Identity Element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures such as groups and rings. The term ''identity element'' is often shortened to ''identity'' (as in the case of additive identity and multiplicative identity) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with. Definitions Let be a set  equipped with a binary operation ∗. Then an element  of  is called a if for all  in , and a if for all  in . If is both a left identity and a right identity, then it is called a , or simply an . An identity with respect to addition is called an (often denoted as 0) and an identity with respect to multiplication is called a (often denoted as 1). These need not be ordinary add ...
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Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulate