additive inverse and multiplicative inverse
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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 fractiona/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 self-inverse.
The term reciprocal was in common use at least as far back as the third edition of Encyclopaedia Britannica(1797) to describe two numbers whose product is 1; geometrical quantities in inverse proportion are described as reciprocall in a 1570 translation ofEuclid'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 â‰ ba; then "inverse" typically implies that an element is both a left and right inverse.
Examples and counterexamples
Zero does not have a finite reciprocal because no real number multiplied by 0 produces 1. With the exception of zero, reciprocals of every complex number are complex, reciprocals of every real number are real, and reciprocals of every rational number are rational. The imaginary units, Â± = Â± are the only numbers with additive inverse equal to multiplicative inverse. For example, additive and multiplicative inverses of are −() = − and 1/ = −, respectively.
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 xâ‰ 0. There must instead 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 become arbitrarily small.
In modular arithmetic, the modular multiplicative inverse of a is also defined: it is the number x such that ax ≡ 1 (mod n). This multiplicative inverse exists if and only ifa and n are coprime. For example, the inverse of 3 modulo 11 is 4 because 4 · 3 ≡ 1 (mod 11). The extended Euclidean algorithm may be used to compute it.
The sedenions are an algebra in which every nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, i.e. nonzero elements x, y such that xy = 0.
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).
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.
It is important to distinguish the reciprocal of a function ƒ in the multiplicative sense, given by 1/ƒ, from the reciprocal or inverse functionwith respect to composition, denoted by ƒ−1 and defined by ƒ o ƒ−1 = id. Only for linear maps are they strongly related (see above), while they are completely different for all other cases. 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 application rÃ©ciproque).
The multiplicative inverse has innumerable applications in algorithms
For example, the additive inverse of 7 is âˆ’7, because 7 + (âˆ’7) = 0, and the additive inverse of âˆ’0.3 is 0.3, because âˆ’0.3 + 0.3 = 0.
In other words, the additive inverse of a number is the number's negative. For example, the additive inverse of 8 is −8, the additive inverse of 10002 is −10002 and the additive inverse of xÂ² is −(xÂ²).
Integers, rational numbers, real numbers, and complex number all have additive inverses, as they contain negative as well as positive numbers. Natural numbers, cardinal numbers, and ordinal numbers, on the other hand, do not have additive inverses within their respective sets. Thus, for example, we can say that natural numbers do have additive inverses, but because these additive inverses are not themselves natural numbers, the set of natural numbers is not closed under taking additive inverses.
The notation '+' is reserved for commutative binary operations, i.e. such that x + y = y + x, for all x,y. If such an operation admits an identity elemento (such that x + o (= o + x) = x for all x), then this element is unique (o' = o' + o = o). If then, for a given x, there exists x' such that x + x' (= x' + x) = o, then x' is called an additive inverse of x.
If '+' is associative ((x+y)+z = x+(y+z) for all x,y,z), then an additive inverse is unique
- ( x" = x" + o = x" + (x + x') = (x" + x) + x' = o + x' = x' )
â€“ y instead of x + (â€“y).
For example, since addition of real numbers is associative, each real number has a unique additive inverse.
All the following examples are in fact abelian groups:
- addition of real valued functions: here, the additive inverse of a function f is the function â€“f defined by (â€“ f)(x) = â€“ f(x), for all x, such that f + (â€“f) = o, the zero function (o(x) = 0 for all x).
- more generally, what precedes applies to all functions with values in an abelian group ('zero' meaning then the identity element of this group):
- * complex valued functions,
- * vector space valued functions (not necessarily linear),
- sequences, matrices and nets are also special kinds of functions.
- In a vector space additive inversion corresponds to scalar multiplication by âˆ’1. For Euclidean space, it is inversion in the origin.
- In modular arithmetic, the modular additive inverse of x is also defined: it is the number a such that a+x â‰¡ 0 (mod n). This additive inverse does always exist. For example, the inverse of 3 modulo 11 is 8 because it is the solution to 3+x â‰¡ 0 (mod 11).
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Answers:1) 0.4 Additive inverse = -0.4 Multiplicative inverse = 1/0.4 = 2.5 2) -1.6 Additive inverse = 1.6 Multiplicative inverse = 1/(-1.6) = -0.625 3) -11/16 Additive inverse = 11/16 Multiplicative inverse = -16/11 4) 5 and 5/6 Since 5 and 5/6 = 35/6: Additive inverse = -35/6 Multiplicative inverse = 6/35 .
Answers:The additive inverse is that number which when the original number is added to it produces a sum of 0. 1) .4..............Additive inverse -.4 The multiplicative inverse of a number is that number which when multiplied by the original number produces 1. 1)...............(0.4 * (2.5)) = 1....................Multiplicative inverse is (2.5) 2) Additive inverse is +1.6 Multiplicative inverse is (1 / - 1.6) = -(5/8) .