# Rational Numbers

## Contents

1. Class rational synopsis
2. Rationale
3. Background
4. Integer Type Requirements
5. Interface
   - Utility functions
   - Constructors
   - Arithmetic operations
   - Input and Output
   - In-place assignment
   - Conversions
   - Numerator and Denominator
6. Performance
7. Exceptions
8. Internal representation
9. Design notes
   - Minimal Implementation
   - Limited-range integer types
   - Conversion from floating point
   - Absolute Value
10. References
11. History and Acknowledgements

## Class rational synopsis

    #include <boost/rational.hpp>

    namespace boost {

    class bad_rational;

    template<typename I> class rational {
        typedef implementation-defined bool_type;

    public:
        typedef I int_type;

        // Constructors
        rational();          // Zero;               constexpr since C++11
        rational(I n);       // Equal to n/1;       constexpr since C++11
        rational(I n, I d);  // General case (n/d); constexpr since C++14
        template<typename J>
        explicit rational(const rational<J> &r);  // Cross-instantiation; constexpr since C++11

        // Normal copy constructors and assignment operators

        // Assignment from I
        rational& operator=(I n); // constexpr since C++14

        // Assign in place
        rational& assign(I n, I d); // constexpr since C++14

        // Representation
        I numerator() const;   // constexpr since C++11
        I denominator() const; // constexpr since C++11

        // In addition to the following operators, all of the "obvious" derived
        // operators are available - see operators.hpp

        // Arithmetic operators
        rational& operator+= (const rational& r); // constexpr since C++14
        rational& operator-= (const rational& r); // constexpr since C++14
        rational& operator*= (const rational& r); // constexpr since C++14
        rational& operator/= (const rational& r); // constexpr since C++14

        // Arithmetic with integers
        rational& operator+= (I i); // constexpr since C++14
        rational& operator-= (I i); // constexpr since C++14
        rational& operator*= (I i); // constexpr since C++14
        rational& operator/= (I i); // constexpr since C++14

        // Increment and decrement
        const rational& operator++(); // constexpr since C++14
        const rational& operator--(); // constexpr since C++14

        // Operator not
        bool operator!() const; // constexpr since C++11

        // Boolean conversion
        operator bool_type() const; // constexpr since C++11

        // Comparison operators
        bool operator< (const rational& r) const;  // constexpr since C++14
        bool operator== (const rational& r) const; // constexpr since C++11

        // Comparison with integers
        bool operator< (I i) const;  // constexpr since C++14
        bool operator> (I i) const;  // constexpr since C++14
        bool operator== (I i) const; // constexpr since C++11
    };

    // Unary operators
    template <typename I> rational<I> operator+ (const rational<I>& r); // constexpr since C++11
    template <typename I> rational<I> operator- (const rational<I>& r); // constexpr since C++14

    // Reversed order operators for - and / between (types convertible to) I and rational
    template <typename I, typename II> inline rational<I> operator- (II i, const rational<I>& r); // constexpr since C++14
    template <typename I, typename II> inline rational<I> operator/ (II i, const rational<I>& r); // constexpr since C++14

    // Absolute value
    template <typename I> rational<I> abs (const rational<I>& r); // constexpr since C++14

    // Input and output
    template <typename I> std::istream& operator>> (std::istream& is, rational<I>& r);
    template <typename I> std::ostream& operator<< (std::ostream& os, const rational<I>& r);

    // Type conversion
    template <typename T, typename I> T rational_cast (const rational<I>& r); // constexpr since C++11

## Rationale

Numbers come in many different forms. The most basic forms are natural numbers (non-negative "whole" numbers), integers and real numbers. These types are approximated by the C++ built-in types unsigned int, int, and float (and their various equivalents in different sizes).

The C++ Standard Library extends the range of numeric types available by providing the complex type.

This library provides a further numeric type, the rational numbers.

The rational class is actually a implemented as a template, in a similar manner to the standard complex class.

## Background

The mathematical concept of a rational number is what is commonly thought of as a fraction - that is, a number which can be represented as the ratio of two integers. This concept is distinct from that of a real number, which can take on many more values (for example, the square root of 2, which cannot be represented as a fraction).

