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Multiplicative Inverse
In mathematics , a MULTIPLICATIVE INVERSE or RECIPROCAL for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity , 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The RECIPROCAL FUNCTION, the function f(x) that maps x to 1/x, is one of the simplest examples of a function which is its own inverse (an involution ). The term reciprocal was in common use at least as far back as the third edition of Encyclopædia Britannica (1797) to describe two numbers whose product is 1; geometrical quantities in inverse proportion are described as reciprocall in a 1570 translation of Euclid
Euclid
's Elements . In the phrase multiplicative inverse, the qualifier multiplicative is often omitted and then tacitly understood (in contrast to the additive inverse ). Multiplicative inverses can be defined over many mathematical domains as well as numbers. In these cases it can happen that ab ≠ ba; then "inverse" typically implies that an element is both a left and right inverse . The notation f −1 is sometimes also used for the inverse function of the function f, which is not in general equal to the multiplicative inverse
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Rectangular Hyperbola
In mathematics , a HYPERBOLA (plural hyperbolas or hyperbolae) is a type of smooth curve lying in a plane , defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows . The hyperbola is one of the three kinds of conic section , formed by the intersection of a plane and a double cone . (The other conic sections are the parabola and the ellipse . A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola. Hyperbolas arise in many ways: * as the curve representing the function f ( x ) = 1 / x {displaystyle f(x)=1/x} in the Cartesian plane , * as the path followed by the shadow of the tip of a sundial , * as the shape of an open orbit (as distinct from a closed elliptical orbit), such as the orbit of a spacecraft during a gravity assisted swing-by of a planet or more generally any spacecraft exceeding the escape velocity of the nearest planet, * as the path of a single-apparition comet (one travelling too fast ever to return to the solar system), * as the scattering trajectory of a subatomic particle (acted on by repulsive instead of attractive forces but the principle is the same),and so on
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Mathematics
MATHEMATICS (from Greek μάθημα _máthēma_, “knowledge, study, learning”) is the study of topics such as quantity (numbers ), structure , space , and change . There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics . 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 , and the systematic study of the shapes and motions of physical objects. 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 Euclid 's _Elements _. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century , it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions
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Multiplication
MULTIPLICATION (often denoted by the cross symbol " ×
×
", by a point "⋅ ", by juxtaposition, or, on computers, by an asterisk "∗") is one of the four elementary mathematical operations of arithmetic ; with the others being addition , subtraction and division . The multiplication of whole numbers may be thought as a repeated addition ; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the multiplicand, as the value of the other one, the multiplier. Normally, the multiplier is written first and multiplicand second, though this can vary, especially in languages with different grammatical structures, such as Japanese , Japanese elementary schools teach writing the multiplicand first, and answers that reverse that order are marked as incorrect. The distinction is not very meaningful: a b = b + + b a = a + + a b = b a {displaystyle atimes b=underbrace {b+cdots +b} _{a}=underbrace {a+cdots +a} _{b}=btimes a} For example, 4 multiplied by 3 (often written as 3 4 {displaystyle 3times 4} and said as "3 times 4") can be calculated by adding 3 copies of 4 together: 3 4 = 4 + 4 + 4 = 12 {displaystyle 3times 4=4+4+4=12} Here 3 and 4 are the "factors" and 12 is the "product"
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Multiplicative Identity
