In mathematics, the word constant conveys 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:
For example, a general quadratic function is commonly written as:
ax2+bx+c,
where, and are constants (coefficients or parameters), and a variable—a placeholder for the argument of the function being studied. A more explicit way to denote this function is
x\mapstoax2+bx+c,
which makes the function-argument status of (and by extension the constancy of, and) clear. 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 . More generally, any polynomial term or expression of degree zero (no variable) is a constant.[5]
See main article: Constant function.
A constant may be used to define a constant function that ignores its arguments and always gives the same value.[6] A constant function of a single variable, such as
f(x)=5
The context-dependent nature of the concept of "constant" can be seen in this example from elementary calculus:
| \begin{align} | d |
| dx |
2x&=\limh\to
| 2x+h-2x | |
| h |
=\limh\to
| ||||
| 2 |
\\[8pt] &=2x\limh\to
| 2h-1 | |
| h |
&&sincexisconstant(i.e.doesnotdependonh)\\[8pt] &=2x ⋅ constant,&&whereconstantmeansnotdependingonx. \end{align}
See 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:
1+\sqrt{5}\over2
In calculus, constants are treated in several different ways depending on the operation. For example, the derivative (rate of change) of a constant function is zero. This is because 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, a 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 is the inverse (opposite) of the derivative meaning that the aim of integration is to recover the original function before differentiation. The derivative 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.
If is the constant function such that
f(x)=72
\begin{align} f'(x)&=0\\ \intf(x)dx&=72x+c\\ \limx\rarr0f(x)&=72 \end{align}