1.
1 (number)
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1, is a number, a numeral, and the name of the glyph representing that number. It represents a single entity, the unit of counting or measurement, for example, a line segment of unit length is a line segment of length 1. It is also the first of the series of natural numbers. The word one can be used as a noun, an adjective and it comes from the English word an, which comes from the Proto-Germanic root *ainaz. The Proto-Germanic root *ainaz comes from the Proto-Indo-European root *oi-no-, compare the Proto-Germanic root *ainaz to Old Frisian an, Gothic ains, Danish een, Dutch een, German eins and Old Norse einn. Compare the Proto-Indo-European root *oi-no- to Greek oinos, Latin unus, Old Persian aivam, Old Church Slavonic -inu and ino-, Lithuanian vienas, Old Irish oin, One, sometimes referred to as unity, is the first non-zero natural number. It is thus the integer before two and after zero, and the first positive odd number, any number multiplied by one is that number, as one is the identity for multiplication. As a result,1 is its own factorial, its own square, its own cube, One is also the result of the empty product, as any number multiplied by one is itself. It is also the natural number that is neither composite nor prime with respect to division. The Gupta wrote it as a line, and the Nagari sometimes added a small circle on the left. The Nepali also rotated it to the right but kept the circle small and this eventually became the top serif in the modern numeral, but the occasional short horizontal line at the bottom probably originates from similarity with the Roman numeral I. Where the 1 is written with an upstroke, the number 7 has a horizontal stroke through the vertical line. While the shape of the 1 character has an ascender in most modern typefaces, in typefaces with text figures, many older typewriters do not have a separate symbol for 1 and use the lowercase letter l instead. It is possible to find cases when the uppercase J is used,1 cannot be used as the base of a positional numeral system, as the only digit that would be permitted in such a system would be 0. Since the base 1 exponential function always equals 1, its inverse does not exist, there are two ways to write the real number 1 as a recurring decimal, as 1.000. and as 0.999. There is only one way to represent the real number 1 as a Dedekind cut, in a multiplicative group or monoid, the identity element is sometimes denoted 1, but e is also traditional. However,1 is especially common for the identity of a ring. When such a ring has characteristic n not equal to 0,1 is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number, to name just a few

2.
0.999...
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In mathematics, the repeating decimal 0. 999… denotes a real number that can be shown to be the number one. In other words, the symbols 0. 999… and 1 represent the same number, more generally, every nonzero terminating decimal has an equal twin representation with infinitely many trailing 9s. The terminating decimal representation is preferred, contributing to the misconception that it is the only representation. The same phenomenon occurs in all other bases or in any representation of the real numbers. The equality of 0. 999… and 1 is closely related to the absence of nonzero infinitesimals in the number system. Some alternative number systems, such as the hyperreals, do contain nonzero infinitesimals, the equality 0. 999… =1 has long been accepted by mathematicians and is part of general mathematical education. Nonetheless, some find it sufficiently counterintuitive that they question or reject it. Such skepticism is common enough that the difficulty of convincing them of the validity of this identity has been the subject of studies in mathematics education. Algebraic proofs showing that 0. 999… represents the number 1 use concepts such as fractions, long division, however, these proofs are not rigorous as they do not include a careful analytic definition of 0. 999…. One reason that infinite decimals are an extension of finite decimals is to represent fractions. Using long division, a division of integers like 1⁄9 becomes a recurring decimal,0. 111…. This decimal yields a quick proof for 0. 999… =1, If 0. 999… is to be consistent, it must equal 9⁄9 =1. 0.333 … =390.888 … =890.999 … =99 =1 When a number in decimal notation is multiplied by 10, the digits do not change but each digit moves one place to the left. Thus 10 ×0. 999… equals 9. 999…, which is 9 greater than the original number, in introductory arithmetic, such proofs help explain why 0. 999… =1 but 0. 333… <0.34. In introductory algebra, the proofs help explain why the method of converting between fractions and repeating decimals works. Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can demonstrate that the decimals 0. 999… and 1. 000… both represent the same real number, it is built into the definition. Since the question of 0. 999… does not affect the development of mathematics

