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In mathematics, a quadratic equation is a polynomial equation of the second degree.

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The constants a, b, and c, are called respectively, the quadratic coefficient, the linear coefficient and the constant term or free term. The term "quadratic" comes from quadratus, which is the Latin word for "square". Quadratic equations can be solved by factoring, completing the square, graphing, Newton's method, and using the quadratic formula (given below).
One common use of quadratic equations is computing trajectories in projectile motion. Another common use is in electronic amplifier design for control of step response and stability.
Quadratic formula
A quadratic equation with real or complex coefficients has two solutions, called roots.
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A quadratic function, in mathematics, is a polynomial function. The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis.
The expression ax2 + bx + c in the definition of a quadratic function is a polynomial of degree 2 or second order, or a 2nd degree polynomial, because the highest exponent of x is 2.
If the quadratic function is set equal to zero, then the result is a quadratic equation. The solutions to the equation are called the roots of the equation.

Origin of word

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The adjective quadratic comes from the Latin word quadratum for square. A term like x2 is called a square in algebra because it is the area of a square with side x.
In general, a prefix quadr(i)- indicates the number 4. Examples are quadrilateral and quadrant. Quadratum is the Latin word for square because a square has four sides.
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In mathematics, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. A sequence is a discrete function.

For example, (C, R, Y) is a sequence of letters that differs from (Y, C, R), as the ordering matters. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers (2, 4, 6,...). Finite sequences are sometimes known as strings or words and infinite sequences as streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.

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A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.

In mathematics, given an infinite sequence of numbers { an }, a series is informally the result of adding all those terms together: a1 + a2 + a3 + · · ·. These can be written more compactly using the summation symbol ∑. An example is the famous series from Zeno's dichotomy
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The logarithm of a product of two numbers equals the sum of their logarithms:

This equation forms the basis for multiplying two numbers on a slide rule or using a logarithm table. Since adding is generally easier than multiplying, this led to the rapid adoption of logarithms for calculations after their invention by John Napier in the early 17th century. For such computational purposes, the logarithm to base b = 10 (common logarithm) was primarily used. The natural logarithm uses the constant e (approximately 2.718) as its base, and is especially widespread in calculus. The binary logarithm uses base b = 2 and primarily aids computing applications.

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Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, the Richter scale uses the common logarithm to measure the amplitude of seismic events. Logarithms are commonplace in scientific formulas, measure the complexity of algorithms and of fractals, and appear in formulas counting prime numbers. They describe musical intervals, inform some models in psychophysics and can aid in forensic accounting.
The complex logarithm is the inverse of the exponential function applied to complex numbers and generalizes the logarithm to complex numbers. The discrete logarithm is another variant; it has applications in public-key cryptography.

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Three-dimensional space is a geometric model of the physical universe in which we live. The three dimensions are commonly called length, width, and depth (or height), although any three directions can be chosen, provided that they do not lie in the same plane.
In physics, our three-dimensional space is viewed as embedded in 4-dimensional space-time, called Minkowski space (see special relativity). The idea behind space-time is that time is hyperbolic-orthogonal to each of the three spatial dimensions.
In mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, usually each perpendicular to the other two at the origin, the point at which they cross. They are usually labeled x, y, and z. Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of real numbers, each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes.
Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates, though there are an infinite number of possible methods.

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Another mathematical way of viewing three-dimensional space is found in linear algebra, where the idea of independence is crucial. Space has three dimensions because the length of a box is independent of its width or breadth. In the technical language of linear algebra, space is three dimensional because every point in space can be described by a linear combination of three independent vectors. In this view, space-time is four dimensional because the location of a point in time is independent of its location in space.
Three-dimensional space has a number of properties that distinguish it from spaces of other dimension numbers. For example, at least 3 dimensions are required to tie a knot in a piece of string. The understanding of three-dimensional space in humans is thought to be learned during infancy using unconscious inference, and is closely related to hand-eye coordination. The visual ability to perceive the world in three dimensions is called depth perception.
In mathematics, a relation is used to describe certain properties of things. That way, certain things may be connected in some way; this is called a relation. Formally, a relation is a set of n-tuples of equal degree. Thus a binary relation is a set of pairs, a ternary relation a set of 3-tuples, and so forth. A ternary relation however is always expressable as two binary relations. Specifically in the context of functions, this is known as currying.
Particularly concerning binary relations, the set of all the starting point is called the domain and the sets of the ending points is the range. The domain is the x's , and the range is the y's.
An example for such a relation might be a function. Functions associate keys with values. The set of all functions is a subset of the set of all relations - a function is a relation where the first value of every tuple is unique through the set.
Other well-known relations are the Equivalence relation and the Order relation. That way, sets of things can be ordered: Take the first element of a set, it is either equal to the element looked for, or there is an order relation that can be used to classify it. That way, the whole set can be classified (compared to some arbitrarily chosen element).
Relations can be transitive. One example of a transitive relation is "smaller-than". If X "is smaller than" Y, and Y is "smaller than" Z, then X "is smaller than" Z
Relations can be symmetric. One example of a symmetric relation is "is equal to".
Relations can be reflexive.
A reflexive relation is "smaller than or equal".

