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Encyclopedia > Multiplicative inverse
The reciprocal function: y = 1/x. For every x except 0, y represents its multiplicative inverse.
The reciprocal function: y = 1/x. For every x except 0, y represents its multiplicative inverse.

In mathematics, the reciprocal, or multiplicative inverse, of a number x is the number which, when multiplied by x, yields 1. The multiplicative inverse for the real numbers, for example, is 1/x. To avoid confusion by writing the inverse using set specific notation, we generally write x-1. Image File history File links Hyperbola_one_over_x. ... Image File history File links Hyperbola_one_over_x. ... Look up one in Wiktionary, the free dictionary. ...


One computes the reciprocal of a (real) number by dividing 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), the reciprocal of 0.25 is 1 divided by 0.25, or 4.


Number can mean here any element of a unital ring, but the term reciprocal, as well as the notation 1/x, is usually restricted to commutative fields. In the non-abelian case, "inverse" implies both, left and right inverse. In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... Abelian, in mathematics, is used in many different definitions: In group theory: Abelian group, a group in which the binary operation is commutative Category of abelian groups Ab has abelian groups as objects and group homomorphisms as morphisms Metabelian group is a group where the commutator subgroup is contained in... In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ...


The qualifier multiplicative is often omitted and then tacitly self-understood (in contrast to the additive inverse). The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. ...

Contents

Examples and counter-examples

Zero does not have a reciprocal, as division by 0 is undefined. Every complex number except zero has a reciprocal that is a complex number. If a number is real, then so is its reciprocal, and if it is rational, then so is its reciprocal. The two complex solutions to the equation i2 = − 1 represent the only numbers whose additive inverse is also its multiplicative inverse. To approximate the reciprocal of x, using only multiplication and subtraction, one can guess a number y, and then repeatedly replace y with 2yxy2. Once the change in y becomes (and stays) sufficiently small, y is an approximation of the reciprocal of x. 0 (zero) is both a number — or, more precisely, a numeral representing a number — and a numerical digit. ... In mathematics, a division is called a division by zero if the divisor is zero. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...


In constructive mathematics, for a real number x to have a reciprocal, it is not sufficient that it be false that x = 0. Instead, there must be given a rational number r such that 0 < r < |x|. In terms of the approximation algorithm in the previous paragraph, this is needed to prove that the change in y will eventually get arbitrarily small. In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ... Logic (from ancient Greek &#955;&#8057;&#947;&#959;&#962; (logos), meaning reason) is the study of arguments. ... Flowcharts are often used to graphically represent algorithms. ...


In modular arithmetic, the multiplicative inverse of x is also defined: it is the number a such that (a·x) mod n = 1. However, this multiplicative inverse exists only if a and n are relatively prime. For example, the inverse of 3 modulo 11 is 4 because it is the solution to (3x) mod 11 = 1. The extended Euclidean algorithm may be used to compute the multiplicative inverse modulo of a number. Modular arithmetic (sometimes called modulo arithmetic) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... Coprime - Wikipedia /**/ @import /skins-1. ... The extended Euclidean algorithm is an extension to the Euclidean algorithm for finding the greatest common divisor (GCD) of a and b: it also finds the integers x and y in Bezouts identity The extended Euclidean algorithm is particularly useful when a and b are coprime, since x is...


The sedenions are an algebra in which every nonzero element has a multiplicative inverse, but which has nonetheless divisors of zero, i.e. nonzero elements x,y such that xy=0. The sedenions form a 16-dimensional algebra over the reals obtained by applying the Cayley-Dickson construction to the octonions. ...


A square matrix has an inverse if and only if its determinant has an inverse in the coefficient ring. The linear map that has the matrix A−1 with respect to some base is then the reciprocal function of the map having A as matrix in the same base. Thus, the two distinct notions of the inverse of a function are strongly related in this case, while they must be carefully distinguished in the general case (see below). For the square matrix section, see square matrix. ... It has been suggested that this article or section be merged with Logical biconditional. ... In algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...


The trigonometric functions are related by the reciprocal identity: the cotangent is the reciprocal of the tangent; the secant is the reciprocal of the cosine; the cosecant is the reciprocal of the sine. Wikibooks has a book on the topic of Trigonometry Trigonometry (from the Greek trigonon = three angles and metron = measure [1]) is a branch of mathematics which deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right triangles). ... Trigonometry In trigonometry, the cotangent is a function (see trigonometric function) defined as: or An interpretation of the cotangent of an angle x is as follows. ... In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ... Secant is a term in mathematics. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...


