Additive Identity
The additive identity is zero: adding zero to any number leaves that number unchanged.
Formula
a + 0 = a
Definition
The additive identity is zero: adding zero to any number leaves the number unchanged. Formally, the additive identity of a number system is the element $e$ such that $a + e = e + a = a$ for all $a$; in the integers, rationals, reals, and complex numbers this element is $0$, and it is unique in any group, since if both $e$ and $e'$ satisfied the identity property, $e = e + e' = e'$. In a group $(G,+)$, this identity satisfies $a+0=0+a=a$ for all $a$; a ring additionally requires $0 \cdot a = 0$ for all $a$ (the absorptive property), and every group homomorphism must map one identity to the other.
Example
$7 + 0 = 7$ and $0 + 100 = 100$: no matter what number you start with, adding zero never changes it. In $\mathbb{Z}/5\mathbb{Z}$: $3 + 0 = 3$ and $0 + 3 = 3$, zero is still the additive identity, and in a vector space the zero vector plays the same role, $\vec{v} + \vec{0} = \vec{v}$. The additive identity generates the trivial subgroup $\{0\}$ in any group, and every homomorphism must send the identity of one group to the identity of the other.
Key Insight
Zero is the "do-nothing" number for addition: the result is identical to what you started with. The additive identity is unique in every abelian group, by the same short proof each time. It plays a central role in homomorphism theory: the additive identity is essentially the kernel of the identity map, anchoring the structure of every group and ring built on top of it.