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In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's instantaneous velocity. Conversely, the integral of the object's velocity over time is how much the object's position changes from the time when the integral begins to the time when the integral ends.

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The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. In higher dimensions, the derivative of a function at a point is a linear transformation called the linearization.The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus states that antidifferentiation is the same as integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.

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Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis. The term "mathematical statistics" is closely related to the term "statistical theory" but also embraces modelling for actuarial science and non-statistical probability theory, particularly in Scandinavia.
Statistics deals with gaining information from data. In practice, data often contain some randomness or uncertainty. Statistics handles such data using methods of probability theory.
Statistical science is concerned with the planning of studies, especially with the design of randomized experiments and with the planning of surveys using random sampling. The initial analysis of the data from properly randomized studies often follows the study protocol.
Of course, the data from a randomized study can be analyzed to consider secondary hypotheses or to suggest new ideas. A secondary analysis of the data from a planned study uses tools from data analysis.

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Data analysis is divided into:
descriptive statistics - the part of statistics that describes data, i.e. summarises the data and their typical properties.
inferential statistics - the part of statistics that draws conclusions from data (using some model for the data): For example, inferential statistics involves selecting a model for the data, checking whether the data fulfill the conditions of a particular model, and with quantifying the involved uncertainty (e.g. using confidence intervals).
While the tools of data analysis work best on data from randomized studies, they are also applied to other kinds of data --- for example, from natural experiments and observational studies, in which case the inference is dependent on the model chosen by the statistician, and so subjective.
Mathematical statistics has been inspired by and has extended many procedures in applied statistics.

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Probability is a way of expressing knowledge or belief that an event will occur or has occurred. The concept has an exact mathematical meaning in probability theory, which is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, Artificial intelligence/Machine learning and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.

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Interpretations
The word probability does not have a consistent direct definition. In fact, there are two broad categories of probability interpretations, whose adherents possess different (and sometimes conflicting) views about the fundamental nature of probability:
Frequentists talk about probabilities only when dealing with experiments that are random and well-defined. The probability of a random event denotes the relative frequency of occurrence of an experiment's outcome, when repeating the experiment. Frequentists consider probability to be the relative frequency "in the long run" of outcomes.
Bayesians, however, assign probabilities to any statement whatsoever, even when no random process is involved. Probability, for a Bayesian, is a way to represent an individual's degree of belief in a statement, or an objective degree of rational belief, given the evidence.

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In mathematics, a relation is used to describe certain properties of things. That way, certain things may be connected in some way; this is called a relation. It is clear, that things are either related, or they are not, there are no in-betweens.

Formally, a relation is a set of n-tuples of equal degree. Thus a binary relation is a set of pairs, a ternary relation a set of 3-tuples, and so forth. A ternary relation however is always expressable as two binary relations. Specifically in the context of functions, this is known as currying.

Particularly concerning binary relations, the set of all the starting point is called the domain and the sets of the ending points is the range. The domain is the x's , and the range is the y's.

An example for such a relation might be a function. Functions associate keys with values. The set of all functions is a subset of the set of all relations - a function is a relation where the first value of every tuple is unique through the set.

Other well-known relations are the Equivalence relation and the Order relation. That way, sets of things can be ordered: Take the first element of a set, it is either equal to the element looked for, or there is an order relation that can be used to classify it. That way, the whole set can be classified (compared to some arbitrarily chosen element).

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A function, in a mathematical sense, expresses the idea that one quantity (the argument of the function, also known as the input) completely determines another quantity (the value, or the output). A function assigns exactly one value to each input of a specified type. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain and the codomain of the function. An example of a function with the real numbers as both its domain and codomain is the function f(x) = 2x, which assigns to every real number the real number with twice its value. In this case, it is written that f(5) = 10.

