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Include the standard header <ratio> to define constants and templates that are used to store and manipulate rational numbers at compile time.

#include <ratio>

template <intmax_t N, intmax_t D = 1>
struct ratio
    static constexpr intmax_t num;
    static constexpr intmax_t den;
    typedef ratio<num, den> type;

The ratio Structure defines the static constants num and den such that num / den == N / D and num and den have no common factors. num / den is the value that is represented by the template class. Therefore, type designates the instantiation ratio<N0, D0> for which num == N0 and den == D0.

<ratio> also defines specializations of ratio that have the following form.

template <class R1, class R2> struct ratio_specialization

Each specialization takes two template parameters that must also be specializations of ratio. The value of type is determined by an associated logical operation.


type Value


R1 + R2


R1 / R2


R1 == R2


R1 > R2


R1 >= R2


R1 < R2


R1 <= R2


R1 * R2


!(R1 == R2)


R1 - R2

typedef ratio<1,        1000000000000000000> atto;
typedef ratio<1,           1000000000000000> femto;
typedef ratio<1,              1000000000000> pico;
typedef ratio<1,                 1000000000> nano;
typedef ratio<1,                    1000000> micro;
typedef ratio<1,                       1000> milli;
typedef ratio<1,                        100> centi;
typedef ratio<1,                         10> deci;
typedef ratio<                        10, 1> deca;
typedef ratio<                       100, 1> hecto;
typedef ratio<                      1000, 1> kilo;
typedef ratio<                   1000000, 1> mega;
typedef ratio<                1000000000, 1> giga;
typedef ratio<             1000000000000, 1> tera;
typedef ratio<          1000000000000000, 1> peta;
typedef ratio<       1000000000000000000, 1> exa;
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