Constant (mathematics)

In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition to variable (i.e. variable quantity), which is a symbol that stands for a value that may vary.

In polynomials, polynomial function of degree 2 is generally denoted as

$a x^2 + b x + c,\$

where a, b and c are constants (or parameters), while x is the 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 clear, and thereby implicitly the constant status of a, b and c. In this example those constants are actually coefficients of the polynomial, and 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.

Context-dependence

The context-dependent nature of the concept of "constant" can be seen in this example from elementary calculus:

$\frac{d}{dx} 2^x = \lim_{h\to 0} \frac{2^{x+h} - 2^x}{h} = \lim_{h\to 0} 2^x\frac{2^h - 1}{h}$

${\color{white}\frac{d}{dx} 2^x} = 2^x \lim_{h\to 0} \frac{2^h - 1}{h}$ since x is constant (i.e. does not depend on h)

${\color{white}\frac{d}{dx} 2^x} = 2^x \cdot\text{constant,}$ where constant means "not depending on x".

"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.

See also

Variable Expression Mathematical constant Physical constant

Constant Coefficient Constant of integration Constant term of a polynomial