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  • ...>\vec{0}</math> such that <math>\vec{x}+\vec{0}=\vec{x}</math> ([[Additive identity]]) ...tor <math>\vec{y}</math> such that <math>\vec{x}+\vec{y}=\vec{0}</math> ([[Additive inverse]])
    11 KB (1,876 words) - 18:01, 29 August 2024
  • * For some <math>0\in R</math>, <math>0+a=a+0=a</math> (existance of additive identity); ...h>-a\in R</math> for which <math>a+ (-a) = (-a)+a = 0</math> (existance of additive inverses);
    6 KB (994 words) - 05:16, 8 April 2015
  • ...e is an additive [[identity]] (usually denoted <math>\mathbf 0</math>) and additive [[inverse with respect to an operation | inverses]]. ...lar multiplication by the multiplicative identity of <math>F</math> is the identity transformation, so <math>\forall {\mathbf x} \in V</math>, <math>1\cdot{\ma
    3 KB (561 words) - 23:47, 20 March 2009
  • Identity Element Identity Element
    2 KB (346 words) - 17:30, 14 June 2020
  • * Identity: <math>a+0=a</math> for any complex number <math>a</math>. * Inverse: The sum of a number and its [[additive inverse]], <math>a+(-a)</math>, is equal to [[Zero (constant)|zero]].
    2 KB (309 words) - 19:34, 4 July 2019
  • '''Zero''', or 0, is the name traditionally given to the additive [[identity]] in number systems such as [[abelian group]]s, [[ring]]s and [[field]]s (e
    3 KB (408 words) - 17:36, 11 December 2024
  • ...for all <math>a \in S</math>. (Existence of additive and multiplicative [[identity | identities]].) ...\cdot a = 1</math> and <math>a + (-a) = (-a) + a = 0</math>. (Existence of additive and multiplicative inverses.)
    1 KB (253 words) - 18:05, 9 September 2008
  • In the additive group <math>\mathbb{Z}/4\mathbb{Z}</math>, shown below, ...\{0, 2\}</math>, shown below. This last subgroup is [[isomorphic]] to the additive group <math>\mathbb{Z}/2\mathbb{Z}</math>.
    2 KB (311 words) - 19:37, 7 May 2008
  • ...' is a construction of a structure from a set of smaller structures with [[identity]] elements. The construction can be applied to [[group]]s, [[ring]]s, and ...h>(A_i)_{i\in I}</math> be a family of structures of the same species with identity elements. The direct sum of the family <math>(A_i)_{i\in I}</math>, denote
    2 KB (278 words) - 11:53, 6 June 2008
  • ...e [[closed]] under the ring [[operation]]s of multiplication, addition and additive inverse-taking. ...it contains the additive identity <math>(0, 0)</math>, the multiplicative identity <math>(1, 1)</math> and is closed under multiplication and addition.
    2 KB (353 words) - 15:37, 16 June 2008
  • * The additive negation is an involution because <math>--x=x</math>. * The identity function <math>I_x</math> is an involution because <math>I_x:X \rightarrow
    2 KB (332 words) - 07:05, 9 May 2024
  • ...ing has additive identity <math>0=(0,0,0,\ldots)</math> and multiplicative identity <math>1 = (1,0,0,\ldots)</math>. ...mials. (In fact, it follows from the theory of [[coset]]s, applied to the additive groups involved, that ''every'' function that is associated with a polynomi
    12 KB (2,010 words) - 23:10, 2 August 2020
  • additive and multiplicative structure of the [[trivial group]]. Technically ...math> and <math>0_R</math> since AoPS defines rings to have multiplicative identity (some sources vary here). It can instead be shown that the [[integers]], <
    1 KB (221 words) - 17:44, 31 January 2022
  • *'''Identity Property of Addition''' *'''Additive Inverse Property'''
    2 KB (273 words) - 20:41, 5 July 2018
  • ...gative numbers were introduced: so that each positive number would have an additive inverse. ... The fact that the product of two negatives is positive is ther == Pascal's Identity ==
    35 KB (5,884 words) - 09:25, 7 December 2024
  • We will apply the following logarithmic identity: ...rentheses, we can find the simplified expanded forms of each sum using the additive property of logarithms:
    6 KB (805 words) - 21:48, 23 November 2023
  • ...e [[group]], or <math>0</math> if <math>1</math> has infinite order in the additive group.
    290 bytes (40 words) - 16:52, 14 March 2022