Note: The following is not a definition of ''actual'' FreeVariable''''''s as seen in mathematics, but that of top level lexically scoped variables.


A ''free variable'' is a variable used within a function, which is neither a formal parameter to the function nor defined in the function's body (and in scope at the point of the variable's use).

For example, in this C function, ''printf'' and ''phase_shift'' are free variables.  The first is a standard library function; the second had better be defined by the programmer; else this function is in error.  ''angle'', being a formal parameter, is not a free variable; nor is ''x''.  
The first use of ''sin'' is a FreeVariable; it too is a standard library function (one which returns the sine of the angle).   The second use of ''sin'' (in the if statement) is not, as a local definition for ''sin'' is provided (which shadows the library function).  

We could argue that "double", "if", and "int" are FreeVariable''''''s as well; but those are keywords so we'll ignore them.

 void silly (double angle)
 {
     double x = sin (angle + phase_shift);
     printf ("The sine of the angle, phase shifted, is %f\n", x);
     if (angle > 0)
     {
          int sin = 7;
          printf ("There are %d deadly sins, %d if you count "
                    "pointer arithmetic\n", sin, sin+1);
     }
 }


Programming languages handle free variables via one of the ScopingRules.