In addition to elementary functions on numbers, functions include maps between algebraic structures like groups and maps between geometric objects like manifolds. In the abstract set-theoretic approach, a function is a relation between the domain and the codomain that associates each element in the domain with exactly one element in the codomain. An example of a function with domain {A,B,C} and codomain {1,2,3} associates A with 1, B with 2, and C with 3.

There are many ways to describe or represent functions: by a formula, by an algorithm that computes it, by a plot or a graph. A table of values is a common way to specify a function in statistics, physics, chemistry, and other sciences. A function may also be described through its relationship to other functions, for example, as the inverse function or a solution of a differential equation. There are uncountably many different functions from the set of natural numbers to itself, most of which cannot be expressed with a formula or an algorithm. Functions with numerical outputs may be added and multiplied, yielding new functions. Collections of functions with certain properties, such as continuous functions and differentiable functions, usually required to be closed under certain operations, are called function spaces and are studied as objects in their own right, in such disciplines as real analysis and complex analysis. An important operation on functions, which distinguishes them from numbers, is the composition of functions.
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In mathematics, a matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers, such as

An item in a matrix is called an entry or an element. The example has entries 1, 9, 13, 20, 55, and 6. Entries are often denoted by a variable with two subscripts, as shown on the right; thus in the example above, a2,1 = 20. Matrices of the same size can be added and subtracted entrywise and matrices of compatible sizes can be multiplied. These operations have many of the properties of ordinary arithmetic, except that matrix multiplication is not commutative, that is, AB and BA are not equal in general. Matrices consisting of only one column or row define the components of vectors, while higher-dimensional (e.g., three-dimensional) arrays of numbers define the components of a generalization of a vector called a tensor. Matrices with entries in other fields or rings are also studied.
Matrices are a key tool in linear algebra. One use of matrices is to represent linear transformations, which are higher-dimensional analogs of linear functions of the form f(x) = cx, where c is a constant; matrix multiplication corresponds to composition of linear transformations. Matrices can also keep track of the coefficients in a system of linear equations. For a square matrix, the determinant and inverse matrix (when it exists) govern the behavior of solutions to the corresponding system of linear equations, and eigenvalues and eigenvectors provide insight into the geometry of the associated linear transformation.

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Matrices find many applications. Physics makes use of matrices in various domains, for example in geometrical optics and matrix mechanics; the latter led to studying in more detail matrices with an infinite number of rows and columns. Graph theory uses matrices to keep track of distances between pairs of vertices in a graph. Computer graphics uses matrices to project 3-dimensional space onto a 2-dimensional screen. Matrix calculus generalizes classical analytical notions such as derivatives of functions or exponentials to matrices. The latter is a recurring need in solving ordinary differential equations. Serialism and dodecaphonism are musical movements of the 20th century that use a square mathematical matrix to determine the pattern of music intervals.
A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old but still an active area of research. Matrix decomposition methods simplify computations, both theoretically and practically. For sparse matrices, specifically tailored algorithms can provide speedups; such matrices arise in the finite element method, for example.



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Integral is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral.

The term integral may also refer to the notion of antiderivative, a function F whose derivative is the given function ƒ. In this case, it is called an indefinite integral, while the integrals discussed in this article are termed definite integrals. Some authors maintain a distinction between antiderivatives and indefinite integrals.

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Integrals and derivatives became the basic tools of calculus, with numerous applications in science and engineering. A rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. A line integral is defined for functions of two or three variables, and the interval of integration [a, b] is replaced by a certain curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space. Integrals of differential forms play a fundamental role in modern differential geometry. These generalizations of integrals first arose from the needs of physics, and they play an important role in the formulation of many physical laws, notably those of electrodynamics. There are many modern concepts of integration, among these, the most common is based on the abstract mathematical theory known as Lebesgue integration, developed by Henri Lebesgue.
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In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives and integrals.
The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory.
In formulas, limit is usually abbreviated as lim as in lim(an) = a or represented by the right arrow (→) as in an → a.
Limit of a function

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In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.
Informally, a function f assigns an output f(x) to every input x. The function has a limit L at an input p if f(x) is "close" to L whenever x is "close" to p. In other words, f(x) becomes closer and closer to L as x moves closer and closer to p. More specifically, when f is applied to each input sufficiently close to p, the result is an output value that is arbitrarily close to L. If the inputs "close" to p are taken to values that are very different, the limit is said to not exist.
Definitions

The following definitions (known as (ε, δ)-definitions) are the generally accepted ones for the limit of a function in various contexts.