Computers cannot represent mathematical concepts exactly - there are always compromises to be made. Machine integers have a limited range of values (often 32 bits), and machine approximations to reals are limited in precision. The compromises have differing motivations - machine integers allow exact calculation, but with a limited range, whereas machine reals allow a much greater range, but at the expense of exactness.

The rational number class provides an alternative compromise. Calculations with rationals are exact, but there are limitations on the available range. To be precise, rational numbers are exact as long as the numerator and denominator (which are always held in normalized form, with no common factors) are within the range of the underlying integer type. When values go outside these bounds, overflow occurs and the results are undefined.

The rational number class is a template to allow the programmer to control the overflow behaviour somewhat. If an unlimited precision integer type is available, rational numbers based on it will never overflow (modulo resource limits) and will provide exact calculations in all circumstances.

## Integer Type Requirements

The rational type takes a single template type parameter I. This is the underlying integer type for the rational type. Any of the built-in integer types provided by the C++ implementation are supported as values for I. User-defined types may also be used, but users should be aware that the performance characteristics of the rational class are highly dependent upon the performance characteristics of the underlying integer type (often in complex ways - for specific notes, see the Performance section below). Note: Should the boost library support an unlimited-precision integer type in the future, this type will be fully supported as the underlying integer type for the rational class.

A user-defined integer type which is to be used as the underlying integer type for the rational type must be a model of the following concepts.

- Assignable
- Default Constructible
- Equality Comparable
- LessThan Comparable

Furthermore, I must be an integer-like type, that is the following expressions must be valid for any two values n and m of type I, with the "expected" semantics.

- `n + m`
- `n - m`
- `n * m`
- `n / m` (must truncate; must be nonnegative if n and m are positive)
- `n % m` (must be nonnegative if n and m are positive)
- Assignment versions of the above
- `+n`, `-n`
- `!n` (must be `true` iff n is zero)

There must be zero and one values available for I. It should be possible to generate these as `I(0)` and `I(1)`, respectively. Note: This does not imply that I needs to have an implicit conversion from integer - an `explicit` constructor is adequate.

It is valid for I to be an unsigned type. In that case, the derived rational class will also be unsigned. Underflow behaviour of subtraction, where results would otherwise be negative, is unpredictable in this case.

- The implementation of rational_cast<T>(rational<I>) relies on the ability to static_cast from type I to type T, and on the expression x/y being valid for any two values of type T.
- The input and output operators rely on the existence of corresponding input and output operators for type I.

The `std::numeric_limits<I>` specialization must exist (and be visible before `boost::rational<I>` needs to be specified). The value of its `is_specialized` static data member must be true and the value of its `is_signed` static data member must be accurate.

## Interface

### Utility functions

Two utility function templates may be provided, that should work with any type that can be used with the `boost::rational<>` class template.

| `gcd(n, m)` |  | The greatest common divisor of n and m
| `lcm(n, m)` |  | The least common multiple of n and m

These function templates now forward calls to their equivalents in the Boost.Integer library. Their presence can be controlled at compile time with the `BOOST_CONTROL_RATIONAL_HAS_GCD` preprocessor constant.

### Constructors

Rationals can be constructed from zero, one, or two integer arguments; representing default construction as zero, conversion from an integer posing as the numerator with an implicit denominator of one, or a numerator and denominator pair in that order, respectively. An integer argument should be of the rational's integer type, or implicitly convertible to that type. (For the two-argument constructor, any needed conversions are evaluated independently, of course.) The components are stored in normalized form.

Rationals can also be constructed from another rational. When the source and destination underlying integer types match, the automatically-defined copy- or move-constructor is used. Otherwise, a converting constructor template is used. The constructor does member-wise initialization of the numerator and denominator. Component-level conversions that are marked `explicit` are fine. When the conversion ends up value-preserving, it is already normalized; but a check for normalization is performed in case value-preservation is violated.