3.
Hapax legomenon
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In corpus linguistics, a hapax legomenon is a word that occurs only once within a context, either in the written record of an entire language, in the works of an author, or in a single text. The term is sometimes used to describe a word that occurs in just one of an authors works. Hapax legomenon is a transliteration of Greek ἅπαξ λεγόμενον, meaning said once, the related terms dis legomenon, tris legomenon, and tetrakis legomenon respectively refer to double, triple, or quadruple occurrences, but are far less commonly used. Hapax legomena are quite common, as predicted by Zipfs law, for large corpora, about 40% to 60% of the words are hapax legomena, and another 10% to 15% are dis legomena. Thus, in the Brown Corpus of American English, about half of the 50,000 words are hapax legomena within that corpus, Hapax legomenon refers to a words appearance in a body of text, not to either its origin or its prevalence in speech. It thus differs from a word, which may never be recorded, may find currency and may be widely recorded, or may appear several times in the work which coins it. Hapax legomena in ancient texts are usually difficult to decipher, since it is easier to infer meaning from multiple contexts than from just one, for example, many of the remaining undeciphered Mayan glyphs are hapax legomena, and Biblical hapax legomena sometimes pose problems in translation. Hapax legomena also pose challenges in natural language processing, some scholars consider Hapax legomena useful in determining the authorship of written works. He argued that the number of hapax legomena in a putative authors corpus indicates his or her vocabulary and is characteristic of the author as an individual, harrisons theory has faded in significance due to a number of problems raised by other scholars. Workman found the numbers of hapax legomena in each Pauline Epistle. At first glance, the last three totals are not out of line with the others, although the Pastoral Epistles have more hapax legomena per page, Workman found the differences to be moderate in comparison to the variation among other Epistles. This was reinforced when Workman looked at several plays by Shakespeare, text topic, if the author writes on different subjects, of course many subject-specific words will occur only in limited contexts. Text audience, if the author is writing to a rather than a student, or their spouse rather than their employer. Time, over the course of years, both the language and a knowledge and use of language will change. There are also subjective questions over whether two forms amount to the word, dog vs. dogs, clue vs. clueless, sign vs. signature. The Jewish Encyclopedia points out that, although there are 1,500 hapaxes in the Hebrew Bible and it would not be especially difficult for a forger to construct a work with any percentage of hapax legomena desired. However, it seems unlikely that much before the 20th century would have conceived such a ploy. In other words, hapax legomena are not a reliable indicator, authorship studies now usually use a wide range of measures to look for patterns rather than rely upon single measurements

4.
Identity function
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In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument. In equations, the function is given by f = x, formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies f = x for all elements x in M. In other words, the value f in M is always the same input element x of M. The identity function on M is clearly a function as well as a surjective function. The identity function f on M is often denoted by idM, in set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or diagonal of M. If f, M → N is any function, then we have f ∘ idM = f = idN ∘ f, in particular, idM is the identity element of the monoid of all functions from M to M. Since the identity element of a monoid is unique, one can define the identity function on M to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, the identity function is a linear operator, when applied to vector spaces. The identity function on the integers is a completely multiplicative function. In an n-dimensional vector space the identity function is represented by the identity matrix In, in a metric space the identity is trivially an isometry. An object without any symmetry has as symmetry group the group only containing this isometry. In a topological space, the identity function is always continuous

5.
Legendre's constant
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Legendres constant is a mathematical constant occurring in a formula conjectured by Adrien-Marie Legendre to capture the asymptotic behavior of the prime-counting function π. Its value is now known to be exactly 1, examination of available numerical evidence for known primes led Legendre to suspect that π satisfies an approximate formula. Legendre conjectured in 1808 that π = x ln − B where lim x → ∞ B =1.08366. A228211 Or similarly, lim n → ∞ = B where B is Legendres constant. He guessed B to be about 1.08366, but regardless of its exact value, pafnuty Chebyshev proved in 1849 that if the limit B exists, it must be equal to 1. An easier proof was given by Pintz in 1980, being evaluated to such a simple number has made the term Legendres constant mostly only of historical value, with it often being used to refer to Legendres first guess 1.08366. Pierre Dusart proved in 2010 x ln x −1 < π for x ≥5393 and this is of the same form as π = x ln − A with 1 ≤ A <1.1

