# mpq data type¶

The mpq data type can store rational numbers whose denominator and numerator have arbitrary size.

Rational numbers are converted in canonical form on input (meaning that the denominator and the numerator have no common factors) and all the operators will return a number in canonical form.

PostgreSQL integer types (int16, int32, int64), numeric and mpz can be converted to mpq without loss of precision and without surprise. Floating point types (float4, float8) are converted without loss as well... but with some surprise, as many fractions with finite decimal expansion have no finite expansion in binary.

=# select 10.1::numeric::mpq as "numeric",
-#        10.1::float4::mpq as "single",
-#        10.1::float8::mpq as "double";
numeric |     single     |              double
---------+----------------+----------------------------------
101/10  | 5295309/524288 | 5685794529555251/562949953421312


mpq values can be converted to integer types (both PostgreSQL’s and mpz): the result will be truncated. Conversion to float4 and float8 will round the values to the precision allowed by the types (in case of overflow the value will be Infinity). Conversion to numeric will perform a rounding to the precision set for the target type.

=# select mpq('4/3')::integer as "integer",
-#        mpq('4/3')::float4 as "single",
-#        mpq('4/3')::decimal(10,3) as "decimal";
integer | single  | decimal
---------+---------+---------
1 | 1.33333 |   1.333


mpq values can be compared using the regular PostgreSQL comparison operators. Indexes on mpq columns can be created using the btree or the hash method.

## mpq textual input/output¶

mpq(text)
mpq(text, base)

Convert a textual representation into an mpq number. The form text::mpq is equivalent to mpq(text).

The string can be an integer like 41 or a fraction like 41/152. The fraction will be converted in canonical form, so common factors between denominator and numerator will be removed.

The numerator and optional denominator are parsed the same as in mpz. White space is allowed in the string, and is simply ignored. The base can vary from 2 to 62, or if base is 0 then the leading characters are used: 0x or 0X for hex, 0b or 0B for binary, 0 for octal, or decimal otherwise. Note that this is done separately for the numerator and denominator, so for instance 0xEF/100 is 239/100, whereas 0xEF/0x100 is 239/256.

Note

The maximum base accepted by GMP 4.1 is 36, not 62.

text(q)
text(q, base)

Convert the mpq q into a string. The form q::text is equivalent to text(q).

The string will be of the form num/den, or if the denominator is 1 then just num.

base may vary from 2 to 62 or from −2 to −36. For base in the range 2..36, digits and lower-case letters are used; for −2..−36, digits and upper-case letters are used; for 37..62, digits, upper-case letters, and lower-case letters (in that significance order) are used. If base is not specified, 10 is assumed.

Note

The maximum base accepted by GMP 4.1 is 36, not 62.

## mpq conversions¶

mpq(num, den)

Return an mpq from its numerator and denominator.

Note

The function signature accepts mpz values. PostgreSQL integers are implicitly converted to mpz so invoking the function as mpq(30,17) will work as expected. However if the numbers become too big for an int8 they will be interpreted by PostgreSQL as numeric and, because the cast from numeric to mpz is not implicit, the call will fail. Forcing a cast to mpz (e.g. mpq(30::mpz,17::mpz)) will work for numbers of every size.

num(q)
den(q)

Return the numerator or the denominator of q as mpz.

## Arithmetic Operators and Functions¶

All the arithmetic operators and functions return their their output in canonical form.

Arithmetic operators
Operator Description Example Return
- Unary minus - '4/3'::mpq -4/3
+ Unary plus + '4/3'::mpq 4/3
+ Addition '2/3'::mpq + '5/6'::mpq 3/2
- Subtraction '1/3'::mpq - '5/6'::mpq -1/2
* Multiplication '2/3'::mpq * '5/6'::mpq 5/9
/ Division '2/3'::mpq / '5/6'::mpq 4/5
<< Multiplication by $$2^n$$ '2/3'::mpq << 3 16/3
>> Division by $$2^n$$ '2/3'::mpq >> 3 1/12
abs(q)

Return the absolute value of q.

inv(q)

Return 1/q.

limit_den(q, max_den=1000000)

Return the closest rational to q with denominator at most max_den.

The function is useful for finding rational approximations to a given floating-point number:

=# select limit_den(pi(), 10);
22/7


or for recovering a rational number that’s represented as a float:

=# select mpq(cos(pi()/3));
4503599627370497/9007199254740992
=# select limit_den(cos(pi()/3));
1/2
=# select limit_den(10.1::float4);
101/10


This function is not part of the GMP library: it is ported instead from the Python library.

## Aggregation functions¶

sum(q)

Return the sum of q across all input values.

prod(q)

Return the product of q across all input values.

max(q)

Return the maximum value of q across all input values.

min(q)

Return the minimum value of q across all input values.