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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
2.
Euclidean geometry
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Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry, the Elements. Euclids method consists in assuming a set of intuitively appealing axioms. Although many of Euclids results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in school as the first axiomatic system. It goes on to the geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, for more than two thousand years, the adjective Euclidean was unnecessary because no other sort of geometry had been conceived. Euclids axioms seemed so obvious that any theorem proved from them was deemed true in an absolute, often metaphysical. Today, however, many other self-consistent non-Euclidean geometries are known, Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms to propositions without the use of coordinates. This is in contrast to analytic geometry, which uses coordinates, the Elements is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones. There are 13 total books in the Elements, Books I–IV, Books V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as numbers and rational and irrational numbers are introduced. The infinitude of prime numbers is proved, a typical result is the 1,3 ratio between the volume of a cone and a cylinder with the same height and base. Euclidean geometry is a system, in which all theorems are derived from a small number of axioms. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. Although Euclids statement of the only explicitly asserts the existence of the constructions. The Elements also include the five common notions, Things that are equal to the same thing are also equal to one another
3.
Pythagorean theorem
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In mathematics, the Pythagorean theorem, also known as Pythagorass theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the two sides. There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework, Mesopotamian, Indian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases. The theorem has been given numerous proofs – possibly the most for any mathematical theorem and they are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The Pythagorean theorem was known long before Pythagoras, but he may well have been the first to prove it, in any event, the proof attributed to him is very simple, and is called a proof by rearrangement. The two large squares shown in the figure each contain four triangles, and the only difference between the two large squares is that the triangles are arranged differently. Therefore, the space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean theorem and that Pythagoras originated this very simple proof is sometimes inferred from the writings of the later Greek philosopher and mathematician Proclus. Several other proofs of this theorem are described below, but this is known as the Pythagorean one, If the length of both a and b are known, then c can be calculated as c = a 2 + b 2. If the length of the c and of one side are known. The Pythagorean equation relates the sides of a triangle in a simple way. Another corollary of the theorem is that in any triangle, the hypotenuse is greater than any one of the other sides. A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation. This theorem may have more known proofs than any other, the book The Pythagorean Proposition contains 370 proofs, Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. Draw the altitude from point C, and call H its intersection with the side AB, point H divides the length of the hypotenuse c into parts d and e. By a similar reasoning, the triangle CBH is also similar to ABC, the proof of similarity of the triangles requires the triangle postulate, the sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the leads to the equality of ratios of corresponding sides. The first result equates the cosines of the angles θ, whereas the second result equates their sines, the role of this proof in history is the subject of much speculation
4.
Distance
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Distance is a numerical description of how far apart objects are. In physics or everyday usage, distance may refer to a physical length, in most cases, distance from A to B is interchangeable with distance from B to A. In mathematics, a function or metric is a generalization of the concept of physical distance. A metric is a function that behaves according to a set of rules. The circumference of the wheel is 2π × radius, and assuming the radius to be 1, in engineering ω = 2πƒ is often used, where ƒ is the frequency. Chessboard distance, formalized as Chebyshev distance, is the number of moves a king must make on a chessboard to travel between two squares. Distance measures in cosmology are complicated by the expansion of the universe, the term distance is also used by analogy to measure non-physical entities in certain ways. In computer science, there is the notion of the distance between two strings. For example, the dog and dot, which vary by only one letter, are closer than dog and cat. In this way, many different types of distances can be calculated, such as for traversal of graphs, comparison of distributions and curves, distance cannot be negative, and distance travelled never decreases. Distance is a quantity or a magnitude, whereas displacement is a vector quantity with both magnitude and direction. Directed distance is a positive, zero, or negative scalar quantity, the distance covered by a vehicle, person, animal, or object along a curved path from a point A to a point B should be distinguished from the straight-line distance from A to B. For example, whatever the distance covered during a trip from A to B and back to A. In general the straight-line distance does not equal distance travelled, except for journeys in a straight line, directed distances are distances with a directional sense. They can be determined along straight lines and along curved lines, for instance, just labelling the two endpoints as A and B can indicate the sense, if the ordered sequence is assumed, which implies that A is the starting point. A displacement is a kind of directed distance defined in mechanics. A directed distance is called displacement when it is the distance along a line from A and B. This implies motion of the particle, the distance traveled by a particle must always be greater than or equal to its displacement, with equality occurring only when the particle moves along a straight path
5.