6.
Monogon
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In geometry a monogon is a polygon with one edge and one vertex. Since a monogon has only one side and only one vertex, in Euclidean geometry a monogon is a degenerate polygon because its endpoints must coincide, unlike any Euclidean line segment. Most definitions of a polygon in Euclidean geometry do not admit the monogon, in spherical geometry, a monogon can be constructed as a vertex on a great circle. This forms a dihedron, with two hemispherical monogonal faces which share one 360° edge and one vertex and its dual, a hosohedron, has two antipodal vertices at the poles, one 360 degree lune face, and one edge between the two vertices. Digon Herbert Busemann, The geometry of geodesics, new York, Academic Press,1955 Coxeter, H. S. M, Regular Polytopes

7.
One-dimensional space
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In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space. When n =1, the set of all locations is called a one-dimensional space. An example of a space is the number line, where the position of each point on it can be described by a single number. In algebraic geometry there are structures which are technically one-dimensional spaces. For a field k, it is a vector space over itself. Similarly, the line over k is a one-dimensional space. In particular, if k = ℂ, the complex plane, then the complex projective line P1 is one-dimensional with respect to ℂ. More generally, a ring is a module over itself. Similarly, the line over a ring is a one-dimensional space over the ring. In case the ring is an algebra over a field, these spaces are one-dimensional with respect to the algebra, the only regular polytope in one dimension is the line segment, with the Schläfli symbol. The hypersphere in 1 dimension is a pair of points, sometimes called a 0-sphere as its surface is zero-dimensional and its length is L =2 r where r is the radius. The most popular systems are the number line and the angle

8.
One-form
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In linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree multilinear functionals on the space, in differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a mapping of the total space of the tangent bundle of M to R whose restriction to each fibre is a linear functional on the tangent space. Symbolically, α, T M → R, α x = α | T x M, T x M → R where αx is linear, often one-forms are described locally, particularly in local coordinates. From this perspective, a one-form has a covariant transformation law on passing from one system to another. Thus a one-form is an order 1 covariant tensor field, many real-world concepts can be described as one-forms, Indexing into a vector, The second element of a three-vector is given by the one-form. That is, the element of is · = y. Mean, The mean element of an n-vector is given by the one-form and that is, mean = ⋅ v. Sampling, Sampling with a kernel can be considered a one-form, where the one-form is the kernel shifted to the appropriate location. Net present value of a net flow, R, is given by the one-form w, = −t where i is the discount rate. That is, N P V = ⟨ w, R ⟩ = ∫ t =0 ∞ R t d t, the most basic non-trivial differential one-form is the change in angle form d θ. This is defined as the derivative of the angle function θ, integrating this derivative along a path gives the total change in angle over the path, and integrating over a closed loop gives the winding number. In the language of geometry, this derivative is a one-form, and it is closed but not exact. This is the most basic example of such a form, let U ⊆ R be open, and consider a differentiable function f, U → R, with derivative f. The differential df of f, at a point x 0 ∈ U, is defined as a linear map of the variable dx. Specifically, d f, d x ↦ f ′ d x, hence the map x ↦ d f sends each point x to a linear functional df. This is the simplest example of a differential form, in terms of the de Rham complex, one has an assignment from zero-forms to one-forms i. e. f ↦ d f. Two-form Reciprocal lattice Tensor Inner product

9.
Root of unity
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In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power n. Roots of unity are used in branches of mathematics, and are especially important in number theory, the theory of group characters. In field theory and ring theory the notion of root of unity also applies to any ring with an identity element. Any algebraically closed field has exactly n nth roots of unity if n is not divisible by the characteristic of the field, an nth root of unity, where n is a positive integer, is a number z satisfying the equation z n =1. Without further specification, the roots of unity are complex numbers, however the defining equation of roots of unity is meaningful over any field F, and this allows considering roots of unity in F. Whichever is the field F, the roots of unity in F are either numbers, if the characteristic of F is 0, or, otherwise. Conversely, every element in a finite field is a root of unity in that field. See Root of unity modulo n and Finite field for further details, an nth root of unity is primitive if it is not a kth root of unity for some smaller k, z k ≠1. Every nth root of unity z is a primitive ath root of unity for some a where 1 ≤ a ≤ n. In fact, if z1 =1 then z is a primitive first root of unity, otherwise if z2 =1 then z is a second root of unity. And, as z is a root of unity, one finds a first a such that za =1. If z is an nth root of unity and a ≡ b then za = zb, Therefore, given a power za of z, it can be assumed that 1 ≤ a ≤ n. Any integer power of an nth root of unity is also an nth root of unity, n = z k n = k =1 k =1. In particular, the reciprocal of an nth root of unity is its complex conjugate, let z be a primitive nth root of unity. Zn−1, zn = z0 =1 are all distinct, assume the contrary, that za = zb where 1 ≤ a < b ≤ n. But 0 < b − a < n, which contradicts z being primitive. Since an nth-degree polynomial equation can only have n distinct roots, from the preceding, it follows that if z is a primitive nth root of unity, z a = z b ⟺ a ≡ b. If z is not primitive there is only one implication, a ≡ b ⟹ z a = z b