1 (ONE, also called UNIT, UNITY, and (MULTIPLICATIVE) IDENTITY) is a number , numeral , and glyph . 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 infinite sequence of natural numbers , followed by 2. CONTENTS * 1 Etymology * 2 As a number * 3 As a digit * 4 Mathematics * 4.1 Table of basic calculations * 5 In technology * 6 In science * 7 In philosophy * 8 In literature * 9 In comics * 10 In sports * 11 In other fields * 12 See also * 13 References * 14 External links ETYMOLOGYThe word ONE can be used as a noun, an adjective and a pronoun. 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- (which means one, single ) to Greek OINOS (which means "ace" on dice ), Latin UNUS (one ), Old Persian AIVAM, Old Church Slavonic -INU and INO-, Lithuanian VIENAS, Old Irish OIN and Breton UN (one ). AS A NUMBEROne, 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
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Rational Number
In mathematics , a RATIONAL NUMBER is any number that can be expressed as the quotient or fraction p/q of two integers , a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as "THE RATIONALS", the FIELD OF RATIONALS or the FIELD OF RATIONAL NUMBERS is usually denoted by a boldface Q (or blackboard bold Q {displaystyle mathbb {Q} } , Unicode ℚ); it was thus denoted in 1895 by Giuseppe Peano
Giuseppe Peano
after quoziente, Italian for "quotient ". The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10 , but also for any other integer base (e.g. binary , hexadecimal ). A real number that is not rational is called irrational . Irrational numbers include √2 , π , e , and φ . The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is countable , and the set of real numbers is uncountable , almost all real numbers are irrational. Rational numbers can be formally defined as equivalence classes of pairs of integers (p, q) such that q ≠ 0, for the equivalence relation defined by (p1,q1) ~ (p2,q2) if, and only if, p1q2 = p2q1
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Involution (mathematics)
In mathematics , an (ANTI-)INVOLUTION, or an INVOLUTORY FUNCTION, is a function f that is its own inverse , f(f(x)) = x for all x in the domain of f. CONTENTS * 1 General properties * 2 Involution throughout the fields of mathematics * 2.1 Pre-calculus * 2.2 Euclidean geometry * 2.3 Projective geometry * 2.4 Linear algebra * 2.5 Quaternion algebra, groups, semigroups * 2.6 Ring theory * 2.7 Group theory * 2.8 Mathematical logic * 2.9 Computer science * 3 See also * 4 References * 5 Further reading GENERAL PROPERTIESAny involution is a bijection . The identity map is a trivial example of an involution. Common examples in mathematics of nontrivial involutions include multiplication by −1 in arithmetic , the taking of reciprocals , complementation in set theory and complex conjugation . Other examples include circle inversion , rotation by a half-turn, and reciprocal ciphers such as the ROT13
ROT13
transformation and the Beaufort polyalphabetic cipher . The number of involutions, including the identity involution, on a set with n = 0, 1, 2, ... elements is given by a recurrence relation found by Heinrich August Rothe in 1800: a0 = a1 = 1; an = an − 1 + (n − 1)an − 2, for n > 1
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Encyclopædia Britannica
The _ENCYCLOPæDIA BRITANNICA_ ( Latin
Latin
for "British Encyclopaedia"), published by Encyclopædia Britannica, Inc. , is a general knowledge English-language encyclopaedia . It is written by about 100 full-time editors and more than 4,000 contributors, who have included 110 Nobel Prize winners and five American presidents . The 2010 version of the 15th edition, which spans 32 volumes and 32,640 pages, was the last printed edition; digital content and distribution has continued since then. The _Britannica_ is the oldest English-language encyclopaedia still in production. It was first published between 1768 and 1771 in the Scottish capital of Edinburgh
Edinburgh
, as three volumes. The encyclopaedia grew in size: the second edition was 10 volumes, and by its fourth edition (1801–1810) it had expanded to 20 volumes. Its rising stature as a scholarly work helped recruit eminent contributors, and the 9th (1875–1889) and 11th editions (1911) are landmark encyclopaedias for scholarship and literary style. Beginning with the 11th edition and following its acquisition by an American firm, the _Britannica_ shortened and simplified articles to broaden its appeal to the North American market. In 1933, the _Britannica_ became the first encyclopaedia to adopt "continuous revision", in which the encyclopaedia is continually reprinted, with every article updated on a schedule
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Euclid