These imply that the following statements are valid:

        I n, d;
        rational<I> zero;
        rational<I> r1(n);
        rational<I> r2(n, d);
        rational<J> r3(r2);  // assuming J(n) and J(d) are well-formed

In C++11, the no-argument constructor, single-argument constructor, and cross-version constructor template are marked as `constexpr`, making them viable in constant-expressions when the initializers (if any) are also constant expressions (and the necessary operations from the underlying integer type(s) are `constexpr`-enabled). Since C++14, all constructors are `constexpr`-enabled.

The single-argument constructor is not declared as explicit, so there is an implicit conversion from the underlying integer type to the rational type. The two-argument constructor can be considered an implicit conversion with C++11's uniform initialization syntax, since it is also not declared explicit. The cross-version constructor template is declared explicit, so the direction of conversion between two rational instantiations must be specified.

### Arithmetic operations

All of the standard numeric operators are defined for the rational class. These include:

        +    +=
        -    -=
        *    *=
        /    /=
        ++   --    (both prefix and postfix)
        ==   !=
        <    >
        <=   >=

        Unary: + - !

Since C++14, all of these operations are `constexpr`-enabled. In C++11, only `operator==`, `operator!=`, unary `operator+`, and `operator!` are.

### Input and Output

Input and output operators `<<` and `>>` are provided. The external representation of a rational is two integers, separated by a slash (`/`). On input, the format must be exactly an integer, followed with no intervening whitespace by a slash, followed (again with no intervening whitespace) by a second integer. The external representation of an integer is defined by the underlying integer type.

### In-place assignment

For any `rational<I> r`, `r.assign(n, m)` provides an alternate to `r = rational<I>(n, m);`, without a user-specified construction of a temporary. While this is probably unnecessary for rationals based on machine integer types, it could offer a saving for rationals based on unlimited-precision integers, for example.

The function will throw if the given components cannot be formed into a valid rational number. Otherwise, it could throw only if the component-level move assignment (in C++11; copy-assignment for earlier C++ versions) can throw. The strong guarantee is kept if throwing happens in the first part, but there is a risk of neither the strong nor basic guarantees happening if an exception is thrown during the component assignments.

### Conversions

There is a conversion operator to an unspecified Boolean type (most likely a member pointer). This operator converts a rational to `false` if it represents zero, and `true` otherwise. This conversion allows a rational for use as the first argument of operator `?:`; as either argument of operators `&&` or `||` without forfeiting short-circuit evaluation; as a condition for a `do`, `if`, `while`, or `for` statement; and as a conditional declaration for `if`, `while`, or `for` statements. The nature of the type used, and that any names for that nature are kept private, should prevent any inappropriate non-Boolean use like numeric or pointer operations or as a `switch` condition.

There are no other implicit conversions from a rational type. Besides the explicit cross-version constructor template, there is an explicit type-conversion function, `rational_cast<T>(r)`. This can be used as follows:

        rational<int> r(22,7);
        double nearly_pi = boost::rational_cast<double>(r);

The `rational_cast<T>` function's behaviour is undefined if the source rational's numerator or denominator cannot be safely cast to the appropriate floating point type, or if the division of the numerator and denominator (in the target floating point type) does not evaluate correctly. Also, since this function has a custom name, it cannot be called in generic code for trading between two instantiations of the same class template, unlike the cross-version constructor.

In essence, all required conversions should be value-preserving, and all operations should behave "sensibly". If these constraints cannot be met, a separate user-defined conversion will be more appropriate.

Boolean conversion and `rational_cast` are `constexpr`-enabled.

Implementation note:

The implementation of the rational_cast function was

        template <typename Float, typename Int>
        Float rational_cast(const rational<Int>& src)
        {
            return static_cast<Float>(src.numerator()) / src.denominator();
        }

Programs should not be written to depend upon this implementation, however, especially since this implementation is now obsolete. (It required a mixed-mode division between types Float and Int, contrary to the Integer Type Requirements.)

### Numerator and Denominator

Finally, access to the internal representation of rationals is provided by the two member functions `numerator()` and `denominator()`. These functions are `constexpr`-enabled.