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In mathematics and physics, vector can refer to:
Euclidean vector, a geometric entity endowed with both length and direction; an element of a Euclidean vector space. In physics, euclidean vectors are used to represent physical quantities which have both magnitude and direction, such as force, in contrast to scalar quantities, which have no direction.
Vector product, or cross product, an operation on two vectors in a three-dimensional Euclidean space, producing a third three-dimensional Euclidean vector
Vector projection, also known as the vector resolute, a mapping of one vector onto another

The vector part of a quaternion, a term used in 19th century mathematical literature on quaternions
Burgers vector, a vector that represents the magnitude and direction of the lattice distortion of dislocation in a crystal lattice
Displacement vector, a vector that specifies the change in position of a point relative to a previous position
Gradient vector, one vector in a vector field
Laplace–Runge–Lenz vector, a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another
Normal vector, or surface normal, a vector which is perpendicular to a surface
Null vector, or zero vector, a vector whose components are all zero
Orbital state vectors, which define the state of an orbiting body
Position (vector), a vector which represents the position of an object in space in relation to an arbitrary reference point
Poynting vector, in physics, a vector representing the energy flux of an electromagnetic field
Tangent vector (disambiguation), a vector that follows the direction of a curve or a surface at a given point
Wave vector, a vector representation of a wave
Gyrovector, a hyperbolic geometry version of a vector
Axial vector, or pseudovector, a quantity that transforms like a vector under a proper rotation
Basis vector, one of a set of vectors (a "basis") that, in linear combination, can represent every vector in a given vector space
Coordinate vector, in linear algebra, an explicit representation of an element of any abstract vector space
Darboux vector, the areal velocity vector of the Frenet frame of a space curve
Four-vector, in the theory of relativity, a vector in a four-dimensional real vector space called Minkowski space
Interval vector, in musical set theory, an array that expresses the intervallic content of a pitch-class set
P-vector, the tensor obtained by taking linear combinations of the wedge product of p tangent vectors
Probability vector, in statistics, a vector with non-negative entries that add up to one
Row vector or column vector, a one-dimensional matrix often representing the solution of a system of linear equations
Spin vector, or Spinor, is an element of a complex vector space introduced to expand the notion of spatial vector
Tuple, an ordered list of numbers, sometimes used to represent a vector
Unit vector, a vector in a normed vector space whose length is 1
Vector, an element of a vector space
Vector fields
Vector field, a construction in vector calculus which associates a vector to every point in a subset of Euclidean space
Conservative vector field, a vector field which is the gradient of a scalar potential field
Hamiltonian vector field, a vector field defined for any energy function or Hamiltonian
Killing vector field, a vector field on a Riemannian manifold
Solenoidal vector field, a vector field with zero divergence
Vector potential, a vector field whose curl is a given vector field
Vector flow, a set of closely related concepts of the flow determined by a vector field

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Vector spaces
Vector space, a mathematical structure made up of vectors, objects which may be added with another vector or multiplied by a scalar value
Euclidean vector space, an n-dimensional space with notions of distance and angle that obey the Euclidean relationships
Dual vector space, a vector space consisting of all linear functionals on another, given vector space
Graded vector space, a type of vector space that includes the extra structure of gradation
Normed vector space, a vector space on which a norm is defined
Ordered vector space, a vector space equipped with a partial order
Super vector space, name for a Z2-graded vector space
Symplectic vector space, a vector space V equipped with a non-degenerate, skew-symmetric, bilinear form
Topological vector space, a blend of topological structure with the algebraic concept of a vector space
Manipulation of vectors, fields, and spaces
Vector bundle, a topological construction which makes precise the idea of a family of vector spaces parameterized by another space
Vector calculus, a branch of mathematics concerned with differentiation and integration of vector fields
Vector Analysis, a free, online book on vector calculus first published in 1901 by Edwin Bidwell Wilson
Vector decomposition, refers to decomposing a vector of Rn to several vectors, each linearly independent
Vector differential, or del, is a vector differential operator represented by the nabla symbol:
Vector Laplacian, the vector Laplace operator, denoted by is a differential operator defined over a vector field
Vector notation, common notations used when working with vectors
Vector operator, a type of differential operator used in vector calculus
Vector product, or cross product, an operation on two vectors in a three-dimensional Euclidean space, producing a third three-dimensional Euclidean vector
Vector projection, also known as the vector resolute, a mapping of one vector onto another
Vector-valued function, a mathematical function that maps real numbers to vectors
Vectorization (mathematics), a linear transformation which converts a matrix into a column vector
Other uses in mathematics and physics
Vector autoregression, an econometric model used to capture the evolution and the interdependencies between multiple time series
Vector boson, a boson with the spin quantum number equal to 1
Vector measure, a function defined on a family of sets and taking vector values satisfying certain properties
Vector meson, a meson with total spin 1 and odd parity
Vector quantization, a quantization technique used in signal processing
Vector soliton, a solitary wave with multiple components coupled together that maintains its shape during propagation
Vector synthesis, a type of audio synthesis
Witt vector, an infinite sequence of elements of a commutative ring

