary to manipulate powers of 10. In the late 16th century, Jost Bürgi used Roman numerals for exponents, early in the 17th century, the first form of our modern exponential notation was introduced by Rene Descartes in his text titled La Géométrie, there, the notation is introduced in Book I. Nicolas Chuquet used a form of notation in the 15th century. The word exponent was coined in 1544 by Michael Stifel, samuel Jeake introduced the term indices in 1696. In the 16th century Robert Recorde used the square, cube, zenzizenzic, sursolid, zenzicube, second sursolid. Biquadrate has been used to refer to the power as well. Some mathematicians used exponents only for greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as ax + bxx + cx3 + d, another historical synonym, involution, is now rare and should not be confused with its more common meaning. In 1748 Leonhard Euler wrote consider exponentials or powers in which the exponent itself is a variable and it is clear that quantities of this kind are not algebraic functions, since in those the exponents must be constant. With this introduction of transcendental functions, Euler laid the foundation for the introduction of natural logarithm as the inverse function for y = ex. The expression b2 = b ⋅ b is called the square of b because the area of a square with side-length b is b2, the expression b3 = b ⋅ b ⋅ b is called the cube of b because the volume of a cube with side-length b is b3. The exponent indicates how many copies of the base are multiplied together, for example,35 =3 ⋅3 ⋅3 ⋅3 ⋅3 =243. The base 3 appears 5 times in the multiplication, because the exponent is 5
12.
Derivative
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The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a tool of calculus. For example, the derivative of the position of an object with respect to time is the objects velocity. The derivative of a function of a variable at a chosen input value. The tangent line is the best linear approximation of the function near that input value, for this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives may be generalized to functions of real variables. In this generalization, the derivative is reinterpreted as a transformation whose graph is the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables and it can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of variables, the Jacobian matrix reduces to the gradient vector. The process of finding a derivative is called differentiation, the reverse process is called antidifferentiation. The fundamental theorem of calculus states that antidifferentiation is the same as integration, differentiation and integration constitute the two fundamental operations in single-variable calculus. Differentiation is the action of computing a derivative, the derivative of a function y = f of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. It is called the derivative of f with respect to x, If x and y are real numbers, and if the graph of f is plotted against x, the derivative is the slope of this graph at each point. The simplest case, apart from the case of a constant function, is when y is a linear function of x. This formula is true because y + Δ y = f = m + b = m x + m Δ x + b = y + m Δ x. Thus, since y + Δ y = y + m Δ x and this gives an exact value for the slope of a line. If the function f is not linear, however, then the change in y divided by the change in x varies, differentiation is a method to find an exact value for this rate of change at any given value of x. The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the value of the ratio of the differences Δy / Δx as Δx becomes infinitely small
13.
Projectively extended real line
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In real analysis, the projectively extended real line, is the extension of the number line by a point denoted ∞. It is thus the set R ∪ with the arithmetic operations extended where possible. The added point is called the point at infinity, because it is considered as a neighbour of both ends of the real line, more precisely, the point at infinity is the limit of every sequence of real numbers whose absolute values are increasing and unbounded. The projectively extended real line may be identified with the line over the reals in which three points have been assigned specific values. The projectively extended real line must not be confused with the real number line, in which +∞. Unlike most mathematical models of the concept of number, this structure allows division by zero. In particular 1/0 = ∞, and moreover 1/∞ =0, making reciprocal, 1/x, the structure, however, is not a field, and none of the binary arithmetic operations are total, as witnessed for example by 0⋅∞ being undefined despite the reciprocal being total. It has usable interpretations, however – for example, in geometry, the projectively extended real line extends the field of real numbers in the same way that the Riemann sphere extends the field of complex numbers, by adding a single point called conventionally ∞. In contrast, the real number line distinguishes between + ∞ and − ∞. The order relation cannot be extended to R ^ in a meaningful way, given a number a ≠ ∞, there is no convincing argument to define either a > ∞ or that a < ∞. Since ∞ cant be compared with any of the other elements, However, order on R is used in definitions in R ^. Fundamental to the idea that ∞ is a point no different from any other is the way the real line is a homogeneous space. For example the general group of 2×2 real invertible matrices has a transitive action on it. The group action may be expressed by Möbius transformations, with the understanding that when the denominator of the linear transformation is 0. This cannot be extended to 4-tuples of points, because the cross-ratio is invariant, the terminology projective line is appropriate, because the points are in 1-to-1 correspondence with one-dimensional linear subspaces of R2. The arithmetic operations on this space are an extension of the operations on reals. A motivation for the new definitions is the limits of functions of real numbers. Consequently, they are undefined, ∞ + ∞ ∞ − ∞ ∞ ⋅00 ⋅ ∞ ∞ / ∞0 /0 The following equalities mean
14.
