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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.
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Constant number
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 by convention usually denoted by lower-case letters from the beginning of the English alphabet , such as a, b, and c.
Mathematical constants and Physical 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 = mc^{2}.
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 that gives rest energy in terms of mass.
Constant term
A constant term is a number that appear 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) = x^{3} + (sinx)^{2} + 4
and
- g(x) = x^{3} - (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.
Constant function
A constant function is a function that only consists of a single constant term:
- f(x,y,z) = c.
It assigns the same value c to each possible combination of arguments. The range of a constant function must be the set {c}.
The empty function
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.)
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.