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In geometry, a rectangle is defined as a quadrilateral where all four of its angles are right angles. A rectangle with vertices ABCD would be denoted as ABCD.
Rectangles hold the following properties:



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Opposite sides are of equal length (congruent)
Adjacent sides meet at right angles
All four angles are of equal measure (congruent)
All four angles are right angles
Adjacent angles are supplementary
Opposite sides are parallel
Parallelogram
Diagonals are of equal measure (congruent)
Diagonals bisect one another

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In geometry, a four-sided figure with one pair of parallel sides is referred to as a trapezoid in American English and as a trapezium in English outside North America. A trapezoid with vertices ABCD is denoted ABCD.

This article uses the term trapezoid in the sense that is current in the United States (and sometimes in some other English-speaking countries). Readers in the United Kingdom and Australia should read trapezium for each use of trapezoid in the following paragraphs. In all other languages using a word derived from the Greek for this figure, the form closest to trapezium (e.g. French 'trapèze', Italian 'trapezio', German 'Trapez', Russian 'трапеция') is used.

The term trapezium has been in use in English since 1570, from Late Latin trapezium, from Greek trapezion, literally "a little table", diminutive of trapeza "table", itself from tra- "four" + peza "foot, edge". The first recorded use of the Greek word translated trapezoid (τραπεζοειδη, table-like) was by Marinus Proclus (412 to 485 AD) in his Commentary on the first book of Euclid’s Elements.

There is also some disagreement on the allowed number of parallel sides in a trapezoid. At issue is whether parallelograms, which have two pairs of parallel sides, should be counted as trapezoids. Some authors define a trapezoid as a quadrilateral having exactly one pair of parallel sides, thereby excluding parallelograms. Other authors define a trapezoid as a quadrilateral with at least one pair of parallel sides, making the parallelogram a special type of trapezoid (along with the rhombus, the rectangle and the square). The latter definition is consistent with its uses in higher mathematics such as calculus. The former definition would make such concepts as the trapezoidal approximation to a definite integral be ill-defined.

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In North America, the term trapezium is used to refer to a quadrilateral with no parallel sides. The term trapezoid has been defined as a quadrilateral without any parallel sides in Britain and elsewhere, but this does not reflect current usage (the Oxford English Dictionary says “Often called by English writers in the 19th century”).
According to the Oxford English Dictionary, the trapezoid as a figure with no sides parallel is the sense for which Proclus introduced the term; it is retained in the French "trapézoïde", German "trapezoïd", and in other languages. A trapezium in Proclus' sense is a quadrilateral having one pair of its opposite sides parallel. This was the specific sense in England in 17th and 18th centuries, and again the prevalent one in recent use. A trapezium as any quadrilateral more general than a parallelogram is the sense of the term in Euclid. The sense of a trapezium as an irregular quadrilateral having no sides parallel was the usual sense in England from c1800 to c1875, but is now rare. This article uses the term trapezoid in the sense that is current in the USA and some other English-speaking countries. Readers in the UK should read trapezium for each use of trapezoid in the following paragraphs.
There is also some disagreement on the allowed number of parallel sides in a trapezoid. At issue is whether parallelograms, which have two pairs of parallel sides, should be counted as trapezoids. Some authors define a trapezoid as a quadrilateral having exactly one pair of parallel sides, thereby excluding parallelograms. Other authors define a trapezoid as a quadrilateral with at least one pair of parallel sides, making a parallelogram a special type of trapezoid.

