# Rational Numbers¶

AUTHORS:

• William Stein (2005): first version
• William Stein (2006-02-22): floor and ceil (pure fast GMP versions).
• Gonzalo Tornaria and William Stein (2006-03-02): greatly improved python/GMP conversion; hashing
• William Stein and Naqi Jaffery (2006-03-06): height, sqrt examples, and improve behavior of sqrt.
• David Harvey (2006-09-15): added nth_root
• Pablo De Napoli (2007-04-01): corrected the implementations of multiplicative_order, is_one; optimized __nonzero__ ; documented: lcm,gcd
• John Cremona (2009-05-15): added support for local and global logarithmic heights.
• Travis Scrimshaw (2012-10-18): Added doctests for full coverage.
• Vincent Delecroix (2013): continued fraction

TESTS:

sage: a = -2/3
True

class sage.rings.rational.Q_to_Z

Bases: sage.categories.map.Map

A morphism from $$\QQ$$ to $$\ZZ$$.

TESTS:

sage: type(ZZ.convert_map_from(QQ))
<type 'sage.rings.rational.Q_to_Z'>

section()

Return a section of this morphism.

EXAMPLES:

sage: sage.rings.rational.Q_to_Z(QQ, ZZ).section()
Natural morphism:
From: Integer Ring
To:   Rational Field

class sage.rings.rational.Rational

A rational number.

Rational numbers are implemented using the GMP C library.

EXAMPLES:

sage: a = -2/3
sage: type(a)
<type 'sage.rings.rational.Rational'>
sage: parent(a)
Rational Field
sage: Rational('1/0')
Traceback (most recent call last):
...
TypeError: unable to convert 1/0 to a rational
sage: Rational(1.5)
3/2
sage: Rational('9/6')
3/2
sage: Rational((2^99,2^100))
1/2
sage: Rational(("2", "10"), 16)
1/8
sage: Rational(QQbar(125/8).nth_root(3))
5/2
sage: Rational(AA(209735/343 - 17910/49*golden_ratio).nth_root(3) + 3*AA(golden_ratio))
53/7
sage: QQ(float(1.5))
3/2
sage: QQ(RDF(1.2))
6/5


Conversion from PARI:

sage: Rational(pari('-939082/3992923'))
-939082/3992923
sage: Rational(pari('Pol([-1/2])'))  #9595
-1/2

absolute_norm()

Returns the norm from Q to Q of x (which is just x). This was added for compatibility with NumberFields

EXAMPLES:

sage: (6/5).absolute_norm()
6/5

sage: QQ(7/5).absolute_norm()
7/5


Return the additive order of self.

OUTPUT: integer or infinity

EXAMPLES:

sage: QQ(0).additive_order()
1
+Infinity

ceil()

Return the ceiling of this rational number.

OUTPUT: Integer

If this rational number is an integer, this returns this number, otherwise it returns the floor of this number +1.

EXAMPLES:

sage: n = 5/3; n.ceil()
2
sage: n = -17/19; n.ceil()
0
sage: n = -7/2; n.ceil()
-3
sage: n = 7/2; n.ceil()
4
sage: n = 10/2; n.ceil()
5

charpoly(var)

Return the characteristic polynomial of this rational number. This will always be just var - self; this is really here so that code written for number fields won’t crash when applied to rational numbers.

INPUT:

• var - a string

OUTPUT: Polynomial

EXAMPLES:

sage: (1/3).charpoly('x')
x - 1/3


AUTHORS:

• Craig Citro
conjugate()

Return the complex conjugate of this rational number, which is the number itself.

EXAMPLES:

sage: n = 23/11
sage: n.conjugate()
23/11

content(other)

Return the content of self and other, i.e. the unique positive rational number $$c$$ such that self/c and other/c are coprime integers.

other can be a rational number or a list of rational numbers.

EXAMPLES:

sage: a = 2/3
sage: a.content(2/3)
2/3
sage: a.content(1/5)
1/15
sage: a.content([2/5, 4/9])
2/45

continued_fraction()

Return the continued fraction of that rational.

EXAMPLES:

sage: (641/472).continued_fraction()
[1; 2, 1, 3, 1, 4, 1, 5]

sage: a = (355/113).continued_fraction(); a
[3; 7, 16]
sage: a.n(digits=10)
3.141592920
sage: pi.n(digits=10)
3.141592654


It’s almost pi!

continued_fraction_list(type='std')

Return the list of partial quotients of this rational number.

