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:

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

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

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 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  , 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.

Graph of  .

Context-dependence edit

The context-dependent nature of the concept of "constant" can be seen in this example from elementary calculus:


"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 i2 = −1.[8]
  •   (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,  .[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   for every x then


See also edit

References edit

  1. ^ "Definition of CONSTANT". Retrieved 2021-11-09.
  2. ^ Weisstein, Eric W. "Constant". 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". Retrieved 2021-11-09.
  6. ^ Arndt, Jörg; Haenel, Christoph (2001). Pi – Unleashed. Springer. p. 240. ISBN 978-3540665724.
  7. ^ Weisstein, Eric W. "e". Retrieved 2021-11-09.
  8. ^ Weisstein, Eric W. "i". Retrieved 2021-11-09.
  9. ^ Weisstein, Eric W. "Pythagoras's Constant". Retrieved 2021-11-09.
  10. ^ Weisstein, Eric W. "Golden Ratio". Retrieved 2021-11-09.

External links edit