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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
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
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Sedenion
In abstract algebra , the SEDENIONS form a 16-dimensional noncommutative and nonassociative algebra over the reals obtained by applying the Cayley–Dickson construction to the octonions . Unlike the octonions, the sedenions are not an alternative algebra . The set of sedenions is denoted by S {displaystyle mathbb {S} } . The term sedenion is also used for other 16-dimensional algebraic structures, such as a tensor product of two copies of the biquaternions , or the algebra of 4 by 4 matrices over the reals, or that studied by Smith (1995) . CONTENTS * 1 Arithmetic * 2 Applications * 3 See also * 4 Notes * 5 References ARITHMETIC A visualization of a 4D extension to the cubic Octonion , showing the 35 triads as hyperplanes through the Real ( e 0 {displaystyle e_{0}} ) vertex of the sedenion example given. Like octonions , multiplication of sedenions is neither commutative nor associative
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Square Matrix
In mathematics , a SQUARE MATRIX is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied. Square matrices are often used to represent simple linear transformations , such as shearing or rotation . For example, if R is a square matrix representing a rotation (rotation matrix ) and V is a column vector describing the position of a point in space, the product RV yields another column vector describing the position of that point after that rotation. If V is a row vector , the same transformation can be obtained using VRT, where RT is the transpose of R
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Determinant
In linear algebra , the DETERMINANT is a useful value that can be computed from the elements of a square matrix . The determinant of a matrix A is denoted det(A), det A, or A. It can be viewed as the scaling factor of the transformation described by the matrix. In the case of a 2 × 2 matrix, the specific formula for the determinant is A = a b c d = a d b c . {displaystyle {begin{aligned}A={begin{vmatrix}a&b\c width:24.893ex; height:6.176ex;" alt="{displaystyle {begin{aligned}A={begin{vmatrix}a&b\c padding-right:0.1em;">A: A = a b c d e f g h i = a e f h i b d f g i + c d e g h = a e i + b f g + c d h c e g b d i a f h
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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
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Coprime
In number theory , two integers a and b are said to be RELATIVELY PRIME, MUTUALLY PRIME, or COPRIME (also spelled CO-PRIME) if the only positive integer that divides both of them is 1. That is, the only common positive factor of the two numbers is 1. This is equivalent to their greatest common divisor being 1. The numerator and denominator of a reduced fraction are coprime. In addition to gcd ( a , b ) = 1 {displaystyle gcd(a,b)=1} and ( a , b ) = 1 , {displaystyle (a,b)=1,} the notation a b {displaystyle aperp b} is sometimes used to indicate that a and b are relatively prime. For example, 14 and 15 are coprime, being commonly divisible by only 1, but 14 and 21 are not, because they are both divisible by 7. The numbers 1 and −1 are the only integers coprime to every integer, and they are the only integers to be coprime with 0. A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm
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Integer
An INTEGER (from the Latin
Latin
integer meaning "whole") is a number that can be written without a fractional component . For example, 21, 4, 0, and −2048 are integers, while 9.75, ​5 1⁄2, and √2 are not. The set of integers consists of zero (0 ), the positive natural numbers (1 , 2 , 3 , …), also called whole numbers or counting numbers, and their additive inverses (the NEGATIVE INTEGERS, i.e., −1
−1
, −2, −3, …). This is often denoted by a boldface Z ("Z") or blackboard bold Z {displaystyle mathbb {Z} } (Unicode U+2124 ℤ) standing for the German word Zahlen ( , "numbers"). Z is a subset of the set of all rational numbers Q, in turn a subset of the real numbers R. Like the natural numbers, Z is countably infinite . The integers form the smallest group and the smallest ring containing the natural numbers
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Zero Of A Function
In mathematics , a ZERO, also sometimes called a ROOT, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) VANISHES at x; that is, x is a solution of the equation f(x) = 0. In other words, a "zero" of a function is an input value that produces an output of zero (0). A ROOT of a polynomial is a zero of the corresponding polynomial function . The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree and that the number of roots and the degree are equal when one considers the complex roots (or more generally the roots in an algebraically closed extension ) counted with their multiplicities . For example, the polynomial f of degree two, defined by f ( x ) = x 2 5 x + 6 {displaystyle f(x)=x^{2}-5x+6} has the two roots 2 and 3, since f ( 2 ) = 2 2 5 2 + 6 = 0 a n d f ( 3 ) = 3 2 5 3 + 6 = 0
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If And Only If
In logic and related fields such as mathematics and philosophy , IF AND ONLY IF (shortened IFF) is a biconditional logical connective between statements. In that it is biconditional , the connective can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false). It is controversial whether the connective thus defined is properly rendered by the English "if and only if", with its pre-existing meaning. There is nothing to stop one from stipulating that we may read this connective as "only if and if", although this may lead to confusion
