Constant (mathematics)

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:

• A fixed and well-defined number or other non-changing mathematical object. The terms mathematical constant or physical constant are sometimes used to distinguish this meaning.^[1]
• 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.

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

${\displaystyle ax^{2}+bx+c\,,}$

where a, b and c are constants (coefficients 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

${\displaystyle x\mapsto ax^{2}+bx+c\,,}$

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 (no variable) is a constant.^[3]: 18

Constant function

A constant may be used to define a constant function that ignores its arguments and always gives the same value.^[4] A constant function of a single variable, such as ${\displaystyle f(x)=5}$, has a graph of a horizontal line parallel to the x-axis.^[5] Such a function always takes the same value (in this case 5), because the variable 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.

Notable mathematical constants

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.^[6]
• e, approximately equal to 2.718281828459045235360287.^[7]
• i, the imaginary unit such that i² = −1.^[8]
• ${\displaystyle {\sqrt {2}}}$ (square root of 2), the length of the diagonal of a square with unit sides, approximately equal to 1.414213562373095048801688.^[9]
• φ (golden ratio), approximately equal to 1.618033988749894848204586, or algebraically, ${\displaystyle 1+{\sqrt {5}} \over 2}$.^[10]

Constants in calculus

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

; 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

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

See also

• Constant (disambiguation)
• Expression
• Level set
• List of mathematical constants
• Physical constant

References

External links

• Media related to Constants at Wikimedia Commons