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 formtext::mpq
is equivalent tompq(text)
.The string can be an integer like
41
or a fraction like41/152
. 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
or0X
for hex,0b
or0B
for binary,0
for octal, or decimal otherwise. Note that this is done separately for the numerator and denominator, so for instance0xEF/100
is 239/100, whereas0xEF/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 formq::text
is equivalent totext(q)
.The string will be of the form
num/den
, or if the denominator is 1 then justnum
.base may vary from 2 to 62 or from 2 to 36. For base in the range 2..36, digits and lowercase letters are used; for 2..36, digits and uppercase letters are used; for 37..62, digits, uppercase letters, and lowercase 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 tompz
so invoking the function asmpq(30,17)
will work as expected. However if the numbers become too big for anint8
they will be interpreted by PostgreSQL asnumeric
and, because the cast fromnumeric
tompz
is not implicit, the call will fail. Forcing a cast tompz
(e.g.mpq(30::mpz,17::mpz)
) will work for numbers of every size.
Arithmetic Operators and Functions¶
All the arithmetic operators and functions return their their output in canonical form.
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 floatingpoint 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.