Zero
ArithmeticZero 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." It is neither positive nor negative and is the starting point of the number line.
Example
If you have $5$ cookies and eat all $5$, you have $0$ cookies. Adding $0$ to any number leaves it unchanged: $7 + 0 = 7$.
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.
Definition
Zero is the unique additive identity in any number system: $a + 0 = 0 + a = a$ for all $a$. Zero is an even integer. Division by zero is undefined. Zero is not positive and not negative.
Example
$0 \times 17 = 0$ (zero times anything is zero). $0 / 5 = 0$ (zero divided by any non-zero number is zero). $5 / 0$ = undefined (cannot divide by zero).
Key Insight
Zero is the only real number that is its own additive inverse: $0 + 0 = 0$. It is also the only number that is simultaneously non-negative and non-positive.
Definition
In ring theory, $0$ is the additive identity element: the unique element such that $0 + a = a$ for all $a$. In any ring, $0 \cdot a = 0$ (absorption). A field has no zero divisors: if $a \cdot b = 0$ then $a = 0$ or $b = 0$. Division by $0$ is excluded from field axioms because no element $z$ satisfies $0 \cdot z = 1$.
Example
In modular arithmetic, $0 \bmod n$ plays the role of the identity for addition mod $n$. The zero ring $\{0\}$ is the unique ring in which the additive and multiplicative identities coincide ($0 = 1$).
Key Insight
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.