Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Identity"

m (reorganized and wikified a bit)
m
Line 17: Line 17:
  
 
This usage of the word identity is common in [[abstract algebra]].
 
This usage of the word identity is common in [[abstract algebra]].
 +
 +
See also the article [[Inverse with respect to an operation]].

Revision as of 08:15, 18 July 2006

This article is a stub. Help us out by expanding it.

There are at least two possible meanings in mathematics for the word identity.


Equations

An identity is "a general relationship which always holds, usually over some choice of variables." For example, $(x+1)^2=x^2+2x+1$ is an identity, since it holds regardless of choice of variable. We therefore sometimes write $(x+1)^2\equiv x^2+2x+1$.


Abstract Algebra

Given a binary operation G on a set S, $G: S \times S \to S$, an identity for G is an element $e\in S$ such that for all $a \in S$, $G(e, a) = G(a, e) = a$. For example, in the real numbers, if we take G to be the operation of multiplication $G(a, b) = a\cdot b$, the number 1 will be the identity for G. If instead we took G to be addition ($G(a, b) = a + b$), 0 would be the identity.

Identities in this sense are unique. Imagine we had two identities, $e$ and $e'$, for some operation $G$. Then $e = G(e, e') = e'$, so $e = e'$, and so e and e' are in fact equal.

This usage of the word identity is common in abstract algebra.

See also the article Inverse with respect to an operation.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Identity&oldid=7870"