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To approximate the reciprocal of x, using only multiplication and subtraction, one can guess a number y, and then repeatedly replace y with 2y-xy2. Once the change in y becomes (and stays) sufficiently small, y is an approximation of the reciprocal of x.
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 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 (3 * x) mod 11 = 1 The extended Euclidean algorithm may be used to compute the multiplicative inverse modulo a number.
The multiplicative group of every finite fieldIn abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theo is cyclic. For all a in GF(n), a = an. For all nonzero a, it follows that a-1 = an-1. The multiplicative inverse of a can thus be found by raising a to a positive exponent, which can be done quickly through exponentiating by squaring.
The trigonometric functions are related by the reciprocal identity. The cotangentTrigonometry 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 a right triangle with one angle equal to x, cot x is the ratio of the length of th is the reciprocal of the tangentIn mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. Geometry In plane geometry, a straight line is tangent to a curve, at some point, if both line and curve pass through the poi. The secantA secant line of a curve is that line which intersects two (or more) points upon the curve. The word secant comes from the Latin secare for to cut''. It can be used to approximate the tangent to a curve, at some point P''. If the secant to a curve is defi is the reciprocal of the cosine. And the cosecant is the reciprocal of the sine.
See also: Additive inverse, Division, Fraction, group (mathematics), ring (mathematics)
In navigation a reciprocal bearing is the bearing that will take you in the reverse direction to that of the original bearing.
In the humanities and social sciences, an interaction between actors is said to be reciprocal when each action or favour given by one party is matched by another in return. See also the principle of reciprocity in international negotiations .
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