An identity element leaves every compatible element unchanged under an operation. For integer addition, zero is the identity because a + 0 = 0 + a = a. section 2.5.1
Identity elements are one property used to characterize algebraic systems. Alongside closure, associativity, commutativity, and inverses, they help distinguish structures such as groups; matrix multiplication shares most of these properties but is not commutative. section 2.5.1
Recognize the identity element by checking that it works on both sides when the operation permits both orders:
- Addition: zero leaves every integer unchanged. section 2.5.1
- An inverse is defined relative to the identity: adding an element to its opposite produces zero. section 2.5.1
- Matrix transformations use multiplication and composition; their algebraic behavior can be compared with integer addition even though matrix multiplication is not commutative. section 4.3.2 section 2.5.1
A quick test
To test whether 0 is an identity for integer addition, substitute any integer a on either side: a + 0 = a and 0 + a = a. The element changes nothing, so the test succeeds. section 2.5.1
