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 (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

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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}}

Fram

From Wikipedia, the free encyclopedia

Fram may refer to:

Ships[edit]

Fram (ship), an arctic exploration vessel from Norway
MS Fram, expedition cruise ship owned by Hurtigruten Group

Places and geography[edit]

Fram, Paraguay, a town in Itapúa, Paraguay
Fram Formation, a sequence of rock strata on Ellesmere Island, Canada
Fram, Rače–Fram, a settlement in Slovenia
Fram (river), a stream in Slovenia
Framlingham, town in England referred to by locals as "Fram"
Framingham, Massachusetts, a city often nicknamed "Fram"
- Framingham State University, a university in that city also often nicknamed "Fram"
Fram Strait, in the Arctic Ocean
Fram Islands, in Antarctica
Fram Mesa, in Antarctica
Fram (crater), crater on Mars explored by the Opportunity rover

Arts and culture[edit]

Fram (play), by Tony Harrison
Fram (Middle-earth), fictional character from the stories of J. R. R. Tolkien
Fram, the Polar Bear, children's book by Romanian writer Cezar Petrescu
Fram Museum, a museum in Oslo, Norway

Sports[edit]

Knattspyrnufélagið Fram, Icelandic association football club
IF Fram Tórshavn, Faroese association football club
IF Fram, Finnish association football club from Åland
IF Fram Larvik, Norwegian sports club
IL Fram, Norwegian sports club
IL Hjelset-Fram, Norwegian sports club
SK Fram (Oslo), Norwegian skiing and social club
SK Fram, Norwegian sports club now a part of Selsbakk IF
SK Fram, Norwegian sports club now a part of Brumunddal IL
Framlingham Town F.C. (known as Fram), English association football club
Fram Stadion, a football stadium in Larvik, Norway

People[edit]

Fram (name), list of people with the name

As an acronym[edit]

Fleet Rehabilitation and Modernization, United States Navy and Coast Guard program to modernize older ships in their respective fleets
Fellow of the Royal Academy of Music in the University of London
Ferroelectric RAM, electronic device using the ferroelectric effect to produce low density random access memory
Flight Releasable Attachment Mechanism for ORUs on External stowage platforms of the International Space Station
Functional Resonance Analysis Method, Modelling Complex Socio-technical Systems, by Erik Hollnagel

Other uses[edit]

FRAM (brand), an American brand of automotive filters
Fram (bicycle company), a Swedish bicycle manufacturer

This disambiguation page lists articles associated with the title Fram.
If an internal link led you here, you may wish to change the link to point directly to the intended article.

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Saturday, June 8, 2024

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Constant (mathematics)

Constant function[edit]

Context-dependence[edit]

Notable mathematical constants[edit]

Constants in calculus[edit]

Examples[edit]

See also[edit]

Fram

Ships[edit]

Places and geography[edit]

Arts and culture[edit]

Sports[edit]

People[edit]

As an acronym[edit]

Other uses[edit]

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