10.
Turn (geometry)
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A turn is a unit of plane angle measurement equal to 2π radians, 360° or 400 gon. A turn is also referred to as a revolution or complete rotation or full circle or cycle or rev or rot, a turn can be subdivided in many different ways, into half turns, quarter turns, centiturns, milliturns, binary angles, points etc. A turn can be divided in 100 centiturns or 1000 milliturns, with each corresponding to an angle of 0. 36°. A protractor divided in centiturns is normally called a percentage protractor, binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32 compass points, the binary degree, also known as the binary radian, is 1⁄256 turn. The binary degree is used in computing so that an angle can be represented to the maximum possible precision in a single byte, other measures of angle used in computing may be based on dividing one whole turn into 2n equal parts for other values of n. The notion of turn is used for planar rotations. Two special rotations have acquired appellations of their own, a rotation through 180° is commonly referred to as a half-turn, the word turn originates via Latin and French from the Greek word τόρνος. In 1697, David Gregory used π/ρ to denote the perimeter of a divided by its radius. However, earlier in 1647, William Oughtred had used δ/π for the ratio of the diameter to perimeter, the first use of the symbol π on its own with its present meaning was in 1706 by the Welsh mathematician William Jones. Euler adopted the symbol with that meaning in 1737, leading to its widespread use, percentage protractors have existed since 1922, but the terms centiturns and milliturns were introduced much later by Sir Fred Hoyle. The German standard DIN1315 proposed the unit symbol pla for turns, since 2011, the HP 39gII and HP Prime support the unit symbol tr for turns. In 2016, support for turns was also added to newRPL for the HP 50g, one turn is equal to 2π radians. In 1958, Albert Eagle proposed the Greek letter tau τ as a symbol for 1/2π and his proposal used a pi with three legs symbol to denote the constant. In 2010, Michael Hartl proposed to use tau to represent Palais circle constant, τ=2π. First, τ is the number of radians in one turn, which allows fractions of a turn to be expressed directly, for instance. Second, τ visually resembles π, whose association with the constant is unavoidable. Hartls Tau Manifesto gives many examples of formulas that are simpler if tau is used instead of pi, however, a rebuttal was given in The Pi Manifesto, stating a variety of reasons tau should not supplant pi

11.
Unary numeral system
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The unary numeral system is the bijective base-1 numeral system. It is the simplest numeral system to represent natural numbers, in order to represent a number N, for examples, the numbers 1,2,3,4,5. Would be represented in this system as 1,11,111,1111,11111 and these numbers should be distinguished from repunits, which are also written as sequences of ones but have their usual decimal numerical interpretation. This system is used in tallying, for example, using the tally mark |, the number 3 is represented as |||. In East Asian cultures, the three is represented as “三”, a character that is drawn with three strokes. Addition and subtraction are particularly simple in the system, as they involve little more than string concatenation. The Hamming weight or population count operation that counts the number of bits in a sequence of binary values may also be interpreted as a conversion from unary to binary numbers. However, multiplication is more cumbersome and has often used as a test case for the design of Turing machines. Compared to standard positional numeral systems, the system is inconvenient. It occurs in some decision problem descriptions in theoretical computer science, therefore, while the run-time and space requirement in unary looks better as function of the input size, it does not represent a more efficient solution. In computational complexity theory, unary numbering is used to distinguish strongly NP-complete problems from problems that are NP-complete, for such a problem, there exist hard instances for which all parameter values are at most polynomially large. Unary is used as part of data compression algorithms such as Golomb coding. It also forms the basis for the Peano axioms for formalizing arithmetic within mathematical logic, a form of unary notation called Church encoding is used to represent numbers within lambda calculus. Sloanes A000042, Unary representation of natural numbers, the On-Line Encyclopedia of Integer Sequences