EUCLID (/ˈjuːklᵻd/ ; Greek : Εὐκλείδης, _Eukleidēs_ Ancient Greek: ; fl. 300 BCE), sometimes called EUCLID OF ALEXANDRIA to distinguish him from Euclides of Megara , was a Greek mathematician , often referred to as the "father of geometry". He was active in Alexandria during the reign of Ptolemy I (323–283 BCE). His _Elements _ is one of the most influential works in the history of mathematics , serving as the main textbook for teaching mathematics (especially geometry ) from the time of its publication until the late 19th or early 20th century. In the _Elements_, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms . Euclid also wrote works on perspective , conic sections , spherical geometry , number theory , and rigor . Euclid is the anglicized version of the Greek name Εὐκλείδης, which means "renowned, glorious". CONTENTS * 1 Life * 2 _Elements_ * 3 Other works * 3.1 Lost works * 4 See also * 5 Notes * 6 References * 7 Further reading * 8 External links LIFEVery few original references to Euclid survive, so little is known about his life. The date, place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him
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Euclid's Elements
The ELEMENTS ( Ancient Greek
Ancient Greek
: Στοιχεῖα Stoicheia) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid
Euclid
in Alexandria
Alexandria
, Ptolemaic Egypt circa 300 BC. It is a collection of definitions, postulates, propositions (theorems and constructions ), and mathematical proofs of the propositions. The books cover plane and solid Euclidean geometry , elementary number theory , and incommensurable lines. Elements is the oldest extant large-scale deductive treatment of mathematics . It has proven instrumental in the development of logic and modern science , and its logical rigor was not surpassed until the 19th century. Euclid's Elements
Euclid's Elements
has been referred to as the most successful and influential textbook ever written. It was one of the very earliest mathematical works to be printed after the invention of the printing press and has been estimated to be second only to the Bible
Bible
in the number of editions published since the first printing in 1482, with the number reaching well over one thousand. For centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclid's Elements
Euclid's Elements
was required of all students
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Additive Inverse
In mathematics, the ADDITIVE INVERSE of a number a is the number that, when added to a, yields zero . This number is also known as the OPPOSITE (number), SIGN CHANGE, and NEGATION. For a real number , it reverses its sign : the opposite to a positive number is negative, and the opposite to a negative number is positive. Zero is the additive inverse of itself. The additive inverse of a is denoted by unary minus : −a (see the discussion below ). For example, the additive inverse of 7 is −7, because 7 + (−7) = 0, and the additive inverse of −0.3 is 0.3, because −0.3 + 0.3 = 0 . The additive inverse is defined as its inverse element under the binary operation of addition (see the discussion below ), which allows a broad generalization to mathematical objects other than numbers. As for any inverse operation, double additive inverse has no net effect : −(−_x_) = _x_. These complex numbers, two of eight values of 8√1 , are mutually opposite CONTENTS* 1 Common examples * 1.1 Relation to subtraction * 1.2 Other properties * 2 Formal definition * 3 Other examples * 4 Non-examples * 5 See also * 6 Footnotes * 7 References COMMON EXAMPLESFor a number and, generally, in any ring , the additive inverse can be calculated using multiplication by −1 ; that is, −n = −1 × n . Examples of rings of numbers are integers , rational numbers , real numbers , and complex numbers
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Inverse Element
In abstract algebra , the idea of an INVERSE ELEMENT generalises concepts of a negation (sign reversal) in relation to addition , and a reciprocal in relation to multiplication . The intuition is of an element that can 'undo' the effect of combination with another given element. While the precise definition of an inverse element varies depending on the algebraic structure involved, these definitions coincide in a group . The word 'inverse' is derived from Latin : inversus that means 'turned upside down', 'overturned'. CONTENTS* 1 Formal definitions * 1.1 In a unital magma * 1.2 In a semigroup * 1.3 U-semigroups * 1.4 Rings and semirings * 2 Examples * 2.1 Real numbers * 2.2 Functions and partial functions * 2.3 Galois connections * 2.4 Matrices * 3 See also * 4 Notes * 5 References FORMAL DEFINITIONSIN A UNITAL MAGMALet S {displaystyle S} be a set closed under a binary operation {displaystyle *} (i.e., a magma ). If e {displaystyle e} is an identity element of ( S , ) {displaystyle (S,*)} (i.e., S is a unital magma) and a b = e {displaystyle a*b=e} , then a {displaystyle a} is called a LEFT INVERSE of b {displaystyle b} and b {displaystyle b} is called a RIGHT INVERSE of a {displaystyle a}