INPUT:

• type - either “std” (the default) for the standard continued fractions or “hj” for the Hirzebruch-Jung ones.

EXAMPLES:

sage: (13/9).continued_fraction_list()
[1, 2, 4]
sage: 1 + 1/(2 + 1/4)
13/9

sage: (225/157).continued_fraction_list()
[1, 2, 3, 4,  5]
sage: 1 + 1/(2 + 1/(3 + 1/(4 + 1/5)))
225/157

sage: (fibonacci(20)/fibonacci(19)).continued_fraction_list()
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]

sage: (-1/3).continued_fraction_list()
[-1, 1, 2]


Check that the partial quotients of an integer n is simply [n]:

sage: QQ(1).continued_fraction_list()
[1]
sage: QQ(0).continued_fraction_list()
[0]
sage: QQ(-1).continued_fraction_list()
[-1]


Hirzebruch-Jung continued fractions:

sage: (11/19).continued_fraction_list("hj")
[1, 3, 2, 3, 2]
sage: 1 - 1/(3 - 1/(2 - 1/(3 - 1/2)))
11/19

sage: (225/137).continued_fraction_list("hj")
[2, 3, 5, 10]
sage: 2 - 1/(3 - 1/(5 - 1/10))
225/137

sage: (-23/19).continued_fraction_list("hj")
[-1, 5, 4]
sage: -1 - 1/(5 - 1/4)
-23/19

denom()

Returns the denominator of this rational number.

EXAMPLES:

sage: x = 5/13
sage: x.denom()
13
sage: x = -9/3
sage: x.denom()
1

denominator()

Returns the denominator of this rational number.

EXAMPLES:

sage: x = -5/11
sage: x.denominator()
11
sage: x = 9/3
sage: x.denominator()
1

factor()

Return the factorization of this rational number.

OUTPUT: Factorization

EXAMPLES:

sage: (-4/17).factor()
-1 * 2^2 * 17^-1


Trying to factor 0 gives an arithmetic error:

sage: (0/1).factor()
Traceback (most recent call last):
...
ArithmeticError: Prime factorization of 0 not defined.

floor()

Return the floor of this rational number as an integer.

OUTPUT: Integer

EXAMPLES:

sage: n = 5/3; n.floor()
1
sage: n = -17/19; n.floor()
-1
sage: n = -7/2; n.floor()
-4
sage: n = 7/2; n.floor()
3
sage: n = 10/2; n.floor()
5

gamma(prec=None)

Return the gamma function evaluated at self. This value is exact for integers and half-integers, and returns a symbolic value otherwise. For a numerical approximation, use keyword prec.

EXAMPLES:

sage: gamma(1/2)
sqrt(pi)
sage: gamma(7/2)
15/8*sqrt(pi)
sage: gamma(-3/2)
4/3*sqrt(pi)
sage: gamma(6/1)
120
sage: gamma(1/3)
gamma(1/3)


This function accepts an optional precision argument:

sage: (1/3).gamma(prec=100)
2.6789385347077476336556929410
sage: (1/2).gamma(prec=100)
1.7724538509055160272981674833

global_height(prec=None)

Returns the absolute logarithmic height of this rational number.

INPUT:

• prec (int) – desired floating point precision (default: default RealField precision).

OUTPUT:

(real) The absolute logarithmic height of this rational number.

ALGORITHM:

The height is the sum of the total archimedean and non-archimedean components, which is equal to $$\max(\log(n),\log(d))$$ where $$n,d$$ are the numerator and denominator of the rational number.

EXAMPLES:

sage: a = QQ(6/25)
sage: a.global_height_arch() + a.global_height_non_arch()
3.21887582486820
sage: a.global_height()
3.21887582486820
sage: (1/a).global_height()
3.21887582486820
sage: QQ(0).global_height()
0.000000000000000
sage: QQ(1).global_height()
0.000000000000000

global_height_arch(prec=None)

Returns the total archimedean component of the height of this rational number.

INPUT:

• prec (int) – desired floating point precision (default: default RealField precision).