12.
Unit sphere
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Usually a specific point has been distinguished as the origin of the space under study and it is understood that a unit sphere or unit ball is centered at that point. Therefore one speaks of the ball or the unit sphere. For example, a sphere is the surface of what is commonly called a circle, while such a circles interior. Similarly, a sphere is the surface of the Euclidean solid known colloquially as a sphere, while the interior. A unit sphere is simply a sphere of radius one, the importance of the unit sphere is that any sphere can be transformed to a unit sphere by a combination of translation and scaling. In this way the properties of spheres in general can be reduced to the study of the unit sphere. In Euclidean space of n dimensions, the sphere is the set of all points which satisfy the equation x 12 + x 22 + ⋯ + x n 2 =1. The volume of the ball in n dimensions, which we denote Vn. It is V n = π n /2 Γ = { π n /2 /, I f n ≥0 i s e v e n, π ⌊ n /2 ⌋2 ⌈ n /2 ⌉ / n. I f n ≥0 i s o d d, where n. is the double factorial, the surface areas and the volumes for some values of n are as follows, where the decimal expanded values for n ≥2 are rounded to the displayed precision. The An values satisfy the recursion, A0 =0 A1 =2 A2 =2 π A n =2 π n −2 A n −2 for n >2. The Vn values satisfy the recursion, V0 =1 V1 =2 V n =2 π n V n −2 for n >1. The surface area of a sphere with radius r is An rn−1. For instance, the area is A = 4π r 2 for the surface of the ball of radius r. The volume is V = 4π r 3 /3 for the ball of radius r. More precisely, the unit ball in a normed vector space V. It is the interior of the unit ball of. The latter is the disjoint union of the former and their common border, the shape of the unit ball is entirely dependent on the chosen norm, it may well have corners, and for example may look like n, in the case of the norm l∞ in Rn

13.
Unit circle
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In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S1, the generalization to higher dimensions is the unit sphere, if is a point on the unit circles circumference, then | x | and | y | are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation x 2 + y 2 =1. The interior of the circle is called the open unit disk. One may also use other notions of distance to define other unit circles, such as the Riemannian circle, see the article on mathematical norms for additional examples. The unit circle can be considered as the complex numbers. In quantum mechanics, this is referred to as phase factor, the equation x2 + y2 =1 gives the relation cos 2 + sin 2 =1. The unit circle also demonstrates that sine and cosine are periodic functions, triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OA from the origin to a point P on the circle such that an angle t with 0 < t < π/2 is formed with the positive arm of the x-axis. Now consider a point Q and line segments PQ ⊥ OQ, the result is a right triangle △OPQ with ∠QOP = t. Because PQ has length y1, OQ length x1, and OA length 1, sin = y1 and cos = x1. Having established these equivalences, take another radius OR from the origin to a point R on the circle such that the same angle t is formed with the arm of the x-axis. Now consider a point S and line segments RS ⊥ OS, the result is a right triangle △ORS with ∠SOR = t. It can hence be seen that, because ∠ROQ = π − t, R is at in the way that P is at. The conclusion is that, since is the same as and is the same as, it is true that sin = sin and it may be inferred in a similar manner that tan = −tan, since tan = y1/x1 and tan = y1/−x1. A simple demonstration of the above can be seen in the equality sin = sin = 1/√2, when working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π/2. However, when defined with the circle, these functions produce meaningful values for any real-valued angle measure – even those greater than 2π