Point at infinity
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In geometry, a point at infinity or ideal point is an idealized limiting point at the end of each line. In the case of a plane, there is one ideal point for each pencil of parallel lines of the plane. Adjoining these points produces a plane, in which no point can be distinguished. This holds for a geometry over any field, and more generally over any division ring, in the real case, a point at infinity completes a line into a topologically closed curve. In higher dimensions, all the points at infinity form a subspace of one dimension less than that of the whole projective space to which they belong. A point at infinity can also be added to the line, thereby turning it into a closed surface known as the complex projective line, CP1. In the case of a space, each line has two distinct ideal points. Here, the set of ideal points takes the form of a quadric, in an affine or Euclidean space of higher dimension, the points at infinity are the points which are added to the space to get the projective completion. As a projective space over a field is an algebraic variety. Similarly, if the field is the real or the complex field. In artistic drawing and technical perspective, the projection on the plane of the point at infinity of a class of parallel lines is called their vanishing point. In hyperbolic geometry, points at infinity are typically named ideal points, all points at infinity together form the Cayley absolute or boundary of a hyperbolic plane. This construction can be generalized to topological spaces, projective line is the Alexandroff extension of the corresponding field. Thus the circle is the one-point compactification of the line. Division by zero Midpoint § Generalizations Asymptote § Algebraic curves
15.
Natural logarithm
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The natural logarithm of a number is its logarithm to the base of the mathematical constant e, where e is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is written as ln x, loge x, or sometimes, if the base e is implicit. Parentheses are sometimes added for clarity, giving ln, loge or log and this is done in particular when the argument to the logarithm is not a single symbol, to prevent ambiguity. The natural logarithm of x is the power to which e would have to be raised to equal x. The natural log of e itself, ln, is 1, because e1 = e, while the natural logarithm of 1, ln, is 0, since e0 =1. The natural logarithm can be defined for any real number a as the area under the curve y = 1/x from 1 to a. The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, like all logarithms, the natural logarithm maps multiplication into addition, ln = ln + ln . However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, for instance, the binary logarithm is the natural logarithm divided by ln, the natural logarithm of 2. Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity, for example, logarithms are used to solve for the half-life, decay constant, or unknown time in exponential decay problems. They are important in many branches of mathematics and the sciences and are used in finance to solve problems involving compound interest, by Lindemann–Weierstrass theorem, the natural logarithm of any positive algebraic number other than 1 is a transcendental number. The concept of the natural logarithm was worked out by Gregoire de Saint-Vincent and their work involved quadrature of the hyperbola xy =1 by determination of the area of hyperbolic sectors. Their solution generated the requisite hyperbolic logarithm function having properties now associated with the natural logarithm, the notations ln x and loge x both refer unambiguously to the natural logarithm of x. log x without an explicit base may also refer to the natural logarithm. This usage is common in mathematics and some scientific contexts as well as in many programming languages, in some other contexts, however, log x can be used to denote the common logarithm. Historically, the notations l. and l were in use at least since the 1730s, finally, in the twentieth century, the notations Log and logh are attested. The graph of the logarithm function shown earlier on the right side of the page enables one to glean some of the basic characteristics that logarithms to any base have in common. Chief among them are, the logarithm of the one is zero. What makes natural logarithms unique is to be found at the point where all logarithms are zero. At that specific point the slope of the curve of the graph of the logarithm is also precisely one
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Continuous function
