1.
Number line
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In basic mathematics, a number line is a picture of a graduated straight line that serves as abstraction for real numbers, denoted by R. Every point of a line is assumed to correspond to a real number. Often the integers are shown as specially-marked points evenly spaced on the line, although this image only shows the integers from −9 to 9, the line includes all real numbers, continuing forever in each direction, and also numbers not marked that are between the integers. It is often used as an aid in teaching simple addition and subtraction, a number line is usually represented as being horizontal, but in a Cartesian coordinate plane the vertical axis is also a number line. Another convention uses only one arrowhead which indicates the direction in which numbers grow, if a particular number is farther to the right on the number line than is another number, then the first number is greater than the second. The distance between them is the magnitude of their difference—that is, it measures the first number minus the second one, taking this difference is the process of subtraction. Thus, for example, the length of a segment between 0 and some other number represents the magnitude of the latter number. Two numbers can be added by picking up the length from 0 to one of the numbers and this gives a result that is 3 combined lengths of 5 each, since the process ends at 15, we find that 5 ×3 =15. This puts the right end of the length 2 at the end of the length from 0 to 6. Since three lengths of 2 filled the length 6,2 goes into 6 three times, the section of the number line between two numbers is called an interval. If the section includes both numbers it is said to be an interval, while if it excludes both numbers it is called an open interval. If it includes one of the numbers but not the other one, all the points extending forever in one direction from a particular point are together known as a ray. If the ray includes the point, it is a closed ray. Sometimes it is convenient to scale the numbers on the line with a logarithmic scale. This approach is useful, for example, in illustrating a sequence of events in the history of the universe or of evolution, a line drawn through the origin at right angles to the real number line can be used to represent the imaginary numbers. This line, called line, extends the number line to a complex number plane. Together these lines form what is known as the Cartesian coordinate system, further, the Cartesian coordinate system can itself be extended by visualizing a third number line coming out of the screen, measuring a third variable called z. Positive numbers are closer to the eyes than the screen is, while negative numbers are behind the screen
2.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
3.
Line (geometry)
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The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects, the straight line is that which is equally extended between its points. In modern mathematics, given the multitude of geometries, the concept of a line is tied to the way the geometry is described. When a geometry is described by a set of axioms, the notion of a line is left undefined. The properties of lines are determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry, thus in differential geometry a line may be interpreted as a geodesic, while in some projective geometries a line is a 2-dimensional vector space. This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line, to avoid this vicious circle certain concepts must be taken as primitive concepts, terms which are given no definition. In geometry, it is frequently the case that the concept of line is taken as a primitive, in those situations where a line is a defined concept, as in coordinate geometry, some other fundamental ideas are taken as primitives. When the line concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy, in a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. In this circumstance it is possible that a description or mental image of a notion is provided to give a foundation to build the notion on which would formally be based on the axioms. Descriptions of this type may be referred to, by some authors and these are not true definitions and could not be used in formal proofs of statements. The definition of line in Euclids Elements falls into this category, when geometry was first formalised by Euclid in the Elements, he defined a general line to be breadthless length with a straight line being a line which lies evenly with the points on itself. These definitions serve little purpose since they use terms which are not, themselves, in fact, Euclid did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed. In an axiomatic formulation of Euclidean geometry, such as that of Hilbert, for example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point. In two dimensions, i. e. the Euclidean plane, two lines which do not intersect are called parallel, in higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not. Any collection of many lines partitions the plane into convex polygons. Lines in a Cartesian plane or, more generally, in affine coordinates, in two dimensions, the equation for non-vertical lines is often given in the slope-intercept form, y = m x + b where, m is the slope or gradient of the line. B is the y-intercept of the line, X is the independent variable of the function y = f
4.
Point (geometry)
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In modern mathematics, a point refers usually to an element of some set called a space. More specifically, in Euclidean geometry, a point is a primitive notion upon which the geometry is built, being a primitive notion means that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms, in particular, the geometric points do not have any length, area, volume, or any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a location in Euclidean space. Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects, Euclid originally defined the point as that which has no part. This idea is easily generalized to three-dimensional Euclidean space, where a point is represented by a triplet with the additional third number representing depth. Further generalizations are represented by an ordered tuplet of n terms, many constructs within Euclidean geometry consist of an infinite collection of points that conform to certain axioms. This is usually represented by a set of points, As an example, a line is a set of points of the form L =. Similar constructions exist that define the plane, line segment and other related concepts, a line segment consisting of only a single point is called a degenerate line segment. In addition to defining points and constructs related to points, Euclid also postulated a key idea about points, in spite of this, modern expansions of the system serve to remove these assumptions. There are several inequivalent definitions of dimension in mathematics, in all of the common definitions, a point is 0-dimensional. The dimension of a space is the maximum size of a linearly independent subset. In a vector space consisting of a point, there is no linearly independent subset. The zero vector is not itself linearly independent, because there is a non trivial linear combination making it zero,1 ⋅0 =0, if no such minimal n exists, the space is said to be of infinite covering dimension. A point is zero-dimensional with respect to the covering dimension because every open cover of the space has a refinement consisting of a open set. The Hausdorff dimension of X is defined by dim H , = inf, a point has Hausdorff dimension 0 because it can be covered by a single ball of arbitrarily small radius. Although the notion of a point is considered fundamental in mainstream geometry and topology, there are some systems that forgo it, e. g. noncommutative geometry. More precisely, such structures generalize well-known spaces of functions in a way that the operation take a value at this point may not be defined
5.
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
6.
Set (mathematics)
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In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. For example, the numbers 2,4, and 6 are distinct objects when considered separately, Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a part of mathematics. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, the German word Menge, rendered as set in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite. A set is a collection of distinct objects. The objects that make up a set can be anything, numbers, people, letters of the alphabet, other sets, Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements. Cantors definition turned out to be inadequate, instead, the notion of a set is taken as a notion in axiomatic set theory. There are two ways of describing, or specifying the members of, a set, one way is by intensional definition, using a rule or semantic description, A is the set whose members are the first four positive integers. B is the set of colors of the French flag, the second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets, one often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance, A = C and B = D, there are two important points to note about sets. First, in a definition, a set member can be listed two or more times, for example. However, per extensionality, two definitions of sets which differ only in one of the definitions lists set members multiple times, define, in fact. Hence, the set is identical to the set. The second important point is that the order in which the elements of a set are listed is irrelevant and we can illustrate these two important points with an example, = =. For sets with many elements, the enumeration of members can be abbreviated, for instance, the set of the first thousand positive integers may be specified extensionally as, where the ellipsis indicates that the list continues in the obvious way. Ellipses may also be used where sets have infinitely many members, thus the set of positive even numbers can be written as
7.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
8.
