fmpq – rational numbers¶

class flint.fmpq(*args)¶

The fmpq type represents multiprecision rational numbers.

>>> fmpq(1,7) + fmpq(50,51)
401/357

static bernoulli(ulong n, bool cache=False)¶

Returns the Bernoulli number \(B_n\) as an fmpq.

>>> [fmpq.bernoulli(n) for n in range(8)]
[1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0]
>>> fmpq.bernoulli(50)
495057205241079648212477525/66

If cache is set to True, all the Bernoulli numbers up to n are computed and cached for fast subsequent retrieval. This feature should be used with caution if n is large. Calling ctx.cleanup() frees cached Bernoulli numbers.

ceil(self)¶

Ceiling function.

>>> fmpq(3,2).ceil()
2

static dedekind_sum(n, k)¶

Dedekind sum.

>>> fmpq.dedekind_sum(10, 3)
1/18

denom(self)¶: Returns the denominator of self as an fmpz.

property denominator¶

fmpq.denom(self)

Returns the denominator of self as an fmpz.

floor(self)¶

Floor function.

>>> fmpq(3,2).floor()
1

gcd(s, t)¶

GCD of two rational numbers.

>>> fmpq(1,2).gcd(fmpq(3,4))
1/4

The GCD is defined as the GCD of the numerators divided by the LCM of the denominators. This is consistent with fmpz.gcd() but not with fmpq_poly.gcd().

static harmonic(ulong n)¶

Returns the harmonic number \(H_n\) as an fmpq.

>>> [fmpq.harmonic(n) for n in range(6)]
[0, 1, 3/2, 11/6, 25/12, 137/60]
>>> fmpq.harmonic(50)
13943237577224054960759/3099044504245996706400

height_bits(self, bool signed=False)¶

Returns the bit length of the maximum of the numerator and denominator. With signed=True, returns the negative value if the number is negative.

>>> fmpq(1001,5).height_bits()
10
>>> fmpq(-5,1001).height_bits(signed=True)
-10

is_zero(self)¶

next(s, bool signed=True, bool minimal=True)¶

Returns the next rational number after s as ordered by minimal height (if minimal is True) or following the Calkin-Wilf sequence (if minimal is False). If signed is set to False, only the nonnegative rational numbers are considered.

>>> fmpq(23456789,98765432).next()
-23456789/98765432
>>> fmpq(23456789,98765432).next(signed=False)
98765432/23456789
>>> fmpq(23456789,98765432).next(signed=False, minimal=False)
98765432/75308643
>>> a, b, c, d = [fmpq(0)], [fmpq(0)], [fmpq(0)], [fmpq(0)]
>>> for i in range(20):
...     a.append(a[-1].next())
...     b.append(b[-1].next(signed=False))
...     c.append(c[-1].next(minimal=False))
...     d.append(d[-1].next(signed=False, minimal=False))
...
>>> a
[0, 1, -1, 1/2, -1/2, 2, -2, 1/3, -1/3, 3, -3, 2/3, -2/3, 3/2, -3/2, 1/4, -1/4, 4, -4, 3/4, -3/4]
>>> b
[0, 1, 1/2, 2, 1/3, 3, 2/3, 3/2, 1/4, 4, 3/4, 4/3, 1/5, 5, 2/5, 5/2, 3/5, 5/3, 4/5, 5/4, 1/6]
>>> c
[0, 1, -1, 1/2, -1/2, 2, -2, 1/3, -1/3, 3/2, -3/2, 2/3, -2/3, 3, -3, 1/4, -1/4, 4/3, -4/3, 3/5, -3/5]
>>> d
[0, 1, 1/2, 2, 1/3, 3/2, 2/3, 3, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, 4, 1/5, 5/4, 4/7, 7/3, 3/8]

numer(self)¶: Returns the numerator of self as an fmpz.

property numerator¶

fmpq.numer(self)

Returns the numerator of self as an fmpz.

property p¶

fmpq.numer(self)

Returns the numerator of self as an fmpz.

property q¶

fmpq.denom(self)

Returns the denominator of self as an fmpz.

repr(self)¶

round(self, ndigits=None)¶

Rounding function.

>>> fmpq(3,2).round()
2
>>> fmpq(-3,2).round()
-2

sqrt(self)¶

Return exact rational square root of self or raise an error.

>>> fmpq(9, 4).sqrt()
3/2
>>> fmpq(8).sqrt()
Traceback (most recent call last):
    ...
flint.utils.flint_exceptions.DomainError: not a square number

str(self, **kwargs)¶

Converts self to a string, forwarding optional keyword arguments to fmpz.str().

>>> fmpq.bernoulli(12).str()
'-691/2730'
>>> fmpq.bernoulli(100).str(base=2, condense=10)
'-110001110{...257 digits...}0011011111/1000001000110010'

trunc(self)¶

Truncation function.

>>> fmpq(3,2).trunc()
1
>>> fmpq(-3,2).trunc()
-1