These functions allow user code to implement any additional required functionality. In particular, it should be noted that there may be cases where the above rational_cast operation is inappropriate - particularly in cases where the rational type is based on an unlimited-precision integer type. In this case, a specially-written user-defined conversion to floating point will be more appropriate.

## Performance

The rational class has been designed with the implicit assumption that the underlying integer type will act "like" the built in integer types. The behavioural aspects of this assumption have been explicitly described above, in the Integer Type Requirements section. However, in addition to behavioural assumptions, there are implicit performance assumptions.

No attempt will be made to provide detailed performance guarantees for the operations available on the rational class. While it is possible for such guarantees to be provided (in a similar manner to the performance specifications of many of the standard library classes) it is by no means clear that such guarantees will be of significant value to users of the rational class. Instead, this section will provide a general discussion of the performance characteristics of the rational class.

There now follows a list of the fundamental operations defined in the <boost/rational.hpp> header and an informal description of their performance characteristics. Note that these descriptions are based on the current implementation, and as such should be considered subject to change.

- Construction of a rational is essentially just two constructions of the underlying integer type, plus a normalization.
- Increment and decrement operations are essentially as cheap as addition and subtraction on the underlying integer type.
- (In)equality comparison is essentially as cheap as two equality operations on the underlying integer type.
- I/O operations are not cheap, but their performance is essentially dominated by the I/O time itself.
- An (implicit) GCD routine call is essentially a repeated modulus operation. Its other significant operations are construction, assignment, and comparison against zero of IntType values. These latter operations are assumed to be trivial in comparison with the modulus operation.
- The (implicit) LCM operation is essentially a GCD plus a multiplication, division, and comparison.
- The addition and subtraction operations are complex. They will require approximately two gcd operations, 3 divisions, 3 multiplications and an addition on the underlying integer type.
- The multiplication and division operations require two gcd operations, two multiplications, and four divisions.
- The compare-with-integer operation does a single integer division & modulus pair, at most one extra integer addition and decrement, and at most three integer comparisons.
- The compare-with-rational operation does two double-sized GCD operations, two extra additions and decrements, and three comparisons in the worst case. (The GCD operations are double-sized because they are done in piecemeal and the interim quotients are retained and compared, whereas a direct GCD function only retains and compares the remainders.)
- The final fundamental operation is normalizing a rational. This operation is performed whenever a rational is constructed (and assigned in place). All other operations are careful to maintain rationals in a normalized state. Normalization costs the equivalent of one gcd and two divisions.

Note that it is implicitly assumed that operations on IntType have the "usual" performance characteristics - specifically, that the expensive operations are multiplication, division, and modulo, with addition and subtraction being significantly cheaper. It is assumed that construction (from integer literals 0 and 1, and copy construction) and assignment are relatively cheap, although some effort is taken to reduce unnecessary construction and copying. It is also assumed that comparison (particularly against zero) is cheap.

Integer types which do not conform to these assumptions will not be particularly effective as the underlying integer type for the rational class. Specifically, it is likely that performance will be severely sub-optimal.

## Exceptions

Rationals can never have a denominator of zero. (This library does not support representations for infinity or NaN). Should a rational result ever generate a denominator of zero, or otherwise fail during normalization, the exception `boost::bad_rational` (a subclass of `std::domain_error`) is thrown. This should only occur if the user attempts to explicitly construct a rational with a denominator of zero, to divide a rational by a zero value, or generate a negative denominator too large to be normalized. The exception can be thrown during a cross-instantiation conversion, when at least one of the components ends up not being value-preserved and the new combination is not considered normalized.

In addition, if operations on the underlying integer type can generate exceptions, these will be propagated out of the operations on the rational class. No particular assumptions should be made - it is only safe to assume that any exceptions which can be thrown by the integer class could be thrown by any rational operation. In particular, the rational constructor may throw exceptions from the underlying integer type as a result of the normalization step. The only exception to this rule is that the rational destructor will only throw exceptions which can be thrown by the destructor of the underlying integer type (usually none).