14.
Unit disk
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In mathematics, the open unit disk around P, is the set of points whose distance from P is less than 1, D1 =. The closed unit disk around P is the set of points whose distance from P is less than or equal to one, D ¯1 =. Unit disks are special cases of disks and unit balls, as such, they contain the interior of the circle and, in the case of the closed unit disk. Without further specifications, the unit disk is used for the open unit disk about the origin, D1. It is the interior of a circle of radius 1, centered at the origin and this set can be identified with the set of all complex numbers of absolute value less than one. When viewed as a subset of the plane, the unit disk is often denoted D. The function f = z 1 − | z |2 is an example of a real analytic and bijective function from the unit disk to the plane. Considered as a real 2-dimensional analytic manifold, the unit disk is therefore isomorphic to the whole plane. In particular, the unit disk is homeomorphic to the whole plane. There is however no conformal bijective map between the unit disk and the plane. Considered as a Riemann surface, the unit disk is therefore different from the complex plane. There are conformal bijective maps between the unit disk and the open upper half-plane. So considered as a Riemann surface, the unit disk is isomorphic to the upper half-plane. One bijective conformal map from the unit disk to the open upper half-plane is the Möbius transformation g = i 1 + z 1 − z which is the inverse of the Cayley transform. The unit disk and the upper half-plane are not interchangeable as domains for Hardy spaces, contributing to this difference is the fact that the unit circle has finite Lebesgue measure while the real line does not. The open unit disk is used as a model for the hyperbolic plane, by introducing a new metric on it. Using the above-mentioned conformal map between the unit disk and the upper half-plane, this model can be turned into the Poincaré half-plane model of the hyperbolic plane. Another model of space is also built on the open unit disk

15.
Unit hyperbola
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In geometry, the unit hyperbola is the set of points in the Cartesian plane that satisfies x 2 − y 2 =1. In the study of orthogonal groups, the unit hyperbola forms the basis for an alternative radial length r = x 2 − y 2. Whereas the unit circle surrounds its center, the unit hyperbola requires the conjugate hyperbola y 2 − x 2 =1 to complement it in the plane and this pair of hyperbolas share the asymptotes y = x and y = −x. When the conjugate of the hyperbola is in use, the alternative radial length is r = y 2 − x 2. The unit hyperbola is a case of the rectangular hyperbola, with a particular orientation, location. As such, its eccentricity equals 2, the unit hyperbola finds applications where the circle must be replaced with the hyperbola for purposes of analytic geometry. A prominent instance is the depiction of spacetime as a pseudo-Euclidean space, there the asymptotes of the unit hyperbola form a light cone. Further, the attention to areas of hyperbolic sectors by Gregoire de Saint-Vincent led to the logarithm function, generally asymptotic lines to a curve are said to converge toward the curve. In algebraic geometry and the theory of algebraic curves there is a different approach to asymptotes, the curve is first interpreted in the projective plane using homogeneous coordinates. Then the asymptotes are lines that are tangent to the curve at a point at infinity, thus circumventing any need for a distance concept. In a common framework are homogeneous coordinates with the line at infinity determined by the equation z =0. Both P, Q are simple on F, with tangents x + y =0, x − y =0, the Minkowski diagram is drawn in a spacetime plane where the spatial aspect has been restricted to a single dimension. The units of distance and time on such a plane are units of 30 centimetres length and nanoseconds, or astronomical units and intervals of 8 minutes and 20 seconds, or light years and years. Each of these scales of coordinates results in photon connections of events along diagonal lines of slope plus or minus one, the plane with the axes refers to a resting frame of reference. The diameter of the unit hyperbola represents a frame of reference in motion with rapidity a where tanh a = y/x and is the endpoint of the diameter on the unit hyperbola, the conjugate diameter represents the spatial hyperplane of simultaneity corresponding to rapidity a. Space is represented by planes perpendicular to the time axis, the here and now is a singularity in the middle. The vertical time axis convention stems from Minkowski in 1908, and is illustrated on page 48 of Eddingtons The Nature of the Physical World. A direct way to parameterizing the unit hyperbola starts with the hyperbola xy =1 parameterized with the exponential function and this hyperbola is transformed into the unit hyperbola by a linear mapping having the matrix A =12, A = =

