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. A function whose value remains unchanged (i.e., a constant function). 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: 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. Constant function and Nullary A constant may be used to define a constant function that ignores its arguments and always gives the same value. A constant function of a single variable, such as , has a graph of a horizontal line parallel to the x-axis. Such a function always takes the same value (in this case 5), because the variable does not appear in the expression defining the function. 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.

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In mathematics, value may refer to several, strongly related notions. In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as pi or an integer such as 42. The value of a variable or a constant is any number or other mathematical object assigned to it. The value of a mathematical expression is the result of the computation described by this expression when the variables and constants in it are assigned values.

A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an alphabet letter), 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 pi occurring in such diverse contexts as geometry, number theory, statistics, and calculus. Some constants arise naturally by a fundamental principle or intrinsic property, such as the ratio between the circumference and diameter of a circle (pi).

In mathematics, a constant term (sometimes referred to as a free term) is a term in an algebraic expression that does not contain any variables and therefore is constant. For example, in the quadratic polynomial the 3 is a constant term. After like terms are combined, an algebraic expression will have at most one constant term. Thus, it is common to speak of the quadratic polynomial where is the variable, as having a constant term of If the constant term is 0, then it will conventionally be omitted when the quadratic is written out.

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