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A triangle or trigon is a two dimensional geometric object that has the specific qualities of having three straight sides that intersect at three vertices.
The sum of the internal angles that exist at the vertices always total the same number for every triangle—180 degrees, or π radians.
In Euclidean geometry, any three non-collinear points determine a unique triangle and a unique plane.
Types of triangles
By relative lengths of sides
Triangles can be classified according to the relative lengths of their sides:
In an equilateral triangle, all sides are the same length. An equilateral triangle is also a regular polygon with all angles 60°.
In an isosceles triangle, at least two sides are equal in length. An isosceles triangle also has two equal angles: the angles opposite the two equal sides.
In a scalene triangle, all sides and internal angles are different from one another.
By internal angles
Triangles can also be classified according to their internal angles, measured here in degrees.



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A triangle that does not contain a right angle is called an oblique triangle. One that does is a right triangle.
There are two types of oblique triangles, those with all the internal angles smaller than 90°, and those with one angle larger than 90°.
The obtuse triangle contains the larger than 90° angle, known as an obtuse angle. The acute triangle is composed of three acute angles, the same as saying that all three of its angles are smaller than 90°.
A right triangle (or right-angled triangle) has one 90° internal angle (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side in the right triangle. Right triangles conform to the Pythagorean theorem: the sum of the squares of the two legs is equal to the square of the hypotenuse; i.e., a2 + b2 = c2, where a and b are the legs and c is the hypotenuse

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A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.




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Definition
Georg Cantor, the founder of set theory, gave the following definition of a set at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:
Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements.
As discussed below, the definition given above turned out to be inadequate for formal mathematics; instead, the notion of a "set" is taken as an undefined primitive in axiomatic set theory, and its properties are defined by the Zermelo–Fraenkel axioms.
The most basic properties are that a set "has" elements, and that two sets are equal (one and the same) if and only if they have the same elements.

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In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values. There are two common types of operations: unary and binary. Unary operations involve only one value, such as negation and trigonometric functions. Binary operations, on the other hand, take two values, and include addition, subtraction, multiplication, division, and exponentiation.


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Operations can involve mathematical objects other than numbers. The logical values true and false can be combined using logic operations, such as and, or, and not. Vectors can be added and subtracted. Rotations can be combined using the function composition operation, performing the first rotation and then the second. Operations on sets include the binary operations union and intersection and the unary operation of complementation. Operations on functions include composition and convolution.

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Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. The major difference between algebra and arithmetic is the inclusion of variables. While in arithmetic only numbers and their arithmetical operations (such as +, −, ×, ÷) occur, in algebra, one also uses symbols such as x and y, or a and b to denote variables.




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The purpose of using variables, symbols that denote numbers, is to allow the making of generalizations in mathematics.
It allows reference to numbers which are not known. It allows the exploration of mathematical relationships between quantities (such as "if you sell x tickets, then your profit will be 3x − 10 dollars").
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The values combined are called operands, arguments, or inputs, and the value produced is called the value, result, or output. Operations can have fewer or more than two inputs.
An operation is like an operator, but the point of view is different. For instance, one often speaks of "the operation of addition" or "addition operation" when focusing on the operands and result, but one says "addition operator" (rarely "operator of addition") when focusing on the process, or from the more abstract viewpoint, the function +: S×S → S.


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An operation ω is a function of the form ω : V → Y, where V ⊂ X1 × … × Xk. The sets Xk are called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k (the number of arguments) is called the type or arity of the operation. Thus a unary operation has arity one, and a binary operation has arity two. An operation of arity zero, called a nullary operation, is simply an element of the codomain Y. An operation of arity k is called a k-ary operation. Thus a k-ary operation is a (k+1)-ary relation that is functional on its first k domains.

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Operations may not be defined for every possible value. For example, in the real numbers one cannot divide by zero or take square roots of negative numbers. The values for which an operation is defined from a set called its domain. The set which contains the values produced is called the codomain, but the set of actual values attained by the operation is its range. For example, in the real numbers, the squaring operation only produces nonnegative numbers; the codomain is the set of real numbers but the range is the nonnegative numbers.
Operations can involve dissimilar objects.


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A vector can be multiplied by a scalar to form another vector. And the inner product operation on two vectors produces a scalar. An operation may or may not have certain properties, for example it may be associative, commutative, anticommutative, idempotent, and so on.
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The describes what is usually called a finitary operation, referring to the finite number of arguments (the value k). Often, use of the term operation implies that the domain of the function is a power of the codomain (i.e. the Cartesian product of one or more copies of the codomain), although this is by no means universal, as in the example of multiplying a vector by a scalar.




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Thus, since k can be 1, in the most general sense given here, operation is synonymous with function, map and mapping, that is, a relation, for which each element of the domain (input set) is associated with exactly one element of the codomain (set of possible outputs).
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