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In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function, a continuous function with a continuous inverse function is called a homeomorphism. Continuity of functions is one of the concepts of topology. The introductory portion of this focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the general case of functions between two metric spaces. In order theory, especially in theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article, as an example, consider the function h, which describes the height of a growing flower at time t. By contrast, if M denotes the amount of money in an account at time t, then the function jumps at each point in time when money is deposited or withdrawn. A form of the definition of continuity was first given by Bernard Bolzano in 1817. Cauchy defined infinitely small quantities in terms of quantities. The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s but the work wasnt published until the 1930s, all three of those nonequivalent definitions of pointwise continuity are still in use. Eduard Heine provided the first published definition of continuity in 1872. This is not a definition of continuity since the function f =1 x is continuous on its whole domain of R ∖ A function is continuous at a point if it does not have a hole or jump. A “hole” or “jump” in the graph of a function if the value of the function at a point c differs from its limiting value along points that are nearby. Such a point is called a discontinuity, a function is then continuous if it has no holes or jumps, that is, if it is continuous at every point of its domain. Otherwise, a function is discontinuous, at the points where the value of the function differs from its limiting value, there are several ways to make this definition mathematically rigorous. These definitions are equivalent to one another, so the most convenient definition can be used to determine whether a function is continuous or not. In the definitions below, f, I → R. is a function defined on a subset I of the set R of real numbers and this subset I is referred to as the domain of f
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Division by zero
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In mathematics, division by zero is division where the divisor is zero. Such a division can be expressed as a/0 where a is the dividend. In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by 0, gives a, and so division by zero is undefined. Since any number multiplied by zero is zero, the expression 0/0 also has no defined value, in computing, a program error may result from an attempt to divide by zero. When division is explained at the elementary level, it is often considered as splitting a set of objects into equal parts. As an example, consider having ten cookies, and these cookies are to be distributed equally to five people at a table, each person would receive 105 =2 cookies. Similarly, if there are ten cookies, and only one person at the table, so, for dividing by zero, what is the number of cookies that each person receives when 10 cookies are evenly distributed amongst 0 people at a table. Certain words can be pinpointed in the question to highlight the problem, the problem with this question is the when. There is no way to evenly distribute 10 cookies to nobody, in mathematical jargon, a set of 10 items cannot be partitioned into 0 subsets. So 100, at least in elementary arithmetic, is said to be either meaningless, similar problems occur if one has 0 cookies and 0 people, but this time the problem is in the phrase the number. A partition is possible, but since the partition has 0 parts, vacuously every set in our partition has a number of elements, be it 0,2,5. If there are, say,5 cookies and 2 people, in any integer partition of a 5-set into 2 parts, one of the parts of the partition will have more elements than the other. But the problem with 5 cookies and 2 people can be solved by cutting one cookie in half, the problem with 5 cookies and 0 people cannot be solved in any way that preserves the meaning of divides. Another way of looking at division by zero is that division can always be checked using multiplication. Considering the 10/0 example above, setting x = 10/0, if x equals ten divided by zero, then x times zero equals ten, but there is no x that, when multiplied by zero, gives ten. If instead of x=10/0 we have x=0/0, then every x satisfies the question what number x, multiplied by zero, the Brahmasphutasiddhanta of Brahmagupta is the earliest known text to treat zero as a number in its own right and to define operations involving zero. The author could not explain division by zero in his texts, according to Brahmagupta, A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is zero or is expressed as a fraction with zero as numerator