Constant
A constant is a fixed value in a mathematical expression that does not change, such as the number 7 in 3x + 7.
Definition
A constant is a number in an expression or equation that always stays the same, no matter what: a term containing only a number and no variable, its value fixed throughout the problem. More broadly, a constant is any element of an expression whose value does not depend on any variable in scope, whether it is an arbitrary number in a specific problem or a mathematical constant like $\pi$ or $e$ that appears universally across many formulas. In calculus, a constant of integration (the $C$ in $\int f + C$) is different again: it represents an entire family of solutions rather than one fixed number, similar to how physical constants such as $c$ (the speed of light) are empirically fixed values.
Example
In $5x + 9$, the number $9$ is a constant, it will always be $9$ no matter the value of $x$. In the formula $C = 2\pi r$, $\pi$ is a mathematical constant with value approximately $3.14159$. Integrating $2x\,dx$ gives $x^2 + C$, where $C$ is an arbitrary constant representing the infinite family of antiderivatives; setting an initial condition pins $C$ to one specific value.
Key Insight
Think of a constant as a fact that is already known, while the variable is the mystery still to be uncovered. Mathematical constants like $\pi$ and $e$ differ from arbitrary constants chosen for a specific problem, and the constant of integration reflects the fundamental non-uniqueness of antidifferentiation, a concept central to the Fundamental Theorem of Calculus.