If the component-level assignment operator(s) can throw, then a rational object's invariants may be violated if an exception happens during the second component's assignment. (The `assign` member function counts here too.) This violates both the strong and basic guarantees.

## Internal representation

Note: This information is for information only. Programs should not be written in such a way as to rely on these implementation details.

Internally, rational numbers are stored as a pair (numerator, denominator) of integers (whose type is specified as the template parameter for the rational type). Rationals are always stored in fully normalized form (ie, gcd(numerator,denominator) = 1, and the denominator is always positive).

## Design notes

### Minimal Implementation

The rational number class is designed to keep to the basics. The minimal operations required of a numeric class are provided, along with access to the underlying representation in the form of the numerator() and denominator() member functions. With these building-blocks, it is possible to implement any additional functionality required.

Areas where this minimality consideration has been relaxed are in providing input/output operators, and rational_cast. The former is generally uncontroversial. However, there are a number of cases where rational_cast is not the best possible method for converting a rational to a floating point value (notably where user-defined types are involved). In those cases, a user-defined conversion can and should be implemented. There is no need for such an operation to be named rational_cast, and so the rational_cast function does not provide the necessary infrastructure to allow for specialisation/overloading.

### Limited-range integer types

The rational number class is designed for use in conjunction with an unlimited precision integer class. With such a class, rationals are always exact, and no problems arise with precision loss, overflow or underflow.

Unfortunately, the C++ standard does not offer such a class (and neither does boost, at the present time). It is therefore likely that the rational number class will in many cases be used with limited-precision integer types, such as the built-in `int` type.

When used with a limited precision integer type, the rational class suffers from many of the precision issues which cause difficulty with floating point types. While it is likely that precision issues will not affect simple uses of the rational class, users should be aware that such issues exist.

As a simple illustration of the issues associated with limited precision integers, consider a case where the C++ `int` type is a 32-bit signed representation. In this case, the smallest possible positive rational<int> is `1/0x7FFFFFFF`. In other words, the "granularity" of the rational<int> representation around zero is approximately 4.66e-10. At the other end of the representable range, the largest representable rational<int> is `0x7FFFFFFF/1`, and the next lower representable rational<int> is `0x7FFFFFFE/1`. Thus, at this end of the representable range, the granularity ia 1. This type of magnitude-dependent granularity is typical of floating point representations. However, it does not "feel" natural when using a rational number class.

Limited-precision integer types may raise issues with the range sizes of their allowable negative values and positive values. If the negative range is larger, then the extremely-negative numbers will not have an additive inverse in the positive range, making them unusable as denominator values since they cannot be normalized to positive values (unless the user is lucky enough that the input components are not relatively prime pre-normalization).

It is up to the user of a rational type based on a limited-precision integer type to be aware of, and code in anticipation of, such issues.

### Conversion from floating point

The library does not offer a conversion function from floating point to rational. A number of requests were received for such a conversion, but extensive discussions on the boost list reached the conclusion that there was no "best solution" to the problem. As there is no reason why a user of the library cannot write their own conversion function which suits their particular requirements, the decision was taken not to pick any one algorithm as "standard".

The key issue with any conversion function from a floating point value is how to handle the loss of precision which is involved in floating point operations. To provide a concrete example, consider the following code:

        // These two values could in practice be obtained from user input,
        // or from some form of measuring instrument.
        double x = 1.0;
        double y = 3.0;

        double z = x/y;

        rational<I> r = rational_from_double(z);

The fundamental question is, precisely what rational should r be? A naive answer is that r should be equal to 1/3. However, this ignores a multitude of issues.

In the first instance, z is not exactly 1/3. Because of the limitations of floating point representation, 1/3 is not exactly representable in any of the common representations for the double type. Should r therefore not contain an (exact) representation of the actual value represented by z? But will the user be happy with a value of 33333333333333331/100000000000000000 for r?

Before even considering the above issue, we have to consider the accuracy of the original values, x and y. If they came from an analog measuring instrument, for example, they are not infinitely accurate in any case. In such a case, a rational representation like the above promises far more accuracy than there is any justification for.

