Table of contents |

2 Constant term 3 Constant function 4 Constants vs variables |

The most widely mentioned sort of *constant* is a fixed, but possibly unspecified, number.
Usually the term *constant* is used in connection with mathematical functions of one or more variable arguments.
These arguments, or other variables, are often called *x*, *y*, or *z*, using lower-case letters from the end of the English alphabet.
Constants are usually denoted by lower-case letters from the beginning of the English alphabet, such as *a*, *b*, and *c*.
Of course, some constants have special symbols, because they *are* specified, such as 1 or &pi.

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*=*m**c*^{2}.

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

*f*(*x*) = sin*x*+*c*.

In a polynomial (or a generalisation of a polynomial, such as a Taylor expansion 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.

A *constant function* is a function that only consists of a single constant term:

*f*(*x*,*y*,*z*) =*c*.

Strictly speaking, the above comments are contradictory, in the case where the domain of the function is the empty set {}.
There is only one function with that domain (given any codomain), the empty function, and any formula can be used to define the empty function, since the formula won't apply to anything and will therefore never be wrong.
This includes a constant formula *c*; but the range of the empty function is not {*c*} but instead {}.
Most authors will not care, when defining the term "constant function" precisely, whether or not the empty function qualifies, and will use whatever definition is most convenient.
Sometimes, however, it is best not to consider the empty function to be constant, and a definition that makes reference to the range is preferable in those situations.
(This is much along the same lines of not considering an empty topological space to be connected, or not considering the trivial group to be simple.)

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*.

*F*(*g*) =*g*(π/2).

*F*(*f*) =*c*+ 1.

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.