Space (mathematics)
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In mathematics, a space is a set with some added structure. Mathematical spaces often form a hierarchy, i. e. one space may inherit all the characteristics of a parent space, modern mathematics treats space quite differently compared to classical mathematics. In the ancient mathematics, space was an abstraction of the three-dimensional space observed in the everyday life. The axiomatic method had been the research tool since Euclid. The method of coordinates was adopted by René Descartes in 1637, two equivalence relations between geometric figures were used, congruence and similarity. Translations, rotations and reflections transform a figure into congruent figures, homotheties — into similar figures, for example, all circles are mutually similar, but ellipses are not similar to circles. The relation between the two geometries, Euclidean and projective, shows that objects are not given to us with their structure. Rather, each mathematical theory describes its objects by some of their properties, distances and angles are never mentioned in the axioms of the projective geometry and therefore cannot appear in its theorems. The question what is the sum of the three angles of a triangle is meaningful in the Euclidean geometry but meaningless in the projective geometry. A different situation appeared in the 19th century, in some geometries the sum of the three angles of a triangle is well-defined but different from the classical value. The non-Euclidean hyperbolic geometry, introduced by Nikolai Lobachevsky in 1829, eugenio Beltrami in 1868 and Felix Klein in 1871 obtained Euclidean models of the non-Euclidean hyperbolic geometry, and thereby completely justified this theory. This discovery forced the abandonment of the pretensions to the truth of Euclidean geometry. It showed that axioms are not obvious, nor implications of definitions, to what extent do they correspond to an experimental reality. This important physical problem no longer has anything to do with mathematics, even if a geometry does not correspond to an experimental reality, its theorems remain no less mathematical truths. These Euclidean objects and relations play the non-Euclidean geometry like contemporary actors playing an ancient performance, relations between the actors only mimic relations between the characters in the play. Likewise, the relations between the chosen objects of the Euclidean model only mimic the non-Euclidean relations. It shows that relations between objects are essential in mathematics, while the nature of the objects is not, according to Nicolas Bourbaki, the period between 1795 and 1872 can be called the golden age of geometry. Analytic geometry made a progress and succeeded in replacing theorems of classical geometry with computations via invariants of transformation groups
9.
Euclidean space
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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria, the term Euclidean distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions, classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are used to define rational numbers. It means that points of the space are specified with collections of real numbers and this approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. In one dimension, this is the line, in two dimensions, it is the Cartesian plane, and in higher dimensions it is a coordinate space with three or more real number coordinates. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance, for example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that point is shifted in the same direction. The other is rotation about a point in the plane. In order to all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation. Even when used in theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments. The standard way to such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. The reason for working with vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner. Once the Euclidean plane has been described in language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. Intuitively, the distinction says merely that there is no choice of where the origin should go in the space
10.
Dimension
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In physics and mathematics, the dimension of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one only one coordinate is needed to specify a point on it – for example. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces, in classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a space but not the one that was found necessary to describe electromagnetism. The four dimensions of spacetime consist of events that are not absolutely defined spatially and temporally, Minkowski space first approximates the universe without gravity, the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Ten dimensions are used to string theory, and the state-space of quantum mechanics is an infinite-dimensional function space. The concept of dimension is not restricted to physical objects, high-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics, in mathematics, the dimension of an object is an intrinsic property independent of the space in which the object is embedded. This intrinsic notion of dimension is one of the ways the mathematical notion of dimension differs from its common usages. The dimension of Euclidean n-space En is n, when trying to generalize to other types of spaces, one is faced with the question what makes En n-dimensional. One answer is that to cover a ball in En by small balls of radius ε. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension, for example, the boundary of a ball in En looks locally like En-1 and this leads to the notion of the inductive dimension. While these notions agree on En, they turn out to be different when one looks at more general spaces, a tesseract is an example of a four-dimensional object. The rest of this section some of the more important mathematical definitions of the dimensions. A complex number has a real part x and an imaginary part y, a single complex coordinate system may be applied to an object having two real dimensions. For example, an ordinary two-dimensional spherical surface, when given a complex metric, complex dimensions appear in the study of complex manifolds and algebraic varieties. The dimension of a space is the number of vectors in any basis for the space. This notion of dimension is referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension
11.
Vector space
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A vector space is a collection of objects called vectors, which may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers. The operations of addition and scalar multiplication must satisfy certain requirements, called axioms. Euclidean vectors are an example of a vector space and they represent physical quantities such as forces, any two forces can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces and these vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are commonly used. This is particularly the case of Banach spaces and Hilbert spaces, historically, the first ideas leading to vector spaces can be traced back as far as the 17th centurys analytic geometry, matrices, systems of linear equations, and Euclidean vectors. Today, vector spaces are applied throughout mathematics, science and engineering, furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques, Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and abstract algebra. The concept of space will first be explained by describing two particular examples, The first example of a vector space consists of arrows in a fixed plane. This is used in physics to describe forces or velocities, given any two such arrows, v and w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the sum of the two arrows and is denoted v + w, when a is negative, av is defined as the arrow pointing in the opposite direction, instead. Such a pair is written as, the sum of two such pairs and multiplication of a pair with a number is defined as follows, + = and a =. The first example above reduces to one if the arrows are represented by the pair of Cartesian coordinates of their end points. A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below, elements of V are commonly called vectors. Elements of F are commonly called scalars, the second operation, called scalar multiplication takes any scalar a and any vector v and gives another vector av. In this article, vectors are represented in boldface to distinguish them from scalars
12.