16.
Dirac delta function
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It was introduced by theoretical physicist Paul Dirac. From a purely mathematical viewpoint, the Dirac delta is not strictly a function, because any extended-real function that is equal to zero everywhere, the delta function only makes sense as a mathematical object when it appears inside an integral. From this perspective the Dirac delta can usually be manipulated as though it were a function, the formal rules obeyed by this function are part of the operational calculus, a standard tool kit of physics and engineering. Formally, the function must be defined as the distribution that corresponds to a probability measure supported at the origin. In many applications, the Dirac delta is regarded as a kind of limit of a sequence of functions having a spike at the origin. The approximating functions of the sequence are thus approximate or nascent delta functions, in the context of signal processing the delta function is often referred to as the unit impulse symbol. Its discrete analog is the Kronecker delta function, which is defined on a discrete domain. The graph of the function is usually thought of as following the whole x-axis. Despite its name, the function is not truly a function. For example, the objects f = δ and g =0 are equal everywhere except at x =0 yet have integrals that are different. According to Lebesgue integration theory, if f and g are functions such that f = g almost everywhere, then f is integrable if and only if g is integrable, rigorous treatment of the Dirac delta requires measure theory or the theory of distributions. The Dirac delta is used to model a tall narrow spike function, for example, to calculate the dynamics of a baseball being hit by a bat, one can approximate the force of the bat hitting the baseball by a delta function. Later, Augustin Cauchy expressed the theorem using exponentials, f =12 π ∫ − ∞ ∞ e i p x d p. Cauchy pointed out that in some circumstances the order of integration in this result was significant. A rigorous interpretation of the form and the various limitations upon the function f necessary for its application extended over several centuries. Namely, it is necessary that these functions decrease sufficiently rapidly to zero in order to ensure the existence of the Fourier integral, for example, the Fourier transform of such simple functions as polynomials does not exist in the classical sense. The extension of the classical Fourier transformation to distributions considerably enlarged the class of functions that could be transformed, and leading to the formal development of the Dirac delta function. An infinitesimal formula for a tall, unit impulse delta function explicitly appears in an 1827 text of Augustin Louis Cauchy. Siméon Denis Poisson considered the issue in connection with the study of propagation as did Gustav Kirchhoff somewhat later

17.
Unit interval
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In mathematics, the unit interval is the closed interval, that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. In addition to its role in analysis, the unit interval is used to study homotopy theory in the field of topology. In the literature, the unit interval is sometimes applied to the other shapes that an interval from 0 to 1 could take. However, the notation I is most commonly reserved for the closed interval, the unit interval is a complete metric space, homeomorphic to the extended real number line. As a topological space it is compact, contractible, path connected, the Hilbert cube is obtained by taking a topological product of countably many copies of the unit interval. In mathematical analysis, the interval is a one-dimensional analytical manifold whose boundary consists of the two points 0 and 1. Its standard orientation goes from 0 to 1, the unit interval is a totally ordered set and a complete lattice. The size or cardinality of a set is the number of elements it contains, the unit interval is a subset of the real numbers R. However, it has the same size as the whole set, the cardinality of the continuum. Moreover, it has the number of points as a square of area 1, as a cube of volume 1. The number of elements in all the sets is uncountable. The interval, with two, demarcated by the positive and negative units, occurs frequently, such as in the range of the trigonometric functions sine and cosine. This interval may be used for the domain of inverse functions, for instance, when θ is restricted to then sin is in this interval and arcsine is defined there. Sometimes, the unit interval is used to refer to objects that play a role in various branches of mathematics analogous to the role that plays in homotopy theory. For example, in the theory of quivers, the interval is the graph whose vertex set is. One can then define a notion of homotopy between quiver homomorphisms analogous to the notion of homotopy between continuous maps. In logic, the interval can be interpreted as a generalization of the Boolean domain, in which case rather than only taking values 0 or 1. Algebraically, negation is replaced with 1 − x, conjunction is replaced with multiplication, interpreting these values as logical truth values yields a multi-valued logic, which forms the basis for fuzzy logic and probabilistic logic. In these interpretations, a value is interpreted as the degree of truth – to what extent a proposition is true, or the probability that the proposition is true