OUTPUT:

(real) The total archimedean component of the height of this rational number.

ALGORITHM:

Since $$\QQ$$ has only one infinite place this is just the value of the local height at that place. This separate function is included for compatibility with number fields.

EXAMPLES:

sage: a = QQ(6/25)
sage: a.global_height_arch()
0.000000000000000
sage: (1/a).global_height_arch()
1.42711635564015
sage: (1/a).global_height_arch(100)
1.4271163556401457483890413081

global_height_non_arch(prec=None)

Returns the total non-archimedean component of the height of this rational number.

INPUT:

• prec (int) – desired floating point precision (default: default RealField precision).

OUTPUT:

(real) The total non-archimedean component of the height of this rational number.

ALGORITHM:

This is the sum of the local heights at all primes $$p$$, which may be computed without factorization as the log of the denominator.

EXAMPLES:

sage: a = QQ(5/6)
sage: a.support()
[2, 3, 5]
sage: a.global_height_non_arch()
1.79175946922805
sage: [a.local_height(p) for p in a.support()]
[0.693147180559945, 1.09861228866811, 0.000000000000000]
sage: sum([a.local_height(p) for p in a.support()])
1.79175946922805

height()

The max absolute value of the numerator and denominator of self, as an Integer.

OUTPUT: Integer

EXAMPLES:

sage: a = 2/3
sage: a.height()
3
sage: a = 34/3
sage: a.height()
34
sage: a = -97/4
sage: a.height()
97


AUTHORS:

• Naqi Jaffery (2006-03-05): examples

Note

For the logarithmic height, use global_height().

imag()

Returns the imaginary part of self, which is zero.

EXAMPLES:

sage: (1/239).imag()
0

is_S_integral(S=[])

Determine if the rational number is S-integral.

x is S-integral if x.valuation(p)>=0 for all p not in S, i.e., the denominator of x is divisible only by the primes in S.

INPUT:

• S – list or tuple of primes.

OUTPUT: bool

Note

Primality of the entries in S is not checked.

EXAMPLES:

sage: QQ(1/2).is_S_integral()
False
sage: QQ(1/2).is_S_integral([2])
True
sage: [a for a in range(1,11) if QQ(101/a).is_S_integral([2,5])]
[1, 2, 4, 5, 8, 10]

is_S_unit(S=None)

Determine if the rational number is an S-unit.

x is an S-unit if x.valuation(p)==0 for all p not in S, i.e., the numerator and denominator of x are divisible only by the primes in $$S$$.

INPUT:

• S – list or tuple of primes.

OUTPUT: bool

Note

Primality of the entries in S is not checked.

EXAMPLES:

sage: QQ(1/2).is_S_unit()
False
sage: QQ(1/2).is_S_unit([2])
True
sage: [a for a in range(1,11) if QQ(10/a).is_S_unit([2,5])]
[1, 2, 4, 5, 8, 10]

is_integer()

Determine if a rational number is integral (i.e is in $$\ZZ$$).

OUTPUT: bool

EXAMPLES:

sage: QQ(1/2).is_integral()
False
sage: QQ(4/4).is_integral()
True

is_integral()

Determine if a rational number is integral (i.e is in $$\ZZ$$).

OUTPUT: bool

EXAMPLES:

sage: QQ(1/2).is_integral()
False
sage: QQ(4/4).is_integral()
True

is_norm(L, element=False, proof=True)

Determine whether self is the norm of an element of L.

INPUT:

• L – a number field
• element – (default: False) boolean whether to also output an element of which self is a norm
• proof – If True, then the output is correct unconditionally. If False, then the output assumes GRH.

OUTPUT:

If element is False, then the output is a boolean B, which is True if and only if self is the norm of an element of L. If element is False, then the output is a pair (B, x), where B is as above. If B is True, then x an element of L such that self == x.norm(). Otherwise, x is None.

ALGORITHM:

Uses PARI’s bnfisnorm. See _bnfisnorm().

