Identity element

In mathematics, an identity element or neutral element is a special number or symbol that doesn’t change other numbers when you use a certain math operation with them.

For example, when you add zero to any number, the number stays the same. That’s why zero is called the identity element for addition.

This idea is important in many areas of math, especially in algebraic structures like groups and rings. It helps mathematicians understand how different operations work and how numbers relate to each other.

The term identity element is often just called identity when it’s clear which operation we’re talking about, like saying “additive identity” for zero in addition or “multiplicative identity” for one in multiplication. But remember, the identity element depends on the operation you’re using.

Definitions

Imagine you have numbers and a way to combine them, like adding or multiplying. An identity element is a special number that doesn’t change any other number when you combine them.

For example, when you add zero to any number, the number stays the same. Zero is called the additive identity. When you multiply any number by one, it stays the same. One is called the multiplicative identity.

These ideas help us understand how numbers and operations work together in math.

Examples

In math, an identity element is a special number that doesn’t change other numbers when you use it in a math operation. For example, when you add zero to any number, the number stays the same. This idea helps us understand how different math rules work together.

Set	Operation	Identity
Real numbers, complex numbers	+ (addition)	0
Real numbers, complex numbers, excluding 0	· (multiplication)	1
Positive integers	Least common multiple	1
Non-negative integers	Greatest common divisor	0 (under most definitions of GCD)
Vectors	Vector addition	Zero vector
	Scalar multiplication	1
m-by-n matrices	Matrix addition	Zero matrix
n-by-n square matrices	Matrix multiplication	In (identity matrix)
m-by-n matrices	○ (Hadamard product)	Jm, n (matrix of ones)
All functions from a set, M, to itself	∘ (function composition)	Identity function
All distributions on a group, G	∗ (convolution)	δ (Dirac delta)
Extended real numbers	Minimum/infimum	+∞
	Maximum/supremum	−∞
Subsets of a set M	∩ (intersection)	M
	∪ (union)	∅ (empty set)
Strings, lists	Concatenation	Empty string, empty list
A Boolean algebra	∧ {\textstyle \land } (conjunction)	⊤ {\textstyle \top } (truth)
	↔ {\textstyle \leftrightarrow } (equivalence)	⊤ {\textstyle \top } (truth)
	∨ {\textstyle \vee } (disjunction)	⊥ {\textstyle \bot } (falsity)
	↮ {\textstyle \nleftrightarrow } (nonequivalence)	⊥ {\textstyle \bot } (falsity)
Knots	Knot sum	Unknot
Compact surfaces	# (connected sum)	S2
Abstract groups	Direct product	Trivial group
Two elements, {e, f}	∗ defined by e ∗ e = f ∗ e = e and f ∗ f = e ∗ f = f	Both e and f are left identities, but there is no right identity and no two-sided identity
Homogeneous relations on a set X	Relative product	Identity relation
Relational algebra	Natural join (⨝)	The unique relation degree zero and cardinality one
A unital magma	Its operation	Its identity element

Properties

In some special math groups, there can be many left identity elements or many right identity elements. But if a group has both a left identity and a right identity, they must be the same, creating one two-sided identity.

For example, even numbers multiplied together never give an identity element. Also, when vectors are multiplied using the cross product, there is no identity element because the result is always at a right angle to the original vectors. Similarly, adding only positive natural numbers does not have an identity element.