|  | Safe Numerics | 
A type is Numeric if it has the properties of a number.
More specifically, a type T is Numeric if there exists a
    specialization of std::numeric_limits<T>. See the
    documentation for the standard library class numeric_limits.
    The standard library includes such specializations for all the built-in
    numeric types. Note that this concept is distinct from the C++ standard
    library type traits is_integral and
    is_arithmetic. These latter fulfill the requirement of the
    concept Numeric. But there are types T which fulfill this concept for
    which is_arithmetic<T>::value == false. For example see
    safe_signed_integer<int>.
| std::numeric_limits<T> | The numeric_limits class template provides a C++ program with information about various properties of the implementation's representation of the arithmetic types. See C++ standard 18.3.2.2. | 
In addition to the expressions defined in Assignable the following expressions must be valid. Any operations which result in integers which cannot be represented as some Numeric type will throw an exception.
Table 1. General
| Expression | Return Type | Return Value | 
|---|---|---|
| std::numeric_limits<T>::is_bounded
               | bool | trueorfalse | 
| std::numeric_limits<T>::is_integer | bool | trueorfalse | 
| std::numeric_limits<T>::is_signed | bool | trueorfalse | 
| std::numeric_limits<T>::is_specialized
               | bool | true | 
Table 2. Unary Operators
| Expression | Return Type | Semantics | 
|---|---|---|
| -t | T | Invert sign | 
| +t | T | unary plus - a no op | 
| t-- | T | post decrement | 
| t++ | T | post increment | 
| --t | T | pre decrement | 
| ++t | T | pre increment | 
Table 3. Binary Operators
| Expression | Return Type | Semantics | 
|---|---|---|
| t - u | V | subtract u from t | 
| t + u | V | add u to t | 
| t * u | V | multiply t by u | 
| t / u | T | divide t by u | 
| t % u | T | t modulus u | 
| t < u | bool | trueif t less than u,falseotherwise | 
| t <= u | bool | trueif t less than or equal to u,falseotherwise | 
| t > u | bool | trueif t greater than u,falseotherwise | 
| t >= u | bool | trueif t greater than or equal to u,falseotherwise | 
| t == u | bool | trueif t equal to u,falseotherwise | 
| t != u | bool | trueif t not equal to u,falseotherwise | 
| t = u |  | assign value of u to t | 
| t += u |  | add u to t and assign to t | 
| t -= u |  | subtract u from t and assign to t | 
| t *= u |  | multiply t by u and assign to t | 
| t /= u |  | divide t by u and assign to t | 
We define the word "Numeric" in terms of the operations which are
    supported by "Numeric" types. This is in line with the current and
    historical usage of the word "concept" in the context of C++. It is also
    common to define compile time predicates such as
    "is_numeric<T>" to permit one to include expressions in
    his code which will generated a compile time error if the specified type
    (T) does not support the operations required. But this is not always true.
    In the C++ standard library there is a predicate
    is_arithmetic<T> whose name might suggest that it
    should return true for any type which supports the operations
    above. But this is not the case. The standard defines
    is_arithmetic<T> as true for any of the
    builtin types int, long, float,
    double, etc and false for any other types. So
    even if a user defined type U were to support the operations above,
    is_arithmetic<U> would still return false.
    This is quite unintuitive and not a good match for our purposes. Hence we
    define our own term "Numeric" to designate any type T which:
Supports the operations above
Specializes the standard type numeric_limits
while following the C++ standard in using
    is_arithmetic<T>, is_integral<T> to
    detect specific types only. The standard types are useful in various
    aspects of the implementation - which of course is done in terms of the
    standard types.
This in turn raises another question: Is it "legal" to specialize
    std::numeric_limits for one's own types such as
    safe<int>. In my view the standard is ambiguous on
    this. See various interpretations: 
In any case, it seems pretty clear that no harm will come of it. In spite of the consideration given to this issue, it turns out that the found no real need to implement these predicates. For example, there is no "is_numeric<T>" implemented as part of the safe numerics library. This may change in the future though. Even if not used, defining and maintaining these type requirements in this document has been very valuable in keeping the concepts and code more unified and understandable.
Remember that above considerations apply to other numeric types used in this library even though we don't explicitly repeat this information for every case.