EXAMPLES:

sage: K = NumberField(x^2 - 2, 'beta')
sage: (1/7).is_norm(K)
True
sage: (1/10).is_norm(K)
False
sage: 0.is_norm(K)
True
sage: (1/7).is_norm(K, element=True)
(True, 1/7*beta + 3/7)
sage: (1/10).is_norm(K, element=True)
(False, None)
sage: (1/691).is_norm(QQ, element=True)
(True, 1/691)


The number field doesn’t have to be defined by an integral polynomial:

sage: B, e = (1/5).is_norm(QuadraticField(5/4, 'a'), element=True)
sage: B
True
sage: e.norm()
1/5


A non-Galois number field:

sage: K.<a> = NumberField(x^3-2)
sage: B, e = (3/5).is_norm(K, element=True); B
True
sage: e.norm()
3/5

sage: 7.is_norm(K)
Traceback (most recent call last):
...
NotImplementedError: is_norm is not implemented unconditionally for norms from non-Galois number fields
sage: 7.is_norm(K, proof=False)
False


AUTHORS:

• Craig Citro (2008-04-05)
• Marco Streng (2010-12-03)
is_nth_power(n)

Returns True if self is an $$n$$-th power, else False.

INPUT:

• n - integer (must fit in C int type)

Note

Use this function when you need to test if a rational number is an $$n$$-th power, but do not need to know the value of its $$n$$-th root. If the value is needed, use nth_root().

AUTHORS:

• John Cremona (2009-04-04)

EXAMPLES:

sage: QQ(25/4).is_nth_power(2)
True
sage: QQ(125/8).is_nth_power(3)
True
sage: QQ(-125/8).is_nth_power(3)
True
sage: QQ(25/4).is_nth_power(-2)
True

sage: QQ(9/2).is_nth_power(2)
False
sage: QQ(-25).is_nth_power(2)
False

is_one()

Determine if a rational number is one.

OUTPUT: bool

EXAMPLES:

sage: QQ(1/2).is_one()
False
sage: QQ(4/4).is_one()
True


Determines whether this rational number is a square in $$\QQ_p$$ (or in $$R$$ when p = infinity).

INPUT:

• p - a prime number, or infinity

EXAMPLES:

sage: QQ(2).is_padic_square(7)
True
True
False


TESTS:

sage: QQ(5/7).is_padic_square(int(2))
False

is_perfect_power(expected_value=False)

Returns True if self is a perfect power.

INPUT:

• expected_value - (bool) whether or not this rational is expected be a perfect power. This does not affect the correctness of the output, only the runtime.

If expected_value is False (default) it will check the smallest of the numerator and denominator is a perfect power as a first step, which is often faster than checking if the quotient is a perfect power.

EXAMPLES:

sage: (4/9).is_perfect_power()
True
sage: (144/1).is_perfect_power()
True
sage: (4/3).is_perfect_power()
False
sage: (2/27).is_perfect_power()
False
sage: (4/27).is_perfect_power()
False
sage: (-1/25).is_perfect_power()
False
sage: (-1/27).is_perfect_power()
True
sage: (0/1).is_perfect_power()
True


The second parameter does not change the result, but may change the runtime.

sage: (-1/27).is_perfect_power(True)
True
sage: (-1/25).is_perfect_power(True)
False
sage: (2/27).is_perfect_power(True)
False
sage: (144/1).is_perfect_power(True)
True


This test makes sure we workaround a bug in GMP (see trac ticket #4612):

sage: [ -a for a in srange(100) if not QQ(-a^3).is_perfect_power() ]
[]
sage: [ -a for a in srange(100) if not QQ(-a^3).is_perfect_power(True) ]
[]

is_square()

Return whether or not this rational number is a square.

OUTPUT: bool

EXAMPLES:

sage: x = 9/4
sage: x.is_square()
True
sage: x = (7/53)^100
sage: x.is_square()
True
sage: x = 4/3
sage: x.is_square()
False
sage: x = -1/4
sage: x.is_square()
False

lcm(other)

Return the least common multiple of self and other.

One way to define this notion is the following:

Note that each rational positive rational number can be written as a product of primes with integer (positive or negative) powers in a unique way.

Then, the LCM of two rational numbers $$x,y$$ can be defined by specifying that the exponent of every prime $$p$$ in lcm(x,y) is the supremum of the exponents of $$p$$ in $$x$$, and the exponent of $$p$$ in $$y$$ (The primes that does not appear in the decomposition of $$x$$ or $$y$$ are considered to have exponent zero).

This definition is consistent with the definition of the LCM in the rational integers. Our hopefully interesting notion of LCM for rational numbers is illustrated in the examples below.

