In mathematics, the wordconstant can have multiple meanings. As an adjective, it refers to non-variance ; as a noun, it has two different meanings:
A fixed and well-defined number or other mathematical object. The terms mathematical constant or physical constant are sometimes used to distinguish this meaning.
A function whose value remains unchanged, referred to as a constant function. Such a constant is commonly represented by a variable which does not depend on the main variable of the studied problem. This is the case, for example, for a constant of integration, which is an arbitrary constant function added to a particular antiderivative to get all the antiderivatives of the given function.
For example, a general quadratic function is commonly written as: where, and are constants, while is the variable, a placeholder for the argument of the function being studied. A more explicit way to denote this function is which makes the function-argument status of clear, and thereby implicitly the constant status of, and. In this example, and are coefficients of the polynomial. Since occurs in a term that does not involve, it is called the constant term of the polynomial and can be thought of as the coefficient of ; any polynomial term or expression of degreezero is a constant.
Constant function
A constant may be used to define a constant function that ignores its arguments and always gives the same value. A constant function of a single variable, such as, has a graph that is a horizontal straight line, parallel to the x-axis. Such a function always takes the same value because its argument does not appear in the expression defining the function.
Context-dependence
The context-dependent nature of the concept of "constant" can be seen in this example from elementary calculus: "Constant" means not depending on some variable; not changing as that variable changes. In the first case above, it means not depending on h; in the second, it means not depending on x.
Some values occur frequently in mathematics and are conventionally denoted by a specific symbol. These standard symbols and their values are called mathematical constants. Examples include:
In calculus, constants are treated in several different ways depending on the operation. For example, the derivative of a constant function is zero. This is because the derivative measures the rate of change of a function with respect to a variable, and since constants, by definition, do not change, their derivative is therefore zero. Conversely, when integrating a constant function, the constant is multiplied by the variable of integration. During the evaluation of a limit, the constant remains the same as it was before and after evaluation. Integration of a function of one variable often involves a constant of integration. This arises because of the integral operator's nature as the inverse of the differential operator, meaning the aim of integration is to recover the original function before differentiation. The differential of a constant function is zero, as noted above, and the differential operator is a linear operator, so functions that only differ by a constant term have the same derivative. To acknowledge this, a constant of integration is added to an indefinite integral; this ensures that all possible solutions are included. The constant of integration is generally written as 'c' and represents a constant with a fixed but undefined value.
Examples
If is the constant function such that for every then
