Scheme numbers are either exact or inexact. A number is exact if it was written as an exact constant or was derived from exact numbers using only exact operations. A number is inexact if it was written as an inexact constant, if it was derived using inexact ingredients, or if it was derived using inexact operations. Thus inexactness is a contagious property of a number.
If two implementations produce exact results for a computation that did not involve inexact intermediate results, the two ultimate results will be mathematically equivalent. This is generally not true of computations involving inexact numbers since approximate methods such as floating point arithmetic may be used, but it is the duty of each implementation to make the result as close as practical to the mathematically ideal result.
Rational operations such as
+ should always produce exact results
when given exact arguments. If the operation is unable to produce an
exact result, then it may either report the violation of an
implementation restriction or it may silently coerce its result to an
inexact value. See Implementation restrictions.
With the exception of
exact, the operations described in
this section must generally return inexact results when given any
inexact arguments. An operation may, however, return an exact result if
it can prove that the value of the result is unaffected by the
inexactness of its arguments. For example, multiplication of any number
by an exact zero may produce an exact zero result, even if the other
argument is inexact.