Affine space
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A Euclidean space is an affine space over the reals, equipped with a metric, the Euclidean distance. Therefore, in Euclidean geometry, a property is a property that may be proved in affine spaces. In an affine space, there is no distinguished point that serves as an origin, hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead displacement vectors, also called translation vectors or simply translations, thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, Any vector space may be considered as an affine space, and this amounts to forgetting the special role played by the zero vector. In this case, the elements of the space may be viewed either as points of the affine space or as displacement vectors or translations. When considered as a point, the vector is called the origin. Adding a fixed vector to the elements of a subspace of a vector space produces an affine subspace. One commonly says that this affine subspace has been obtained by translating the linear subspace by the translation vector, in finite dimensions, such an affine subspace is the solution set of an inhomogeneous linear system. The displacement vectors for that space are the solutions of the corresponding homogeneous linear system. Linear subspaces, in contrast, always contain the origin of the vector space, the dimension of an affine space is defined as the dimension of the vector space of its translations. An affine space of one is an affine line. An affine space of dimension 2 is an affine plane, an affine subspace of dimension n –1 in an affine space or a vector space of dimension n is an affine hyperplane. The following characterization may be easier to understand than the formal definition. Imagine that Alice knows that a point is the actual origin. Two vectors, a and b, are to be added, similarly, Alice and Bob may evaluate any linear combination of a and b, or of any finite set of vectors, and will generally get different answers. However, if the sum of the coefficients in a combination is 1, then Alice. If Alice travels to λa + b then Bob can similarly travel to p + λ + = λa + b, under this condition, for all coefficients λ + =1, Alice and Bob describe the same point with the same linear combination, despite using different origins
13.
Metric space
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In mathematics, a metric space is a set for which distances between all members of the set are defined. Those distances, taken together, are called a metric on the set, a metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces. The most familiar metric space is 3-dimensional Euclidean space, in fact, a metric is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the line segment connecting them. Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel, since for any x, y ∈ M, The function d is also called distance function or simply distance. Often, d is omitted and one just writes M for a space if it is clear from the context what metric is used. Ignoring mathematical details, for any system of roads and terrains the distance between two locations can be defined as the length of the shortest route connecting those locations, to be a metric there shouldnt be any one-way roads. The triangle inequality expresses the fact that detours arent shortcuts, many of the examples below can be seen as concrete versions of this general idea. The real numbers with the function d = | y − x | given by the absolute difference. The rational numbers with the distance function also form a metric space. The positive real numbers with distance function d = | log | is a metric space. Any normed vector space is a space by defining d = ∥ y − x ∥. Examples, The Manhattan norm gives rise to the Manhattan distance, the maximum norm gives rise to the Chebyshev distance or chessboard distance, the minimal number of moves a chess king would take to travel from x to y. The British Rail metric on a vector space is given by d = ∥ x ∥ + ∥ y ∥ for distinct points x and y. The name alludes to the tendency of railway journeys to proceed via London irrespective of their final destination, If is a metric space and X is a subset of M, then becomes a metric space by restricting the domain of d to X × X. The discrete metric, where d =0 if x = y and d =1 otherwise, is a simple but important example and this, in particular, shows that for any set, there is always a metric space associated to it. Using this metric, any point is a ball, and therefore every subset is open. A finite metric space is a metric space having a number of points
14.
Topological space
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Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion, the branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology. The utility of the notion of a topology is shown by the fact there are several equivalent definitions of this structure. Thus one chooses the axiomatisation suited for the application, the most commonly used, and the most elegant, is that in terms of open sets, but the most intuitive is that in terms of neighbourhoods and so this is given first. Note, A variety of other axiomatisations of topological spaces are listed in the Exercises of the book by Vaidyanathaswamy and this axiomatization is due to Felix Hausdorff. Let X be a set, the elements of X are usually called points, let N be a function assigning to each x in X a non-empty collection N of subsets of X. The elements of N will be called neighbourhoods of x with respect to N, the function N is called a neighbourhood topology if the axioms below are satisfied, and then X with N is called a topological space. If N is a neighbourhood of x, then x ∈ N, in other words, each point belongs to every one of its neighbourhoods. If N is a subset of X and includes a neighbourhood of x, I. e. every superset of a neighbourhood of a point x in X is again a neighbourhood of x. The intersection of two neighbourhoods of x is a neighbourhood of x, any neighbourhood N of x includes a neighbourhood M of x such that N is a neighbourhood of each point of M. The first three axioms for neighbourhoods have a clear meaning, the fourth axiom has a very important use in the structure of the theory, that of linking together the neighbourhoods of different points of X. A standard example of such a system of neighbourhoods is for the real line R, given such a structure, we can define a subset U of X to be open if U is a neighbourhood of all points in U. A topological space is a pair, where X is a set and τ is a collection of subsets of X, satisfying the following axioms, The empty set. Any union of members of τ still belongs to τ, the intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X, given X =, the collection τ = of only the two subsets of X required by the axioms forms a topology of X, the trivial topology. Given X =, the collection τ = of six subsets of X forms another topology of X, given X = and the collection τ = P, is a topological space. τ is called the discrete topology, using de Morgans laws, the above axioms defining open sets become axioms defining closed sets, The empty set and X are closed. The intersection of any collection of closed sets is also closed, the union of any finite number of closed sets is also closed
15.
Measure (mathematics)
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In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, for instance, the Lebesgue measure of the interval in the real numbers is its length in the everyday sense of the word – specifically,1. Technically, a measure is a function that assigns a real number or +∞ to subsets of a set X. It must further be countably additive, the measure of a subset that can be decomposed into a finite number of smaller disjoint subsets, is the sum of the measures of the smaller subsets. In general, if one wants to associate a consistent size to each subset of a set while satisfying the other axioms of a measure. This problem was resolved by defining measure only on a sub-collection of all subsets, the so-called measurable subsets and this means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are complicated in the sense of being badly mixed up with their complement. Indeed, their existence is a consequence of the axiom of choice. Measure theory was developed in stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon. The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorovs axiomatisation of probability theory, probability theory considers measures that assign to the whole set the size 1, and considers measurable subsets to be events whose probability is given by the measure. Ergodic theory considers measures that are invariant under, or arise naturally from, let X be a set and Σ a σ-algebra over X. A function μ from Σ to the real number line is called a measure if it satisfies the following properties, Non-negativity. Countable additivity, For all countable collections i =1 ∞ of pairwise disjoint sets in Σ, μ = ∑ k =1 ∞ μ One may require that at least one set E has finite measure. Then the empty set automatically has measure zero because of countable additivity, because μ = μ = μ + μ + μ + …, which implies that μ =0. If only the second and third conditions of the definition of measure above are met, the pair is called a measurable space, the members of Σ are called measurable sets. If and are two spaces, then a function f, X → Y is called measurable if for every Y-measurable set B ∈ Σ Y. See also Measurable function#Caveat about another setup, a triple is called a measure space. A probability measure is a measure with total measure one – i. e, a probability space is a measure space with a probability measure
16.
