Constant (mathematics)

For a narrower treatment related to this subject, see Mathematical constant.

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, or the symbol denoting it. ^{[1] [2]} The terms mathematical constant or physical constant are sometimes used to distinguish this meaning. ^[3]
• A function whose value remains unchanged (i.e., a constant function ). ^[4] 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. ^{[5] : 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. ^[6] 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. ^[7] Such a function always takes the same value (in this case 5), because the variable does not appear in the expression defining the function.

Graph of ${\displaystyle f(x)=5}$.

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

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. ^[8]
• e , approximately equal to 2.718281828459045235360287. ^[9]
• i, the imaginary unit such that i² = −1. ^[10]
• ${\displaystyle {\sqrt {2}}}$ (square root of 2), the length of the diagonal of a square with unit sides, approximately equal to 1.414213562373095048801688. ^[11]
• φ (golden ratio), approximately equal to 1.618033988749894848204586, or algebraically, ${\displaystyle 1+{\sqrt {5}} \over 2}$. ^[12]

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 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.

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

1. Sobolev, S.K. (originator). "Individual constant". Encyclopedia of Mathematics . Springer. ISBN   1402006098 . Retrieved 2024-09-05.
2. Sobolev, S.K. (originator). "Constant". Encyclopedia of Mathematics . Springer. ISBN   1402006098 . Retrieved 2024-09-05.
3. "Definition of CONSTANT". www.merriam-webster.com. Retrieved 2021-11-09.
4. Weisstein, Eric W. "Constant". mathworld.wolfram.com. Retrieved 2020-08-08.
5. Foerster, Paul A. (2006). Algebra and Trigonometry: Functions and Applications, Teacher's Edition (Classics ed.). Upper Saddle River, NJ: Prentice Hall. ISBN   0-13-165711-9.
6. Tanton, James (2005). Encyclopedia of mathematics. New York: Facts on File. ISBN   0-8160-5124-0. OCLC   56057904.
7. "Algebra". tutorial.math.lamar.edu. Retrieved 2021-11-09.
8. Arndt, Jörg; Haenel, Christoph (2001). Pi – Unleashed . Springer. p.  240. ISBN   978-3540665724.
9. Weisstein, Eric W. "e". mathworld.wolfram.com. Retrieved 2021-11-09.
10. Weisstein, Eric W. "i". mathworld.wolfram.com. Retrieved 2021-11-09.
11. Weisstein, Eric W. "Pythagoras's Constant". mathworld.wolfram.com. Retrieved 2021-11-09.
12. Weisstein, Eric W. "Golden Ratio". mathworld.wolfram.com. Retrieved 2021-11-09.

External links

• Media related to Constants at Wikimedia Commons