18.
Unit length
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In mathematics, a unit vector in a normed vector space is a vector of length 1. A unit vector is denoted by a lowercase letter with a circumflex, or hat. The term direction vector is used to describe a unit vector being used to represent spatial direction, two 2D direction vectors, d1 and d2 are illustrated. 2D spatial directions represented this way are equivalent numerically to points on the unit circle, the same construct is used to specify spatial directions in 3D. As illustrated, each direction is equivalent numerically to a point on the unit sphere. The normalized vector or versor û of a vector u is the unit vector in the direction of u, i. e. u ^ = u ∥ u ∥ where ||u|| is the norm of u. The term normalized vector is used as a synonym for unit vector. Unit vectors are often chosen to form the basis of a vector space, every vector in the space may be written as a linear combination of unit vectors. By definition, in a Euclidean space the dot product of two vectors is a scalar value amounting to the cosine of the smaller subtended angle. In three-dimensional Euclidean space, the product of two arbitrary unit vectors is a 3rd vector orthogonal to both of them having length equal to the sine of the smaller subtended angle. Unit vectors may be used to represent the axes of a Cartesian coordinate system and they are often denoted using normal vector notation rather than standard unit vector notation. In most contexts it can be assumed that i, j, the notations, or, with or without hat, are also used, particularly in contexts where i, j, k might lead to confusion with another quantity. When a unit vector in space is expressed, with Cartesian notation, as a combination of i, j, k. The value of each component is equal to the cosine of the angle formed by the vector with the respective basis vector. This is one of the used to describe the orientation of a straight line, segment of straight line, oriented axis. It is important to note that ρ ^ and φ ^ are functions of φ, when differentiating or integrating in cylindrical coordinates, these unit vectors themselves must also be operated on. For a more complete description, see Jacobian matrix, to minimize degeneracy, the polar angle is usually taken 0 ≤ θ ≤180 ∘. It is especially important to note the context of any ordered triplet written in spherical coordinates, here, the American physics convention is used

19.
Unit square
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In mathematics, a unit square is a square whose sides have length 1. Often, the unit square refers specifically to the square in the Cartesian plane with corners at the four points, and. In a Cartesian coordinate system with coordinates the unit square is defined as the square consisting of the points where x and y lie in a closed unit interval from 0 to 1. That is, the square is the Cartesian product I × I. The unit square can also be thought of as a subset of the complex plane, in this view, the four corners of the unit square are at the four complex numbers 0,1, i, and 1 + i. It is not known whether any point in the plane is a distance from all four vertices of the unit square. However, no point is on an edge of the square. Unit circle Unit sphere Unit cube Weisstein, Eric W. Unit square

20.
Unit vector
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In mathematics, a unit vector in a normed vector space is a vector of length 1. A unit vector is denoted by a lowercase letter with a circumflex, or hat. The term direction vector is used to describe a unit vector being used to represent spatial direction, two 2D direction vectors, d1 and d2 are illustrated. 2D spatial directions represented this way are equivalent numerically to points on the unit circle, the same construct is used to specify spatial directions in 3D. As illustrated, each direction is equivalent numerically to a point on the unit sphere. The normalized vector or versor û of a vector u is the unit vector in the direction of u, i. e. u ^ = u ∥ u ∥ where ||u|| is the norm of u. The term normalized vector is used as a synonym for unit vector. Unit vectors are often chosen to form the basis of a vector space, every vector in the space may be written as a linear combination of unit vectors. By definition, in a Euclidean space the dot product of two vectors is a scalar value amounting to the cosine of the smaller subtended angle. In three-dimensional Euclidean space, the product of two arbitrary unit vectors is a 3rd vector orthogonal to both of them having length equal to the sine of the smaller subtended angle. Unit vectors may be used to represent the axes of a Cartesian coordinate system and they are often denoted using normal vector notation rather than standard unit vector notation. In most contexts it can be assumed that i, j, the notations, or, with or without hat, are also used, particularly in contexts where i, j, k might lead to confusion with another quantity. When a unit vector in space is expressed, with Cartesian notation, as a combination of i, j, k. The value of each component is equal to the cosine of the angle formed by the vector with the respective basis vector. This is one of the used to describe the orientation of a straight line, segment of straight line, oriented axis. It is important to note that ρ ^ and φ ^ are functions of φ, when differentiating or integrating in cylindrical coordinates, these unit vectors themselves must also be operated on. For a more complete description, see Jacobian matrix, to minimize degeneracy, the polar angle is usually taken 0 ≤ θ ≤180 ∘. It is especially important to note the context of any ordered triplet written in spherical coordinates, here, the American physics convention is used