R
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R is the 18th letter of the modern English alphabet and the ISO basic Latin alphabet. The original Semitic letter may have inspired by an Egyptian hieroglyph for tp. It was used for /r/ by Semites because in their language, the Lapis Satricanus inscription shows the form of the Latin alphabet around 500 BC. Here, the rounded, closing Π shape of the p, the descending stroke of the Latin letter R has fully developed by the 3rd century BC, as seen in the Tomb of the Scipios sarcophagus inscriptions of that era. From around 50 AD, the letter P would be written with its loop fully closed, the minuscule form developed through several variations on the capital form. Along with Latin minuscule writing in general, it developed ultimately from Roman cursive via the uncial script of Late Antiquity into the Carolingian minuscule of the 9th century. In handwriting, it was not to close the bottom of the loop but continue into the leg. The loop-leg stroke shortened into the simple arc used in the Carolingian minuscule, a calligraphic minuscule r, known as r rotunda, was used in the sequence or, bending the shape of the r to accommodate the bulge of the o. Later, the variant was also used where r followed other lower case letters with a rounded loop towards the right. Use of r rotunda was mostly tied to blackletter typefaces, insular script used a minuscule which retained two downward strokes, but which did not close the loop, this variant survives in the Gaelic type popular in Ireland until the mid 20th century. The name of the letter in Latin was er, following the pattern of letters representing continuants. This name is preserved in French and many other languages, in Middle English, the name of the letter changed from /ɛr/ to /ar/, following a pattern exhibited in many other words such as farm, and star. The letter R is sometimes referred to as the littera canina and this phrase has Latin origins, the Latin R was trilled to sound like a growling dog. A good example of a trilling R is the Spanish word for dog, in William Shakespeares Romeo and Juliet, such a reference is made by Juliets nurse in Act 2, scene 4, when she calls the letter R the dogs name. The reference is found in Ben Jonsons English Grammar. The letter ⟨r⟩ is the eighth most common letter in English, the letter ⟨r⟩ is used to form the ending -re, which is used in certain words such as centre in some varieties of English spelling, such as British English. Canadian English also uses the -re ending, unlike American English, ⟨r⟩ represents a rhotic consonant in many languages, as shown in the table below. Other languages may use the letter ⟨r⟩ in their alphabets to represent rhotic consonants different from the alveolar trill, in Haitian Creole, it represents a sound so weak that it is often written interchangeably with ⟨w⟩, e. g. Kweyol for Kreyol
17.
Blackboard bold
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Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol are doubled. The symbols usually denote number sets, one way of producing blackboard bold is to double-strike a character with a small offset on a typewriter. Thus they are referred to as double struck. e. by using the edge rather than point of the chalk. It then made its way back in print form as a style from ordinary bold, possibly starting with the original 1965 edition of Gunning. Some mathematicians do not recognize blackboard bold as a style from bold. Jean-Pierre Serre uses double-struck letters when writing bold on the blackboard, donald Knuth also prefers boldface to blackboard bold, and consequently did not include blackboard bold in the Computer Modern fonts he created for the TeX mathematical typesetting system. The Chicago Manual of Style in 1993 advises, blackboard bold should be confined to the classroom whereas in 2003 it states that open-faced symbols are reserved for systems of numbers. In Unicode, a few of the common blackboard bold characters are encoded in the Basic Multilingual Plane in the Letterlike Symbols area. The rest, however, are encoded outside the BMP, from U+1D538 to U+1D550, U+1D552 to U+1D56B, being outside the BMP, these are relatively new and not widely supported. The following table shows all available Unicode blackboard bold characters, the symbols are nearly universal in their interpretation, unlike their normally-typeset counterparts, which are used for many different purposes. The first column shows the letter as typically rendered by the ubiquitous LaTeX markup system, the second column shows the Unicode codepoint. The third column shows the symbol itself, the fourth column describes known typical usage in mathematical texts. In addition, a blackboard-bold Greek letter mu is used by number theorists. Mathematical alphanumeric symbols Set notation Weisstein, Eric W. Doublestruck, http, //www. w3. org/TR/MathML2/double-struck. html shows blackboard bold symbols together with their Unicode encodings. Encodings in the Basic Multilingual Plane are highlighted in yellow
18.
Topology
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In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, important topological properties include connectedness and compactness. Topology developed as a field of study out of geometry and set theory, through analysis of such as space, dimension. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs, Leonhard Eulers Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the fields first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, by the middle of the 20th century, topology had become a major branch of mathematics. It defines the basic notions used in all branches of topology. Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to geometry and together they make up the geometric theory of differentiable manifolds. Geometric topology primarily studies manifolds and their embeddings in other manifolds, a particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots, Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler and his 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750 Euler wrote to a friend that he had realised the importance of the edges of a polyhedron and this led to his polyhedron formula, V − E + F =2. Some authorities regard this analysis as the first theorem, signalling the birth of topology, further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term Topologie in Vorstudien zur Topologie, written in his native German, in 1847, the term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator. Their work was corrected, consolidated and greatly extended by Henri Poincaré, in 1895 he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a case of a general topological space. In 1914, Felix Hausdorff coined the term topological space and gave the definition for what is now called a Hausdorff space, currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski
19.
