Constant (mathematics)

In mathematics, the word constant can have multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other value); as a noun, it has two different meanings:

A fixed and well-defined number or other non-varying mathematical object.[1] The terms mathematical constant or physical constant are sometimes used to distinguish this meaning.
A function whose value remains unchanged (i.e., a constant function).[2] Such a constant is commonly represented by a variable which does not depend on the main variable(s) in question. This is the case, for example, for a constant of integration, which is an arbitrary constant function (i.e., one that does not depend on the variable of integration) added to a particular antiderivative to get all the antiderivatives of the given function.

For example, a general quadratic function is commonly written as:

\( a x^2 + b x + c\, , \)

where a, b and c are constants (or parameters), and x a variable—a placeholder for the argument of the function being studied. A more explicit way to denote this function is

\( x\mapsto a x^2 + b x + c \, , \)

which makes the function-argument status of x (and by extension the constancy of a, b and c) clear. In this example a, b and c are coefficients of the polynomial. Since c occurs in a term that does not involve x, it is called the constant term of the polynomial and can be thought of as the coefficient of x0. More generally, any polynomial term or expression of degree zero is a constant.[3]:18

Constant function
Main articles: Constant function and Nullary

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 f(x)=5, has a graph of a horizontal straight line parallel to the x-axis. Such a function always takes the same value (in this case, 5), 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:

\( {\displaystyle {\begin{aligned}{\frac {d}{dx}}2^{x}&=\lim _{h\to 0}{\frac {2^{x+h}-2^{x}}{h}}=\lim _{h\to 0}2^{x}{\frac {2^{h}-1}{h}}\\[8pt]&=2^{x}\lim _{h\to 0}{\frac {2^{h}-1}{h}}&&{\text{since }}x{\text{ is constant (i.e. does not depend on }}h{\text{)}}\\[8pt]&=2^{x}\cdot \mathbf {constant,} &&{\text{where }}\mathbf {constant} {\text{ means not depending on }}x.\end{aligned}}} \)

"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. A constant in a narrower context could be regarded as a variable in a broader context.[1]
Notable mathematical constants
Main article: Mathematical constant

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:

0 (zero).
1 (one), the natural number after zero.
π (pi), the constant representing the ratio of a circle's circumference to its diameter, approximately equal to 3.141592653589793238462643.[4]
e, approximately equal to 2.718281828459045235360287.
i, the imaginary unit such that i2 = −1.
\( {\sqrt {2}} \) (square root of 2), the length of the diagonal of a square with unit sides, approximately equal to 1.414213562373095048801688.
φ (golden ratio), approximately equal to 1.618033988749894848204586, or algebraically, \( {1+{\sqrt {5}} \over 2} \).[1]

Constants in calculus

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 hence 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 due to the fact that the integral operator is the inverse of the differential operator, meaning that 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 f is the constant function such that \( {\displaystyle f(x)=72} \) for every x then

\( {\displaystyle {\begin{aligned}f'(x)&=0\\\int f(x)\,dx&=72x+c\end{aligned}}} \)