EXAMPLES:

sage: lcm(2/3,1/5)
2


This is consistent with the definition above, since:

$2/3 = 2^1 * 3^{-1}*5^0$
$1/5 = 2^0 * 3^0 *5^{-1}$

and hence,

$lcm(2/3,1/5)= 2^1*3^0*5^0 = 2.$
sage: lcm(2/3,7/5)
14


In this example:

$2/3 = 2^1*3^{-1}*5^0 * 7^0$
$7/5 = 2^0*3^0 *5^{-1} * 7^1$
$lcm(2/3,7/5) = 2^1*3^0*5^0*7^1 = 14$
sage: lcm(1/3,1/5)
1


In this example:

$1/3 = 3^{-1}*5^0$
$1/5 = 3^0 * 5^{-1}$
$lcm(1/3,1/5)=3^0*5^0=1$
sage: lcm(1/3,1/6)
1/3


In this example:

$1/3 = 2^0*3^{-1}$
$1/6 = 2^{-1}*3^{-1}$
$lcm(1/3,1/6)=2^0*3^{-1}=1/3$
list()

Return a list with the rational element in it, to be compatible with the method for number fields.

OUTPUT:

• list - the list [self]

EXAMPLES:

sage: m = 5/3
sage: m.list()
[5/3]

local_height(p, prec=None)

Returns the local height of this rational number at the prime $$p$$.

INPUT:

• p – a prime number
• prec (int) – desired floating point precision (default: default RealField precision).

OUTPUT:

(real) The local height of this rational number at the prime $$p$$.

EXAMPLES:

sage: a = QQ(25/6)
sage: a.local_height(2)
0.693147180559945
sage: a.local_height(3)
1.09861228866811
sage: a.local_height(5)
0.000000000000000

local_height_arch(prec=None)

Returns the Archimedean local height of this rational number at the infinite place.

INPUT:

• prec (int) – desired floating point precision (default: default RealField precision).

OUTPUT:

(real) The local height of this rational number $$x$$ at the unique infinite place of $$\QQ$$, which is $$\max(\log(|x|),0)$$.

EXAMPLES:

sage: a = QQ(6/25)
sage: a.local_height_arch()
0.000000000000000
sage: (1/a).local_height_arch()
1.42711635564015
sage: (1/a).local_height_arch(100)
1.4271163556401457483890413081

minpoly(var='x')

Return the minimal polynomial of this rational number. This will always be just x - self; this is really here so that code written for number fields won’t crash when applied to rational numbers.

INPUT:

• var - a string

OUTPUT: Polynomial

EXAMPLES:

sage: (1/3).minpoly()
x - 1/3
sage: (1/3).minpoly('y')
y - 1/3


AUTHORS:

• Craig Citro
mod_ui(n)

Return the remainder upon division of self by the unsigned long integer n.

INPUT:

• n - an unsigned long integer

OUTPUT: integer

EXAMPLES:

sage: (-4/17).mod_ui(3)
1
sage: (-4/17).mod_ui(17)
Traceback (most recent call last):
...
ArithmeticError: The inverse of 0 modulo 17 is not defined.

multiplicative_order()

Return the multiplicative order of self.

OUTPUT: Integer or infinity

EXAMPLES:

sage: QQ(1).multiplicative_order()
1
sage: QQ('1/-1').multiplicative_order()
2
sage: QQ(0).multiplicative_order()
+Infinity
sage: QQ('2/3').multiplicative_order()
+Infinity
sage: QQ('1/2').multiplicative_order()
+Infinity

norm()

Returns the norm from $$\QQ$$ to $$\QQ$$ of $$x$$ (which is just $$x$$). This was added for compatibility with NumberFields.

OUTPUT:

• Rational - reference to self

EXAMPLES:

sage: (1/3).norm()
1/3


AUTHORS:

• Craig Citro
nth_root(n)

Computes the $$n$$-th root of self, or raises a ValueError if self is not a perfect $$n$$-th power.