Real analysis
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Real analysis is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. The theorems of real analysis rely intimately upon the structure of the number line. The real number system consists of a set, together with two operations and an order, and is, formally speaking, an ordered quadruple consisting of these objects, there are several ways of formalizing the definition of the real number system. The synthetic approach gives a list of axioms for the numbers as a complete ordered field. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense there is a model for the axioms. Any one of these models must be constructed, and most of these models are built using the basic properties of the rational number system as an ordered field. These constructions are described in detail in the main article. In addition to these notions, the real numbers, equipped with the absolute value function as a metric. Many important theorems in real analysis remain valid when they are restated as statements involving metric spaces and these theorems are frequently topological in nature, and placing them in the more abstract setting of metric spaces may lead to proofs that are shorter, more natural, or more elegant. The real numbers have several important lattice-theoretic properties that are absent in the complex numbers, most importantly, the real numbers form an ordered field, in which addition and multiplication preserve positivity. Moreover, the ordering of the numbers is total. These order-theoretic properties lead to a number of important results in analysis, such as the monotone convergence theorem, the intermediate value theorem. However, while the results in analysis are stated for real numbers. In particular, many ideas in analysis and operator theory generalize properties of the real numbers – such generalizations include the theories of Riesz spaces. Also, mathematicians consider real and imaginary parts of complex sequences, a sequence is a function whose domain is a countable, totally ordered set, usually taken to be the natural numbers or whole numbers. Occasionally, it is convenient to consider bidirectional sequences indexed by the set of all integers. Of interest in analysis, a real-valued sequence, here indexed by the natural numbers, is a map a, N → R, n ↦ a n. Each a = a n is referred to as a term of the sequence, a sequence that tends to a limit is said to be convergent, otherwise it is divergent
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Algebra
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Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols, as such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine, abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are unknown or allowed to take on many values. For example, in x +2 =5 the letter x is unknown, in E = mc2, the letters E and m are variables, and the letter c is a constant, the speed of light in a vacuum. Algebra gives methods for solving equations and expressing formulas that are easier than the older method of writing everything out in words. The word algebra is used in certain specialized ways. A special kind of object in abstract algebra is called an algebra. A mathematician who does research in algebra is called an algebraist, the word algebra comes from the Arabic الجبر from the title of the book Ilm al-jabr wal-muḳābala by Persian mathematician and astronomer al-Khwarizmi. The word entered the English language during the century, from either Spanish, Italian. It originally referred to the procedure of setting broken or dislocated bones. The mathematical meaning was first recorded in the sixteenth century, the word algebra has several related meanings in mathematics, as a single word or with qualifiers. As a single word without an article, algebra names a broad part of mathematics, as a single word with an article or in plural, an algebra or algebras denotes a specific mathematical structure, whose precise definition depends on the author. Usually the structure has an addition, multiplication, and a scalar multiplication, when some authors use the term algebra, they make a subset of the following additional assumptions, associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word refers to a generalization of the above concept. With a qualifier, there is the distinction, Without an article, it means a part of algebra, such as linear algebra, elementary algebra. With an article, it means an instance of some abstract structure, like a Lie algebra, sometimes both meanings exist for the same qualifier, as in the sentence, Commutative algebra is the study of commutative rings, which are commutative algebras over the integers
21.
Field (mathematics)
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In mathematics, a field is a set on which are defined addition, subtraction, multiplication, and division, which behave as they do when applied to rational and real numbers. A field is thus an algebraic structure, which is widely used in algebra, number theory. The best known fields are the field of numbers. In addition, the field of numbers is widely used, not only in mathematics. Finite fields are used in most cryptographic protocols used for computer security, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Formally, a field is a set together with two operations the addition and the multiplication, which have the properties, called axioms of fields. An operation is a mapping that associates an element of the set to every pair of its elements, the result of the addition of a and b is called the sum of a and b and denoted a + b. Similarly, the result of the multiplication of a and b is called the product of a and b, associativity of addition and multiplication For all a, b and c in F, one has a + = + c and a · = · c. Commutativity of addition and multiplication For all a and b in F one has a + b = b + a and a · b = b · a. Existence of additive and multiplicative identity elements There exists an element 0 in F, called the identity, such that for all a in F. There is an element 1, different from 0 and called the identity, such that for all a in F. Existence of additive inverses and multiplicative inverses For every a in F, there exists an element in F, denoted −a, such that a + =0. For every a ≠0 in F, there exists an element in F, denoted a−1, 1/a, or 1/a, distributivity of multiplication over addition For all a, b and c in F, one has a · = +. The elements 0 and 1 being required to be distinct, a field has, at least, for every a in F, one has − a = ⋅ a. Thus, the inverse of every element is known as soon as one knows the additive inverse of 1. A subtraction and a division are defined in every field by a − b = a +, a subfield E of a field F is a subset of F that contains 1, and is closed under addition, multiplication, additive inverse and multiplicative inverse of a nonzero element. It is straightforward to verify that a subfield is indeed a field, two groups are associated to every field. The field itself is a group under addition, when considering this group structure rather the field structure, one talks of the additive group of the field
22.
Least-upper-bound property
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In mathematics, the least-upper-bound property is a fundamental property of the real numbers and certain other ordered sets. A set X has the property if and only if every non-empty subset of X with an upper bound has a least upper bound in X. The least-upper-bound property is one form of the axiom for the real numbers. It is usually taken as an axiom in synthetic constructions of the real numbers, in order theory, this property can be generalized to a notion of completeness for any partially ordered set. A linearly ordered set that is dense and has the least upper bound property is called a linear continuum, let S be a non-empty set of real numbers. A real number x is called a bound for S if x ≥ s for all s ∈ S. A real number x is the least upper bound for S if x is a bound for S and x ≤ y for every upper bound y of S. The least-upper-bound property states that any non-empty set of numbers that has an upper bound must have a least upper bound in real numbers. More generally, one may define upper bound and least upper bound for any subset of an ordered set X. In this case, we say that X has the property if every non-empty subset of X with an upper bound has a least upper bound. For example, the set Q of rational numbers does not have the property under the usual order. For instance, the set = Q ∩ has a bound in Q. The construction of the numbers using Dedekind cuts takes advantage of this failure by defining the irrational numbers as the least upper bounds of certain subsets of the rationals. The least-upper-bound property is equivalent to forms of the completeness axiom. It is possible to prove the least-upper-bound property using the assumption that every Cauchy sequence of real numbers converges, let S be a nonempty set of real numbers, and suppose that S has an upper bound B1. Since S is nonempty, there exists a real number A1 that is not a bound for S. Define sequences A1, A2, A3. and B1, B2. Recursively as follows, Check whether ⁄2 is a bound for S. If it is, let An+1 = An and let Bn+1 = ⁄2, otherwise there must be an element s in S so that s> ⁄2
23.
