Constant (mathematics)

In mathematics, a constant is a non-varying value, i.e. a value that is 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.

By convention, constants in formulas are denoted by letters from the beginning of the alphabet (e.g. a, b, c), while letters at the end of the alphabet (e.g. x, y, z) are reserved for variables. For example, a general quadratic function is conventionally written 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 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.

Constant function

Main articles: Constant function and Nullary

A сonstant is often considered as a function, which does not depend on its arguments and is nullary in fact. A constant is, therefore, usually plotted as a single-variable function, e.g. $f(x)=5$ whereas the graph is a straight line, parallel to the x-axis. It always takes on the same value (in this case, 5) because there is no variable in the function expression.

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.

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:

• π (pi), the constant representing the ratio of a circle's circumference to its diameter, approximately equal to 3.141592653589793238462643...
• e, approximately equal to 2.718281828459045235360287...
• 0 (zero).
• 1 (one), the natural number after zero.
• $Square root of 2$ (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, $1+ \sqrt{5} \over 2$.

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 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, so 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

$f(x)=72 \Rightarrow f'(x)=0$
$f(x)=72 \Rightarrow \int 72 \, dx = 72x+c$
$f(x)=72 \Rightarrow \lim_{x \to \infty} 72 = 72$

See also

• Expression
• Physical constant
• Constant (disambiguation)

• Constant of integration
• Constant term of a polynomial

References

• Arndt & Haenel 2006, p. 240°—°