INPUT:

• n - integer (must fit in C int type)

AUTHORS:

• David Harvey (2006-09-15)

EXAMPLES:

sage: (25/4).nth_root(2)
5/2
sage: (125/8).nth_root(3)
5/2
sage: (-125/8).nth_root(3)
-5/2
sage: (25/4).nth_root(-2)
2/5

sage: (9/2).nth_root(2)
Traceback (most recent call last):
...
ValueError: not a perfect 2nd power

sage: (-25/4).nth_root(2)
Traceback (most recent call last):
...
ValueError: cannot take even root of negative number

numer()

Return the numerator of this rational number.

EXAMPLE:

sage: x = -5/11
sage: x.numer()
-5

numerator()

Return the numerator of this rational number.

EXAMPLE:

sage: x = 5/11
sage: x.numerator()
5

sage: x = 9/3
sage: x.numerator()
3

ord(p)

Return the power of p in the factorization of self.

INPUT:

• p - a prime number

OUTPUT:

(integer or infinity) Infinity if self is zero, otherwise the (positive or negative) integer $$e$$ such that self = $$m*p^e$$ with $$m$$ coprime to $$p$$.

Note

See also val_unit() which returns the pair $$(e,m)$$. The function ord() is an alias for valuation().

EXAMPLES:

sage: x = -5/9
sage: x.valuation(5)
1
sage: x.ord(5)
1
sage: x.valuation(3)
-2
sage: x.valuation(2)
0


Some edge cases:

sage: (0/1).valuation(4)
+Infinity
sage: (7/16).valuation(4)
-2

period()

Return the period of the repeating part of the decimal expansion of this rational number.

ALGORITHM:

When a rational number $$n/d$$ with $$(n,d)=1$$ is expanded, the period begins after $$s$$ terms and has length $$t$$, where $$s$$ and $$t$$ are the smallest numbers satisfying $$10^s=10^{s+t} \mod d$$. In general if $$d=2^a3^bm$$ where $$m$$ is coprime to 10, then $$s=\max(a,b)$$ and $$t$$ is the order of 10 modulo $$d$$.

EXAMPLES:

sage: (1/7).period()
6
sage: RR(1/7)
0.142857142857143
sage: (1/8).period()
1
sage: RR(1/8)
0.125000000000000
sage: RR(1/6)
0.166666666666667
sage: (1/6).period()
1
sage: x = 333/106
sage: x.period()
13
sage: RealField(200)(x)
3.1415094339622641509433962264150943396226415094339622641509

prime_to_S_part(S=[])

Returns self with all powers of all primes in S removed.

INPUT:

• S - list or tuple of primes.

OUTPUT: rational

Note

Primality of the entries in $$S$$ is not checked.

EXAMPLES:

sage: QQ(3/4).prime_to_S_part()
3/4
sage: QQ(3/4).prime_to_S_part([2])
3
sage: QQ(-3/4).prime_to_S_part([3])
-1/4
sage: QQ(700/99).prime_to_S_part([2,3,5])
7/11
sage: QQ(-700/99).prime_to_S_part([2,3,5])
-7/11
sage: QQ(0).prime_to_S_part([2,3,5])
0
sage: QQ(-700/99).prime_to_S_part([])
-700/99

real()

Returns the real part of self, which is self.

EXAMPLES:

sage: (1/2).real()
1/2

relative_norm()

Returns the norm from Q to Q of x (which is just x). This was added for compatibility with NumberFields

EXAMPLES:

sage: (6/5).relative_norm()
6/5

sage: QQ(7/5).relative_norm()
7/5

round(mode='away')

Returns the nearest integer to self, rounding away from 0 by default, for consistency with the builtin Python round.

INPUT:

• self - a rational number
• mode - a rounding mode for half integers:
• ‘toward’ rounds toward zero
• ‘away’ (default) rounds away from zero
• ‘up’ rounds up
• ‘down’ rounds down
• ‘even’ rounds toward the even integer
• ‘odd’ rounds toward the odd integer

OUTPUT: Integer

EXAMPLES:

sage: (9/2).round()
5
sage: n = 4/3; n.round()
1
sage: n = -17/4; n.round()
-4
sage: n = -5/2; n.round()
-3
sage: n.round("away")
-3
sage: n.round("up")
-2
sage: n.round("down")
-3
sage: n.round("even")
-2
sage: n.round("odd")
-3

sign()

Returns the sign of this rational number, which is -1, 0, or 1 depending on whether this number is negative, zero, or positive respectively.