Countable set
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In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A countable set is either a set or a countably infinite set. Some authors use countable set to mean countably infinite alone, to avoid this ambiguity, the term at most countable may be used when finite sets are included and countably infinite, enumerable, or denumerable otherwise. Georg Cantor introduced the term countable set, contrasting sets that are countable with those that are uncountable, today, countable sets form the foundation of a branch of mathematics called discrete mathematics. A set S is countable if there exists a function f from S to the natural numbers N =. If such an f can be found that is also surjective, in other words, a set is countably infinite if it has one-to-one correspondence with the natural number set, N. As noted above, this terminology is not universal, some authors use countable to mean what is here called countably infinite, and do not include finite sets. Alternative formulations of the definition in terms of a function or a surjective function can also be given. In 1874, in his first set theory article, Cantor proved that the set of numbers is uncountable. In 1878, he used one-to-one correspondences to define and compare cardinalities, in 1883, he extended the natural numbers with his infinite ordinals, and used sets of ordinals to produce an infinity of sets having different infinite cardinalities. A set is a collection of elements, and may be described in many ways, one way is simply to list all of its elements, for example, the set consisting of the integers 3,4, and 5 may be denoted. This is only effective for small sets, however, for larger sets, even in this case, however, it is still possible to list all the elements, because the set is finite. Some sets are infinite, these sets have more than n elements for any integer n, for example, the set of natural numbers, denotable by, has infinitely many elements, and we cannot use any normal number to give its size. Nonetheless, it out that infinite sets do have a well-defined notion of size. To understand what this means, we first examine what it does not mean, for example, there are infinitely many odd integers, infinitely many even integers, and infinitely many integers overall. However, it out that the number of even integers. This is because we arrange things such that for every integer, or, more generally, n→2n, see picture. However, not all sets have the same cardinality
24.
Subset
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In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is contained inside B, that is, all elements of A are also elements of B. The relationship of one set being a subset of another is called inclusion or sometimes containment, the subset relation defines a partial order on sets. The algebra of subsets forms a Boolean algebra in which the relation is called inclusion. For any set S, the inclusion relation ⊆ is an order on the set P of all subsets of S defined by A ≤ B ⟺ A ⊆ B. We may also partially order P by reverse set inclusion by defining A ≤ B ⟺ B ⊆ A, when quantified, A ⊆ B is represented as, ∀x. So for example, for authors, it is true of every set A that A ⊂ A. Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively and this usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may or may not equal y, but if x < y, then x definitely does not equal y, and is less than y. Similarly, using the convention that ⊂ is proper subset, if A ⊆ B, then A may or may not equal B, the set A = is a proper subset of B =, thus both expressions A ⊆ B and A ⊊ B are true. The set D = is a subset of E =, thus D ⊆ E is true, any set is a subset of itself, but not a proper subset. The empty set, denoted by ∅, is also a subset of any given set X and it is also always a proper subset of any set except itself. These are two examples in both the subset and the whole set are infinite, and the subset has the same cardinality as the whole. The set of numbers is a proper subset of the set of real numbers. In this example, both sets are infinite but the set has a larger cardinality than the former set. Another example in an Euler diagram, Inclusion is the partial order in the sense that every partially ordered set is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example—if each ordinal n is identified with the set of all ordinals less than or equal to n, then a ≤ b if and only if ⊆. For the power set P of a set S, the partial order is the Cartesian product of k = |S| copies of the partial order on for which 0 <1. This can be illustrated by enumerating S = and associating with each subset T ⊆ S the k-tuple from k of which the ith coordinate is 1 if and only if si is a member of T
25.
Rational number
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In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number, the decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number and these statements hold true not just for base 10, but also for any other integer base. A real number that is not rational is called irrational, irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating, since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be defined as equivalence classes of pairs of integers such that q ≠0, for the equivalence relation defined by ~ if. The rational numbers together with addition and multiplication form field which contains the integers and is contained in any field containing the integers, finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the numbers form a dense subset of the real numbers. The real numbers can be constructed from the numbers by completion, using Cauchy sequences, Dedekind cuts. The term rational in reference to the set Q refers to the fact that a number represents a ratio of two integers. In mathematics, rational is often used as a noun abbreviating rational number, the adjective rational sometimes means that the coefficients are rational numbers. However, a curve is not a curve defined over the rationals. Any integer n can be expressed as the rational number n/1, a b = c d if and only if a d = b c. Where both denominators are positive, a b < c d if and only if a d < b c. If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations, − a − b = a b, two fractions are added as follows, a b + c d = a d + b c b d. A b − c d = a d − b c b d, the rule for multiplication is, a b ⋅ c d = a c b d. Where c ≠0, a b ÷ c d = a d b c, note that division is equivalent to multiplying by the reciprocal of the divisor fraction, a d b c = a b × d c. Additive and multiplicative inverses exist in the numbers, − = − a b = a − b and −1 = b a if a ≠0
26.
Disjoint sets
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In mathematics, two sets are said to be disjoint if they have no element in common. Equivalently, disjoint sets are sets whose intersection is the empty set, for example, and are disjoint sets, while and are not. This definition of disjoint sets can be extended to any family of sets, a family of sets is pairwise disjoint or mutually disjoint if every two different sets in the family are disjoint. For example, the collection of sets is pairwise disjoint, two sets are said to be almost disjoint sets if their intersection is small in some sense. For instance, two sets whose intersection is a finite set may be said to be almost disjoint. In topology, there are notions of separated sets with more strict conditions than disjointness. For instance, two sets may be considered to be separated when they have disjoint closures or disjoint neighborhoods, similarly, in a metric space, positively separated sets are sets separated by a nonzero distance. Disjointness of two sets, or of a family of sets, may be expressed in terms of their intersections, two sets A and B are disjoint if and only if their intersection A ∩ B is the empty set. It follows from definition that every set is disjoint from the empty set. A family F of sets is pairwise disjoint if, for two sets in the family, their intersection is empty. If the family more than one set, this implies that the intersection of the whole family is also empty. However, a family of one set is pairwise disjoint, regardless of whether that set is empty. Additionally, a family of sets may have an empty intersection without being pairwise disjoint, for instance, the three sets have an empty intersection but are not pairwise disjoint. In fact, there are no two disjoint sets in this collection, also the empty family of sets is pairwise disjoint. A Helly family is a system of sets within which the only subfamilies with empty intersections are the ones that are pairwise disjoint. For instance, the intervals of the real numbers form a Helly family, if a family of closed intervals has an empty intersection and is minimal. A partition of a set X is any collection of mutually disjoint non-empty sets whose union is X, every partition can equivalently be described by an equivalence relation, a binary relation that describes whether two elements belong to the same set in the partition. A disjoint union may mean one of two things, most simply, it may mean the union of sets that are disjoint
27.
