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:See also Variable#Constant|constants in computer
programming.
----
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.
Usually the term constant is used in connection
with mathematical function (mathematics)|functions
of one or more variable parameter|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 by convention usually denoted by
lower-case letters from the beginning of the
English alphabet, such as a, b, and
c.

== Specified constants ==

Of course, some constants have special symbols,
because they are specified, such as num|1 or
pi|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, 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 (mathematics)|function that gives rest
energy in terms of mass.

:(See  mathematical constant and physical
constant.)

== Constant term ==

A constant term is a number that appear as an
addend in a formula, such as
:$f(x)\; =\; \backslash sin\; x\; +\; c.$
Here the constant c is the constant term of the
function (mathematics)|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
:$f(x)\; =\; x^3+(\backslash sin\; x)^2\; +\; 4$
and
:$g(x)\; =\; x^3-(\backslash cos\; x)^2\; +\; 5$
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 0 (number)|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
(mathematics)|function f defined by
:f(x) = sin x + c.
Now consider a functional (mathematics)|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.

== See also ==

* Astronomical constant
* Mathematical constant
* Physical constant