OUTPUT: Integer

EXAMPLES:

sage: (2/3).sign()
1
sage: (0/3).sign()
0
sage: (-1/6).sign()
-1

sqrt(prec=None, extend=True, all=False)

The square root function.

INPUT:

• prec – integer (default: None): if None, returns an exact square root; otherwise returns a numerical square root if necessary, to the given bits of precision.
• extend – bool (default: True); if True, return a square root in an extension ring, if necessary. Otherwise, raise a ValueError if the square is not in the base ring.
• all – bool (default: False); if True, return all square roots of self, instead of just one.

EXAMPLES:

sage: x = 25/9
sage: x.sqrt()
5/3
sage: sqrt(x)
5/3
sage: x = 64/4
sage: x.sqrt()
4
sage: x = 100/1
sage: x.sqrt()
10
sage: x.sqrt(all=True)
[10, -10]
sage: x = 81/5
sage: x.sqrt()
9*sqrt(1/5)
sage: x = -81/3
sage: x.sqrt()
3*sqrt(-3)

sage: n = 2/3
sage: n.sqrt()
sqrt(2/3)
sage: n.sqrt(prec=10)
0.82
sage: n.sqrt(prec=100)
0.81649658092772603273242802490
sage: n.sqrt(prec=100)^2
0.66666666666666666666666666667
sage: n.sqrt(prec=53, all=True)
[0.816496580927726, -0.816496580927726]
sage: n.sqrt(extend=False, all=True)
Traceback (most recent call last):
...
ValueError: square root of 2/3 not a rational number
sage: sqrt(-2/3, all=True)
[sqrt(-2/3), -sqrt(-2/3)]
sage: sqrt(-2/3, prec=53)
0.816496580927726*I
sage: sqrt(-2/3, prec=53, all=True)
[0.816496580927726*I, -0.816496580927726*I]


AUTHORS:

• Naqi Jaffery (2006-03-05): some examples
squarefree_part()

Return the square free part of $$x$$, i.e., an integer z such that $$x = z y^2$$, for a perfect square $$y^2$$.

EXAMPLES:

sage: a = 1/2
sage: a.squarefree_part()
2
sage: b = a/a.squarefree_part()
sage: b, b.is_square()
(1/4, True)
sage: a = 24/5
sage: a.squarefree_part()
30

str(base=10)

Return a string representation of self in the given base.

INPUT:

• base – integer (default: 10); base must be between 2 and 36.

OUTPUT: string

EXAMPLES:

sage: (-4/17).str()
'-4/17'
sage: (-4/17).str(2)
'-100/10001'


Note that the base must be at most 36.

sage: (-4/17).str(40)
Traceback (most recent call last):
...
ValueError: base (=40) must be between 2 and 36
sage: (-4/17).str(1)
Traceback (most recent call last):
...
ValueError: base (=1) must be between 2 and 36

support()

Return a sorted list of the primes where this rational number has non-zero valuation.

OUTPUT: The set of primes appearing in the factorization of this rational with nonzero exponent, as a sorted list.

EXAMPLES:

sage: (-4/17).support()
[2, 17]


Trying to find the support of 0 gives an arithmetic error:

sage: (0/1).support()
Traceback (most recent call last):
...
ArithmeticError: Support of 0 not defined.

trace()

Returns the trace from $$\QQ$$ to $$\QQ$$ of $$x$$ (which is just $$x$$). This was added for compatibility with NumberFields.

OUTPUT:

• Rational - reference to self

EXAMPLES:

sage: (1/3).trace()
1/3


AUTHORS:

• Craig Citro
trunc()

Round this rational number to the nearest integer toward zero.

EXAMPLES:

sage: (5/3).trunc()
1
sage: (-5/3).trunc()
-1
sage: QQ(42).trunc()
42
sage: QQ(-42).trunc()
-42

val_unit(p)

Returns a pair: the $$p$$-adic valuation of self, and the $$p$$-adic unit of self, as a Rational.

We do not require the $$p$$ be prime, but it must be at least 2. For more documentation see Integer.val_unit().

