Zero
Zero is the integer that represents the absence of quantity; it is the additive identity and separates positive from negative numbers.
Formula
a + 0 = a
Definition
Zero ($0$) is the number that means "none," neither positive nor negative and the starting point of the number line. It is the unique additive identity in any number system, $a + 0 = 0 + a = a$ for all $a$, is an even integer, and division by zero is undefined. In ring theory, $0$ is the unique element such that $0 + a = a$ for all $a$, and in any ring $0 \cdot a = 0$ (absorption); a field has no zero divisors, and division by $0$ is excluded from field axioms because no element $z$ satisfies $0 \cdot z = 1$.
Example
If you have $5$ cookies and eat all $5$, you have $0$ cookies, and adding $0$ to any number leaves it unchanged: $7 + 0 = 7$. Likewise $0 \times 17 = 0$ and $0 / 5 = 0$, while $5 / 0$ is undefined. In modular arithmetic, $0 \bmod n$ plays the role of the identity for addition mod $n$, and the zero ring $\{0\}$ is the unique ring in which the additive and multiplicative identities coincide ($0 = 1$).
Key Insight
Zero was one of humanity's greatest mathematical inventions: without it, place value, algebra, and computers as we know them would not exist. It is the only real number that is its own additive inverse ($0 + 0 = 0$) and the only number that is simultaneously non-negative and non-positive. The historical introduction of a symbol for zero by Indian mathematicians (circa 7th century) was transformative: the positional number system, and therefore all digital computation, depends entirely on zero as a placeholder.