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Inverse Function
In mathematics , an INVERSE FUNCTION is a function that "reverses" another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, and vice versa. i.e., f(x) = y if and only if g(y) = x. As a simple example, consider the real-valued function of a real variable given by f(x) = 5x − 7. Thinking of this as a step-by-step procedure (namely, take a number x, multiply it by 5, then subtract 7 from the result), to reverse this and get x back from some output value, say y, we should undo each step in reverse order. In this case that means that we should add 7 to y and then divide the result by 5. In functional notation this inverse function would be given by, g ( y ) = y + 7 5 . {displaystyle g(y)={frac {y+7}{5}}.} With y = 5x − 7 we have that f(x) = y and g(y) = x. Not all functions have inverse functions. In order for a function f: X → Y to have an inverse, it must have the property that for every y in Y there must be one, and only one x in X so that f(x) = y. This property ensures that a function g: Y → X will exist having the necessary relationship with f
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Cosecant
In mathematics , the TRIGONOMETRIC FUNCTIONS (also called CIRCULAR FUNCTIONS, ANGLE FUNCTIONS or GONIOMETRIC FUNCTIONS ) are functions of an angle . They relate the angles of a triangle to the lengths of its sides. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine , cosine , and tangent . In the context of the standard unit circle (a circle with radius 1 unit), where a triangle is formed by a ray starting at the origin and making some angle with the x-axis, the sine of the angle gives the length of the y-component (the opposite to the angle or the rise) of the triangle, the cosine gives the length of the x-component (the adjacent of the angle or the run), and the tangent function gives the slope (y-component divided by the x-component). More precise definitions are detailed below. Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle . More modern definitions express them as infinite series or as solutions of certain differential equations , allowing their extension to arbitrary positive and negative values and even to complex numbers . Trigonometric functions have a wide range of uses including computing unknown lengths and angles in triangles (often right triangles )
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Inverse Trigonometric Functions
In mathematics , the INVERSE TRIGONOMETRIC FUNCTIONS (occasionally also called ARCUS FUNCTIONS, ANTITRIGONOMETRIC FUNCTIONS or CYCLOMETRIC FUNCTIONS ) are the inverse functions of the trigonometric functions (with suitably restricted domains ). Specifically, they are the inverses of the sine , cosine , tangent , cotangent , secant , and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering , navigation , physics , and geometry
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French Language
Phonological history * Oaths of Strasbourg * Ordinance of Villers-Cotterêts * Anglo-Norman GRAMMAR * Adverbs * Articles and determiners * Pronouns (personal )* Verbs * (conjugation * morphology ) ORTHOGRAPHY * Alphabet * Reforms * Circumflex * Braille PHONOLOGY * Elision * Liaison * Aspirated h * Help:IPA for French * v * t * e FRENCH (_le français_ (_ listen ) or la langue française_ ) is a Romance language of the Indo-European family . It descended from the Vulgar Latin of the Roman Empire , as did all Romance languages. French has evolved from Gallo-Romance, the spoken Latin in Gaul, and more specifically in Northern Gaul. Its closest relatives are the other langues d\'oïl —languages historically spoken in northern France and in southern Belgium, which French ( Francien ) has largely supplanted. French was also influenced by native Celtic languages of Northern Roman Gaul like Gallia Belgica and by the (Germanic ) Frankish language of the post-Roman Frankish invaders. Today, owing to France's past overseas expansion, there are numerous French-based creole languages , most notably Haitian Creole . A French-speaking person or nation may be referred to as "FRANCOPHONE" in both English and French
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