INPUT:

• p - a prime

OUTPUT:

• int - the $$p$$-adic valuation of this rational
• Rational - $$p$$-adic unit part of self

EXAMPLES:

sage: (-4/17).val_unit(2)
(2, -1/17)
sage: (-4/17).val_unit(17)
(-1, -4)
sage: (0/1).val_unit(17)
(+Infinity, 1)


AUTHORS:

• David Roe (2007-04-12)
valuation(p)

Return the power of p in the factorization of self.

INPUT:

• p - a prime number

OUTPUT:

(integer or infinity) Infinity if self is zero, otherwise the (positive or negative) integer $$e$$ such that self = $$m*p^e$$ with $$m$$ coprime to $$p$$.

Note

See also val_unit() which returns the pair $$(e,m)$$. The function ord() is an alias for valuation().

EXAMPLES:

sage: x = -5/9
sage: x.valuation(5)
1
sage: x.ord(5)
1
sage: x.valuation(3)
-2
sage: x.valuation(2)
0


Some edge cases:

sage: (0/1).valuation(4)
+Infinity
sage: (7/16).valuation(4)
-2

class sage.rings.rational.Z_to_Q

A morphism from $$\ZZ$$ to $$\QQ$$.

section()

Return a section of this morphism.

EXAMPLES:

sage: f = QQ.coerce_map_from(ZZ).section(); f
Generic map:
From: Rational Field
To:   Integer Ring


This map is a morphism in the category of sets with partial maps (see trac ticket #15618):

sage: f.parent()
Set of Morphisms from Rational Field to Integer Ring in Category of sets with partial maps

class sage.rings.rational.int_to_Q

A morphism from int to $$\QQ$$.

sage.rings.rational.integer_rational_power(a, b)

Compute $$a^b$$ as an integer, if it is integral, or return None. The positive real root is taken for even denominators.

INPUT:

a -- an Integer
b -- a positive Rational


OUTPUT:

a^b as an Integer or None


EXAMPLES:

sage: from sage.rings.rational import integer_rational_power
sage: integer_rational_power(49, 1/2)
7
sage: integer_rational_power(27, 1/3)
3
sage: integer_rational_power(-27, 1/3) is None
True
sage: integer_rational_power(-27, 2/3) is None
True
sage: integer_rational_power(512, 7/9)
128

sage: integer_rational_power(27, 1/4) is None
True
sage: integer_rational_power(-16, 1/4) is None
True

sage: integer_rational_power(0, 7/9)
0
sage: integer_rational_power(1, 7/9)
1
sage: integer_rational_power(-1, 7/9) is None
True
sage: integer_rational_power(-1, 8/9) is None
True
sage: integer_rational_power(-1, 9/8) is None
True

sage.rings.rational.is_Rational(x)

Return true if x is of the Sage rational number type.

EXAMPLES:

sage: from sage.rings.rational import is_Rational
sage: is_Rational(2)
False
sage: is_Rational(2/1)
True
sage: is_Rational(int(2))
False
sage: is_Rational(long(2))
False
sage: is_Rational('5')
False

sage.rings.rational.make_rational(s)

Make a rational number from s (a string in base 32)

INPUT:

• s - string in base 32

OUTPUT: Rational

EXAMPLES:

sage: (-7/15).str(32)
'-7/f'
sage: sage.rings.rational.make_rational('-7/f')
-7/15

sage.rings.rational.rational_power_parts(a, b, factor_limit=100000)

Compute rationals or integers $$c$$ and $$d$$ such that $$a^b = c*d^b$$ with $$d$$ small. This is used for simplifying radicals.

INPUT:

- a -- a rational or integer
- b -- a rational
- factor_limit -- the limit used in factoring a


EXAMPLES:

sage: from sage.rings.rational import rational_power_parts
sage: rational_power_parts(27, 1/2)
(3, 3)
sage: rational_power_parts(-128, 3/4)
(8, -8)
sage: rational_power_parts(-4, 1/2)
(2, -1)
sage: rational_power_parts(-4, 1/3)
(1, -4)
sage: rational_power_parts(9/1000, 1/2)
(3/10, 1/10)


TESTS:

Check if trac ticket #8540 is fixed:

sage: rational_power_parts(3/4, -1/2)
(2, 3)
sage: t = (3/4)^(-1/2); t
2/3*sqrt(3)
sage: t^2
4/3


#### Previous topic

Field $$\QQ$$ of Rational Numbers

Finite Fields