It is important to distinguish the reciprocal of a function f in the multiplicative sense, given by 1/f, from the reciprocal or inverse function w.r.t. composition, rather denoted by f−1, defined by f o f−1 = id. Only for linear maps they are strongly related (see above), while they are completely different for all other cases. The terminology reciprocal vs inverse is not sufficient to make this distinction, since many authors prefer the opposite naming convention, probably for historical reasons (for example in French, the inverse function is preferably called application réciproque). In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...


A ring or an algebra in which every nonzero element has a multiplicative inverse is called a division ring resp. division algebra. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In ring theory, an algebra over a base ring is a generalization of the concept of associative algebra. ... In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i. ... In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division is possible. ...


Practical applications

The multiplicative inverse has innumerable applications in algorithms of computer science, particularly those related to number theory, since many such algorithms rely heavily on the theory of modular arithmetic. As a simple example, consider the exact division problem where you have a list of odd word-sized numbers each divisible by k and you wish to divide them all by k. One solution is as follows:

  1. Use the extended Euclidean algorithm to compute k-1, the multiplicative inverse of k mod 2w, where w is the number of bits in a word. This inverse will exist since the numbers are odd and the modulus has no odd factors.
  2. For each number in the list, multiply it by k-1 and take the least significant word of the result.

On many machines, particularly those without hardware support for division, division is a slower operation than multiplication, so this approach can yield a considerable speedup. The first step is relatively slow but only needs to be done once.


Pseudo-random number generation

The decimal expansion of the reciprocal 1/q can also act as a source of pseudo-random numbers, if q is a “suitable” prime number. A stream of random numbers oaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa A pseudorandom process is a process that appears random but is not. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ...

Superscript textf length q - 1 will be produced by the decimal expansion if q is such that q = 2S + 1 where S is a Sophie Germain prime, such that both S and 2S + 1 are prime, with S being of the form 3, 9 or 11 mod 20. Thus “suitable” prime numbers q are 7, 23, 47, 59, 167, 179, etc (corresponding to S = 3, 11, 23, 29, 83, 89, etc.). The result is a stream of length q-1 digits (including leading zeros). So, for example, using q = 23 generates the random digits 0,4,3,4,7,8,2,6.....3,9,1,3 A prime number p is called a Sophie Germain prime if 2p + 1 is also prime. ...

Further remarks

An element which has a multiplicative inverse cannot be a zero divisor if the multiplication is associative. To see this, it is sufficient to multiply the equation x y = 0 by the inverse of x (on the left), and then simplify using associativity. The sedenions provide a counter example. In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. ... The sedenions form a 16-dimensional algebra over the reals obtained by applying the Cayley-Dickson construction to the octonions. ...


Conversely, an element which is not a zero divisors needs not to have a multiplicative inverse. The nonzero integers provide an example. (They are not zero divisors but have no inverse in Z.) If the ring or algebra is finite, however, then all elements a which are not zero divisors do have a (left and right) inverse. This can be seen by observing that the map x→ax must be injective (ax=ay => a(x-y)=0 => x-y=0), thus surjective, thus there is x such that ax=1. In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...


Reference

Maximally Periodic Reciprocals, Matthews R.A.J. Bulletin of the Institute of Mathematics and its Applications vol 28 pp 147-148 1992


See also


  Results from FactBites:
 
Multiplicative inverse - Wikipedia, the free encyclopedia (986 words)
In mathematics, the reciprocal, or multiplicative inverse, of a number x is the number which, when multiplied by x, yields 1.
In modular arithmetic, the multiplicative inverse of x is also defined: it is the number a such that (a·x) mod n = 1.
It is important to distinguish the reciprocal of a function f in the multiplicative sense, given by 1/f, from the reciprocal or inverse function w.r.t.
Inverse element - Wikipedia, the free encyclopedia (428 words)
In mathematics, the idea of inverse element generalises the concepts of negation, in relation to addition, and reciprocal, in relation to multiplication.
A square matrix M with entries in a field K is invertible (in the set of all square matrices of the same size, under matrix multiplication) if and only if its determinant is different from zero.
If the determinant of M is zero, it is impossible for it to have a one-sided inverse; therefore a left inverse or right inverse implies the existence of the other one.
  More results at FactBites »

 
 

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