Empty set
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In mathematics, and more specifically set theory, the empty set is the unique set having no elements, its size or cardinality is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, in other theories, many possible properties of sets are vacuously true for the empty set. Null set was once a synonym for empty set, but is now a technical term in measure theory. The empty set may also be called the void set, common notations for the empty set include, ∅, and ∅. The latter two symbols were introduced by the Bourbaki group in 1939, inspired by the letter Ø in the Norwegian, although now considered an improper use of notation, in the past,0 was occasionally used as a symbol for the empty set. The empty-set symbol ∅ is found at Unicode point U+2205, in LaTeX, it is coded as \emptyset for ∅ or \varnothing for ∅. In standard axiomatic set theory, by the principle of extensionality, hence there is but one empty set, and we speak of the empty set rather than an empty set. The mathematical symbols employed below are explained here, in this context, zero is modelled by the empty set. For any property, For every element of ∅ the property holds, There is no element of ∅ for which the property holds. Conversely, if for some property and some set V, the two statements hold, For every element of V the property holds, There is no element of V for which the property holds. By the definition of subset, the empty set is a subset of any set A. That is, every element x of ∅ belongs to A. Indeed, since there are no elements of ∅ at all, there is no element of ∅ that is not in A. Any statement that begins for every element of ∅ is not making any substantive claim and this is often paraphrased as everything is true of the elements of the empty set. When speaking of the sum of the elements of a finite set, the reason for this is that zero is the identity element for addition. Similarly, the product of the elements of the empty set should be considered to be one, a disarrangement of a set is a permutation of the set that leaves no element in the same position. The empty set is a disarrangment of itself as no element can be found that retains its original position. Since the empty set has no members, when it is considered as a subset of any ordered set, then member of that set will be an upper bound. For example, when considered as a subset of the numbers, with its usual ordering, represented by the real number line
28.
Order theory
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Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a framework for describing statements such as this is less than that or this precedes that. This article introduces the field and provides basic definitions, a list of order-theoretic terms can be found in the order theory glossary. Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the order on the natural numbers e. g.2 is less than 3,10 is greater than 5. This intuitive concept can be extended to orders on sets of numbers, such as the integers. The idea of being greater than or less than another number is one of the basic intuitions of number systems in general, other familiar examples of orderings are the alphabetical order of words in a dictionary and the genealogical property of lineal descent within a group of people. The notion of order is very general, extending beyond contexts that have an immediate, in other contexts orders may capture notions of containment or specialization. Abstractly, this type of order amounts to the relation, e. g. Pediatricians are physicians. However, many other orders do not and those orders like the subset-of relation for which there exist incomparable elements are called partial orders, orders for which every pair of elements is comparable are total orders. Order theory captures the intuition of orders that arises from such examples in a general setting and this is achieved by specifying properties that a relation ≤ must have to be a mathematical order. This more abstract approach makes sense, because one can derive numerous theorems in the general setting. These insights can then be transferred to many less abstract applications. Driven by the wide usage of orders, numerous special kinds of ordered sets have been defined. In addition, order theory does not restrict itself to the classes of ordering relations. A simple example of an order theoretic property for functions comes from analysis where monotone functions are frequently found and this section introduces ordered sets by building upon the concepts of set theory, arithmetic, and binary relations. Suppose that P is a set and that ≤ is a relation on P, a set with a partial order on it is called a partially ordered set, poset, or just an ordered set if the intended meaning is clear. By checking these properties, one sees that the well-known orders on natural numbers, integers, rational numbers
29.
Set theory
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Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics, the language of set theory can be used in the definitions of nearly all mathematical objects. The modern study of set theory was initiated by Georg Cantor, Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals. Mathematical topics typically emerge and evolve through interactions among many researchers, Set theory, however, was founded by a single paper in 1874 by Georg Cantor, On a Property of the Collection of All Real Algebraic Numbers. Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1867–71, with Cantors work on number theory, an 1872 meeting between Cantor and Richard Dedekind influenced Cantors thinking and culminated in Cantors 1874 paper. Cantors work initially polarized the mathematicians of his day, while Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. This utility of set theory led to the article Mengenlehre contributed in 1898 by Arthur Schoenflies to Kleins encyclopedia, in 1899 Cantor had himself posed the question What is the cardinal number of the set of all sets. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics, in 1906 English readers gained the book Theory of Sets of Points by William Henry Young and his wife Grace Chisholm Young, published by Cambridge University Press. The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment, the work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory. The work of such as Henri Lebesgue demonstrated the great mathematical utility of set theory. Set theory is used as a foundational system, although in some areas category theory is thought to be a preferred foundation. Set theory begins with a binary relation between an object o and a set A. If o is a member of A, the notation o ∈ A is used, since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, for example, is a subset of, and so is but is not. As insinuated from this definition, a set is a subset of itself, for cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined
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Absolute difference
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The absolute difference of two real numbers x, y is given by |x − y|, the absolute value of their difference. It describes the distance on the line between the points corresponding to x and y. It is a case of the Lp distance for all 1 ≤ p ≤ ∞ and is the standard metric used for both the set of rational numbers Q and their completion, the set of real numbers R. As with any metric, the metric properties hold, |x − y| ≥0, |x − y| =0 if and only if x = y. |x − z| ≤ |x − y| + |y − z|, in the case of the difference, equality holds if. By contrast, simple subtraction is not non-negative or commutative, but it does obey the second and fourth properties above, since x − y =0 if and only if x = y, and x − z = +. The absolute difference is used to other quantities including the relative difference, the L1 norm used in taxicab geometry. This follows since |x − y|2 =2 and squaring is monotonic on the nonnegative reals, Absolute deviation Weisstein, Eric W. Absolute Difference
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Ball (mathematics)