All of this implies that we should be looking for some form of "nearest simple fraction". Algorithms to determine this sort of value do exist. However, not all applications want to work like this. In other cases, the whole point of converting to rational is to obtain an exact representation, in order to prevent accuracy loss during a series of calculations. In this case, a completely precise representation is required, regardless of how "unnatural" the fractions look.

With these conflicting requirements, there is clearly no single solution which will satisfy all users. Furthermore, the algorithms involved are relatively complex and specialised, and are best implemented with a good understanding of the application requirements. All of these factors make such a function unsuitable for a general-purpose library such as this.

### Absolute Value

In the first instance, it seems logical to implement abs(rational<IntType>) in terms of abs(IntType). However, there are a number of issues which arise with doing so.

The first issue is that, in order to locate the appropriate implementation of abs(IntType) in the case where IntType is a user-defined type in a user namespace, Koenig lookup is required. Not all compilers support Koenig lookup for functions at the current time. For such compilers, clumsy workarounds, which require cooperation from the user of the rational class, are required to make things work.

The second, and potentially more serious, issue is that for non-standard built-in integer types (for example, 64-bit integer types such as long long or __int64), there is no guarantee that the vendor has supplied a built in abs() function operating on such types. This is a quality-of-implementation issue, but in practical terms, vendor support for types such as long long is still very patchy.

As a consequence of these issues, it does not seem worth implementing abs(rational<IntType>) in terms of abs(IntType). Instead, a simple implementation with an inline implementation of abs() is used:

        template <typename IntType>
        inline rational<IntType> abs(const rational<IntType>& r)
        {
            if (r.numerator() >= IntType(0))
                return r;

                return rational<IntType>(-r.numerator(), r.denominator());
        }

The same arguments imply that where the absolute value of an IntType is required elsewhere, the calculation is performed inline.

## References

- The rational number header itself: rational.hpp
- Some example code: rational_example.cpp
- The regression test: rational_test.cpp

## History and Acknowledgements

In December, 1999, I implemented the initial version of the rational number class, and submitted it to the boost.org mailing list. Some discussion of the implementation took place on the mailing list. In particular, Andrew D. Jewell pointed out the importance of ensuring that the risk of overflow was minimised, and provided overflow-free implementations of most of the basic operations. The name rational_cast was suggested by Kevlin Henney. Ed Brey provided invaluable comments - not least in pointing out some fairly stupid typing errors in the original code!

David Abrahams contributed helpful feedback on the documentation.

A long discussion of the merits of providing a conversion from floating point to rational took place on the boost list in November 2000. Key contributors included Reggie Seagraves, Lutz Kettner and Daniel Frey (although most of the boost list seemed to get involved at one point or another!). Even though the end result was a decision not to implement anything, the discussion was very valuable in understanding the issues.

Stephen Silver contributed useful experience on using the rational class with a user-defined integer type.

Nickolay Mladenov provided the current implementation of operator+= and operator-=.

Discussion of the issues surrounding Koenig lookup and std::swap took place on the boost list in January 2001.

Daryle Walker provided a Boolean conversion operator, so that a rational can be used in the same Boolean contexts as the built-in numeric types, in December 2005. He added the cross-instantiation constructor template in August 2013.

July 2014: Updated numerator/denominator accessors to return values by constant reference: this gives a performance improvement when using with multiprecision (class) types.

July 2014: Updated to use BOOST_THROW_EXCEPTION uniformly throughout.

July 2014: Added support for C++11 constexpr constructors, plus tests to match.

Nov 2014: Added support for gcd and lcm of rational numbers.

Dec 2016: Reworked constructors and operators to prohibit narrowing implicit conversions, in particular accidental conversion from floating point types.

Oct/Nov 2018: Add more constexpr.

Revised July 14, 2017

© Copyright Paul Moore 1999-2001; © Daryle Walker 2005, 2013. Permission to copy, use, modify, sell and distribute this document is granted provided this copyright notice appears in all copies. This document is provided "as is" without express or implied warranty, and with no claim as to its suitability for any purpose.