Line segment
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In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. A closed line segment includes both endpoints, while a line segment excludes both endpoints, a half-open line segment includes exactly one of the endpoints. Examples of line include the sides of a triangle or square. More generally, when both of the end points are vertices of a polygon or polyhedron, the line segment is either an edge if they are adjacent vertices. When the end points both lie on a such as a circle, a line segment is called a chord. Sometimes one needs to distinguish between open and closed line segments, thus, the line segment can be expressed as a convex combination of the segments two end points. In geometry, it is defined that a point B is between two other points A and C, if the distance AB added to the distance BC is equal to the distance AC. Thus in R2 the line segment with endpoints A = and C = is the collection of points. A line segment is a connected, non-empty set, if V is a topological vector space, then a closed line segment is a closed set in V. However, an open line segment is an open set in V if and only if V is one-dimensional. More generally than above, the concept of a segment can be defined in an ordered geometry. A pair of segments can be any one of the following, intersecting, parallel, skew. The last possibility is a way that line segments differ from lines, in an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else be defined in terms of an isometry of a line. Segments play an important role in other theories, for example, a set is convex if the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of sets to the analysis of a line segment. The Segment Addition Postulate can be used to add congruent segment or segments with equal lengths and consequently substitute other segments into another statement to make segments congruent. A line segment can be viewed as a case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints. A complete orbit of this ellipse traverses the line segment twice, as a degenerate orbit this is a radial elliptic trajectory. In addition to appearing as the edges and diagonals of polygons and polyhedra, some very frequently considered segments in a triangle include the three altitudes, the three medians, the perpendicular bisectors of the sides, and the internal angle bisectors
6.
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
7.
Rational trigonometry
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His ideas are set out in his 2005 book Divine Proportions, Rational Trigonometry to Universal Geometry. According to New Scientist, part of his motivation for an alternative to traditional trigonometry was to some problems that occur when infinite series are used in mathematics. Rational trigonometry avoids direct use of functions like sine and cosine by substituting their squared equivalents. To date, rational trigonometry is largely unmentioned in mainstream mathematical literature, Rational trigonometry follows an approach built on the methods of linear algebra to the topics of elementary geometry. Distance is replaced with its value and angle is replaced with the squared value of the usual sine ratio associated to either angle between two lines. Following this, it is claimed, makes many classical results of Euclidean geometry applicable in rational form over any field not of characteristic two, the book Divine Proportions shows the application of calculus using rational trigonometric functions, including three-dimensional volume calculations. It also deals with rational trigonometrys application to situations involving irrationals, quadrance and distance both measure separation of points in Euclidean space. Following Pythagorass theorem, the quadrance of two points A1 = and A2 = in a plane is defined as the sum of squares of differences in the x and y coordinates. In effect, the triangle inequality d1 + d2 − d3 ≥0 is modified under rational trigonometry to a symmetric form 2 ≥0 equivalent to Pythagorass theorem, spread gives one measure to the separation of two lines as a single dimensionless number in the range for Euclidean geometry. It replaces the concept of angle but has differences from angle. Trigonometric, it is the ratio for the quadrances in a right triangle. Cartesian, as a function of the three co-ordinates used to ascribe the two vectors. Linear algebra, a rational function, the square of the determinant of two vectors divided by the product of their quadrances. Suppose two lines, l1 and l2, intersect at the point A as shown at right, choose a point B ≠ A on l1 and let C be the foot of the perpendicular from B to l2. Then the spread s is s = Q Q = Q R, like angle, spread depends only on the relative slopes of two lines and is invariant under translation. Which becomes,1 − s =24 and this simplifies, in the numerator, to 2, giving,1 − s =2. Then, using the Brahmagupta–Fibonacci identity 2 +2 =, the expression for spread in terms of slopes of two lines becomes s =2. In this form a spread is the ratio of the square of a determinant of two vectors to the product of their quadrances This replaces with, with and the origin with in the previous result, s =2
8.
