Undefined
In mathematics, an expression is undefined when there is no valid value it can take, most commonly seen in division by zero.
Definition
An expression is undefined when it has no answer in mathematics, most commonly seen when dividing by zero. More precisely, an expression is undefined when it has no meaningful value within the given number system: $a/0$ is undefined for all $a$ because no number $q$ satisfies $0 \cdot q = a$ (if $a \neq 0$), or the value is non-unique (if $a = 0$, in which case $0/0$ is called indeterminate rather than simply undefined, since it can approach any value in a limit). Formally, undefined expressions signal that a partial function is evaluated outside its domain: division is defined as a function $\mathbb{R} \times (\mathbb{R}\setminus\{0\}) \to \mathbb{R}$, and the input $(a,0)$ falls outside that domain.
Example
$5/0$ is undefined: you cannot split $5$ things into $0$ groups, and no number times $0$ equals $5$. $0/0$ is indeterminate, since it can approach any value in a limit, $\lim_{x \to 0} x/x = 1$, $\lim_{x \to 0} 2x/x = 2$, $\lim_{x \to 0} x^2/x = 0$, and calculus resolves such forms using L'Hopital's rule or algebraic simplification. In IEEE 754 floating-point arithmetic: $1.0/0.0 = +\text{Inf}$, $-1.0/0.0 = -\text{Inf}$, and $0.0/0.0 = \text{NaN}$, conventions chosen for computational continuity rather than mathematical truth.
Key Insight
"Undefined" is not the same as "zero" or "infinity": it means the question itself has no valid answer in the number system. In extended real arithmetic (or the Riemann sphere $\mathbb{C}\cup\{\infty\}$), some "undefined" operations are given values by convention, but consistency requires sacrificing some algebraic laws. The concept is foundational in computability theory too: a Turing machine may fail to halt on some inputs, producing an undefined output, and partial functions (defined on only a subset of their natural domain) are the model for all computable processes.