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In mathematics, a ball is the space bounded by a sphere. It may be a ball or an open ball. These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, a ball or hyperball in n dimensions is called an n-ball and is bounded by an -sphere. Thus, for example, a ball in the Euclidean plane is the thing as a disk. In Euclidean 3-space, a ball is taken to be the bounded by a 2-dimensional spherical shell. In a one-dimensional space, a ball is a line segment, in other contexts, such as in Euclidean geometry and informal use, sphere is sometimes used to mean ball. In Euclidean n-space, an n-ball of radius r and center x is the set of all points of less than r from x. A closed n-ball of radius r is the set of all points of less than or equal to r away from x. In Euclidean n-space, every ball is bounded by a hypersphere, the ball is a bounded interval when n =1, is a disk bounded by a circle when n =2, and is bounded by a sphere when n =3. The n-dimensional volume of a Euclidean ball of radius R in n-dimensional Euclidean space is, V n = π n 2 Γ R n, where Γ is Leonhard Eulers gamma function. These are, V2 k = π k k. R2 k, V2 k +1 =2 k +1 π k. R2 k +1 =2 k. R2 k +1. In the formula for odd-dimensional volumes, the double factorial. is defined for odd integers 2k +1 as, let be a metric space, namely a set M with a metric d. Note in particular that a ball always includes p itself, since the definition requires r >0, the closure of the open ball Br is usually denoted Br. While it is always the case that Br ⊆ Br ⊆ Br, for example, in a metric space X with the discrete metric, one has B1 = and B1 = X, for any p ∈ X. A unit ball is a ball of radius 1, a subset of a metric space is bounded if it is contained in some ball. A set is bounded if, given any positive radius. The open balls of a space are a basis for a topological space. This space is called the topology induced by the metric d, any normed vector space V with norm | · | is also a metric space, with the metric d = | x − y |
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Metric (mathematics)
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In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set. A set with a metric is called a metric space, a metric induces a topology on a set, but not all topologies can be generated by a metric. A topological space whose topology can be described by a metric is called metrizable, an important source of metrics in differential geometry are metric tensors, bilinear forms that may be defined from the tangent vectors of a differentiable manifold onto a scalar. A metric tensor allows distances along curves to be determined through integration, however, not every metric comes from a metric tensor in this way. The first condition is implied by the others, for sets on which an addition +, X × X → X is defined, d is called a translation invariant metric if d = d for all x, y and a in X. These conditions express intuitive notions about the concept of distance, for example, that the distance between distinct points is positive and the distance from x to y is the same as the distance from y to x. The triangle inequality means that the distance x to z via y is at least as great as from x to z directly. Euclid in his work stated that the shortest distance between two points is a line, that was the triangle inequality for his geometry, if a modification of the triangle inequality 4*. D ≤ d + d is used in the definition then property 1 follows straight from property 4*, properties 2 and 4* give property 3 which in turn gives property 4. The discrete metric, if x = y then d =0, the Euclidean metric is translation and rotation invariant. The taxicab metric is translation invariant, more generally, any metric induced by a norm is translation invariant. If n ∈ N is a sequence of seminorms defining a vector space E. Graph metric, a defined in terms of distances in a certain graph. The Hamming distance in coding theory, Riemannian metric, a type of metric function that is appropriate to impose on any differentiable manifold. For any such manifold, one chooses at each point p a symmetric, positive definite, bilinear form L, Tp × Tp → ℝ on the tangent space Tp at p, a smooth manifold equipped with a Riemannian metric is called a Riemannian manifold. The Fubini–Study metric on complex projective space and this is an example of a Riemannian metric. String metrics, such as Levenshtein distance and other string edit distances, graph edit distance defines a distance function between graphs. For a given set X, two metrics d1 and d2 are called equivalent if the identity mapping id, → is a homeomorphism
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Cauchy sequence
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In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a number of elements of the sequence are less than that given distance from each other. It is not sufficient for each term to become close to the preceding term. For instance, in the harmonic series ∑1 n a difference between consecutive terms decreases as 1 n, however the series does not converge, rather, it is required that all terms get arbitrarily close to each other, starting from some point. More formally, for any given ε >0 there exists an N such that for any m, n > N. The notions above are not as unfamiliar as they might at first appear, the customary acceptance of the fact that any real number x has a decimal expansion is an implicit acknowledgment that a particular Cauchy sequence of rational numbers has the real limit x. In some cases it may be difficult to describe x independently of such a process involving rational numbers. Generalizations of Cauchy sequences in more abstract uniform spaces exist in the form of Cauchy filters, in a similar way one can define Cauchy sequences of rational or complex numbers. Cauchy formulated such a condition by requiring x m − x n to be infinitesimal for every pair of infinite m, n, to define Cauchy sequences in any metric space X, the absolute value |xm - xn| is replaced by the distance d between xm and xn. A metric space X in which every Cauchy sequence converges to an element of X is called complete, the real numbers are complete under the metric induced by the usual absolute value, and one of the standard constructions of the real numbers involves Cauchy sequences of rational numbers. A rather different type of example is afforded by a metric space X which has the discrete metric, any Cauchy sequence of elements of X must be constant beyond some fixed point, and converges to the eventually repeating term. The rational numbers Q are not complete, There are sequences of rationals that converge to irrational numbers, if one considers this as a sequence of real numbers, however, it converges to the real number φ = /2, the Golden ratio, which is irrational. Every Cauchy sequence of numbers is bounded. Every Cauchy sequence of numbers is bounded, hence by Bolzano-Weierstrass has a convergent subsequence, hence is itself convergent. It should be noted, though, that proof of the completeness of the real numbers implicitly makes use of the least upper bound axiom. The alternative approach, mentioned above, of constructing the real numbers as the completion of the rational numbers, makes the completeness of the real numbers tautological. Such a series ∑ n =1 ∞ x n is considered to be convergent if and only if the sequence of sums is convergent. It is a matter to determine whether the sequence of partial sums is Cauchy or not
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Connected space
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In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the topological properties that are used to distinguish topological spaces. A subset of a topological space X is a set if it is a connected space when viewed as a subspace of X. A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets, otherwise, X is said to be connected. A subset of a space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set as a space. For a topological space X the following conditions are equivalent, X is connected, X cannot be divided into two disjoint nonempty closed sets. The only subsets of X which are open and closed are X and the empty set. The only subsets of X with empty boundary are X and the empty set, X cannot be written as the union of two nonempty separated sets. All continuous functions from X to are constant, where is the two-point space endowed with the discrete topology, the maximal connected subsets of a nonempty topological space are called the connected components of the space. The components of any topological space X form a partition of X, they are disjoint, nonempty, Every component is a closed subset of the original space. It follows that, in the case where their number is finite, however, if their number is infinite, this might not be the case, for instance, the connected components of the set of the rational numbers are the one-point sets, which are not open. A space in all components are one-point sets is called totally disconnected. Clearly any totally separated space is totally disconnected, but the converse does not hold, for example take two copies of the rational numbers Q, and identify them at every point except zero. The resulting space, with the quotient topology, is totally disconnected, however, by considering the two copies of zero, one sees that the space is not totally separated. In fact, it is not even Hausdorff, and the condition of being separated is strictly stronger than the condition of being Hausdorff. The closed interval in the subspace topology is connected, although it can, for example, be written as the union of [0, 1) and. The union of is disconnected, both of these intervals are open in the topological space
