constant

In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. This is in contrast to a variable, which is not fixed.

Unspecified constants

The most widely mentioned sort of constant is a fixed, but possibly unspecified number. For example, consider "c" in the Pythagorean Theorem: a² + b² = c². Here, "c" is the hypotenuse of a right triangle. Though the exact value of "c" is unspecified, it always represents the hypotenuse in the Pythagorean Theorem (while a and b are interchangeable [as far as the theorem is concerned] and represent the other two sides of the triangle).

Usually the term constant is used in connection with mathematical functions of one or more variable parameters. These parameters, or other variables, are often called x, y, or z, using lowercase letters from the end of the Latin alphabet. Constants are, by convention, usually denoted by lowercase letters from the beginning of the Latin alphabet, such as a, b, and c.

Specified constants

Of course, some constants have special symbols, because they are specified, such as 1 or π. A special case of this may be found in physics, chemistry, and related fields, where certain features of the natural world that are described by numbers are found to have the same value at all times and places.

For example, in Albert Einstein's special theory of relativity, we have the formula

E=mc².

Here, the letter c stands for the speed of light in a vacuum, a constant physical quantity which is the same in all physical situations (to the best of current knowledge). In contrast, the letter m stands for the mass of an object, which could be anything, so it is a variable. E stands for the object's rest energy, another variable, and the formula defines a function that gives rest energy in terms of mass.

(See mathematical constant and physical constant.)

In computer science, a specified constant is sometimes called an immediate. Immediates are simply a number, rather than a symbol. For example, in the phrase x=45, "45" is an immediate, while "x" is a variable that is assigned the constant value 45.

Constant term

A constant term is a number that appears as an addend in a formula, such as

f(x) = sinx + c.

Here the constant c is the constant term of the function f. The value of c has not been specified in this formula, but it must be a specific value for f to be a specific function.

The constant term may depend on how the formula is written. For example

f(x) = x³ + (sinx)² + 4

and

g(x) = x³ − (cosx)² + 5

are formulae for the same function.

In a polynomial (or a generalisation of a polynomial, such as a Taylor series or Fourier expansion), the constant term is associated to the exponent zero. Note that the constant term may be zero, however. In a sense, any formula has a constant term, if you allow the constant term to be zero.

For some purposes, the constant is taken to be the value of f(0), but this depends on the function being defined at 0; it would not work for f(x)=1-1/x.

Constants vs variables

A number that is constant in one place may be a variable in another. Consider the example above, with a function f defined by

f(x) = sin x + c.

Now consider a functional F, a function whose argument is itself another function, defined by

F(g) = g(π/2).

Then for the function f given above, we have

F(f) = c + 1.

In the formula for f(x), we are fixing c and varying x, so c is a constant. But in the formula for F(f), we are varying both c and f, so c is a variable. Even this statement might be false in the presence of some larger context that gives yet another point of view.

Thus, there is no precise definition of "constant" in mathematics; only phrases such as "constant function" or "constant term of a polynomial" can be defined.

There is a mathematicians' joke to the effect that "variables don't; constants aren't." That is, the term variable is frequently used to mean a value that is fixed in a given equation, albeit unknown; while the term constant is used to mean an arbitrary quantity which may assume any value, as in the constant of integration.