Constant (mathematics)

Last updated February 15, 2023

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:

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

where $a$ , $b$ and $c$ are constants (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

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 $x 0$ . 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 $f(x)=5$ , 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:

{\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 $i 2 = -1$ .^[8]
${\sqrt {2}}$ (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, $1+{\sqrt {5}} \over 2$ .^[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 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 $f(x)=72$ for every $x$ then

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

Related Research Articles

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.

The number $e$ , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithms. It is the limit of $(1 + 1/ n) n$ as $n$ approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series

<span class="mw-page-title-main">Exponential function</span> Mathematical function, denoted exp(x) or e^x

The exponential function is a mathematical function denoted by $or . Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation, but modern definitions allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to opine that the exponential function is "the most important function in mathematics".$

<span class="mw-page-title-main">Mean value theorem</span> On the existence of a tangent to an arc parallel to the line through its endpoints

In mathematics, the mean value theorem states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It is one of the most important results in real analysis. This theorem is used to prove statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.

The natural logarithm of a number is its logarithm to the base of the mathematical constant $e$ , which is an irrational and transcendental number approximately equal to $2.718 281828459$ . The natural logarithm of $x$ is generally written as $ln x$ , $log e x$ , or sometimes, if the base $e$ is implicit, simply $log x$ . Parentheses are sometimes added for clarity, giving $ln(x)$ , $log e (x)$ , or $log(x)$ . This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.

In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve.

<span class="mw-page-title-main">Taylor's theorem</span> Approximation of a function by a truncated power series

In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a polynomial of degree k, called the kth-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order k of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial.

The Heaviside step function, or the unit step function, usually denoted by $H$ or $θ$ , is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive arguments. It is an example of the general class of step functions, all of which can be represented as linear combinations of translations of this one.

In calculus, the constant of integration, often denoted by $, is a constant term added to an antiderivative of a function to indicate that the indefinite integral of, on a connected domain, is only defined up to an additive constant. This constant expresses an ambiguity inherent in the construction of antiderivatives.$

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of definite integrals.

<span class="mw-page-title-main">Quadratic function</span> Polynomial function of degree two

In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomial and its associated polynomial function; so "quadratic polynomial" and "quadratic function" were almost synonymous. This is still the case in many elementary courses, where both terms are often abbreviated as "quadratic".

<span class="mw-page-title-main">Cubic function</span> Polynomial function of degree 3

In mathematics, a cubic function is a function of the form $where the coefficients a, b, c, and d are complex numbers, and the variable x takes real values, and . In other words, it is both a polynomial function of degree three, and a real function. In particular, the domain and the codomain are the set of the real numbers.$

In mathematical analysis, the maximum and minimum of a function, known generically as extremum, are the largest and smallest value of the function, either within a given range, or on the entire domain. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.

<span class="mw-page-title-main">Constant function</span> Type of mathematical function

In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function $y (x) = 4$ is a constant function because the value of $y (x)$ is 4 regardless of the input value $x$ (see image).

In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form

In mathematics, particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, but does not stand for anything else except itself. It may be used as a placeholder in objects such as polynomials and formal power series. In particular:

In mathematics, a variable is a symbol that represents a mathematical object. A variable may represent a number, a vector, a matrix, a function, the argument of a function, a set, or an element of a set.

In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term order has been used as a synonym of degree but, nowadays, may refer to several other concepts.

A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol, or by mathematicians' names to facilitate using it across multiple mathematical problems. Constants arise in many areas of mathematics, with constants such as $e$ and $π$ occurring in such diverse contexts as geometry, number theory, statistics, and calculus.

Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.

References

↑ "Definition of CONSTANT". www.merriam-webster.com. Retrieved 2021-11-09.
↑ Weisstein, Eric W. "Constant". mathworld.wolfram.com. Retrieved 2020-08-08.
↑ 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.
↑ Tanton, James (2005). Encyclopedia of mathematics. New York: Facts on File. ISBN 0-8160-5124-0. OCLC 56057904.
↑ "Algebra". tutorial.math.lamar.edu. Retrieved 2021-11-09.
↑ Arndt, Jörg; Haenel, Christoph (2001). Pi – Unleashed . Springer. p. 240. ISBN 978-3540665724.
↑ Weisstein, Eric W. "e". mathworld.wolfram.com. Retrieved 2021-11-09.
↑ Weisstein, Eric W. "i". mathworld.wolfram.com. Retrieved 2021-11-09.
↑ Weisstein, Eric W. "Pythagoras's Constant". mathworld.wolfram.com. Retrieved 2021-11-09.
↑ Weisstein, Eric W. "Golden Ratio". mathworld.wolfram.com. Retrieved 2021-11-09.

External links

Media related to Constants at Wikimedia Commons

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[1] "Definition of CONSTANT". www.merriam-webster.com. Retrieved 2021-11-09.

[2] Weisstein, Eric W. "Constant". mathworld.wolfram.com. Retrieved 2020-08-08.

[3] 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.

[4] Tanton, James (2005). Encyclopedia of mathematics. New York: Facts on File. ISBN 0-8160-5124-0. OCLC 56057904.

[5] "Algebra". tutorial.math.lamar.edu. Retrieved 2021-11-09.

[6] Arndt, Jörg; Haenel, Christoph (2001). Pi – Unleashed . Springer. p. 240. ISBN 978-3540665724.

[7] Weisstein, Eric W. "e". mathworld.wolfram.com. Retrieved 2021-11-09.

[8] Weisstein, Eric W. "i". mathworld.wolfram.com. Retrieved 2021-11-09.

[9] Weisstein, Eric W. "Pythagoras's Constant". mathworld.wolfram.com. Retrieved 2021-11-09.

[10] Weisstein, Eric W. "Golden Ratio". mathworld.wolfram.com. Retrieved 2021-11-09.

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