Constant
Pre-AlgebraA 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 that always stays the same. It never changes no matter what.
Example
In $5x + 9$, the number $9$ is a constant because it has no variable attached to it. It will always be $9$.
Key Insight
Think of a constant as a fact that is already known, while the variable is the mystery still to be uncovered.
Definition
A constant is a term in an algebraic expression that contains only a number and no variable. Its value is fixed throughout the problem.
Example
In the expression $4x^2 - 7x + 12$, the constant term is $12$. In the formula $C = 2\pi r$, $\pi$ is a mathematical constant with the value approximately $3.14159$.
Key Insight
Mathematical constants like $\pi$ and $e$ appear universally across many formulas. They differ from arbitrary constants chosen in a specific problem.
Definition
A constant is any element of an expression whose value is fixed and does not depend on any variable in scope. In the context of differential equations, constants of integration (e.g., $C$ in $\int f + C$) represent an entire family of solutions. Physical constants such as $c$ (speed of light) are empirically fixed values.
Example
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 a specific value.
Key Insight
The presence of a constant of integration reflects the fundamental non-uniqueness of antidifferentiation, a concept central to the Fundamental Theorem of Calculus.