"Reciprocal (mathematics)" redirects here. Not to be confused with Reciprocation (geometry).
The reciprocal function: y = 1/x. For every x except 0, y represents its multiplicative inverse. The graph forms a rectangular hyperbola.
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).
Multiplying by a number is the same as dividing by its reciprocal and vice versa. For example, multiplication by 4/5 (or 0.8) will give the same result as division by 5/4 (or 1.25). Therefore, multiplication by a number followed by multiplication by its reciprocal yields the original number (since the product of the number and its reciprocal is 1).
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's Elements.[1]
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 for most functions not equal to the multiplicative inverse. For example, the multiplicative inverse 1/(sin x) = (sin x)−1 is the cosecant of x, and not the inverse sine of x denoted by sin−1x or arcsin x. The terminology difference reciprocal versus 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 the bijection réciproque).
^"In equall Parallelipipedons the bases are reciprokall to their altitudes". OED "Reciprocal" §3a. Sir Henry Billingsley translation of Elements XI, 34.
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mathematics, a multiplicativeinverse 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...
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by A, and is called the (multiplicative) inverse of A, denoted by A−1. Matrix inversion is the process of finding the inverse matrix of an invertible matrix...
specifying the operation, such as in additive inverse, multiplicativeinverse, and functional inverse. In this case (associative operation), an invertible...
the multiplication is associative, commutative, and that the class of 1 is the unique multiplicative identity. Finally, given a, the multiplicative inverse...
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000000001 (The elements {53} and {CA} are multiplicativeinverses of one another since their product is 1.) Multiplication in this particular finite field can...
modular multiplicativeinverse of a modulo m. If a ≡ b (mod m) and a−1 exists, then a−1 ≡ b−1 (mod m) (compatibility with multiplicativeinverse, and, if...
multiplicativeinverse. For example, a matrix such that all entries of a row (or a column) are 0 does not have an inverse. If it exists, the inverse of...
yet in) and an additive inverse −n for each nonzero natural number n; the rational numbers, by including a multiplicativeinverse 1 / n {\displaystyle 1/n}...
misunderstood, (f(x))−1 certainly denotes the multiplicativeinverse of f(x) and has nothing to do with the inverse function of f. The notation f ⟨ − 1 ⟩ {\displaystyle...
estimates 1 x {\textstyle {\frac {1}{\sqrt {x}}}} , the reciprocal (or multiplicativeinverse) of the square root of a 32-bit floating-point number x {\displaystyle...
of them Multiplicativeinverse (reciprocal), a number which when multiplied by a given number yields the multiplicative identity, 1 Inverse matrix of...
nonzero element a has a unique modular multiplicativeinverse, a−1 such that aa−1 = a−1a ≡ 1 mod m. This inverse can be found by solving the congruence...
nonzero rational number a b {\displaystyle {\tfrac {a}{b}}} has a multiplicativeinverse, also called its reciprocal, ( a b ) − 1 = b a . {\displaystyle...
respect to multiplication is called a multiplicative identity (often denoted as 1). These need not be ordinary addition and multiplication—as the underlying...
The inverse second or reciprocal second (s−1), also called per second, is a unit defined as the multiplicativeinverse of the second (a unit of time)....
contradicts the equation 2b2 = a2 which implies that a must be even. The multiplicativeinverse (reciprocal) of the square root of two (i.e., the square root of...
we have defined multiplication for formal power series, we can define multiplicativeinverses as follows. The multiplicativeinverse of a formal power...
Multiplicative may refer to: MultiplicationMultiplicative function Multiplicative group Multiplicative identity Multiplicativeinverse Multiplicative...
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