In mathematics, the additive inverse of a number a (sometimes called the opposite of a) is the number that, when added to a, yields zero. The operation...
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omitted and then tacitly understood (in contrast to the additive inverse). Multiplicative inverses can be defined over many mathematical domains as well...
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added for specifying the operation, such as in additive inverse, multiplicative inverse, and functional inverse. In this case (associative operation), an invertible...
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Field (mathematics) (redirect from Additive group of a field)
numbers, including the existence of an additive inverse −a for all elements a, and of a multiplicative inverse b−1 for every nonzero element b. This allows...
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−1 (section Inverse and invertible elements)
(negative one or minus one) is the additive inverse of 1, that is, the number that when added to 1 gives the additive identity element, 0. It is the negative...
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Inequality (mathematics) (section Additive inverse)
inequality relation would be reversed. The rules for the additive inverse, and the multiplicative inverse for positive numbers, are both examples of applying...
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phrase "change of sign" is associated with exchanging an object for its additive inverse (multiplication with −1, negation), an operation which is not restricted...
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is therefore shown. 0 (number) Additive inverse Identity element Multiplicative identity Weisstein, Eric W. "Additive Identity". mathworld.wolfram.com...
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construction. The additive inverse of a number is unique, as is shown by the following proof. As mentioned above, an additive inverse of a number is defined...
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addition operation Additive set-function see Sigma additivity Additive category, a preadditive category with finite biproducts Additive inverse, an arithmetic...
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of a positive natural number (−1, −2, −3, . . .). The negations or additive inverses of the positive natural numbers are referred to as negative integers...
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natural numbers: the integers, by including an additive identity 0 (if not yet in) and an additive inverse −n for each nonzero natural number n; the rational...
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conditional sentence Additive inverse (negation), the inverse of a number that, when added to the original number, yields zero Compositional inverse, a function...
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= 2. Subtraction is the inverse of addition. The function whose value for any real or complex argument is the additive inverse of that argument. For example...
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(for all numbers x), the square of x is the same as the square of its additive inverse −x. That is, the square function satisfies the identity x2 = (−x)2...
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mathematics, −1 is the additive inverse of 1. −1, minus one, or negative one may also refer to: the exponent −1 denoting a multiplicative inverse the superscript...
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{\displaystyle a\times 1=a} for every real number a. Every real number a has an additive inverse denoted − a . {\displaystyle -a.} This means that a + ( − a ) = 0 {\displaystyle...
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(that is, 0 is the additive identity). For each a in R there exists −a in R such that a + (−a) = 0 (that is, −a is the additive inverse of a). R is a monoid...
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is given below. An additive category may then be defined as a semiadditive category in which every morphism has an additive inverse. This then gives the...
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additive semigroup of positive natural numbers. Absorbing element Additive inverse Generalized inverse Identity (equation) Identity function Inverse element...
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axioms that define a ring; they are the additive identity (0), the multiplicative identity (1), and the additive inverse of 1 (−1). The arithmetic operations...
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Semiring (section Additively idempotent semirings)
of a ring, dropping the requirement that each element must have an additive inverse. At the same time, it is a generalization of bounded distributive lattices...
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Replacing i by −i, j by −j, and k by −k sends a vector to its additive inverse, so the additive inverse of a vector is the same as its conjugate as a quaternion...
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idempotent semirings, such as the tropical semiring. The lack of an additive inverse in the semiring is compensated somewhat by the idempotent rule A ⊕...
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determines an addition operation, an additive inverse operation, and an additive identity; for this reason, an additive group can be described as a set that...
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its additive inverse −1 are always units. More generally, any root of unity in a ring R is a unit: if rn = 1, then rn−1 is a multiplicative inverse of...
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unary operations (i.e., operations of arity 1), such as additive inverse and multiplicative inverse. An operation of arity zero, or nullary operation, is...
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Note that steps 2 and 3 together are a valid method to compute the additive inverse − n {\displaystyle -n} of any (positive or negative) integer n {\displaystyle...
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gives the additive inverse: (−1)v = −v. Here, + is addition either in the field or in the vector space, as appropriate; and 0 is the additive identity...
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ones" number (the term "complement" refers to such pairs of mutually additive inverse numbers, here in respect to a non-0 base number). This mathematical...
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