Generalization
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A generalization is the formulation of general concepts from specific instances by abstracting common properties. Generalizations posit the existence of a domain or set of elements, as such, they are the essential basis of all valid deductive inferences. The process of verification is necessary to determine whether a generalization holds true for any given situation, generalization is the process of identifying the parts of a whole, as belonging to the whole. The parts, completely unrelated may be together as a group. It must be stated that, the parts cannot be generalized into a whole until a common relation is established among all the parts, but this does not mean that the parts are unrelated, only that no common relation has been established yet for the generalization. The concept of generalization has broad application in related disciplines, sometimes having a specialized context or meaning. For instance, animal is a generalization of bird because every bird is an animal, the relation of generalization to specialization is reflected in the contrasting words hypernym and hyponym. In contrast, a hyponym is one of the items included in the generic, such as peach and oak which are included in tree, a hypernym is superordinate to a hyponym, and a hyponym is subordinate to a hypernym. An animal is a generalization of a mammal, a bird, a fish, an amphibian, generalization has a long history in cartography as an art of creating maps for different scale and purpose. Cartographic generalization is the process of selecting and representing information of a map in a way that adapts to the scale of the medium of the map. In this way, every map has, to some extent and this includes small cartographic scale maps, which cannot convey every detail of the real world. Cartographers must decide and then adjust the content within their maps to create a suitable, generalization is meant to be context-specific. That is to say, correctly generalized maps are those that emphasize the most important map elements while still representing the world in the most faithful, a polygon is a generalization of a 3-sided triangle, a 4-sided quadrilateral, and so on to n sides. A hypercube is a generalization of a 2-dimensional square, a 3-dimensional cube, a quadric, such as a hypersphere, ellipsoid, paraboloid, or hyperboloid, is a generalization of a conic section to higher dimensions
9.
Euclidean vector
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In mathematics, physics, and engineering, a Euclidean vector is a geometric object that has magnitude and direction. Vectors can be added to other vectors according to vector algebra, a Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by A B →. A vector is what is needed to carry the point A to the point B and it was first used by 18th century astronomers investigating planet rotation around the Sun. The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from A to B. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space. Vectors play an important role in physics, the velocity and acceleration of a moving object, many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances, their magnitude and direction can still be represented by the length, the mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the system include pseudovectors and tensors. The concept of vector, as we know it today, evolved gradually over a period of more than 200 years, about a dozen people made significant contributions. Giusto Bellavitis abstracted the basic idea in 1835 when he established the concept of equipollence, working in a Euclidean plane, he made equipollent any pair of line segments of the same length and orientation. Essentially he realized an equivalence relation on the pairs of points in the plane, the term vector was introduced by William Rowan Hamilton as part of a quaternion, which is a sum q = s + v of a Real number s and a 3-dimensional vector. Like Bellavitis, Hamilton viewed vectors as representative of classes of equipollent directed segments, grassmanns work was largely neglected until the 1870s. Peter Guthrie Tait carried the standard after Hamilton. His 1867 Elementary Treatise of Quaternions included extensive treatment of the nabla or del operator ∇, in 1878 Elements of Dynamic was published by William Kingdon Clifford. Clifford simplified the quaternion study by isolating the dot product and cross product of two vectors from the complete quaternion product and this approach made vector calculations available to engineers and others working in three dimensions and skeptical of the fourth. Josiah Willard Gibbs, who was exposed to quaternions through James Clerk Maxwells Treatise on Electricity and Magnetism, the first half of Gibbss Elements of Vector Analysis, published in 1881, presents what is essentially the modern system of vector analysis. In 1901 Edwin Bidwell Wilson published Vector Analysis, adapted from Gibbs lectures, in physics and engineering, a vector is typically regarded as a geometric entity characterized by a magnitude and a direction. It is formally defined as a line segment, or arrow
10.
Real line
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In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the line is the set R of all real numbers, viewed as a geometric space. It can be thought of as a space, a metric space, a topological space. Just like the set of numbers, the real line is usually denoted by the symbol R. However. This article focuses on the aspects of R as a space in topology, geometry. The real numbers also play an important role in algebra as a field, for more information on R in all of its guises, see real number. The real line is a linear continuum under the standard < ordering, specifically, the real line is linearly ordered by <, and this ordering is dense and has the least-upper-bound property. In addition to the properties, the real line has no maximum or minimum element. It also has a dense subset, namely the set of rational numbers. It is a theorem that any linear continuum with a dense subset. The real line also satisfies the countable chain condition, every collection of mutually disjoint, in order theory, the famous Suslin problem asks whether every linear continuum satisfying the countable chain condition that has no maximum or minimum element is necessarily order-isomorphic to R. This statement has been shown to be independent of the axiomatic system of set theory known as ZFC. The real line forms a space, with the distance function given by absolute difference. The metric tensor is clearly the 1-dimensional Euclidean metric, since the n-dimensional Euclidean metric can be represented in matrix form as the n by n identity matrix, the metric on the real line is simply the 1 by 1 identity matrix, i. e.1. If p ∈ R and ε >0, then the ε-ball in R centered at p is simply the open interval. This real line has several important properties as a space, The real line is a complete metric space. The real line is path-connected, and is one of the simplest examples of a metric space The Hausdorff dimension of the real line is equal to one. The real line carries a standard topology which can be introduced in two different, equivalent ways, first, since the real numbers are totally ordered, they carry an order topology