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Constructive Mathematics
In the philosophy of mathematics , CONSTRUCTIVISM asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. In standard mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. This proof by contradiction is not constructively valid. The constructive viewpoint involves a verificational interpretation of the existential quantifier , which is at odds with its classical interpretation. There are many forms of constructivism. These include the program of intuitionism founded by Brouwer , the finitism of Hilbert and Bernays , the constructive recursive mathematics of Shanin and Markov , and Bishop 's program of constructive analysis . Constructivism also includes the study of constructive set theories such as IZF and the study of topos theory
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Bit Shift
In digital computer programming , a BITWISE OPERATION operates on one or more bit patterns or binary numerals at the level of their individual bits . It is a fast, simple action directly supported by the processor , and is used to manipulate values for comparisons and calculations. On simple low-cost processors, typically, bitwise operations are substantially faster than division, several times faster than multiplication, and sometimes significantly faster than addition. While modern processors usually perform addition and multiplication just as fast as bitwise operations due to their longer instruction pipelines and other architectural design choices, bitwise operations do commonly use less power because of the reduced use of resources
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Cavalieri's Quadrature Formula
In calculus , CAVALIERI\'S QUADRATURE FORMULA, named for 17th-century Italian mathematician Bonaventura Cavalieri , is the integral 0 a x n d x = 1 n + 1 a n + 1 n 0 , {displaystyle int _{0}^{a}x^{n},dx={tfrac {1}{n+1}},a^{n+1}qquad ngeq 0,} and generalizations thereof. This is the definite integral form; the indefinite integral form is: x n d x = 1 n + 1 x n + 1 + C n 1. {displaystyle int x^{n},dx={tfrac {1}{n+1}},x^{n+1}+Cqquad nneq -1.} There are additional forms , listed below. Together with the linearity of the integral, this formula allows one to compute the integrals of all polynomials. The term "quadrature " is a traditional term for area ; the integral is geometrically interpreted as the area under the curve y = xn. Traditionally important cases are y = x2, the quadrature of the parabola , known in antiquity, and y = 1/x, the quadrature of the hyperbola, whose value is a logarithm
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Natural Logarithm
The NATURAL LOGARITHM of a number is its logarithm to the base of the mathematical constant e , where e is an irrational and transcendental number approximately equal to 7000271828182845899♠2.718281828459. The natural logarithm of x is generally written as ln x, loge x, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x), loge(x) or log(x). This is done in particular when the argument to the logarithm is not a single symbol, to prevent ambiguity. The natural logarithm of x is the power to which e would have to be raised to equal x. For example, ln(7.5) is 2.0149..., because e2.0149... = 7.5. The natural log of e itself, ln(e), is 1, because e1 = e, while the natural logarithm of 1, ln(1), is 0, since e0 = 1. The natural logarithm can be defined for any positive real number a as the area under the curve y = 1/x from 1 to a (the area being taken as negative when a < 1). The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural"
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Long Division
In arithmetic , LONG DIVISION is a standard division algorithm suitable for dividing multidigit numbers that is simple enough to perform by hand. It breaks down a division problem into a series of easier steps. As in all division problems, one number, called the dividend , is divided by another, called the divisor , producing a result called the quotient . It enables computations involving arbitrarily large numbers to be performed by following a series of simple steps. The abbreviated form of long division is called short division , which is almost always used instead of long division when the divisor has only one digit. Chunking (also known as the partial quotients method or the hangman method) is a less-efficient form of long division which may be easier to understand. While related algorithms have existed since the 12th century AD, the specific algorithm in modern use was introduced by Henry Briggs c. 1600 AD
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Power Rule
In calculus , the POWER RULE is used to differentiate functions of the form f ( x ) = x r {displaystyle f(x)=x^{r}} , whenever r {displaystyle r} is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated using this rule
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Calculus
* v * t * e CALCULUS (from Latin
Latin
calculus, literally 'small pebble', used for counting and calculations, as on an abacus ) is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations . It has two major branches, differential calculus (concerning rates of change and slopes of curves), and integral calculus (concerning accumulation of quantities and the areas under and between curves). These two branches are related to each other by the fundamental theorem of calculus . Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit . Generally, modern calculus is considered to have been developed in the 17th century by Isaac Newton and Gottfried Wilhelm Leibniz
Gottfried Wilhelm Leibniz

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